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The factorial urban ecology of Greater Vancouver : characteristics of the data base Patterson, John Michael 1974

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THE FACTORIAL URBAN ECOLOGY OF CHARACTERISTICS OF THE GREATER VANCOUVER: DATA BASE by JOHN MICHAEL PATTERSON B.A., University of British Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF ARTS in the Department of Geography We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1974 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I ag ree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y pu rposes may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date fl/j/frw , f ; Wf i i ABSTRACT The methods of factorial urban ecology are followed for Greater Vancouver for 1961. One hundred and ninety-two variables are factor analysed to produce the seven most important underlying dimensions of areal differentiation. The factors include the three constructs of social area analysis. Factor 1, Family Status, factor 3, Housing Tenure, factor 5, Families with Older Children, and factor 6, Rural-Urban Status are a l l required to encompass the family status construct. Factor 2, Socio-Economic Status, and factor 7, Retail and Clerical Workers, are both required for the economic status construct. The ethnic status construct is paralleled by factor 4, Ethnic Status, which includes a wide range of ethnic groups. An examination of the f i r s t four factors for correspondence with theories of urban structure leads to the conclusions that Family Status displays an "interrupted radial" spatial pattern, and that Socio-economic Status displays a sectorial pattern centred on Vancouver City, and a radial pattern centred on New Westminster. Housing Tenure shows tendencies towards a multiple nuclei structure, though the complexity of the City of Vancouver dominates the pattern. Ethnic Status is concluded to have a unique pattern, unlike those predicted by any of the urban models. The data available to factorial ecologists are rarely normally distributed. Procedures are developed with which each of the variables is transformed to a distribution which is normal or very close to normal. The factor analysis is performed using both untransformed data and normalized data. In comparing results, i t is concluded that normalization aids the ab i l i t y of the correlation coefficient to measure linear association among variables. This helps the rotation procedure to achieve simple structure and consequently, the factors based upon normalized data are more easily interpreted. They include variables which seemed more closely interrelated and seem to correspond more closely to theoretically derived hypotheses about urban areal differentiation. This is especially true of the Ethnic Status factor, where a considerably larger number of ethnic groups are incorporated in the factor based upon normalized data. In addition to helping in the search for parsimony and for simple structure, the normalizing of data would seem to produce results which w i l l be more easily comparable, ceteris paribus, from city to city and hopefully among countries as well. It is tentatively claimed that the use of normalization procedures such as the ones developed here w i l l have a beneficial impact on our a b i l i t y to describe and hence understand the ecological structure of the city. The areal unit used in this analysis is the enumeration area, about one tenth the size of census tracts. This provides the opportunity to investigate the importance of the level of data aggregation. The approach used is to consider census tracts as aggregations of enumeration areas and to compare the information on areal differentiation lost through averaging with this aggregation to that lost with one achieved with an optimal analytic grouping procedure. It is concluded that the internal v a r i a b i l i t y of demographic, socio-economic, and housing characteristics of urban dwellers within census tracts is high enough to mask important information about the population. The best data base is one at the lowest level of aggregation suitable for data handling and at which data error is sufficiently small. Census tracts are too large to be considered suitable as the unit of analysis for factorial urban ecologies. TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF MAPS ACKNOWLEDGEMENTS Chapter 1 INTRODUCTION 1.1 The Study of Urban Structure 1.2 Purposes of the Study 2 FACTORIAL URBAN ECOLOGY 2.1 Social Area Anslysis 2.2 Factor Analysis 2.3 Factorial Urban Ecology 3 PROCEDURES OF THE STUDY 3.1 Previous Work on the Urban Ecology of Greater Vancouver 3.2 Structure of the Present Study 3.3 The Data 3.4 The Analysis 3.5 The Results vi TABLE OF CONTENTS (Continued) Chapter Page 4 DATA NORMALIZATION 28 4.1 Introduction 28 4.2 The Assumption of Correlation 28 4.3 A Procedure for Testing Normality 31 4.4 The Effects of Non-Normality on Correlation 34 4.5 Normalizing Procedures - Scaling Data 45 4.6 Transforms Used 47 4.7 Normalizing the Variables 54 4.8 The Effect of Normalization on the Correlation Matrix 60 4.9 The Effect of Normalizing on the Factor Model 63 4.10 Conclusions 71 5 AREAL AGGREGATION 73 5.1 Introduction 73 5.2 Areal Units 74 5.3 Some Characteristics of Data at Different Levels of Aggregation 75 5.4 Ecological Correlation 77 5.5 Ecological Correlation and Within-Region Variance 81 5.6 Enumeration Areas and Census Tracts 91 5.7 The Ward Algorithm Aggregation Procedure 92 5.8 Aggregating Vancouver's EA's 98 5.9 The Optimality of Census Tracts 100 5.10 Conclusion 108 6 THE URBAN STRUCTURE OF GREATER VANCOUVER 111 6.1 Choice of the Number of Factors 111 6.2 The Family Status Construct Factors 112 6.3 The Economic Status Construct Factors 114 6.4 The Ethnic Status Construct Factor 115 6.5 Assessment of the Factoring 116 6.6 The Spatial Patterning of the Factors 118 6.7 Comparison of the Factors with Theories of Urban Structure 131 v i i TABLE OF CONTENTS (Conti nued) Chapter Page 7 SUMMARY AND CONCLUSIONS 146 BIBLIOGRAPHY 148 Appendices A VARIABLE DEFINITIONS 153 B TESTS FOR NORMALITY 165 C VARIABLE CHARACTERISTICS AND TRANSFORMATIONS 171 D PROBABILITY IN AN F-DISTRIBUTION 179 E FACTOR LOADINGS MATRICES 181 v i i i LIST OF TABLES Table Page 2- 1 Social Area Analysis Constructs 7 3- 1 Summary of Rotated Factor Loadings from Peucker and Rase 18 3- 2 Summary of Factors 27 4- 1 Examples of Linear Transforms Versus Removed Non-Normality for the Log Transform 46 4-2 Non-Li near Transforms Used for Normalizing 49 4-3 Summary of Results in Normalizing 192 Variables 58 4-4 Characteristics of Correlation Coefficients Before and After Data Transformation 62 4-5 Characteristics of the Factor Models for Original and for Normalized Data 64 4-6 Characteristics of Factor Scores for Original Data and Normalized Data 67 4- 7 Numbers of Variables for which the Highest Loading is with one of Factors 1 through 7 - for Original Data, for Normalized Data, and for Both 69 5- 1 Relative Importance of Municipality Groups for Different Regionalizations 105 5- 2 Ratios of Within-Region Variance to Individual EA Variance for 21 Factors for Census Tracts and Ward Algorithm Regions 107 6- 1 Class Intervals for Factor Score Maps 119 C-l Variable Characteristics Before Normalization 171 C-2 Normalization and Resulting Characteristics 175 E-l Factor Loadings Matrix - Normalized Data 181 ix LIST OF TABLES (Continued) Table Page E-2 Reordered Factor Loadings Matrix - Normalized Data 185 E-3 Factor Loadings Matrix - Original Data 189 E-4 Reordered Factor Loadings Matrix - Original Data 193 X LIST OF FIGURES Figure Page 2- 1 Factor Analysis Research Design Flow Diagram 9 3- 1 Research Design Flow Diagram 21 4- 1 A Bivariate Normal Distribution 30 4-2 Histograms Showing Various Values of Skewness and Kurtosis 32 4-3 Histograms of Examples of Unusual Distributions which are Normal According to Skewness and Kurtosis Tests 35 4-4 Correlation Coefficient Versus Kurtosis for Normal Variables Correlated with Non-Normal Variables: 5 Cases 37 4-5 Correlation Coefficient Versus Kurtosis for Normal Variables Correlated with Non-Normal Variables: Scattergram 38 4-6 Correlation Coefficient Versus Skewness for Normal Variables Correlated with Non-Normal Variables: 5 Cases 39 4-7 Correlation Coefficient Versus Skewness for Variables having a Moderate Positive Skewness and Kurtosis Correlated with Non-Normal Variables 40 4-8 Correlation Coefficient Versus Skewness for Variables : having a High Positive Skewness and Kurtosis Correlated with Non-Normal Variables 42 4-9 Correlation Coefficient Versus Skewness for Variables having Zero Skewness and Low Kurtosis Correlated with Non-Normal Variables 43 4-10 The Influence of Scaling Upon the Effect of the Logarithm Transform 48 4-11 Ranges of Non-Normality Removed by the Seven Transforms 51 4-12 Curves for Estimating Logarithm Transform Scaling Constant 52 4-13 Distribution of 192 Variables Before Normalizing 55 XI LIST OF FIGURE (Continued) Figure Page 4-14 Distribution of 192 Variables after Normalizing 56 4-15 Distribution of Variable 1 Before and After Normalizing 57 4-16 Example of Unsuccessful Coordinate Transformation to Normality 57 4-17 The Effects of Normalization on the Correlation Matrix 61 4-18 Total Variance Accounted for by Each Factor for Both Original Data and Normalized Data Factor Models 66 4- 19 Comparison of Factor Structures for Original Data - and Normalized Data Factor Models 68 5- 1 The Ratio of Within-Region Correlation to Universal WR WVy WVY Correlation For A l l Values of -r ^ - A n d __! where pp - R RXY VX VY t KXY KXY. 83 5-2 The Value of the Ecological Correlation Coefficient for the Within-Region Correlation Coefficient and for WVy WVX WV V - Where — - = -JL And R Y V = 0.5. " y V X »Y AT 5-3 The Value of the Ecological Correlation Coefficient for the Within-Region Correlation Coefficient and for wvx wvY - — Where R Y V = 0.5 And is Chosen such that when V X AT Vy ER x y = R X Y, WRxy = 1.5 Rxy_ 8 7 5-4 The Value of the Ecological Correlation Coefficient for the Within-Region Correlation Coefficient and for WVY WVY . — - Where — - = 0.2 And R Y V = 0.5. 88 Vy VX '^ 5-6 The Interrelationship Among Figures 5-1, 5-2, 5-3, and 5-4. 90 xi i LIST OF FIGURES (Continued) Figure Page 5-6 The Departure of ER^ y From R y^ and the Equality of Within-Region Variance for X and Y. 97 5- 7 Comparison of Within-Region Variances 109 6- 1 Comparison of Within-Region Variance of Aggregated Enumeration Areas Versus Census Tracts for Aggregation by Individual Factor 134 6-2 Aggregation Error for Factor 1 135 6-3 Aggregation Error for Factor 2 136 6-4 Aggregation Error for Factor 3 137 6-5 Aggregation Error for Factor 4 138 LIST OF MAPS Map . Page 5-1 Census Tracts and Enumeration Area Centres 93 5-2 23 Ward Algorithm Regions - 101 5-3 64 Ward Algorithm Regions 102 5- 4 122 Ward Algorithm Regions 103 Map Legend . 120 6- 1 Factor 1 - Family Status 121 6-2 Factor 2 - Socio-Economic Status 123 6-3 Factor 3 - Housing Tenure > 124 6-4 Factor 4 - Ethnic Status 126 6-5 Factor 5 - Families with Older Children 128 6-6 Factor 6 - Rural-Urban Status 129 6-7 Factor 7 - Retail and Clerical Workers 130 6-8 Grouped Family Status 140 6-9 Grouped Socio-Economic Status 141 6-10 Grouped Housing Tenure 143 6-11 Grouped Ethnic Status 145 xiv ACKNOWLEDGEMENTS Many people have made generous contributions of time and knowledge in the course of the preparation of this thesis. The thesis topic was originally suggested by my advisor, Dr. Ken Denike, who added many valuable insights and gave me continued support throughout the long course of completion. The technical side of the work benefited greatly from the contributions of Dr. Mike Church, with whom I consulted frequently and who c r i t i c a l l y read chapters 4 and 5. I wish to express my appreciation to Dr. Gary Gates for his helpful comments and intriguing i f sometimes startling perspectives as second reader. Fellow graduate students Roy Whitaker, Pete Lewis, Arthur Nowell, John Bottom!ey, Dave Rothwell, P. Q. McAullay, and Warren G i l l a l l added valuable insights and encouragement. Escape to mountain tops with Pete Lewis, John Bottomley, and Gord Mulligan was indispensible in maintaining my sense of perspective. To Hoc Woo and Pete Lewis who introduced me to computer programming, and to UBC Computering Centre staff members John Coulthard, Jason Halm, Dennis O'Reilley, and Marty Goldstein for their generosity in time and expertise, I owe a deep debt of gratitude. The cartography in the thesis has benefited greatly from the expertise of Karen Ewing, who also contributed considerable time to the completion of the maps. The generosity of my employer Terry Johnston in allowing me time to work on the thesis and in putting the resources of the office at my disposal have made completion possible. Mimi Gibb and Joanne McDonnell worked tir e l e s s l y to complete the typing. To a l l of the above, and to the many more who have not been mentioned, I extend my gratitude for their help in making the realization of this thesis a meaningful and rewarding learning experience. 1 CHAPTER 1 - INTRODUCTION Factorial urban ecology can be defined as the study of the inter-relationships and spatial distributions of characteristics of urban dwellers using factor analytic techniques. It is used to describe the basic patterns of residential differentiation within an urban area. This approach has been applied to a large number of ci t i e s varying widely in situation, history, and culture. The result of these applications has been the general accept-ance of several basic postulates. Yet many questions regarding technical and other aspects of the procedure remain unresolved. It is the purpose of this study, not only to produce for Greater Vancouver the concise description of urban structure afforded by factorial ecology, but also to examine two technical problems associated with the approach which are significant and yet are essentially unexplored to date. These are the importance of 1) normality; and 2) level of aggregation, of the data base used for analy-sis to the results obtained. 1.1 The Study of Urban Structure People in urban areas who have similar social, economic, ethnic, or household characteristics are l i k e l y to live near each other in relatively homogeneous areas or neighbourhoods. This structuring or sorting-out phenom-enon has been studied for many years by sociologists, economists, geographers and others. For example, the "Chicago school" of urban sociologists used the term "urban ecology" to refer to their application of the concepts and 2 techniques of human ecology in the study of the ci t y . Here human ecology can be defined as "... the study of the spatial distribution of interrelated social variables [Theodorson, 1961, p. 3]." Land economists, on the other hand, have seen land rents as central to the phenomenon (see Alonso, 1964). Planners have studied i t in the context of entrepreneurial decision making and the workings of the market for housing and other land uses (Weiss, 1966). Transportation analysts have attempted to relate urban structuring to a number of independent variables with predictive modeling. The examination of urban areal differentiation can be seen as the description of "how", and the explanation of "why", areas differ in the characteristics of their human populations. The procedures of Social Area Analysis were developed as an aid in the description of differentiation based on postulates concerning the changing character of modern society. Three basic constructs were derived from these postulates for which indices for census tracts were calculated from a small number of census variables (Shevky, Bell & Williams, 1949; and Shevky & B e l l , 1955). These indices were then used to classify census tracts into social areas. An outgrowth of this descriptive procedure is the use of the s t a t i s t i c a l technique of factor analysis to identify, for a given c i t y , the patterns of interdependences among a selection of census variables. This approach is referred to as 'factorial urban ecology'. The variables are replaced by a smaller number of factors, or composite variables, each uncorrelated from a l l others and each accounting for different amounts of the information contained in the original variables. Thus one of the major criticisms of social area analysis, that the constructs are derived a priori with no subsequent check for their v a l i d i t y , is overcome. For each of the factors an index, or set of-scores, 3 is calculated and mapped to provide a concise and systematic description of areal differentiation. Among procedures used to explore the question of why areas are differentiated is that of comparing observed spatial structure, as can be described using factorial ecology, with patterns predicted by descriptive models of land use and social geography. These models generate either zonal, sectorial, or multiple nuclei forms of growth and structure. Burgess (1925) suggested that socio-economic status varies directly with distance from the city centre in concentric zones. Based on a study of residential rents, Hoyt (1939) proposed that better quality residences move outwards from the city center in a sectoral pattern along transportation routes and along higher ground while areas between sectors f i l l in with lower quality residences. Harris and Ullman (1945) suggested that land use patterns develop around several nuclei within the city rather than around a single center. 1.2 Purposes of the Study The basic purpose here is to follow the methods of factorial urban ecology for Greater Vancouver. A selection of variables is made from the Census of Canada for 1961. These are factor-analysed to produce the most important underlying dimensions of areal differentiation, and these dimensions are compared with those postulated in social area analysis and those found for other North American c i t i e s . The spatial structure is f i r s t displayed by means of maps. Then some investigation is done into the correspondence between this structure and that predicted by the descriptive models. 4 The f i r s t step in factor analyzing an assembled data base is the calculation from the data base of a product-moment correlation matrix, which measures the association between each variable and every other. The a b i l i t y of the correlation coefficient to reflect accurately the true association is impaired by any bivariate non-norma1ity. A further purpose of this study is to develop and apply data normalization techniques, and to examine the hypothesis that data non-normality - almost universal to social s c i e n t i f i c research - can significantly impair the a b i l i t y of the factor analysis model to achieve i t s intended goals. This is done by replicating the analysis for the original data base and for i t s normalized version and by then comparing the results. Implicit in the use of areal data is the assumption "... that the only variance of interest occurs between the observational units; their internal v a r i a b i l i t y is ignored [Berry, 1971, p. 209]." The areal unit used in this analysis is the enumeration area, which is considerably smaller than the census tract or i t s equivalent used in every study to date known to the author. Another purpose of this study is to investigate the importance of the level of data aggregation. Different levels of aggregation w i l l result in different spatial d e t a i l , ecological correlations, and data r e l i a b i l i t y . An examination is made of the hypothesis that the internal v a r i a b i l i t y of demographic, socio-economic and housing characteristics of urban dwellers within census tracts i s high enough to mask important information about the population. The approach used is to consider census tracts as aggregations of enumeration 5 areas and to compare the information on areal differentiation lost through averaging with this aggregation to that lost with one achieved with an optimal analytic grouping procedure. 6 CHAPTER 2 - FACTORIAL URBAN ECOLOGY 2.1 Social Area Analysis Social area analysis originated in sociology as a technique for classifying census tracts so as to define intra-metropolitan community areas. The technique was f i r s t applied to Los Angeles (Shevky, Bell & Williams, 1949) with the major statement appearing in 1955 (Shevky & Bell). It rests upon postulates about social change, derived from the work of Clark (1951), Ogburn (1933), and Wirth (1938), inter a l i a . It suggests that by considering cross-sectional attributes of the postulated elements of change, societies can be stratified along three separate dimensions, termed (by Bell) socio-economic status, family status, and ethnic status. Census characteristics are selected as possible measures of the constructs (see Table 2-1), and derived measures calculated from these, which are then combined into indices. These indices are used to characterize the tract populations within urban areas, and ultimately to produce a typology of social areas. The technique has come under severe criticism primarily because of the lack of a clear connection between the postulates of social change and the rea l i t i e s of areal residential differentiation between census tracts. Hawley and Duncan (1957) claim that no explanation is given for the differences between areas within c i t i e s , and specifically suggest that the Shevky & Bell theory is an ex post facto attempt to accommodate similar 7 Table 2-1 Social Area Analysis Constructs Constructs Sample Statistics Derived Measures I Economic Status (Social Rank) Years of schooling Employment status Class of worker Major occupation group Value of home Rent by dwelling unit Plumbing and repair Persons per room Heating and refrigeration Occupation Schooling Rent II Family Status (Urbanization) Age and sex Owner or tenant House structure Persons in household F e r t i l i t y Women at work Single-family dwelling units III Ethnic Status (Segregation) Race and nativity Country of birth Citizenship Racial and national groups in relative isolation Source: Adapted from Shevky and B e l l , 1955. empirical results. This has been virtual l y admitted by Bell and Moskos (1964). Hawley and Duncan also comment that census data for census tracts are insufficient in scope and detail for use in the analysis of ecological structure (see Chapter 3), and that census tracts may mask too much heterogeneity of population characteristics. This last problem is the topic of Chapter 5. Some researchers, however, have commented more favourably on the approach, primarily because of i t s capabilities in the description of how, as distinct from the explanation of why, areas of the city d i f f e r one from another, and hence i t s value in providing sample frames (Herbert, 1967). 8 Testing the validity and completeness of the hypothesized constructs of social area analysis has been done almost exclusively with the use of factor analysis. Before further discussion of social area analysis and i t s successor factorial ecology, i t is suitable to examine the methods of factor analysis and the characteristics which make i t suited for the investigation of urban ecological structure. 2.2 Factor Analysis The basic purpose of factor analysis is to replace a number of variables, each measured on some collection of subjects, with a smaller number of factors. Each factor is equivalent to a variable. Its data values, or scores, can be calculated for every subject, and i t s degree of association with any other variable can be measured. The factors are established in such a way that each w i l l summarize, or replace, a set of the original variables which are highly interrelated. In short, "... a principal objective of factor analysis is to attain a parsimonious description of observed data [Harman, 1967, p. 5]." The mathematics of factor analysis are complicated, but a knowledge of them is not indispensible to grasping the most important concepts of the technique. The interested reader is referred to thorough treatments extant in the literature, especially Harman (1967), Horst (1965), and Rummel (1970). What follows here is intended to describe the technique and the few of i t s many possible alternative procedures which have received wide use in factorial ecologies. The discussion follows the general outline of procedures given in Figure 2-1. FIGURE 2-1: FACTOR ANALYSIS RESEARCH DESIGN FLOW DIAGRAM 9 Design Goals I Data Distributional Transformations Matrix Transformation Communality -# of Factors Unrotated Factors Orthogonal Rotation Oblique Rotation Higher Order Factors Interpretation New Data Cycle Heavy arrows indicate the route taken by the majority of factorial urban ecologies. Heavy boxes indicate steps which have received special attention in this study. Source: Adapted from Rummel, 1970, p. 158. 10 The articulation of design goals is a necessary f i r s t step in the use of factor analysis in urban ecological investigation. Before collecting data and commencing analysis, the researcher must know the capabilities and limitations of the technique and how these relate to theoretical structures and possible research questions. In urban ecology the basic purposes are: to determine whether the constructs of social area analysis exist for the urban area under study, to see how important they are in accounting for areal differentiation, and to produce factor scores which can be mapped to display the ecological structure and which w i l l be used for any further analysis. The next step in the analysis is the assembly of the data base. Almost a l l factorial ecologies have used published census data for census tracts, often supplemented with some geographical information, such as distance from each tract to the peak land value road intersection (Murdie, 1969). (The heterogeneity of census tract data relative to the enumeration area data used in this study is the subject of Chapter 5.) Next, distributional transformations are applied to the data so as to achieve the normality of distribution assumed for later steps in the analysis and possibly to compensate for data error and for extreme values. "Although the data may already have the required distributions, the effect of improper distributions is so great on factor results as to prescribe the consideration of transformations as a possibility in every design [Rummel, 1970, p. 163]." (The development of transformation techniques and their impact on the model in this study are the topic of Chapter 4.) 11 The matrix transformation required next is the reduction of the data matrix to a form suitable for factoring. Although other procedures are possible (see, for example, Horst, 1965, pp. 286-335), the one almost invariably followed is to extract from the data a matrix of correlations, measuring the association between each variable and a l l others. While i t is reasonable to use any measure of association, such as Spearman's r or the Phi coefficient, the Pearson product-moment correlation coefficient is almost always used in factorial ecologies. Once the correlation matrix is formed, the factoring procedures begin. The diagonal of the correlation matrix is replaced with estimates of commun-al i t y , a decision rule is established for the number of factors, and unrotat-ed factors are extracted from the correlation matrix. Note that the original data is not considered again until factor scores are to be calculated. Two approaches to the calculation of unrotated factors have received almost exclusive use, one being the principal components model, or component analysis, and the other being the principal axes model, or common factor analysis. Central to the difference between them is the concept of communality. The variance in each variable can be considered to consist of two parts - that which the factor model can account for, and that unique to the variable, including measurement error. The communality of a variable is that part common to the factor model. In the principal components approach i t is assumed that the communality of each variable is 1, or that a l l variance is accounted for by the model. One's are entered on the correlation matrix diagonal and the factors are extracted, one for every one of the original 12 variables. The decision rule on number of factors is then used to determine how many of these to rotate. In the principal axes approach i t is assumed that the communality of each variable is less than 1, or that only part of the variance of each variable can be explained by the factor model. However there is no direct solution to the values of the communalities, and so estimates must be made. A number of different estimates have been used but the most common is the squared multiple correlation coefficient of each variable with every other. The extraction of unrotated factors is done by solving the correlation matrix for eigenvectors, vectors orthogonal one to the other which span the space of the matrix, and their associated eigenvalues, measuring the importance of the eigenvectors. The decision rule for the number of factors to extract is usually to use a l l eigenvectors whose eigenvalues are greater than 1. This indicates that they contain at least as much information as a single variable. Since the communality of each variable can now be uniquely determined, this can be used as a new estimate to the final communality. The factoring procedure can then be repeated until two sets of communalities differ from each other by a suitably small amount (this iteration on communalities is available in some computer programs, for example, Halm, 1971). There is a large number of alternative procedures possible at several stages in the i n i t i a l factoring, very few of which have been discussed here. Every different choice can lead to a different result, yet the maximum difference which could occur w i l l always decrease with an increase in the number of variables being factored. For example, i t is the author's experience that the choice of communality estimates for factoring anything 13 over approximately 75 variables has an insignificant effect upon the final communalities, even with iteration. This is primarily because the number of diagonal elements in a correlation matrix is a small proportion of the total number of elements in the matrix. Differences in choice rules for number of factors and in the factoring procedures used w i l l influence the unrotated solution but the normalization of variables may have more impact than any changes in procedure (see Chapter 4). The next step is the rotation of the factors. The purpose of rotation is to ensure that each factor is aligned as closely as possible to a cluster of interrelated variables. Rotation changes the relation between the factors and the variables and also the relative importance of each factor. The com-munality of each variable and the total variance accounted for by the factors are unaffected. Rotation procedures exist which either do or do not maintain the orthogonal quality of the unrotated matrix, wherein each factor is uncor-rected with a l l others. When correlations among factors are permitted, the rotation is termed oblique. Relationships between factors and variables are described in a factor matrix, whose elements are referred to as loadings. Two matrices are needed to f u l l y account for these relationships; the factor pattern matrix containing factor coefficients, and the factor structure matrix containing the correlations between the factors and variables. For orthogonal relations, however, these matrices are identical and no distinction is necessary. (See Harman, 1967, for a complete account.) The desired quality of the rotated factors is termed 'simple structure'. This is the condition where each variable loads only one of the factors but this is rarely achievable. For the general case, Thurstone has 14 developed five c r i t e r i a for simple structure (1947, p. 335). Let the rows of a factor matrix contain the coefficients for variables, and the columns those for factors, and let m be the number of common factors. Then: 1. Each row should have at least one zero. 2. Each column should have at least m zeros. 3. For every pair of columns there should be several variables whose entries vanish in one column but not in the other. 4. For every pair of columns, a large proportion of the variables should have vanishing entries in both columns. 5. For every pair of columns there should be only a small number of variables with non-vanishing entries in both columns. In the foregoing the factor matrix could be either the factor pattern or the factor structure, depending upon the specific rotation used. These c r i t e r i a cannot be incorporated directly into a calculation pro-cedure. Many methods have been devised, however, which try to achieve these objectives using various indirect c r i t e r i a . The orthogonal varimax rotation maximizes the variance of the squared column factor loadings. This variance w i l l attain a maximum when a l l loadings are either zero or one. Among ortho-gonal rotations this criterion is considered the best (Harman, 1967, p. 311; Rummel, 1970, p. 392). Several oblique rotation c r i t e r i a are available, such as maximizing the sum of the fourth powers of row factor loadings in the oblimax method. However no one scheme is clearly better than a l l others (Rummel, 1970, p. 411). Note that the correlation matrix for oblique factors can be subjected to a second factoring to produce higher order factors. 15 The rotation almost universally employed in factorial ecology is the orthogonal varimax. The rationale for not using an oblique rotation is usually that the orthogonality of factors corresponds to the discrete quality of the three constructs of social area analysis (as in Murdie, 1969, p. 72). Some studies using oblique rotations, however, have found that factors corres-ponding to the constructs were highly correlated (as high as 0.73 in B e l l , 1955), yet a recent study of Montreal found a maximum correlation of only 0.23 (Haynes, 1971). As a solution to this problem, Hunter has suggested "... using a variety of different computing algorithms to obtain i n i t i a l sol-utions, finding both orthogonal and oblique derived solutions, comparing the results, and determining which common factors emerge independently of the. factor analytic model used [1972, p. 109]." Once rotated factors have been extracted from the correlation matrix, the next step usually taken in factorial ecologies is the computation of factor scores. For most factoring procedures these scores are generated using regression estimation methods. The scores give, for each factor, a measure of the position of each subject relative to a l l others. They are usually mapped to display the spatial distribution of each factor, and are often used for subsequent analysis. Interpretation of the results includes identifying the rotated factors. This is done by determining from the structure matrix (and from the pattern matrix with oblique rotations) which variables are strongly associated with each factor. Interpretation also includes comparing the revealed structure with that of other factorial ecologies, and often attempting to explain why the structure occurs. _ ... 16 2.3 Factorial Urban Ecology Factorial urban ecology has i t s beginnings in the attempts of Bell (1955) to use factor analysis to test empirically the constructs he had helped to derive. Since that time, in the neighborhood of one hundred factor analytic studies of urban differentiation have been published (a number of these are reviewed in Murdie, 1969; Robson, 1969; and Rees, 1971, 1972). Generalizations concerning the number and nature of resulting factors are d i f f i c u l t to make, however, because of differences between the studies (see Janson, 1969; and Rees, 1972, for an extended discussion). The number of variables used has ranged from the six of social area analysis to around 150, with 60 to 100 occurring most often. The selection of variables varies quite widely, due to changes in their a v a i l a b i l i t y from census to census and from one country to another. Though the census tract is the almost universally used unit of observation, i t varies considerably in average size, in v a r i a b i l i t y in size, and in homogeneity from city to city. Differences in the factor models used also, of course, hamper the search for generalizations. For North America, nevertheless, the three constructs of social area analysis almost invariably emerge as factors. Most studies find more than three factors but these have often been features of either the data input or the place. That the three factors of socio-economic status, family status and ethnic status are universal suggests that these are indeed basic separate components of areal differentiation within c i t i e s . 17 CHAPTER 3 - PROCEDURES OF THE STUDY 3.1 Previous Work on the Urban Ecology of Greater Vancouver Although several authors have examined sociological aspects of Vancouver (for example, P a t i l l o , 1969), only two studies have addressed themselves to the overall structure of the Greater Vancouver area. The f i r s t was a social area analysis by Bell (1965), primarily concerned with tabulating and mapping selected biographical characteristics by census tract. In the tradition of Shevky and Bell he constructed a socio-economic index, comprising income, occupation and education characteristics, and defined social areas based upon occupation and education characteristics as one dimension, and f e r t i l i t y and one-persone households as the second. Intended primarily as a presentation of census data, the work includes no analysis apart from the calculation of indices. The only other assessment of the overall social structure of Greater Vancouver was the factorial ecology of Peucker and Rase (1971). They established four factors from 79 variables, using census tracts as their areal unit. The rotated factors from this study are summarized in Table 3-1. The f i r s t factor, corresponding with socio-economic status, was considered to have a sectorial distribution. Factor two, family status and l i f e cycle, appeared to have a zonal pattern. The third factor included median income and ethnic (Asiatic) status. The fourth factor was referred to as mobility and population growth and, although i t s spatial pattern appeared to be clustered, the authors concluded that i t had a zonal pattern. 18 TABLE 3-1 Summary of Rotated Factor Loadings from Peucker and Rase Correlation Variable with Factor Factor 1: Socio-Economic Status (28.3 percent of Total Variance) Percent with University Background .929 Percent of Male Labour Force Managerial .910 Percent of Male Labour Force Professional and Technical .907 Average Female Income .892 Income Per Family .882 Percent of Female Labour Force with Income over $6,000 .863 • Average Male Income .824 Percent of Male Labour Force with Income over $6,000 .808 Median Value of Owner-Occupied Dwellings .793 Rooms Per Dwelling .707 Percent of Families with Income Under $5,000 - .701 Percent Lutheran - .735 Percent of Male Labour Force in Transport and Communication - .795 Percent of Male Labour Force with Income $3,000 - $6,000 - .817 Percent of Male Labour Force Craftsmen - .863 Factor 2: Family Status and Life Cycle (19.5 percent of Total Variance) Percent of Population of Age 0-11 .964 Children Per Family .957 Percent of Households with 5 or More Persons .946 Percent of Families with 3 or More Children .914 Percent of Households in Single Detached Dwellings .854 Percent of Families with Children Attending School .834 Percent of Population Attending School .816 Percent of Female Population Aged Over 15 in the Labour Force -.748 Percent of Households Tenant Occupied -.763 Percent of Population Born Outside Canada -.799 Percent of Population of Age Over 65 -.865 Percent of Households Apartments -.877 Percent of Families Without Children -.958 19 i TABLE 3-1 (cont'd.) Correlation Variable : with Factor Factor 3: Median Income and Ethnic Status (10.4 percent of Total Variance) Percent of Male Labour Force in Service and Recreation .744 Percent of Population of Asiatic Origin .676 Percent of Male Labour Force Clerical -.712 Percent of Male Labour Force in Sales -.713 Percent of Population Originating from British Isles -.729 Percent of Female Labour Force with Income $3,000 - $6,000 -.733 Percent of Families with Income $5,000 - $10,000 -.787 Percent of Female Labour Force Clerical -.848 Percent of Population with High School Background -.876 Factor 4: Mobility and Population Growth (7.2 percent of Total Variance) Percent of Population Movers Within Canada .937 Percent of Dwellings Occupied 0-2 Years .799 Percent of Families with Head Under 35 Years .624 Percent Population Increase 1961-66 .580 Percent Population Increase 1956-61 .518 Percent of Dwellings Occupied Over 6 Years -.911 Percent of Population Non-Movers -.924 20 The authors noted that ethnic groups other than Asiatic had very l i t t l e association with the four factors. They speculated that this might result from a dispersion of these groups throughout the area, but noted that the use of a data base with greater areal detail might be necessary for the inclusion of the variables in ethnic factors. They concluded that "... It would comfort the authors very much i f they could say that the ethnic minorities are more integrated in Vancouver than in other North American c i t i e s , but such a hypothesis can only be tested by using census enumeration areas as a basis [p. 86]." This thesis uses enumeration areas as the unit of analysis, thus affording the opportunity to test the hypothesis. This w i l l be done in subsequent chapters, along with testing the idea that distributional non-normality may also hinder the inclusion of these groups. 3.2 Structure of the Present Study Before describing the procedures used in this analysis, the interconnectedness of i t s components must be recognized. For instance, data normalization dealt with in Chapter 4 uses the results of the factor analysis discussed in Chapter 6 for interpretation (see Figure 3 - 1 ) . The unit of observation for the data base used for the analysis is the enumeration area. The investigation of the impact of this low level of areal aggregation is based on factor scores. This is dealt with in Chapter 5, in which maps of social areas are also presented. The presentation of results, then, is complicated by this use of factor analysis output in discussions of characteristics of the data base used 21 FIGURE 3-1 RESEARCH DESIGN FLOW DIAGRAM CHAPTER 3 4 4 .4 6 6 5 5 6 Data Data Normalization Interpretation Correlation Matrix Factor Structure Factor Scores Correlation Matrix Factor Structure I Factor Scores Aggregation of Enumeration Areas Testing Census Tracts Spatial Structure 22 for the analysis to investigate the spatial structure of the city. The procedure followed to help achieve a coherent presentation is to provide in this chapter not only a summary of the procedures used but also a brief outline of the results. 3.3 The Data The data used in this analysis were extracted from 1961 Census of Canada computer tapes. These contained 2,694 different variable values for each of the 1,267 enumeration areas in Greater Vancouver. Thirty of the enumeration areas were omitted from the study because their populations resided in hospitals or hotels. For the remaining 1,237 enumeration areas, 192 variables, as defined in Appendix A, were assembled. Some variables required data from other sources. Variable 1, population density, was calculated using a value of net residential land area estimated in the following way: a map of enumeration area boundaries was drawn based on the descriptions of the boundaries given in Canada (1961); the area of each was measured and recorded; the proportion of the area of each in residential land use was estimated by superimposing on the boundary map one of several land use maps acquired from the municipalities in Greater Vancouver; and f i n a l l y the area was multiplied by the residential land use proportion to give net residential land area. In the process of estimating residential land use proportions, enumeration area population centres of gravity were also estimated and recorded on the boundary map. Variable 4 was then assembled by superimposing upon the boundary map a map of isochrones of travel to city centre provided by the Planning 23 Department of the City of Vancouver. The variable value for each enumeration area was then estimated by interpolating between isochrones. The population centres of gravity were then digitized and used as the basis of calculation for Variable 2, population potential. This v/as calculated as follows: Let D.. be the straight-line distance in miles between "ij the i * th and j 1 t h enumeration areas. Let P. be the 3 population of the j'th enumeration area. Then population potential for the i'th enumeration area is given by the expression 1237 n Variable 3, population increase from 1956 to 1961, v/as calculated using the estimates of 1956 population given in Canada (1961). The main reason for choosing such a large number of variables was to take f u l l advantage of the wide range available (published data for census tracts in 1961 includes only 181 variables). Variables were i n i t i a l l y chosen from the f u l l range on the basis that they might be reasonable indicators of areal differentiation. Variable values had to vary widely for different enumeration areas, and variables with a large number of values a l l equal to each other were avoided. Indeed, investigation of the variables during normalization procedures with regard to these c r i t e r i a lead to the elimination of quite a few and the modification of others. An advantage of having such a large number of variables was that certain characteristics could be finely subdivided with l i t t l e fear of weighting the information on differentiation unduly with those characteristics. For instance, 18 variables are used for age of population. This fear was also allayed by the use of the varimax rotation. This rotation has a characteristic referred to as factorial invariance, wherein the factors derived from a given set of variables w i l l change only a small amount with the addition of more variables. This was f i r s t shown by Kaiser (1958) and has since been verified in factorial ecologies by Schmid and Tagashira (1964) and Sweetser (1965). 3.4 The Analysis After the data were assembled, the analysis commenced with the development and application of normalization techniques. The removal of almost a l l univariate non-normality with the techniques ensured that the correlation coefficients calculated as the f i r s t step in the factor analysis more accurately reflected the true associations between variables. The normalized data were then factor analyzed on the University's IBM 360/67 computer using the program FAN (Halm, 1971) modified by the author to suit the special needs of the study. The principal axes model with squared multiple correlations as communality estimates was used in extracting unrotated factors. The 21 factors with eigenvalues greater than 1 were rotated using the varimax criterion and regression 25 estimates of their factor scores computed. The seven most important rotated factors were then chosen for investigation based on interpretability and variance contribution. The same analysis was repeated using untransformed data and a comparison of results was made to assess the impact of the normalization. The factor scores from the normalized data model were then used as data in the analysis of the effects of areal aggregation. Aggregation necessarily increases within-region v a r i a b i l i t y , equivalent to loss of information. The effects of this increase upon ecological correlations was investigated. Since the Ward algorithm hierarchical grouping procedure minimizes at each grouping step the total within-region variance, i t was used to produce an "optimal" aggregation of enumeration areas into 122 regions. The resulting information loss was then compared with the loss for the city's 122 census tracts to check for their optimality. Attention was then turned to the investigation of the city's ecological structure. To display the revealed structure the factors were f i r s t mapped. Since the enumeration areas are so small, i t was assumed for the purposes of mapping that the change in value between adjacent enumeration areas is linear. This leads to isopleth maps, different from the choropleth maps customarily used in factorial ecology. The mapping was done on a digital plotter controlled by a set of computer programs written by the author. (The algorithms for contouring are described in Coulthard, 1972, and Coulthard, Patterson and Herring, 1972.) 26 The procedure followed to compare observed structure with the described models (see Section 1.1) did not include s t a t i s t i c a l analysis, such as used by Murdie (1969). To avoid the assumptions required for such an approach, the detailed and complex patterns revealed in the factor score maps were simplified and a visual comparison made. This simplification was achieved by aggregating enumeration areas for each of the f i r s t four factors, and drawing maps for each showing for each region the average score of included enumeration areas. 3-5 The Results Detailed accounts of results are l e f t to the remaining chapters. A summary of the factors is presented here, however, for the sake of continuity. To f a c i l i t a t e interpretation of the factors, the order of the variables in the table of rotated factor loadings was changed to c l a r i f y the patterns of correlations. Then correlation values less than 0.4 in magnitude were omitted, except i f a variable's highest value with any factor was less than this (see Appendix Table E-2). The interpreted factors (Table 3-2) predictibly include the three constructs of social area analysis. However, factors 1, 3, 5 and 6 are a l l required to describe the family status construct, as factors 2 and 7 are required for the socio-economic status construct (see Table 2-1). Factor 4 corresponds very closely to the ethnic status construct, encompassing a wide range of ethnic groups. Many variables related to the constructs, however, appear in factors of lesser importance. 27 Table 3-2. Summary of Factors. Factor Percent of Common Variance I Percent of Total Variance Number of Variables with Highest Correlation 1. Family Status 26.41 17.25 49 2. Socio-Economic Status 18.83 29.55 48 3. Housing Tenure 9.66 35.86 18 4. Ethnic Status 6.21 39.92 14 5. Families with Older Children 4.77 43.04 10 6. Rural-Urban Status 4.36 45.89 12 7. Retail and Clerical Workers 4.06 48.54 7 28 CHAPTER 4 - DATA NORMALIZATION 4.] Introduction Factor analysis, l i k e most multivariate s t a t i s t i c a l models, makes underlying assumptions about the nature of the data. Of major importance is the assumption of bivariate normality of each variable with every other so that the product-moment correlation matrix, central to the factoring procedure, contains entries which have s t a t i s t i c a l validity. In the course of preparing the data for analysis, procedures were developed by which variables were transformed to a distribution which was normal or close to normal. Factor analyses were performed using both the original data and the normalized data and the results were compared in an attempt to determine the impact of the normalization; that i s , how factor analysis might be sensitive to non-normality of variable distributions. 4-2 The Assumption of Correlation Assumptions about the distribution of variables are made for deriving st a t i s t i c s to test hypotheses and for determining the distributions of these test s t a t i s t i c s . In the model used for this analysis - principal axes with varimax r o t a t i o n - no significance tests are available either for determining the number of factors or for determining the effectiveness of the rotation. However the nature of the elements in the correlation matrix is of major importance to the model, since after the formation of the matrix the original 29 data is abandoned, at least until factor scores are estimated. If each coefficient meets the assumption of product-moment correlation then valid s t a t i s t i c a l tests can be performed on i t , such as determining whether i t is significantly different from zero (Kowalski, 1972). If the assumption is not met then the coefficient loses much of i t s power as an indicator of association (Farlie, 1960; Frechet, 1959), since equal values do not necessarily indicate an equal amount of association. Thus the information about association contained in the matrix w i l l have undesirable inaccuracies which may have significant ramifications throughout the analysis. The assumption made for tests of significance of the product-moment correlation coefficient is that of bivariate normality (Fisher, 1915) - see Figure 4-1. Testing for bivariate normality is cumbersome and in the main unsatisfactory (Kowalski, 1970, 522-527) though reliable tests have been developed (Mardia, 1970). Transforming univariate distributions to optimize this characteristic would, except for very small numbers of variables, be exceedingly d i f f i c u l t i f not impossible. For example with the 192 variables in this analysis there are 18,336 entries on either side of the diagonal in the correlation matrix, each calculated from two variables having a bivariate distribution. Attempting to transform each variable so as to optimize the normality of these bivariate distributions would be a task of monumental proportions and is for a l l practical purposes out of the question. However, i f two variables have a bivariate normal distribution then i t necessarily follows that each of them has a univariate normal (or simply a 1 normal') distribution, although the reverse is not true. So a normal distribution for a l l variables is a necessary condition for bivariate normal Frequency FIGURE 4-1 A BIVARIATE NORMAL DISTRIBUTION 31 pairs, though not a sufficient one, while non-normality definitely precludes bivariate normality. (Note also that bivariate normality necessarily results in the linearity of a relationship between two variables while the reverse is not true.) The normalizing procedures used in this analysis are applied to univariate distributions. Other research has shown that "for a wide variety of non-normal (X_,Y_) distributions, coordinate trans-formations to normality result in bivariate distributions which are 'more bivariate normal' than the original observations (Kowalski, 1970, p. 535), "and by their use, one may increase the power of the normal theory test for independence [Kowalski & Tarter, 1969, p. 144]." While they do not necessarily lead to bivariate normality and hence to s t a t i s t i c a l l y significant correlation coefficients, failure to use transforms to normality would, for the bulk of real-world data, a r b i t r a r i l y preclude significance tests". 4.3 A Procedure for Testing Normality The procedure for testing normality used here is based on the skewness and kurtosis of univariate distributions. (These are test s t a t i s t i c s calculated from the data, and they describe the distribution of a variable, skewness being equivalent to asymmetry, and kurtosis to peakedness). (Mathematical description in Appendix B, along with a computer program which performs the tests.) Figure 4-2 shows hypothetical histograms for various values of skewness and kurtosis. The distributions of these test sta t i s t i c s cannot be expressed in functional form but their characteristics have been developed and tables of c r i t i c a l values calculated (Pearson, 1930; and Pearson & Hartley, 1966). Least squares curves were fi t t e d to the tabulated 32 C) NEGATIVE SKEWNESS E) "NEGATIVE KURTOSIS FIGURE 4-2 HISTOGRAMS SHOWING VARIOUS VALUES OF SKEWNESS AND KURTOSIS 33 values and these were used to interpolate c r i t i c a l values for various numbers of data points. This procedure yields values which give results significantly closer to the tabulated values, never differing by more than 1 in the last significant figure, than the functional approximations commonly used (see, for example, Veldman, 1967, p. 182). Relative to the Chi-square goodness-of-fit test for normality, probably the one most frequently used by social scientists, this procedure has both advantages and disadvantages, in the Chi-square test, class intervals of data values are chosen and frequencies for the normal distribution are calculated. Using the Chi-square s t a t i s t i c , these theoretical frequencies are then compared with observed frequencies in each class to determine whether there is any significant difference between the two distributions (see Kowalski, 1970, pp. 518-521 for discussion of the lack of unanimity on exactly what test s t a t i s t i c to use.) Relative to this test the main advantage enjoyed by the skewness and kurtosis tests is that the numerical values of skewness and kurtosis give a concise, though not complete, description of the nature and magnitude of non-normality in a variable, information not available from the Chi-square test. This characteristic proved to be of considerable value in applying normalizing transforms (as is discussed below). The skewness and kurtosis tests extract more information from the data since each datum value is used in their calculation, whereas the Chi-square test summarizes the data according to class membership. Also these tests are operationally i f not computationally more straightforward. With regard to the Chi-square test there is considerable discussion about choice of class limits and also about dealing 34 with classes for which the theoretical frequencies are very small. The skewness and kurtosis tests do, however, have some disadvantages. Since they use the third and fourth powers of data values, sampling errors and inaccuracies become important since small changes in data values can have a large influence on the results. Together they do not, s t r i c t l y speaking, comprise a test for normality since they deal only with two of an i n f i n i t e number of moments that might be calculated, and which have expected values. Anomalous situations can arise where.particular combinations of data values in non-normal distributions w i l l lead to the conclusion of normality (see Figure 4-3), though this occurs only rarely in practice and can be easily identified by inspection of a histogram. These disadvantages notwithstanding, the skewness and kurtosis tests for normality are considered to be preferable to the Chi-square test. 4. 4 The Effects of Non-Normality on Correlation A knowledge of the influence of various values of skewness and kurtosis on the value of the correlation coefficient, would be useful for predicting the effect of non-normality, and for deciding whether or not to normalize or to use rank-order correlation methods. An investigation of this influence was performed using a sample of over 600 correlation coefficients of variables having 1237 data points. The procedure used was to generate a random normal variable and then to calculate the correlation coefficient " r " of that variable with a transformed verson of i t s e l f , the transform being chosen so as to introduce non-normality SKEWNESS = -0.0411 KURTOSIS = 2.8339 1 1 — 1 1 1 n SKEWNESS = 0.1353 * KURTOSIS = 2.7586 * n i l IT ITUT FIGURE 4-3 HISTOGRAMS OF EXAMPLES OF UNUSUAL DISTRIBUTIONS WHICH ARE NORMAL ACCORDING TO SKEWNESS AND KURTOSIS TESTS .( * 0.05 < p < 0.01; ** p < 0.01) 36 of varying degree. Since the introduction of non-normality into the univariate, and therefore bivariate, distributions invalidates significance tests, the purpose here is only to give a general indication of the nature of the influence of non-normality. In Figure 4-4 the coefficient is plotted against the kurtosis of the transformed variable. The top curve shows that whenever the kurtosis of the non-normal variable departs from 3, the expected kurtosis for a normal distribution, the value of r_ drops. The lower curves are for cases where the transformed variable was f i r s t adjusted so that when i t was normal the r_ value was less than 1, here 0.8, 0.6, 0.4 and 0.2. When this variable was then transformed into non-normality, the r. value again dropped but the decrease was always less for a smaller i n i t i a l _r value. The scatter of observed r_ values for the top curve is shown in Figure 4-5. The transform used introduced not only kurtosis but also skewness, and this figure seems to indicate that negative skewnesses lead to smaller decreases in _r values for the same kurtosis. This is born out by Figure 4-6 where the same values of r_ as in Figure 4-4 are plotted against skewness instead of kurtosis. Positive skewness leads to lower jr values though this effect diminishes as the correlations decrease. When the original variable is slightly non-normal the shape of the correlation-normality line changes. Figure 4-7 shows the correlation of variously distributed variables against variables having moderate positive skewness and kurtosis. Values of r_ equal to 1 are achieved only for variables with nearly the same distribution, and for those variables with higher skewness and kurtosis the _r value is higher than for the case of using a normal variable, and lower for those variables having negative skewness and higher kurtosis. When this procedure is tried for variables with high 1.0 ~"I -0.8 i 0.6 -L 0.4 -0.2 i o.o H 1 1 1 1 1 1 1 r 1 2 3 5 . 1 0 20 50 100 200 KURTOSIS FIGURE 4-4: CORRELATION COEFFICIENT VERSUS KURTOSIS FOR NORMAL VARIABLES CORRELATED WITH NON-NORMAL VARIABLES: 5 CASES. 1 2 3 5 10 20 50 100 KURTOSIS FIGURE 4-5: CORRELATION COEFFICIENT VERSUS KURTOSIS FOR NORMAL VARIABLES CORRELATED WITH NON-NORMAL VARIABLES: SCATTERGRAM 39 FIGURE 4-6: CORRELATION COEFFICIENT VERSUS SKEWNESS FOR NORMAL VARIABLES CORRELATED WITH NON-NORMAL VARIABLES: 5 CASES n 1 1 1 1 1 — i 1 -6 -4 -2 0 2 4 6 8 SKEWNESS 0 #4 H i i i i i i i i i i -8 -6 -4 -2 0 2 4 6 8 10 12 SKEWNESS FIGURE 4-7: CORRELATION COEFFICIENT VERSUS SKEWNESS FOR VARIABLES HAVING A MODERATE POSITIVE SKEWNESS AND KURTOSIS CORRELATED WITH NON-NORMAL VARIABLES 41 positive skewness and kurtosis (Figure 4-8) again r_ values close to 1 are obtained only for variables of nearly the same distribution, but when the variable i s normal i t has negative skewness the r_ value drops considerably. In the case of variables having low kurtosis but zero skewness the effect is to drop the r. values for a l l cases of skewness along with kurtosis from the position for normal variables (Figure 4-9). This investigation into the influence of non-normality on correlation coefficients is intended only to give an indication of what might be encountered. No hypotheses have been tested, and results contrary to those suggested here may not be uncommon. This may result from a number of causes. One of these is that in skewed variables the extreme values may increase the correlation of the variables considerably such that when the skewness is removed by normalizing, the _r value w i l l decrease. Another w i l l occur when the normalizing process leads to a bivariate distribution which is decidedly not bivariate normal, that i s , when the relationship between the variables is markedly non-linear. That aspect of the influence of non-normality on correlation which has been emphasized here is the relationship between the numerical values of skewness and kurtosis and the numerical value of the correlation coefficient. Previous research has been focused on the distribution of the correlation coefficient around i t s mean value for various types of non-normality; that i s , a more detailed look at the scatter around a point, on for example, the curve in Figure 4-5 (see Kowalski, 1972; Heath, 1961; Norris & Hjelm, 1961; and Farlie, 1960). -] I I 1 I I 1 1 1 1 -8 -6 -4 -2 0 2 4 6 8 10 SKEWNESS FIGURE 4-8: CORRELATION COEFFICIENT VERSUS SKEWNESS FOR VARIABLES HAVING A HIGH POSITIVE SKEWNESS AND KURTOSIS CORRELATED WITH NON-NORMAL VARIABLES ™ 44 For some research the rank-order correlation such as Spearman's r (see Farlie, 1960) may be a viable alternative to the product-moment coefficient discussed here. This w i l l be the case when association is the final object of investigation and prediction techniques such as regression are not needed. It was found that the calculation of Spearman's £ gave coefficients slightly lower than the equivalent for completely normal variables-values comparable to those obtained for variables with no skewness and a low kurtosis. In fact when the ranks assigned to data values in the process of calculating Spearman's _r are used in a product-moment calculation the two techniques give identical results, and these ranks are distributed with l i t t l e skewness and with a kurtosis of between 1.7 and 1.8. The location of rank order coefficients on the normality-correlation curves are given on Figures 4-4 and 4-6. Note that these values would be obtained for a l l values of skewness and kurtosis since the transform used to introduce non-normality does not change the ranks of data values. For some investigations the rank-order coefficient w i l l therefore be a useful alternative even with interval data i f that data is markedly non-normal. For factor analysis, however, this is not the case, since i f factor scores are to be calculated and perhaps subsequently analysed, the distribution of the original variables is of importance. This is because the factor scores are calculated as the linear combination of data values and w i l l be normal only i f the original variables are normally distributed. This w i l l be important inasmuch as the normality of factor scores is a desirable attribute for differentiating the subjects and for meeting assumptions which may be required for further analysis of scores. 45 4.5 Normalizing Procedures - Scaling Data The normalizing procedures used in this analysis owe much of their capabilities to a procedure of data scaling applied before transformation. In the process of investigating various transforms for the kind of non-normality they were capable of removing, i t was noticed that i f a linear transformation (that is a transformation of the form "aX + b" where "a" and "b" are constants) was applied before the non-linear transform, the range of skewness and kurtosis values which could be removed became very wide, and the effect of the transform could be adjusted with relative ease by changing the scaling coefficients. An example of this procedure was applied to the log transform is given in Table 4-1. Here the linear transform parameters are adjusted such that the minimum is kept at 1.0 and the standard deviation changed. In example 1 the variable is scaled to give a standard deviation of 0.5 and the log values taken of this scaled variable giving skewness and kurtosis values not significantly different from those values for a normal distribution. For examples 2 and 3 the higher values of skewness and kurtosis in the original distributions are successfully removed by scaling the variables such that they have higher standard deviations. Note that the scaling procedure has no effect on either the skewness or kurtosis of the original variable but rather changes the effect of the non-linear transform applied after the scaling. The transformed data values T = log(Xc-j + c^) where X = the original data values: r = 1og(c,(X + •_!)) c l c = log(c, + log(X + - 1 ) -Table 4-1. Examples of Linear Transforms Versus Removed Non-Normality for the Log Transform Example Number 1 2 3 Original Skewness 0.795** 2.68** 12.6** Original Kurtosis 4.10** 13.9** 214.** New Minimum 1.0 1.0 1.0 New Standard Deviation 0.50 10.0 5000. New Skewness -0.04 0.07 -0.02 New Kurtosis 3.08 2.97 3.04 47 which can be expressed as T = K, + log(X + K?) where K, = log(c,); The influence of this scaling can perhaps be better understood by looking at what happens to individual data values in the process of transformation. Figure 4-10 shows that as the range of data is increased by scaling, the right end of the data points becomes increasingly more compressed after applying the log transform. Thus the right t a i l of a positively skewed variable w i l l become shorter as the range of data values is increased before taking log values. The above discussion considers only the log transform, but the same basic procedure of scaling data before applying a non-linear transform proved to be equally effective in controlling the effect of the distributional change for a l l transforms examined. 4-6 Transforms Used Approximately 40 non-linear transforms were examined in the search for a set suitable for covering the range of types of non-normality. Of these, seven were eventually chosen as covering as wide a range as possible and these are presented in Table 4-2. These were incorporated into a computer program with which the data of this analysis was transformed. Column 2 of the table gives the restrictions imposed upon data values by the mathematical nature of the transforms and column 3 gives the actual constraints used in the program. For the purpose of scaling the data before transforming, two constants are needed and these were expressed in FIGURE 4-10 THE INFLUENCE OF SCALING UPON THE EFFECT OF THE LOGARITHM TRANSFORM' For each example the top line shows the relative position of five data points with minimum and maximum values as shown. The bottom line indicates the relative position of each after taking i t s logarithm value. Table 4-2. Non-Linear Transforms Used for Normalizing 1 Transform 2 Restrictions on Data Values 3 Constraints Used 4 Parameters •] to Estimate 5 Type of Skewness Removed 6 Type of 2 Kurtosis Removed 1. log(X) X>0 minimum=l S positive positive 2. vT X>0 minimum=l S positive positive 3. arctan(X) none none M,S various positive 4. log X 0<X<1 0<X<1 M,S none negative 1 - X 5. arcsin(X) 0<X<1 0<X<1 M,S negative none 6. arcsin(X^) 0<X<1 0<X<1 M,S positive none 7. X e none minimum=l S negative positive M = mean; S = standard deviation positive = >3; negative = <3 50 the form of the new mean and standard deviation of the data after scaling. For transforms 1, 2 and 7 the minimum scaled value was ar b i t r a r i l y set equal to one leaving only one parameter to estimate, and for this the standard deviation was used. The majority of real-world data has positive skewness and kurtosis and this case is covered by the f i r s t three transforms. Another common case is that of negative skewness and positive kurtosis and for this transforms 7 and also 3 handle most variables. Other types of non-normality tend to be less common but most can be covered by transforms 3, 4, 5 and 6. The ranges of non-normality which each of the transforms can deal with is indicated more e x p l i c i t l y in Figure 4-11. The log transform, suitable for positive skewness and kurtosis, is relatively easy to apply, requiring only one parameter to be estimated. An estimating curve (Figure 4-12), generated from a sample of random normal variables, was used to give a starting parameter for scaling, and based upon the results of this f i r s t attempt, iterations were performed to achieve a final solution. I f , for a given kurtosis, the skewness of a variable was not close to that indicated on the graph, i t was found that the log transform would not give satisfactory results. The square root transform was found to be useful for variables where there was only a small amount of non-normality but i t can remove more skewness for a given level of kurtosis than can the log transform. The estimation of the parameter for this and for most of the other transforms was not systematized into a graph as was the case for the log transform but rather the cumulative experience of the writer was the only guide. The arctan transform proved to be very versatile. By suitably adjusting the scaling parameters i t 53 could be used to remove positive kurtosis and almost any skewness encountered, though this skewness could not be much larger in absolute value than that indicated for the log transform. The simultaneous estimation of two scaling parameters did, however, prove to be troublesome since a change in one of them influenced both the skewness and the kurtosis of the transformed variable, but experience did help simplify the process. Transforms 4 , 5 and 6 turned out to be d i f f i c u l t to apply as well, each also having two parameters to estimate, and each being sensitive to data values close to the permissible values of 0 or 1 after scaling. Combinations of skewness and kurtosis not indicated in the table were found to be vi r t u a l l y impossible to normalize though this did not become a major problem since they were found to occur very infrequently. The exponential transform, number 7 , behaved in a fashion very similar to the log transform. The scaling parameters were different but the ratio of skewness to kurtosis removable for a given value of kurtosis was very similar and the arctan transform could be used for variables with different ratios. Iterating from i n i t i a l parameter estimates to final choices was often d i f f i c u l t though the process was simplified by having the transforming programs give results not only for the parameters chosen but also for values on either side of the estimate. Experience gained with the f i r s t variables also proved to be of considerable help with later ones as some knowledge was acquired for the sensitivity of the various transforms to changes in parameters. However the process was by no means straightforward and an average of about 20 iterations was eventually needed for each variable transformed. It is f e l t that automatic iteration procedures could be 54 incorporated into a computer program which would greatly simplify the normalizing process, but this has yet to be carried out. 4.7 Normalizing the Variables Normalizing procedures were applied to a l l 192 variables before the factor analysis was performed. The skewness and kurtosis of each before normalizing are given in Appendix Table C-l, and also plotted in Figure 4-13. When the figure is compared to figure 4-11 i t can be seen that most of the variables seem to f a l l within the capabilities of the transforming procedures, with the log transform appearing to be suitable for many of the variables. The skewness and kurtosis of each after applying the transforms are given in Appendix Table C-2, and plotted in Figure 4-14. The majority f a l l within the 99% confidence interval of both skewness and kurtosis, and a l l but 15 within that of skewness alone. The most extreme case of non-normality removed is that of variable 1, where the original skewness of 9.59 and kurtosis of 135.1 were removed as shown in Figure 4-15. A breakdown of results by transform and by non-normality removed is given in Table 4-3. The arctan transform, that most frequently used, was often applied to variables which appeared to be suited to the log transform but which turned out to have more kurtosis relative to skewness than that transform could handle (see Figure 4-11). The variables for which skewness could not be removed came almost exclusively from a class of variables characterized by a relatively large number of observations having identical values, a situation commonly found in many areas of social s c i e n t i f i c research. The in a b i l i t y to normalize o O L . 00 _l l_ x x x x^x x X x x X 2 3 10 20 50 100 KURTOSIS 200 500 FIGURE 4-13 DISTRIBUTION OF 192 VARIABLES BEFORE NORMALIZING CO - , CvJ —I 00 UJ CO o —\ 4 5 KURTOSIS FIGURE 4-14: DISTRIBUTION OF 192 VARIABLES AFTER NORMALIZING cn 57 n n SKEWNESS = 9.590** KURTOSIS = 135.1** H SKEWNESS = 0.0723 KURTOSIS = 2.828 FIGURE 4-15 DISTRIBUTION OF VARIABLE 1 BEFORE AND AFTER NORMALIZING J l A) ORIGINAL VARIABLE 86 SKEWNESS = 2.445** KURTOSIS = 12.10** i n i 11 i n i — i n J l B) TRANSFORMED VARIABLE 86 SKEWNESS = 1.220** KURTOSIS = 2.494** J l FIGURE 4-16 EXAMPLE OF UNSUCCESSFUL COORDINATE TRANSFORMATION TO NORMALITY 58 Table 4-3. Summary of Results in Normalizing 192 Variables Number of Variables Transform Skewness = 0, Kurtosis = Skewness f 0, Kurtosis f 3 3 <3.5 >2.5 >2.0 >1.0 Totals (p>.01) (p<.01) (p<-01) (p<.01) (p<-01) (P<.01) 1 y = log(x) 45 5 12 17 12 91 2 y = /x 2 2 3 y = arctan(x) 65 1 66 4 y = "log ( y ^ ) 1 1 2 5 y = arcsin (x) 2 1 2 2 7 6 y = arcsin (x^3) 10 2 1 13 7 y = e x 5 1 3 2 11 Totals 130 3 8 17 19 15 192 59 them results from the reliance of these and any other transforming techniques upon their a b i l i t y to change the distances between data values by differing amounts across the range of those values. If a number of observations have the same value, the distances between them " cannot be changed by any transform. If this situation occurs with a sizeable percentage of the observations, the transforms w i l l be severely restricted in doing their task. Transformation of variable 1 (see Figure 4-15) was possible only because each datum value was unique, result-ing from the way in which the variable was constructed. With variable 86, however, a large number of observations have a value of 0.0 (Figure 4-T6) and normality cannot be achieved. Note that the use of rank-order techniques on variables exhibiting this type of behaviour w i l l not solve the problem since the ranking procedure is also limited to changing distances between observations. In the process of choosing variables for inclusion in the analysis, some attention was paid to their distribution and also to their amenability to transformation, considered as an indicator of how well they differentiated the population under study. Because of this some variables were eliminated and others combined or otherwise changed. The remaining variables which could not be transformed to normality were included because of their substantive importance. For a few of the variables which are shown to be normal after transformation, examination of their histograms indicated that normality had not really been achieved but the skewness and kurtosis tests were unable to detect this situation (see discussion above). 60 Transformations were not adjusted, however, since no means was found to remove non-normality not indicated by skewness and kurtosis measures. 4.8 The Effect of Normalization on the Correlation Matrix As suggested in an earlier section of this chapter, the anticipated effect on correlations of normalizing the 192 variables used in the analysis was that there would be a significant increase in absolute correlation coefficient values. Figure 4-17 shows the result. Any values which do not l i e on the diagonal have, of course, been affected by the transforming procedure. It is clear that the expected pattern of increase did not occur. Indeed an examination of the upper left-hand and lower right-hand quadrants reveals a significant number of coefficients which have changed sign - an outcome which was not expected. As is indicated in Table 4-4, numerical differences between the matrices are slight, with an average increase in association about 5%. When the coefficients for the original data are segregated according to sign, however, those which had been negative before transformation become slightly closer to zero, on average. The reasons for a decrease in absolute value of coefficients after transforming are not entirely clear. Two possibilities are suggested here, though neither has been empirically tested. The f i r s t is that since the normalizing of marginal distributions in no way ensures bivariate normality, i t is possible that transformation may produce strong non-linearity between some variables which did not exist before transformation. The second is that with non-normal variables a small percentage of values a long way 61 FIGURE 4-17: THE EFFECTS OF NORMALIZATION ON THE CORRELATION MATRIX Table 4-4. Characteristics of Correlation Coefficients Before and After Data Transformation Characteristic of Correlation Coefficients Below the Diagonal For Original Data For Transformed Data Difference Root Mean Square .18200 .19018 .00818 Average Absolute Value .19484 .20862 .01378 Among the 130 Successfully Transformed Variables .22092 .23524 .01432 Average of Originally Positive Values .20255 .21507 .01251 Average of Originally Negative Values -.18787 -.18504 .00283 63 from the means relative to the rest of the values can give a coefficient whose absolute value is much higher than expected. Successful normal-izing procedures w i l l eliminate these extrema and the coefficient for the transformed distributions may be much lower in absolute value than for the original data, whether or not bivariate normality is achieved. 4.9 The Effect of Normalizing on the Factor Model In spite of the small apparent change in correlation coefficient values, the normalization of data does have a major impact upon the factor model. A variety of indicators lead to the conclusion that the normalization of data has permitted a significant increase not only a more parsimonious description of the data, but also in the a b i l i t y of the rotation procedure used to achieve "simple structure," a prime goal in factor analysis (Harman, 1967; Rummel, 1970). In the model based on normalized data, the average squared multiple correlation of each variable with a l l others, the average communality of each variable, and also the number of factors extracted based on the criterion of eigenvalues greater than 1.0 are a l l considerably lower than in the model based upon the original data (Table 4-5). It is important to note, however, that the relative decrease in the number of factors is proportionately greater than that in the average communality. As a result each factor in the normalized data model, on average, accounts for more of the total variance in the correlation matrix than in the original data model. For each of the f i r s t eight factors, except the fourth, this 64 Table 4-5. Characteristics of the Factor Models for Original and for Normalized Data Characteristic Original Data Normalized Data Average Squared Multiple Correlation 0.89028 0.78329 Average Final Communality 0.72954 0.65321 Number of Factors with Eigenvalues Greater than 1.0 27 21 65 is the case (Figure 4-18), so that i t is the most important factors which benefit most. From factor 9 to factor 21 the original data model factors account for more common variance as well as total variance (Table 4-6), re-emphasizing the amount of information contained in the f i r s t eight factors of the normalized data model. Indeed i t is not until the 14th factor that the original data model's factors account for as much of cumulative total variance (Figure 4-19). These results, along with the decrease in the average squared multiple correlation suggest that normalizing has "tidied up" the correlation matrix - increasing inter-correlation among particular groups of variables and decreasing those among the rest of the variables. This in turn suggests that correlation coefficients based upon normalized data may come closer to measuring true association than do those based on data containing significant non-normality (as do the original data here), a result in keeping with the findings of other research (Kowalski, 1972). An examination of the changes in the pattern of factor loadings reinforces the idea that simpler structure has been achieved by normalizing data. The most important factors are l i t t l e affected, but, as the factors account for less of the variance in the models, the more dramatic are the differences between the models. Factors 1, 2, and 3 are essentially the same for both models (see Table 4-7 and Appendix Tables E-2 and E-4). Factor 2 for the normalized data model, however, includes variables 75, 76, 86 and 182, a l l related to the numbers of people attending university, which appear in the original data model as the only variables whose _j ! , ( , ! ! 1 5 10 15 20 25 27 FACTOR NUMBER FIGURE 4-18: TOTAL VARIANCE ACCOUNTED FOR BY EACH FACTOR FOR BOTH ORIGINAL DATA AND NORMALIZED DATA FACTOR MODELS Table 4-6. Characteristics of Factor Scores for Original Data and Normalized Data Factor % of Common Cumulative % of Number Variance Total Variance Skewness Kurtosis NORMALIZED DATA 1 26.41 17.25 .1935** 3.294* 2 18.83 29.55 :4555** 2.492** 3 9.66 35.86 -.3084** 3.206 4 6.21 39.92 .0031 3.948** 5 4.77 43.04 -.2618** 3.427** 6 4.36 45.89 1.1384** 5.289** 7 4.06 48.54 -1.3219*'* 5.955** 8 3.39 50.75 .2242** 6.296** 9 2.77 52.56 .0923 4.117** 10 2.15 53.96 .2013** 6.096** 11 2.06 55.31 .0200 3.219 12 2.05 56.65 -.0802 3.990** 13 2.01 57.96 -.2005** 3.809** 14 1.90 59.20 -.0345 2.496** 15 1.56 60.22 -.2721** 4.360** 16 1.52 61.21 .2731** 3.362* 17 1.32 62.07 .3664** 6.090** 18 1.28 62.91 .9169** 6.116** 19 1.26 63.73 .0601 5.558** 20 1.25 64.55 -.4515** 4.116** 21 1.21 65.32 -.4739** 4.392** ORIGINAL DATA 1 22.78 16.62 .9926** 4.115** 2 14.36 27.10 -.6708** 9.373** 3 7.94 32.89 .9845** 3.810** 4 7.02 38.01 -.9061** 4.740** 5 4.21 41.08 -1.0422** 6.919** 6 3.96 43.97 .0754 4.046** 7 3.06 46.20 -.0394 4.232** 8 2.82 48.26 .1563* 7.917** 9 2.81 50.31 -3.4172** 27.689** 10 2.74 52.31 2.3698** 12.724** 11 2.74 54.31 -1.0912** 4.721** 12 2.60 56.21 6.5017** 89.722** 13 2.24 57.84 6.5164** 85.964** 14 2.12 59.39 -6.6754** 87.217** 15 1.77 60.68 1.8081** 11.216** 16 1.71 61.93 1.1713** 6.624** 17 1.66 63.14 1.2610** 13.640** 18 1.65 64.34 2.3700** 14.654** 19 1.53 65.46 .4284** 15.805** 20 1.41 66.49 -.9757** 5.822** 21 1.38 67.50 4.4106** 75.626** 22 1.35 68.48 -.6465** 4.117** 23 1.32 69.44 3.3627** 48.382** 24 1.31 70.40 4.6685** 55.210** 25 1.29 71.34 -.0260 4.784** 26 1.13 72.16 .6442** 11.279** 27 1.08 72.95 1.6342** 23.135** FIGURE 4-19: COMPARISON OF FACTOR STRUCTURES FOR ORIGINAL DATA AND NORMALIZED DATA FACTOR MODELS Table 4-7. Numbers of variables for which the highest loading is with one of factors 1 through 7 - for original data, for normalized data, and for both. Factor Number Number of Variables Original Data Normalized Data Both 1 47 49 43 2 37 48 33 3 15 18 14 4 17 14 5 . 5 9 10 1 6 8 12 1 7 5 7 4 70 highest loading is on factor 14. It also includes variables 87, 101, 102 and 114 which appear in factor 4 in the original data model. The fourth factors for the models are, however, considerably different. For the normalized data model the variables whose highest loadings are on this factor include nine of the eighteen variables measuring birth or origin in a foreign country other than Britain, and thus i t is considered to be a comprehensive index of ethnic status. For the original data model the two variables which have the highest loadings on factor four are 53 and 54, Chinese birth and origin, but no other variables measuring place of birth or origin have their highest loadings with this factor. The ethnic variables for Central Europeans and Russians are included in factor 8 and those for Italians and Scandinavians in factor 10. The factor includes variables which in the normalized data model occur in a variety of other factors, two in factor 1, four in factor 2, and the variables 148 and 149 measuring television ownership in a separate factor of their own. Factor 5 of the original data model is almost entirely different from i t s normalized data counterpart. It includes four of the five variables whose highest loadings are with the eighth factor of the normalized data model, plus an assortment of other variables. Factor 5 of the normalized data model is almost identical to factor 6 of the other, while factor 6 includes five of the six variables of factor 9 in the original data model plus seven others which are closely related 71 conceptually. Both factors 7 are similar to each other. So for the f i r s t seven factors, those chosen for interpretation, major differences appear between the two models. For the model using normalized data, several characteristics suggest that the normalization procedure has indeed led to a more simple structure. These factors account for more of the common variance in the models; greater numbers of variables have their highest loadings with these factors; and perhaps most important, the factors seem more easily interpreted; that i s , they include variables which seem more closely interrelated and to correspond more closely to theoretically derived hypotheses about urban areal differentiation. Another characteristic of the factor models enhanced by the normalization of data is the distribution of factor scores. Since these have been estimated from the data using a regression procedure they are linear combinations of the data. For data which are normally distributed, the factor scores w i l l also be normally distributed (Brunk, 1965, p. 229). As shown in Table 4-6 the factor scores are not perfectly normally distributed as measured by skewness and kurtosis, but the improvement over the original data model's score distributions is substantial. 4-10 Conclusions The data available to factorial ecologists are rarely normally distributed. This problem, as indicated in the discussion above, can have serious ramifications in the results of factor analysis. Basically, data non-normality impairs the a b i l i t y of the correlation coefficient 72 to measure true association between variables, and patterns of association are what factor analysis seeks to summarize. In this study i t was found that the masking of interrelationships among variables caused by non-normality affected the results in several ways. More factors were extracted with each accounting for less variance, on average, with the most important accounting for significantly less. Some of the more important factors were very similar but others differed in undesirable ways. They tended to be more specific, to include fewer variables, to differ in order of importance, and several sets of different but closely related variables were separated into their own factors. Further the factor scores generated for them also contained serious non-normality, a characteristic which would interfere with any further analysis using s t a t i s t i c a l techniques which assume normality. In addition to helping in the search for parsimony and for simple structure, the normalizing of data before factor analysis of census data would seem to produce results which w i l l be more easily comparable, ceteris paribus, from city to city and hopefully among countries as well. It is tentatively claimed that the use of normalization procedures such as the ones developed here w i l l have a beneficial impact on our a b i l i t y to describe and hence understand the ecological structure of the city. 73 CHAPTER 5 - AREAL AGGREGATION 5.1 Introduction In the most factorial urban ecologies to date, the areal unit used for a data base has been the census tract or i t s equivalent (Rees, 1971, pp. 223-224). The areal unit used for this study i s the enumeration area - in Greater Vancouver about one tenth the size of a census tract, on average - and the purpose now is to evaluate the significance of the finer detail of information which resulted from using enumeration areas instead of census tracts. The approach taken is to consider census tracts as aggregations of enumeration areas and to compare the information on areal differentiation lost through averaging with this aggregation method to that lost with the Ward algorithm analytic grouping procedure. The direct approach of replicating the analysis with both areal units and comparing the results was not considered suitable since factor analysing census tract data would have necessitated using for census tracts a subset of the variables used for enumeration areas to avoid problems of singularity in the correlation matrix. This would mean that the factors produced for each areal unit would not necessarily measure identical dimensions. Also maps of factor scores for the different areal units would have to be compared for correspondence; techniques exist which are suitable but only where detail in each pair i s similar (A.H. Robinson, 1968) while i t is the loss of detail resulting from aggregation which is being investigated here. 74 5.2 Areal Units The basic units of analysis suitable for this study are areas of urban land each including individual people and households whose character-i s t i c s constitute the data for the area. These characteristics exist (or could be calculated) for any level of area! aggregation, whether single building l o t s , c i t y blocks, enumeration areas, census tracts or even muni-c i p a l i t i e s . Whatever areas are chosen as a data base, they are known as "modifiable units [Yule & Kendall, 1950, p. 312]," since there is no one level of aggregation that is better a priori than a l l others. The basic concern of this enquiry is the spatial coincidence and covariance of the characteristics of modifiable units at some level of aggregation - the essence of "ecological" investigation. This i s distinct from investigation at the level of the individual person where the concern is with individuals' characteristics and their interrelationships, and where any aggregation is undesi rable. Areal units at any level of aggregation can be considered to be "regions" since they are the combinations of units at a lower level of aggregation and themselves can be aggregated to create new units. Aggregation of areal units can be done in a number of different ways. One important distinction among them i s whether a contiguity constraint is imposed - that is whether only spatially contiguous units can be joined together. To ensure spatially compact regions this constraint must be imposed, and so, i t is always assumed. A second distinction is whether 75 analytic methods are used to meet a specific aggregation criterion. This distinction w i l l be elaborated upon in the evaluation of census tracts as an aggregation scheme in section 5.6. 5.3 Some Characteristics of Data at Different Levels of Aggregation An important attribute of data at a low level of areal aggregation is internal homogeneity. In most urban areas the majority of people tend to l i v e next to those with demographic characteristics similar to their own (see Salins, 1971). The smaller the areal aggregations of basic units, the less chance there is for these regions to contain units which have great dissimilarities of characteristics, regardless of how units are aggregated. With regions which are relatively large, however, there is a much greater chance that they can be subdivided into smaller units which are very different from each other, depending of course upon the regional-ization procedure. Since the datum value for a given variable for a region is the average of the values for the units in that region, aggregation constitutes a spatial averaging process where, the more dissimilar the units within each region, the more spatial information is lost, corresponding with a commensurate increase in within-region v a r i a b i l i t y . This loss of information not only implies loss of spatial resolution both for data and for analytical results such as factor scores, but also can have substantial impact upon the numerical values of measures of association among variables (see section 5-5). In the choice of a given level of aggregation for a study the assumption is made, usually i m p l i c i t l y , that, for the purposes of that study, information loss within the regions that constitute the units used i s small enough to be disregarded. 76 There are several desirable characteristics of data at a low level of areal aggregation which merit attention. One is that large numbers of units w i l l result in large data-handling problems. For example, i f data by building lot had been available for this study, each of the 192 variables woulld have had about 180,000 observations. Recent advances in computer techniques have greatly simplified handling large data sets, but simultaneous analysis of that volume of data remains infeasible. A second problem is that of data r e l i a b i l i t y . Error in data can be classified into systematic and random components and each w i l l be affected differently by the size of homogeneity of units. With purely random components larger units are more desirable because the confidence interval around the expected value for a unit is proportional to the square root of the number of individuals within the unit (note that internal homogeneity is not a consideration). With perfectly systematic error components, the size and homogeneity of units i s unimportant. However i f the error in a variable depends on other variables then proportional differences in error between units w i l l be accentuated for units with high internal homogeneity, usually the case with smaller units. For example, i f , in the census of Canada, undercount of population is a constant per-centage of the total population in a unit, then homogeneity and size of units w i l l not affect this error. However i f i t is related to another population characteristic, say income, then i f units are chosen to maximize homogeneity of income, units with high income w i l l have maximally 77 different undercount from units with low income. The more heterogeneous the units, the smaller w i l l be this difference in error between units. In summary, then, units at a low level of aggregation are desirable because of low internal v a r i a b i l i t y , leading to better spatial resolution. They are undesirable because of data-handling problems, because of the relative importance of random components of data error, and because internal homogeneity w i l l accentuate differences in systematic error between units i f that error is not independent of a l l other possible variables. 5.4 Ecological Correlation Central to the technique of factor analysis is the correlation matrix of each variable with every other. Having census data available at different levels of aggregation leads directly to the question of how and why correlation coefficients calculated for these different levels w i l l depart from each other. In the following discussion, i t is assumed that there are units at one level of aggregation and regions at a higher one consisting of discrete numbers of units such that each unit belongs to one and only one region. The units here could be at any level of aggregation but in any case they are modifiable - the true "individual" is undefined. Correlation coefficients calculated with aggregations of units as their data base are referred to as "ecological" correlations (Robinson, 1950), the characteristics of which have been investigated by several writers [W.'S. Robinson, 1950; Duncan, et a l , 1961; and Hannan, 1971 inter a l i a ) . 78 The following discussion departs in several aspects from their work for the purposes of c l a r i t y and of indicating the importance of heterogeneity within the units. Define: R = the total number of regions N = the total number of units N r = the number of units in the region r X^r (i=l N; r=l R) The value of the variable X for unit i which belongs to the region r. v _ i N r The average value of X for a l l N units i belonging to • r N I X i r r ier the region r; the within-region mean. N x = 1 V x T n e average value of X for a l l units i ; the universal N i r mean. , N o y = ' V (X. - X ) The universal variance of X. X ^ •,•=] R N 2^ WVY = 1 v v (X. - X V The within-region variance of X. n r=l ier i R 2 EVY = - y N (X - X ) The between-region or "ecological" variance of X. For a second variable Y, each of the above terms would have an equivalent definition. N C X Y = -ft -j=] ( X i r ~ X ^ ^ Y i r " Y ^ u r n v e r s a ^ covariance of variables. X and Y for a l l i . 79 R N R WC = 1 I I ( X i r - X r ) ( Y - r - Y » The within-region covariance of X * Y N r=l ier and Y. 1 R ECw = TT I N v . ( x v - X )(Y - Y ) The between-region or ecological XY N -j r .r . .r covariance of X and Y. The universal product-moment correlation coefficient of X and Y is given by: RXY " ^  • The within-region correlation of X and Y is given by: w c Y Y W RXY = / — — 5 and the between-region or ecological correlation of X and Y i s : ER^Y = XY / E V „ E V X-'Y The universal covariance can be partitioned into the within-region covariance and the ecological covariance: CXY = W CXY + E CXY ' This has been shown before (as suggested by Kenney & Keeping, 1951, p. 274) but i t s derivation i s given here for the sake of completeness: X i r " X . - X 1 r - X . + X . r - . X . r " ' X i r - X . r ' + < X . r - X . » • Multiplying each side of this expression by the same side of an equivalent expression in Y gives 80 < X i r " X . " Y i r " V.' " [ < X i r " X.r> + < X.r " X . ' ] [ ' V i r " Y . r > + < Y.r " Y.'] • Multiplying out the RHS, summing over a l l units and dividing by N gives C X Y - | Z ( X 1 r - X ) ( Y i r - Y ) = 1 H X i r - X > r ) ( Y 1 r - Y > r ) + T r H x . r - x > ) ( Y > r - Y > ) + ^ ( x l r - x > r ) ( Y > r - Y B ) + I I (X - X )(Y. - Y ). N L .r . i r . r ' Replacing the terms for within- and between-region covariance and rearranging the remaining terms gives C x y - WCxy + EC x y + 1 I (X rY_ - X„J - X_Yjr + X Y r) After summing over N, each expression in the f i r s t remaining summation term is equal to NX Y , so this term is equal to zero. For the second summation term we can f i r s t sum over each unit in a region and then sum over a l l regions CXY = W CXY + E CXY + if 4, ,1 < X i r Y . r + X . r Y n > " 2 X . r Y . r > • W CXY + E CXY + V j , < 2 f ,r X.r Y.r " 2 N r X . r Y . r > • Therefore C X Y = W ^ X Y + EC X Y as claimed. (1) Using this result and the three definitions of correlation coefficients above, 81 R X Y / V Y = W R X Y A V ^ + ^ X Y / W ' and the universal product-moment correlation coefficient i s : RXY = W RXY wv wv Y X ER VX VY X Y E V ^ Y ( 2 ) V Y • Rearranging terms, the ecological correlation coefficient is given by: WVxwVv RXY " W RXY/"OT-ERV.. -  i X Y (3) XXY E V X E V Y  VX VY 5.5 Ecological Correlation and Within-Region Variance Ideally, any aggregation of units w i l l preserve as well as possible the relationships among the variables measured for them; that i s , a "good" aggregation w i l l keep the ecological correlation as nearly as possible equal to the universal correlation for a l l variables X and Y. It is the purpose here to show that minimizing the within-region variance for the variables-is the-most- important -criterion in keeping ER^ y close to R^ y with aggregation. If ER^ y = R^ y then from equation 2 1 - / E V X E V Y ^XY V VX VY (4) R XY / W V XWV Y 82 The universal variance for any variable can be partitioned into within-and between-region components in a fashion similar to that for covariance: V X - WVX • EVX . If the within- and between-region variance are expressed as proportions of the universal variance, they w i l l sum to 1; ^ • f V - 1. (5) VX VX wv wv In examining equation 4, consider the case where X Y . VX~ VY Then, using equation 5, ^ X Y and V " V X Y WR ^ = — ^ - ^ = 1 (6) KXY WVy WVy WR This corresponds to the diagonal line on Figure 5-1 and here XY is at i t s RXY minimum. Its maximum value for any ^ VY occurs when M VX — 0 and i t has a VY VX relative maximum when W X —=» 1. When W X = 1, then E VX = 0, and so VX VX VX EC x y = 0. From equation 1, C x y = WCxy a n ( J S Q WV RXY = W RXY 83 84 and unless R X Y = 0> ER X Y cannot equal RXY. So i f the aggregation of data has resulted in the loss of a l l between-region variance for one variable, the ecological correlation between that variable and a l l others w i l l be zero and relationships at the unit level have been completely hidden by aggregating to regions. Similarly, in the case that = 0, W CXY = °' GXY = E GXY a n d s o VX 'EV Y RXY " E RXY ,/ V, WV and i f Y is also equal to 0, then R X Y = ERXY- This last result indicates v Y that i f within-region internal v a r i a b i l i t y is kept equal to zero in aggregation, relationships among variables w i l l be perfectly maintained. As within-region v a r i a b i l i t y increases from zero (and between-region variance drops from universal variance) with aggregation, the relationship between ER X Y and the other variables in equation 3 becomes more complex. Assuming for the moment that = *^ Y , the relationship is given in VX VY Figure 5-2 for the case that R X Y = 0.5. Since the within-region variance ratios are equal, then for a l l WRXY = R X Y = 0.5, ER X Y = R X Y as was shown above (see Figure 5-1). When WRXY is greater than 0.5;then ER X Y < R X Y and i f WRXY < 0.5, ER X Y > RXY. Of particular interest is the rate of departure of ER X Y from R X Y for a given value of WRXY k 0.5. For small values of the ratios WV then ER X Y is close to R X Y > but as the values 86 increase ER X Y departs more and more rapidly from RXY. When is adjusted such that M RXY = 1.5 for a l l values of ^  , VY R x y Vx a similar pattern is produced (Figure 5-3). The value of WRXY for which ER X Y is equal to R X Y has changed, of course, to 0.75, and the rate of change of ER X Y from R X Y for a given value of WRXY as ^ VY increases from VY zero is considerably less than for the case of = ^ Y . The combination of these effects results in values of ER X Y closer to R X Y for a l l values of WRXY less than about 0.4 than for Figure 5-2. However when the values of ^Y are considered (see Figure 5-1), the rate of change of ER X Y from R X Y for a given value of WRXY is considerably higher, and for a l l values of WRXY less than about 0.7, ER V V is further away from R v v than for W VX = W VY . VX VY With t h e r a t i o for one variable held constant, the effect of, changing^ the ratio for the other variable upon the relationship of ER X Y to WRXY is shown in Figure 5-4. As ^ Y departs from ^  = 0.2, the effect of the W RXY V y V x R x y ratio is to increase the value of WRXY for which ER X Y = R X Y, corresponding to the values of XY on Figure 5-1. RX88 89 In examining the interrelationship among the variables in equation 3 we have considered cases of three different ratios of to W VY for R Y V = 0.5. h VY WR The connection among these and between them and the XY ratio is suggested RXY in Figure 5-5. In this figure, each value of ER^ y would be represented by a curved surface. The lines for ER^ y values on Figures 5-2 through 5-4 are thus the intersections of these surfaces with a vertical plane as suggested in the Figure. For different values of R^ y the vertical spacing between ER X Y surfaces w i l l be the same above any point, but the distance above that WR point to the surface for a given ER X Y value w i l l be governed by the XY RXY ratio. For a l l cases considered, the only situation in which ERXy was equal . WR to R X Y was i f equation 4 was satisfied, requiring that XY be not less than RXY one. However this requirement is a d i f f i c u l t one to meet since, as W.S. Robinson has pointed out, " a l l available exidence is that (whatever properties X and Y denote) the correlation between~X-and Y is certainly not larger [in absolute value] for relatively homogeneous sub-groups of persons than i t is for the population at large [1950, p. 356]." As indicated above this w i l l lead to ER XY greater than one and empirical findings to date substantiate this RXY observation (W.S. Robinson, 1950; Yule & Kendall, 1950; Grinfeld & Griliches., 1960; Cramer, 1964). The conclusion of this section, however, is that whatever the value of WRXy, the departure of ERXy from RXy is minimized 91 whenever the within-region variances for both variables are also minimized. Since aggregation of units can only add to these variances, the best data base is one at the lowest level of aggregation suitable for data handling and at which data error is not too high. 5.6 Enumeration Areas and Census Tracts The smallest areal unit for which data were made available by the Dominion Bureau of Statistics in 1961 is the enumeration area (EA), with the data for each collected by one enumerator. Greater Vancouver is divided into 1267 EA's (30 of which were not used in the analysis because they consist of institutional buildings such as hospitals and hotels), the population in each ranging from 15 to 2619 with an average of 630. Their size may be as small as a single city block and is usually around six blocks in completely built-up areas, but ranges up to 36.1 square miles on the edges of Greater Vancouver. The census tract is the smallest areal unit for which the DBS publishes data (EA data i s provided on computer tapes with print-out of selected variables available). There are 122 census tracts in Greater Vancouver with an average population of about 6500 people, ranging in size from 0.23 to 59.5 square miles and in population from 165 to 15,093. Each tract is the aggregation of a discrete number of EA's with boundaries coinciding with those of EA's and data the sum of that for each included EA. In deciding upon the aggregation from EA's to census tracts, the DBS worked within the following three restrictions: 1) contiguity of EA's for each tract (with the exceptions of military lands and Indian reserves). 92 2) that tract boundaries coincide with municipal boundaries. 3) that a l l census tracts include roughly the same number of people and the same area. While meeting these restrictions, census tracts were chosen such that they were " f a i r l y homogeneous with respect to economic status and living conditions [Canada, 1963, p. 3]," though this characteristic was achieved without the help of analytic techniques. The boundaries of a l l census tracts and the estimated population centres of gravity of a l l EA's are presented in Map 5-1. Since EA's are the smallest unit available, their use as a data base w i l l ensure that internal v a r i a b i l i t y is at a minimum, though the DBS has given no indication of what magnitude i t might have. This also provides an opportunity to test, for the case of Greater Vancouver, how well the DBS has achieved i t s stated goal of minimizing internal v a r i a b i l i t y with respect to "economic status and living conditions" in aggregating from EA's to census tracts. The approach taken is to aggregate EA's so as to minimize within-region variance and to compare this aggregation with census tracts to see how close they come to being optimal. First the aggregation procedure is discussed (sections 5.7 and 5.8) and then the test of optimality is developed and applied (section 5.9). 5.7 The Ward Algorithm Aggregation Procedure The procedure used to aggregate EA's maintaining within-region variance at a minimum is the hierarchical Ward algorithm (Ward, 1963) 93 94 w i t h c o n t i g u i t y c o n s t r a i n t . Each EA i s considered to be a r e g i o n c o n t a i n i n g one u n i t and then cont inuous reg ions are combined, two a t a s t e p , u n t i l a l l EA's belong to one reg ion w i t h a record kept a t each step o f the reg ion t o which each EA belongs. The cho ice a t each s tep o f which two reg ions should be j o i n e d i s made accord ing t o which combinat ion w i l l c o n t r i b u t e the l e a s t t o the t o t a l w i t h i n - r e g i o n sum o f squares , summed over the t o t a l number o f v a r i a b l e s used. I f the number o f v a r i a b l e s i s M, then a m a t r i x o f d i s tances i s e x t r a c t e d f rom raw data c o n s i s t i n g , f o r each p a i r o f observa t ions i and j , o f the value M 2 D . . " = I ( X k , - X . . ) 2 (7) U k = 1 Ki k j where X k l- i s the i ' t h obse rva t i on on the k ' t h v a r i a b l e . The o b j e c t i v e f u n c t i o n used i n the a l g o r i t h m i s the e r r o r sum o f squares •M N » M ESS = X [ I ( X k . - X. j ' ] = N I WV (8) k=l i = l K i r K - r k=l x k which i s to be kept as smal l as p o s s i b l e a t each aggrega t ion s tep where X k i r D e l o n 9 s t 0 t n e r ' t n r e g i o n . Before the f i r s t g roup ing s t e p , the number o f reg ions i s equal t o the number o f u n i t s ( N ) , thus X k p = X k l - r f o r a l l r and ESS =0. When the f i r s t two reg ions i and j are j o i n e d , they are chosen such t h a t 2 D.• i s the s m a l l e s t i n the m a t r i x , s u b j e c t t o the c o n s t r a i n t t h a t i and j are con t i guous . Now two terms i n t h e ESS are non -ze ro , those f o r the two reg ions i and j t h a t are j o i n e d . So 95 ESS = = I k=l M U k j - • (Xkj - _ M ^ ) 2 M 1 M 9 - y j , < X « - X H ) 2 • 7 V 2 Since D.. is a minimum, then ESS is also a minimum and likewise total within-region variance is a minimum (equation 8). The aggregation procedure continues combining contiguous regions in the same fashion, adding the smallest possible increment to the ESS (and so to the sum of the within-region variances) for a l l variables until a l l regions have been joined into one. When regions to be joined contain more than one of the original one-member regions, the procedure is adjusted to take this into account (see Veldman, 1967, p. 310 for description of the adjustment). When aggregation is based on more than one variable, the question arises of how the departure of ecological correlation from universal correlation is affected by the relative sizes of within-region variance for each of the variables. That i s , i s i t preferable to keep a l l values equal as aggregation proceeds or should some of these terms be kept as small as possible and others be allowed to become quite large? The problem was examined by considering r for the -variables-X- and Y, the follow-ing two cases: 96 Case 1 wvx = wvY Case 2: ^ =f= ^ Y VX VY For both Cases: ^ X + = 2K, where 0<K<1 V V X Y The within-region variance ratio for each variable is kept equal for the f i r s t case but not for the second, but in each case the ratios sum to the same value. The problem then is to compare the values of E RXY " RXY for both cases for a l l possible values of R Y V, WRYV, M VX, and W VY. Since AY AY V V VX VY there appears to be no analytic solution to this problem, the approach used was to calculate the values of and for a sample of some 15,000 sets of correlation and variance values, the comparisons of which are given in Figure 5-6. It is clear that for a given value of R X Y, the value of A^  is smaller for values of WRXY most l i k e l y to occur, especially i f R X Y is close to 1 or -1. In conclusion, then, departure of ER X Y from R X Y is in general minimized i f ^ VX i s kept equal to M VY . V V VX VY 2 Since the contribution to the matrix of D.. values of any one i j variable is the sum of the squared differences between each data value and every other (equation 7), i t is clear that the importance of each 97 1.0-WRXY 0.0-1.0 RXY SB A 1 < A 2 T h e D i f f e r e n c e b e t w e e n and A2 depends W V Y W V Y u p o n K a n d u p o n — — a n d — — — f o r case 2. v Y A 1 > A 2 FIGURE 5 - 6 . THE DEPARTURE OF E R W FROM R w AND XY XY THE EQUALITY OF WITHIN-REGION VARIANCE FOR X AND Y. 98 variable relative to any others is related to the spread of i t s data values. A useful measure of this spread is the standard deviation of the variable, SDX = Jy^ • If the standard deviations of a l l M variables used for aggre-gation are set equal, then each w i l l contribute equally in the aggregation process and the within-region variance for each w i l l tend to remain the same at each aggregation step, though this cannot be assured since variables are not treated separately while aggregating, and the situation can be considerably influenced by a contiguity constraint. 5.8 Aggregating Vancouver's EA's The variables on which the EA's were aggregated were factors based on -normalized data. It was thought preferable to use factors instead of a l l the original variables, since they include a l l significant areal differentiation of EA's and exclude information unique to single variables (see Chapter 2) . A case has been made above for keeping the standard deviations of aggregation variables equal to each other but this creates a conflict. If a l l factors are weighted equally then the relative importance of each in the factor model is disregarded.—Since-this importance varies greatly, the procedure—-followed was to weight the standard deviation of each factor in proportion to the information i t contained (Table 4-10). The resulting discrepancies between within-group variances are small relative to the range of weightings, however, as w i l l be discussed in the next section. The aggregation of EA's was done with the computer program CGROUP (Patterson & Whitaker, 1973). The available computer had sufficient capacity to handle only 724 EA's at one time: the only way to aggregate 99 a l l 1237 EA's was to s p l i t them into sub-areas, to aggregate the EA's within these and then combine the results. Three sub-areas were chosen i n i t i a l l y - south of the north arm of the Fraser River including Richmond, Delta, Surrey and White Rock; the north shore including West Vancouver, North Vancouver municipality and North Vancouver City; and the rest, comprising Vancouver, Burnaby, New Westminster, Coquitlam, Port Moody and Fraser M i l l s . This choice seemed reasonable in light of the limited access across the bodies of water separating the sub-areas, and also since the time of i n i t i a l settlement and the nature of development of areas adjacent across the bodies of water are often very different. Unfortunately, this division s t i l l l e f t 942 EA's in the Vancouver-Burnaby-Coquitlam sub-area within which a reasonable dividing line did not appear obvious. To overcome this , an arbitrary s p l i t was made, to the west of which was a l l of Vancouver City except a small triangle in i t s south-east corner. The EA's in this western portion of the area were aggregated and a very dis-tinct boundary appeared which, as the EA's were aggregated step by step until only one region remained, was the second from last to be removed. This False Creek-Ontario Street boundary is indicated on Map 5-2 which shows the aggregation for Greater Vancouver when there are 23 regions remaining. Would this boundary have appeared i f i t had been possible to aggregate a l l EA's in this sub-area simultaneously? To answer this question aggregating was done with two subsets of about 150 EA's each which straddled the original arbitrary boundary. These were then aggregated as one subset and the influence of the boundary determined 100 by comparing the aggregations. Some effect was evident in regions whose EA's touched or were very close to the boundary but no effect could be detected one mile from the boundary except when very few regions remained. Since the False Creek-Ontario Street boundary was never closer than about i three miles from the arbitrary boundary, i t was concluded that the arbitrary boundary had no effect on i t . The EA's to the east of the False Creek-Ontario Street boundary were then aggregated and the results of the four separate aggregations were combined together so that they were in a l l likelihood identical to the results that would have been obtained i f a l l 1237 EA's had been aggregated simultaneously with no contiguities across the north arm of the Fraser or across Burrard Inlet. The boundaries of the regions for different levels of aggregation are presented in Maps 5-2 through 5-4. A visual comparison of Map 5-4 with census tract boundaries (Map 5-1) shows that the regions have a much greater variety of boundary shapes, of size, and of frequency over various parts of the map area. The proportion of regions in various municipality groups is quite similar for a l l three levels of aggregation (Table 5-1) but is considerably different from the proportions for census tracts for municipality groups 2, 3 and 5. 5.9 The Optimality of Census Tracts Regionalization can be considered as a spatial averaging process wherein the more dissimilar the units within each region, the more spatial information is lost. Within-region variance is a good measure of this loss and i t has 101 1 0 2 103 104 been shown that small values of within-region variance are closely linked to high correspondence between values of the ecological correlation coef-ficient and coefficients obtained for a lower level of aggregation. The use of the Ward algorithm for aggregating EA's ensured that the hier-archical regionalization produced had a minimum of within-region variance at each level and in this sense was optimal. A test of the optimality of census tracts can then be made by checking the significance of the differ-ence of within-region variance for census tracts and for Ward algorithm regions. For a given variable X, the term R r - I I 11 (x1r - x_r)2 ] WVY x " r=l ier gives within-region variance where the variance within each region is weighted by the number of units i t contains. 2 N WVY is an unbiased estimator of a , the population variance. N - R o If X i s normally distributed, thenN-WVx has the x distribution with 2~ " a N - R degrees of freedom (as in Brunk, 1965, p. 288). Now the random variable F = A-| / has, by definition (as in Brunk, 1965, p. 260), the F A~2 7 j 2 2 distribution i f A-j and A2 are independent random variables having x dis-tributions with j-j and ^  degrees of freedom respectively. For census tracts and Ward algorithm regions, the WV^  values can be considered inc pendent since the aggregation procedures are independently generated. Table 5-1. Relative Importance of Municipality Groups for Different Regionalizations Percent of Regions Included Ward Ale orithm Regions Municipality Group Census Tracts 122 64 23 1. Vancouver, University Endowment Lands 46.7 47.5 40.6 47.8 2. West Vancouver, North Vancouver, City and Municipality 9.8 18.0 17.2 17.4 3. Burnaby, New Westminster, Coquitlam City and Municipality, Fraser Mills and Port Moody 26.2 15.6 17.2 13.0 4. Richmond, Surrey and Delta 15.6 12.3 12.5 13.0 5. Indian Reserves and Military Lands 1.6 6.6 12.5 8.7 Total 100.0 100.0 100.0 100.0 106 Therefore N-WV XI ( M _ D ) A 2 F = 1 1 1 has the F distribution with N - R, and N - R 0 N-WVX2 1 2 (N - R 2 ) a 2 2 degress of freedom. The test of equality of within-region variances has 2 2 the null hypothesis H : a, = a 0 . Therefore o 1 2 WV (N - R ) F = XI 2/ can be used to test this hypothesis and construct WVX2(N - R-j) confidence intervals. The variables used in the test are each of the 21 factors, summar-izing the significant spatial variation within the study area. The assumption of the test that these variables are normally distributed i s not met for most factors (see Chapter 4), yet the test should be robust enough to accommodate their small departues from normality (Pearson, 1931). The assumption that the two regionalizations give independent WVx's is unlikely to be perfectly met though this was not investigated. Nevertheless the test is considered to be meaningful for the purposes at hand. Application of the test, the results of which are indicated by asterisks in Table 5-2, shows that for 18 of the 21 factors, representing 90.9% of the common variance in the matrix of correlations among the variables, the 122 Ward algorithm regions have WVX values numerically smaller than for census tracts. For seven of these factors, representing 70.6% of the common variance, the difference is significant at the 99% confidence level, Table 5-2. Ratios of Within-Region Variance to Individual EA Variance for 21 Factors for Census Tracts and Ward Algorithm Regions. WVY A 0 / Factor v / o X Number 122 Census Tracts 122 Ward Algorithm Regions 1 34.50 9.62** 2 22.53 12.21** 3 50.44 32.49** 4 37.82 28.90** 5 55.18 46.77** 6 28.62 28.62 7 35.40 33.35 8 53.03 43.61** 9 49.64 50.58 10 63.41 57.08* 11 73.18 68.79 12 69.12 65.72 13 34.74** 40.26 14 83.24 80.17 15 59.98 53.32 16 79.39 79.00 17 64.55 50.24** 18 70.95 64.90 19 72.39 67.08 20 73.36 68.24 21 58.65 52.52 Average , Weighted Average 55.72 49.21** 42.04 29.64** Weighted' Average— of First 7 34.85 19.66** Asterisks indicate lower variance - * p<0.05; ** p<0.01. ^Weighted by the variance accounted for by each factor in the factor model. 108 and for only one factor representing 2% of the common variance do the regions have WVX values significantly greater than for census tracts. Since the Ward regionalization is hierarchical, the WVX for census tracts can be compared with the WVX for any number of regions (Figure 5-7). The f i r s t through third factors require only 15," 28 and 32 regions respectively for these regions to differentiate the city as well as 122 census tracts. The impact on the WV values of scaling the standard deviations of V the factors proportional to the variance they account for in the correlation matrix is clear for the f i r s t four factors but for the remainder, each accounting for less than 5% of the variance, the WV values show l i t t l e V relationship to variance contribution. Yet for a l l but three of these the WV values are less than for census tracts which suggests that they had V some impact upon the aggregation. In any case, while the largest variance contribution is 21.8 times larger than the smallest, the largest WV is V only 8.3 times larger than the smallest. 5.10 Conclusion A low level of data v a r i a b i l i t y internal to units of analysis i s desirable for detail of spatial information and to avoid obscuring relationships which may exist among the data. In the case of Vancouver, EA's are probably as small a unit as w i l l have sufficiently small error in census data. The aggregation of these units minimizing within-region variance (with the Ward algorithm) leads to small information loss for 109 LU CO Z o I -u 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 U°J. d_2 u • i ? - L E G E N D -9 i w v 1 < w v 2 w v 1 = w v 2 w v 1 > w v 2 p=0.01 p=0.05 p=0.05 p=0.01 J2-L d —2 L " T ~ 20 " T " 60 40 80 100 r 120 140 160 180 200 NUMBER OF WARD ALGORITHM REGIONS 220 244 F I G U R E 5-7- C O M P A R I S O N O F W I T H I N - R E G I O N V A R I A N C E S - (1) W A R D A L G O R I T H M , R E G I O N S V E R S U S (2) C E N S U S T R A C T S . the two most important dimensions of areal differentiation (with 122 regions), but for the other 19 factors of the study even this optimal aggregation may obscure relationships existing at the EA level (see Table 5-1 and figures 5-2 through 5-4). Aggregating EA's to census tracts results in considerably larger within-region variances. It is the conclusion of this chapter that census tracts are unsuitable as an areal unit for analysis of census data. m CHAPTER 6 - THE URBAN STRUCTURE OF GREATER VANCOUVER Vancouver's urban structure is concisely described by the factors produced in this study. Before examining the factors in d e t a i l , consideration is given to the choice of the number of factors. Then the factors are discussed in relation to the postulated constructs of social area analysis. Factor score mapping procedures are considered next, followed by an examination of the spatial configurations of the factors, as revealed by the maps, and in comparison with theories of urban structure. 6.1 Choice of the Number of Factors In the normalized data factor model, 21 eigenvectors of the i n i t i a l factor solution had eigenvalues greater than 1.0. A l l of these were rotated, but only seven have been retained for presentation, based upon variance contribution and content interpretability. The variance accounted for drops rapidly.from factor 1 to factor 5 (see Table 4-6 and Figure 4-18), slowly from factor 5 to factor 7, more rapidly from factor 7 to factor 10, and slowly from factor 10 to factor 14. This suggests the choice of either seven or fourteen factors. If only four factors are chosen for interpretation, factors 5 through 7 are omitted, accounting for 13.2 percent of the common variance. In choosing between seven and fourteen factors, factors 8 through 14 include factors that are d i f f i c u l t to interpret, and factors that cover a single substantive component. For example, the only variables whose highest loading is on factor 11 are variables 166 through 169, giving information on the class of female workers. The f i r s t seven factors were chosen for these reasons. 112 6.2 The Family Status Construct Factors There are four factors closely related to the social area analysis construct of family status (see Table 2-1 and appendix Table E-2). Factor 1 measures the presence of young children in an area versus the presence of old people. Since i t seems conceptually closest to the construct, i t has been named Family Status. Variables positively correlated with the factor include children under 16 years of age, single family households, number of persons per household, f e r t i l i t y rate, and family heads 35 to 44 years of age. Variables negatively correlated with the factor include families with no children, people over 50 years of age, and single, widowed and divorced people. The second factor related to the family status construct is factor 3, Housing Tenure. This factor measures whether dewelling units are owned or rented. Variables positively correlated with the factor are the proportions of dwellings which are owned, single family, detached, having a garage, having many rooms, and being occupied for over ten years. Variables negatively correlated include the proportion of dwellings which are rented, the number of tenants per dwelling, dwellings occupied less than two years, and apartments and duplexes. The factor, therefore, includes not only tenure but also length of residence. Factor 5, Families with Older Children, is the third factor related to the family status construct. This factor measures the presence of families with children attending high-school. Highly correlated with the factor are the percent of population between 15 and 19 years of age, the proportion of 113 children aged 15 to 24, the proportion of family heads aged 45 to 54 years, the proportion of the population attending high-school, and the proportion of dwellings constructed between 1920 and 1945. Negatively correlated with the factor are the proportion of the population 30 to 34 years of age, the proportion of family heads 25 to 34 years old, and the proportion of dwellings occupied from three to five years. The last factor related to the family status construct is factor 6, Rural-Urban Status. Positively correlated with the factor are the proportion of the population which i s rural, which originates from Belgium and the Netherlands, and which is occupied in farming, the proportion of the male labour force which is self-employed, and dwellings less than two years old. Travel time from the city centre is positively correlated as contrasted with population potential and population density, which are negatively correlated. The proportion of dwellings having neither sewer nor septic tank is positively correlated, as contrasted with the negatively correlated proportion of dwellings connected to a sewer. Each of these four factors i s , because of the factoring procedure, linearly independent of a l l others. That four of them are required to encompass a l l of the family status construct may result from one or a combination of several causes. These are data normalization, a small areal unit, and a large number of variables. These factors correspond closely with factors 1, 3, 6, and 9 in the original data model, indicating that the cause is not data normalization. The variamax rotation of a principal axes solution is characterized by factor invariance (see section 3.3), 114 suggesting that reducing the number of variables by half would leave the factors basically intact. Therefore i t is tentatively concluded that this subdivision of the family status construct of social area analysis results from the use of a small areal unit. 6.3 The Economic Status Construct Factors Two of the seven factors are closely related to the social area analysis construct of economic status. Factor 2 corresponds almost exactly with the construct, and hence is named Socio-economic Status. Factor 7 identifies r e t a i l and cleri c a l workers of moderate income as an areally independent subset of this construct. Highly positively correlated with factor 2 are the professional, technical, and managerial occupations, the proportion of the population having at least some university training, whether attending or not, average family earnings, the proportion of the labour force in the industries of education, finance, insurance, and wholesale trade, the proportion of the male labour force composed of employers, owned single-family dwellings worth more than $18,000,average gross rent, and the ownership of more than one car. Highly negatively correlated with the factor are the proportion of the population having completed only elementary education, dwellings worth less than $12,000, the proportion of families where the head earns less than $5,000, number of persons per room, wage earners as a proportion of the male labour force, employees in the industries of wood finishing, construction, food and beverages, hotels, restaurants, and transportation and storage, the proportion of the labour force consisting of loggers, and the male-female ratio. 115 A number of ethnicity variables is also included, and these w i l l be discussed in section 6.5. Factor 7 includes the proportion of dwellings of value between $13,000 and $17,000, the proportion of families whose wage earner head earns between $5,000 and $6,000,the proportions of the labour force in cler i c a l occupations and in the industries of r e t a i l trade and telecommunication. Following the same line of reasoning as in the discussion of the family status construct factors, i t appears that the subdivision into two factors of the economic status construct results from the use of a small area! unit. 6.4 The Ethnic Status Construct Factor Factor 4 appears to provide a good measure of the level of segregation of racial and national groups. Highly correlated with the factor are the proportions of the population born in or originating from China, Central Europe, the Scandinavian countries, Italy, Russia, or Japan, the proportion of the population whose o f f i c i a l language is neither English nor French and w h o s e T e l i gion is not -Catholic, -Jewi sh r o r onei;of "the major Protestant " religions. Negatively correlated with the factor are the proportion of Protestants, and the proportion of the labour force in public administration and the Armed Forces, presumably reflecting the hiring policies of these agencies. Also highly negatively correlated with this factor i s the proportion of people originating from Great Britain, although the highest correlation of this variable is with factor 2. Two variables whose strongest association is with factor 6, Rural-Urban Status, also have high correlations with this 116 factor. The travel time from city centre is negatively correlated, while population potential i s positively correlated, reflecting the tendency of ethnic concentrations to occur near the center of the city. 6.5 Assessment of the Factoring It is clear that each factor includes only variables that are conceptually closely related to each other. Attention is now turned to the examination of whether conceptual groupings of variables are maintained in the factoring. In general, groups of variables are maintained very well in the factoring. With those variables connected with the family status construct, only four of the twenty-four age and marital status variables are correlated more highly with other factors, these a l l being factors not chosen for interpretation. Among the 74 household, family, and dwelling characteristics, a l l variables are correlated most highly with the family status construct factors, with a few exceptions. F i r s t , the 14 variables measuring income, housing value, and rent are connected with the economic status factors. Also connected are the variables of average number of persons per room and the proportion of dwellings with more than one car. The proportion of households with more than one family i s more closely connected with the Ethnic Status factor. Of the remainder the three variables measuring dwelling condition, the two variables showing television ownership, and variables 94, 114, and 129 are not included in any of the f i r s t seven factors. 117 For the education variables, related to the economic status construct, six are more highly correlated with the family status construct factors. Four of these, however, measure the presence of very young students and hence of children in general. These reasonably belong to factor 1. The other two are included in factor 5, measuring high-school aged children. Four other education variables, measuring the proportion of people who are not at school and who have attended or completed high-school but not university, are not included in the f i r s t seven factors. Of the employment variables, belonging to the economic status construct, 17 of the 21 variables giving industry and occupation of worker are most highly correlated with the economic status factors. Only two of the nine variables measuring employment class are so correlated. Female labour force classes and male and female unemployment rates are not strongly related to any of the f i r s t seven factors. The variables conceptually related to the ethnic status construct i nclude- country- of bi rth -or-^of~ origi n j ^ period. of---immigration-,~and -ireHgion—= Of the 26 variables giving country of birth or origin, only 9 have their highest correlations with factor 4. These are for ethnic groups that tend to l i v e in spatially compact areas, with each area having a reasonably broad range of socio-economic and family characteristics. Eleven other variables are more strongly associated with other of the f i r s t seven factors. The ethnic groups indicated by these variables w i l l constitute differing proportions of the populations around the city and may or may not be highly segregated. In any case, the groups w i l l tend to be f a i r l y homogeneous 118 in terms of the factor with which they are most closely correlated. The ethnic groups indicated by the remaining variables are not associated with any of the f i r s t seven factors, and are included in lesser order factors which may or may not indicate high levels of areal segregation. Two of the six period of immigration variables are associated with factor 1, with the rest not included in the f i r s t seven factors. This implies that period of immigration indicates l i t t l e more than average age. Two of the four variables indicating regligion are most strongly correlated with the Ethnic Status factor, while the remaining two correspond with socio-economic status. Hence, people of the Jewish and Roman Catholic religions tend to be of similar economic status to others of the same religion, and their spatial distributions tend to reflect that of the economic factor. A close examination of the factor structure has lead to the conclusion that variables of the same type have been included in the same factor. The major exception to this finding i s that people of different ethnic .origins are located with different levels of segregation and include differing ranges of heterogeneity in terms of family and socio-economic characteristics. 6.6 The Spatial Patterning of the Factors The factor scores for the seven factors are presented in Maps 6-1 through 6-7. As discussed in Chapter 3, isopleth maps are used to display the factor scores rather than choropleth maps. The class intervals for the maps are based upon percentiles of EA's, ensuring that each data value class contains the same number of EA's for a l l maps. If the factor scores had been distributed perfectly normally, then the factor score values corresponding 119 to the class boundaries would a l l have been equal to the values indicated in the bottom row of Table 6-1. As shown by the rest of the Table, this was not the case, confirming that the distributions were not perfectly normal. Table 6-1. Class Intervals for Factor Score Maps Percentiles of EA's Factor 10 25 40 60 75 90 1 -1.255 -0.603 -0.267 0.168 0.604 1.299 2 -1.100 -0.772 -0.472 0.106 0.706 1.490 3 -1.414 -0.557 -0.102 0.300 0.645 1.182 4 -1.108 -0.669 -0.292 0.182 0.653 1.271 5 -1.270 -0.602 -0.205 0.284 0.679 1.222 6 -0.979 -0.584 -0.330 -0.005 0.364 1.329 7 -1.337 -0.455 -0.042 0.408 0.658 1.015 Normal Distribution -1.282 -0.674 -0.253 0.253 0.674 1.282 The spatial configuration of factor 1, Family Status, appears to reflect, in part,, the historical development of the region, with the lowest scores occurring in the older areas and the highest scores in newer ones. The high scores occur in the northern parts of North and West Vancouver, in Burnaby except along Kingsway, in Coquitlam and especially Port Coquitlam, in North Delta and northern Surrey, in Richmond, and in the Fraserview area. The lowest scores occur in the West End and the CBD, in Kerrisdale, Marpole, Kitsilano, and West Vancouver. Additionally, large 120 M A P L E G E N D 100% PERCENTILES O F E N U M E R A T I O N A R E A S 121 122 parts of Vancouver City are quite low, with this pattern extending along Kingsway to New Westminster. Factor 2, Socio-Economic Status, i s patterned very differently. The high scores are restricted to West Vancouver and the northern part of North Vancouver, the Shaughnessy, Kerrisdale, Marine Drive, and University Endowment Lands areas of Vancouver, with s l i g h t l y lower scores occurring in Deep Cove, the Dunbar area of Vancouver, central Burnaby, Western Richmond, Tsawwassen, and Sunshine Heights in North Delta. The lowest scores occur in a l l Indian Reserves, on the waterfront in North Vancouver City, in the CBD, on the south and east shores of False Creek, in the Strathcona area and around the PNE, in the Fraserview area, in Queensborough, and across the Fraser River in the south edge of New Westminster, just west of Fraser M i l l s , in South Westminster, just east of the Guildford Town Center, and at the Vancouver Wireless Station in South Delta. Areas below the 25th percentile include most of Vancouver City east of Main Street, and the eastern portions of Surrey. Factor 3, Housing Tenure including a component of length of residence, has i t s highest scores in the Indian Reserves except Semiahmoo, between Dundarave and West Bay in West Vancouver, in Vancouver City just south of Spanish Banks, extending south of 12th Avenue east to Cambie except for the apartments in Kerrisdale and Marpole, around Killarney Park, and in pockets in the Hastings-Sunrise area. Quite high scores occur in loco, in pockets in the east and west sides of North Vancouver City, north of Ambleside, in the southern portion of Vancouver City, west of Burnaby Mountain, in the 123 124 M A P 6 - 3 : F A C T O R 3 - H O U S I N G TENURE 125 n o r t h and south-west corners o f New Westmins ter , i n C l o v e r d a l e , Newton, n o r t h o f Mud Bay and i n White Rock. The lowest scores occur i n the n o r t h o f West Vancouver nex t to Capi lano Creek, on Lonsdale i n North Vancouver C i t y , i n the West End, j u s t south o f K i t s i l a n o Beach, i n the F a i r v i e w a r e a , i n the s tuden t res idence area on the U n i v e r s i t y Endowment Lands, i n Marpo le , the Fraserv iew and Renfrew Heights a r e a s , a long the s o u t h - e a s t edge o f New Westmins ter , a long the Barnet Highway i n Burnaby, on Sea I s l a n d and a t the Vancouver Wi re less S t a t i o n . Genera l l y low areas surround False Creek and occur i n pockets i n Richmond. The h i g h e s t scores f o r f a c t o r 4 , E thn ic S t a t u s , occur i n o n l y f o u r p l a c e s . The f i r s t i s the S t ra thcona Grandview-Woodland a r e a , i n c l u d i n g Chinese, Japanese, and I t a l i a n s . The second i s an area between 16th and 45th Avenues between G r a n v i l l e and Main and inc ludes ma in l y c e n t r a l Europeans. The t h i r d i d e n t i f i e s the Japanese p o p u l a t i o n i n S t e v e s t o n , and the f o u r t h , the Japanese and Chinese p o p u l a t i o n i n Queensborough and Annacis I s l a n d . Low scores f o r t h i s f a c t o r correspond p r i m a r i l y w i t h c o n c e n t r a t i o n s o f people o f t h e P r o t e s t a n t r e l i g i o n s as^de f ined i n A p p e n d i x . T a b l e - A r X , and a l s o o f people born i n or o r i g i n a t i n g f rom Great B r i t a i n . A l l I n d i a n Reserves have very low s c o r e s , r e f l e c t i n g t h e i r P r o t e s t a n t f a i t h and the l a c k o f e t h n i c groups f rom o u t s i d e Canada. Other low scores occur bes ide Lynn Canyon i n North Vancouver, a long the w a t e r f r o n t i n West Vancouver, i n t he Fraserv iew area o f Vancouver C i t y , on Sea I s l a n d , r e f l e c t i n g the Canadian A i r Force , on bo th s ides o f t he Fraser R iver near t h e P a t u l l o B r i d g e , around l o c o , i n t he .Wh i te Rock-Crescent Beach a r e a , , and a t _the.Vancouver.Wire!ess S t a t i o n i n D e l t a . 126 127 Factor 5, Families with Older Children, has i t s highest scores in the central part of West Vancouver, near the PNE, in the University Endowment Lands, in the Kerrisdale and Marine Drive areas, in Fraserview, in the north-west portions of New Westminster, immediately west of Fraser M i l l s , in the north-eastern portions of Richmond, in Steveston, in Ladner, in Cloverdale, and along the eastern edge of Surrey. Its lowest scores occur in the northern part of North Vancouver, in the CBD and West End, in the north-west parts of Coquitlam, in the central and western parts of Richmond, in the north-western end of Surrey and just west of White Rock. Factor 6, Rural-Urban Status, including self-employed male labour force and new housing, has i t s highest values covering almost a l l of Surrey, Delta, Richmond, and Port Coquitlam, as well as central and southern Burnaby, Deep Cove and just east of Lynn Creek in North Vancouver. Its lowest values occur adjacent to Capilano Canyon, in pockets around False Creek and extending south to Fraserview, in the West Point Grey area, in Renfrew, in the Brentwood area and north-west of Burnaby Mountain in Burnaby, as well as in the military bases on Sea" Island and at the r VancouverTdirelesYStation~in"Deltar The Musqueam and Capilano Indian Reserves also have very low scores, perhaps because of few self-employed persons. Factor 7, Retail and Clerical Workers, has i t s highest scores in North Vancouver City, in the Kitsilano and Arbutus Ridge areas of Vancouver City, near Oakridge, just west and north of Fraserview, along the eastern edge of Burnaby extending along Kingsway to New Westminster, and in the west-central portions of Richmond. The factor has i t s lowest values in the 128 M A P 6-5: F A C T O R 5 - FAMILIES W I T H O L D E R C H I L D R E N 130 M A P 6-7: F A C T O R 7 - RETAIL A N D CLERICAL W O R K E R S 131 northern and western parts of West Vancouver, in the Strathcona area and surrounding False Creek, in the University Endowment lands, in the Shaughnessy and Kerrisdale areas, in Queensborough, North Surrey, in Port Kells and Cloverdale, as well as in a l l military areas and Indian Reserves. 6.7 Comparison of the Factors with Theories of Urban Sturcture As outlined in Chapter 1, there are three major theories to account for urban structure. These give rise to zonal, sectorial, and multiple nuclei spatial patterns. Discussions of these models and attempts to verify them occupy a considerable body of literature (for example, see Murdie, 1969, pp. 10-17, and Salins, 1971). It i s beyond the scope of this thesis to enter into an extended coverage of the literature and to attempt rigorous investigations into the applicability of the models to Vancouver. Nevertheless, the completed factor score maps invite interpretation with regard to the models. Empirical validations of models of urban structure have almost always involved the superimposition upon the city of a grid system incorporating concentric circles entered upon the city's CBD and straight lines radiating outward from this centre. Average scores for each element in the grid are calculated, and tests are made of the differences among them as distance increases from the centre or as they change from sector to sector. Many variations of this procedure have been used, but a l l average the data within the boundaries established by the grid, the locations of which cannot coincide with "natural" subdivisions of the urban area because of the grid's arbitrary nature. The definition and use of such natural boundaries is 132 a p p e a l i n g . Each element i n the g r i d would c o n s t i t u t e an area which was i n t e r n a l l y r e l a t i v e l y homogeneous, w i t h much l a r g e r d i f f e r e n c e s e x i s t i n g between elements than w i t h i n them. These boundar ies cou ld be d e f i n e d i n a number o f ways. They cou ld accommodate h i s t o r i c a l p a t t e r n s o f growth and development, t r a n s p o r t a t i o n r o u t e s , p o l i t i c a l i n f l u e n c e s , and s i m i l a r i t i e s perce ived by l o c a l r e s i d e n t s . However, t h e i r use i n a n a l y s i s i s not s t r a i g h t f o r w a r d . Zones and sec to rs are not easy t o d e f i n e . Time t r a v e l o r cogna t i ve d i s t a n c e may be the bes t m e t r i c t o use. Sectors may best be d e f i n e d i n terms o f p h y s i c a l b a r r i e r s o r t r a n s p o r t a t i o n l i n k s . I n s p i t e o f these prob lems, the approach has appea l . Th is i s e s p e c i a l l y t r u e i n Vancouver, where the phys i ca l form o f t he c i t y i s c o n s t r a i n e d by mounta ins , ocean, and p o l i t i c a l boundar ies , so u n l i k e t he i d e a l u n d i f f e r e n t i a t e d p l a i n which forms the bas is o f the r a d i a l - s e c t o r i a l models. The approach used here i s t o d e f i n e the elements and t h e i r " n a t u r a l " boundar ies i n a r i g o r o u s , a n a l y t i c a l f a s h i o n , and then t o assess the a p p l i c a b i l i t y o f the urban models s u b j e c t i v e l y . By t h i s means, no d i s t a n c e o r - d i r e c t i o n . m e t r i c need, be . imposed, and the assessment can accommodate t h e r e a l i t i e s o f the revea led s t r u c t u r e s . The elements were d e f i n e d w i t h an ex tens ion o f the aggrega t ion procedures developed i n Chapter 5. By us ing the Ward a l g o r i t h m p rocedure , elements w i t h the s m a l l e s t i n t e r n a l d i f f e r e n c e s i n c h a r a c t e r i s t i c s were d e f i n e d , w i t h t h e g r e a t e s t d i f f e r e n c e o c c u r r i n g between e lements . Sets o f elements were d e f i n e d f o r each o f the f i r s t f o u r f a c t o r s . Since aggrega t ion was done w i t h respec t to o n l y one c h a r a c t e r i s t i c a t a t ime ( t h e f a c t o r scores 1 3 3 for a single factor), a map of the elements with their average data values identified is equivalent to a smoothed version of the corresponding detailed factor score map. The smoothing eliminates a large amount of detail confusing to the search for spatial patterns, while maintaining as much of the information as possible. To avoid the size limitations of the computer used for the analysis, as discussed in Section 5 . 8 , i t was decided to begin aggregation with 2 4 4 groups of the aggregation based on a l l factors. It would have been preferable to start with the 1 2 3 7 EA's, but using the 2 4 4 groups seems to have had a very small influence. This i s confirmed in Figure 6 - 1 , where the variance comparison between census tracts and each of the f i r s t four factors indicates that no more than fourteen groups are needed for the groups to equal census tracts in information content. The choice of the number of regions to map and interpret was based upon the increment to the ESS at each grouping step. The logarithms of these values were, plotted.-(Figure. 6 r 2 . through. J 5 - 5 ) , resulting-in -the slope- of .the^ curve being equal to the rate of change of the values. Thus breaks in slope indicate suitable numbers of groups for interpretation. The sudden.:drops in error increment are caused by the contiguity constraint. The combination of two groups permits another grouping step which, without the constraint, would have occurred earlier. The results of the grouping are presented in Maps 6 - 8 through 6 - 1 1 , with the average of factor scores for a l l included EA's indicated for each 134 FIGURE 6-1: COMPARISON OF WITHIN-REGION VARIANCE OF AGGREGATED ENUMERATION AREAS VERSUS CENSUS TRACTS FOR AGGREGATION BY INDIVIDUAL FACTOR 2H O h -<=c 3 4H J _ ? L . 1 T I . , 1 T L_, L _ I L 1 1 ' 1 ' ' ' I I I ' . ' ' I I I I I I. I I I I "I .5 10 15 20 25 NUMBER OF REGIONS For iinterpretation, see Figure -5-7. 135 FIGURE 6-2: AGGREGATION ERROR FOR FACTOR 1 NUMBER OF REGIONS FIGURE 6-3: AGGREGATION ERROR FOR FACTOR 2 NUMBER OF REGIONS NUMBER OF REGIONS NUMBER OF REGIONS 139 region. The pattern for factor 1 appears to be "interrupted radial". Lowest or very low scores occur in the West End, CBD and Strathcona, in the southern part of West Vancouver, and in the South Granville area. These are nearly surrounded by low to average scores in most of Vancouver City and North Vancouver City. These in turn are surrounded by very high and highest scores, stretching from Horseshoe Bay across the North Shore, through Coquitlam and Burnaby south to Kingsway and New Westminster. Here a major interruption breaks the pattern. From Kingsway to western Richmond the scores are neither very low nor very high. Very high scores do occur, however, in northern Surrey. It appears as though the transportation route along Kingsway, established relatively early in the region's history, has lead to a displacement of high scores from southern Burnaby and New Westminster into Surrey. The pattern of high scores is resumed again in western Richmond and Sea Island, with a ring of above average scores extending from eastern and southern Richmond through Delta into southern Surrey. A major anomaly occurs in the south-west corner of Surrey, with lowest scores indicating the large retired population in White Rock. The spatial configuration of Socio-economic Status is much less clear. It appears to have a sectorial configuration in Vancouver City and the North Shore, and a radial pattern in municipalities to the south and east, centred on New Westminster. Centred upon the CBD, a series of very high and highest scores extends from the False Creek - Ontario Street boundary wesward to Point Grey, including the west half of the West End, starting again at Horseshoe Bay and extending to the upper parts of North Vancouver. Anomalies include 140 141 6 - 9 : G R O U P E D S O C I O - E C O N O M I C STATUS 1 4 2 o n l y I n d i a n Reserves, the J e r i c h o M i l i t a r y Base, and the eas t h a l f o f Horseshoe Bay. A second s e c t o r encompasses the CBD, the False Creek area and the eas te rn h a l f o f Vancouver C i t y , a l l having very low sco res . A t h i r d s e c t o r , w i t h average and above average s c o r e s , extends f rom Lions Gate Br idge across Nor th Vancouver, e x c l u d i n g the n o r t h - w e s t e r n c o r n e r . A f o u r t h s e c t o r , w i t h average s c o r e s , l i e s between the ve ry low scores o f eas te rn Vancouver and the ve ry h igh scores o f the western p a r t o f the C i t y . The suburban r a d i a l p a t t e r n i s cen t red on a s e r i e s o f ve ry low scores o c c u r r i n g i n Queensborough, the sou th -wes te rn edge o f New Westmins ter , South Westminster and n o r t h - e a s t e r n New Westmins te r , M a i l l a r d v i l l e and Fraser M i l l s . Th is i s surrounded by average scores ex tend ing f rom Coqui t lam th rough Burnaby, Richmond, D e l t a , and S u r r e y , i n c l u d i n g pockets o f above average scores j u s t south o f Kingsway, i n western Richmond, Tsawwassen, and Sunshine H e i g h t s . The s p a t i a l p a t t e r n s o f Fac tor 3 seem t o f i t the models o f urban s t r u c t u r e p o o r l y . Average scores dominate a l l areas bu t Vancouver C i t y , w h e r e . v e r y . h i g h scores are dominant , .anomal ies , are the r u l e , and s u b s t a n t i v e i n t e r p r e t a t i o n s seem most s u i t a b l e . Apartment areas a re c l e a r l y d e l i n e a t e d i n sou th -wes te rn Nor th Vancouver C i t y , south eas te rn West Vancouver, the West End, South G r a n v i l l e , K e r r i s d a l e , Marpo le , and New Westminster . The m i l i t a r y bases a t J e r i c h o , Sea I s l a n d and De l ta are d i s t i n c t , as are a l l I nd ian Reserves except Semiahmoo. The pos t -war CMHC housing developments o f Renfrew Heights and Fraserv iew a re c l e a r l y d e l i n e a t e d , as are the. K i t s i l a n o and West P o i n t Grey m u l t i p l e f a m i l y d w e l l i n g s and the s tuden t res idences a t 143 144 UBC. The very h igh scores i n Vancouver C i t y ' s s i n g l e f a m i l y areas r e f l e c t the age o f the d w e l l i n g s , which might a lso account f o r the h igher than average scores around New Westminster and i n White Rock. A m u l t i p l e n u c l e i model m igh t be hypothes ized f o r t h i s f a c t o r , w i t h the o l d e s t areas o f Vancouver, New Westmins ter , West Vancouver and Nor th Vancouver as the nodes. However, o n l y Vancouver C i t y has a complex s t r u c t u r e su r round ing the n o d e , w h i l e the o t h e r nodes themselves a re a l l t h a t i s d i s t i n c t f rom the sur round ing average a r e a s , and they are no more d i s t i n c t than Ind ian Reserves and m i l i t a r y bases. Fac to r 4 , E t h n i c S t a t u s , a l s o appears t o be composed o f s p a t i a l anomal ies . As d i s t i n c t f rom f a c t o r 3 , however, t h e r e i s no t e m p t a t i o n t o cons ide r a m u l t i p l e n u c l e i mode l , s i n c e most v a r i a t i o n occurs independent o f h i s t o r i c a l c e n t r e s . A c r e s c e n t shaped area o f ve ry h igh va lues extends f rom St ra thcona th rough Shaughnessy ou t t o Queensborough. S t e v e s t o n ' s Japanese p o p u l a t i o n i s un ique. The n o r t h shore i s n e a r l y homogeneously low except ad jacen t t o Capi lano Canyoq. A s e r i e s of. v e r y low scores e x t e n d s . a l o n g - t h e n o r t h s i d e o f the Fraser R ive r f rom Fraserv iew t o Fraser M i l l s , i n c l u d i n g South Westminster . Surrey i s lower than D e l t a , and White Rock has ve ry low sco res . I n t o t a l , the f a c t o r i s u n i q u e l y p a t t e r n e d . 146 CHAPTER 7 - SUMMARY AND CONCLUSIONS I n f o l l o w i n g the methods o f f a c t o r i a l urban ecology f o r Greater Vancouver, i t was found t h a t seven f a c t o r s comprised the most impor tan t independent dimensions o f a rea l d i f f e r e n t i a t i o n s f o r t he r e s i d e n t i a l p o p u l a t i o n i n 1961. The f a c t o r s o f Fami ly S t a t u s , Housing Tenure, Fami l ies w i t h Older C h i l d r e n , and Rural -Urban S ta tus were a l l r e q u i r e d t o encompass t h e f a m i l y s t a t u s c o n s t r u c t o f s o c i a l area a n a l y s i s . The economic s t a t u s c o n s t r u c t was conveyed by Socio-economic S ta tus and R e t a i l and C l e r i c a l Workers. The e t h n i c s t a t u s c o n s t r u c t was p a r a l l e l e d by the E thn ic S ta tus f a c t o r . I n assess ing the f a c t o r i n g , i t was concluded t h a t the correspondence between conceptual groupings o f v a r i a b l e s and the f a c t o r s was very h i g h . The one excep t ion to t h i s f i n d i n g was t h a t people w i t h d i f f e r e n t e t h n i c o r i g i n s were l o c a t e d w i t h d i f f e r e n t l e v e l s o f seg rega t ion and were c h a r a c t e r i z e d by d i f f e r i n g amounts o f h e t e r o g e n e i t y i n terms o f f a m i l y and socio-economic c h a r a c t e r i s t i c s . An examinat ion o f the f i r s t f o u r f a c t o r s f o r correspondence w i t h t h e o r i e s o f urban s t r u c t u r e lead t o the conc lus ions t h a t Fami ly S ta tus d i s p l a y e d an " i n t e r r u p t e d r a d i a l " s p a t i a l p a t t e r n , t h a t Socio-economic S ta tus d i s p l a y e d a s e c t o r i a l p a t t e r n cen t red on Vancouver C i t y , and a r a d i a l p a t t e r n cen t red on New Westminster . Housing Tenure showed tendenc ies towards a m u l t i p l e n u c l e i s t r u c t u r e , though the comp lex i t y o f the C i t y o f Vancouver dominated the p a t t e r n . E thn ic S ta tus was concluded t o have a unique p a t t e r n , u n l i k e any o f t h e urban models. 1 4 7 The f a c t o r a n a l y s i s was performed us ing normal ized d a t a , w i t h the census enumeration area as the u n i t o f o b s e r v a t i o n f o r the data base. I t was concluded t h a t these c h a r a c t e r i s t i c s o f the data base had impor tan t impacts upon the r e s u l t s . In comparing f a c t o r a n a l y s i s r e s u l t s based upon untransformed data and normal ized d a t a , i t was concluded t h a t n o r m a l i z a t i o n a ids the a b i l i t y o f the c o r r e l a t i o n c o e f f i c i e n t t o measure l i n e a r a s s o c i a t i o n among v a r i a b l e s , and p a t t e r n s o f a s s o c i a t i o n a r e what f a c t o r a n a l y s i s seeks t o summarize. Th is helped the r o t a t i o n procedure t o achieve s imple s t r u c t u r e , a prime goal o f the techn ique . Consequent ly , t he f a c t o r s based upon normal ized data were more e a s i l y i n t e r p r e t e d . They i n c l u d e d v a r i a b l e s which seemed more c l o s e l y i n t e r r e l a t e d and seemed t o correspond more c l o s e l y t o t h e o r e t i c a l l y d e r i v e d hypothes is about urban a rea l d i f f e r e n t i a t i o n . Th is was e s p e c i a l l y t r u e o f t h e E thn ic S ta tus f a c t o r , where a c o n s i d e r a b l y l a r g e r number o f e t h n i c groups were i n c o r p o r a t e d i n the f a c t o r based upon normal ized d a t a . In e v a l u a t i n g the s i g n i f i c a n c e o f the f i n e r a rea l d e t a i l o f i n f o r m a t i o n r e s u l t i n g f rom us ing enumerat ion areas i n s t e a d o f census t r a c t s , c u s t o m a r i l y used f o r urban e c o l o g i e s , i t was concluded t h a t the bes t data base i s one a t the lowest l e v e l o f aggrega t ion s u i t a b l e f o r data hand l ing and a t which data e r r o r i s s u f f i c i e n t l y low. In assess ing the s u i t a b i l i t y o f census t r a c t s as a rea l u n i t s , i t was concluded t h a t they were markedly non-opt imal i n terms o f - i n t e r n a l v a r i a b i l i t y o f p o p u l a t i o n c h a r a c t e r i s t i c s . Consequent ly , i t was ... concluded t h a t enumerat ion areas were much more d e s i r a b l e than census t r a c t s as an a rea l u n i t f o r f a c t o r i a l eco logy . 148 BIBLIOGRAPHY Abramowi tz , M. A., & Stegun, I . A. (Eds. ) Handbook o f mathematical  f u n c t i o n s . New York : Dover, 1965. A lonso , W. Loca t ion and land use: Toward a general t heo ry o f land r e n t . Cambridge: Harvard U n i v e r s i t y P r e s s , 1964. B e l l , W. Economic, f a m i l y , and e t h n i c s t a t u s : An e m p i r i c a l t e s t . American S o c i o l o g i c a l Review, 1955, 20 , 45 -52 . B e l l , L. I . M e t r o p o l i t a n Vancouver: An overv iew f o r s o c i a l p l a n n e r s . Vancouver, B. C . ; Research depar tment , Community Chest and Counc i ls o f the Greater Vancouver A r e a , 1965. B e l l , W., & Moskos, C. C. A comment on Udry ' s ' I n c r e a s i n g sca le and s p a t i a l d i f f e r e n t i a t i o n 1 . Soc ia l Fo rces , 1964, 4 2 , 414-417. B e r r y , B. J . L. The l o g i c and l i m i t a t i o n s o f comparat ive f a c t o r i a l eco logy . Economic Geography, 1 9 7 1 , 4 7 , 209-219. Brunk, H. D. An i n t r o d u c t i o n t o mathemat ical s t a t i s t i c s . Waltham, Mass. : B l a i s d e l l , 1965. Burgess, E. W. The growth o f the c i t y . I n R. E. Park , E. W. Burgess, & R. D. McKenzie. The c i t y . Chicago: U n i v e r s i t y o f Chicago Press , 1925. Canada. 1961 Census o f Canada: B r i t i s h Columbia and Yukon enumerat ion a reas . Government o f Canada, Dominion Bureau o f S t a t i s t i c s , 1961. Canada.- Popu la t i on and housing . c h a r a c t e r i s t i e s - b y census - t r a c t s : . Vancouver. Government o f Canada, Dominion Bureau o f S t a t i s t i c s , B u l l e t i n CT-22, 1963. C l a r k , C. The c o n d i t i o n s o f economic p r o g r e s s . (2nd ed . r e v . ) London: M a c M i l l a n , 1951. C o u l t h a r d , W. J . UBC CNTOUR: Contour ing a g r i d . Vancouver: U n i v e r s i t y o f B r i t i s h Columbia Computing C e n t r e , 1972 (Mimeo). C o u l t h a r d , W. J . , P a t t e r s o n , J . M. , & H e r r i n g , W. UBC XPAND: Programs t o generate a g r i d f rom a s e t o f s c a t t e r e d da ta p o i n t s . Vancouver: U n i v e r s i t y o f B r i t i s h Columbia Computing C e n t r e , 1972 (Mimeo); < 149 Cramer, J . S. E f f i c i e n t g r o u p i n g : Regression and c o r r e l a t i o n i n Engel curve a n a l y s i s . Journa l o f the American S t a t i s t i c a l  x A s s o c i a t i o n , 1964, 59, 233-250. Duncan, 0 . D . , C u z z o r t , R. P . , & Duncan, B. S t a t i s t i c a l geography: Problems i n a n a l y z i n g a rea l d a t a . Glencoe, 1 1 1 . : Free Press , 1961. F a r l i e , D. J . G. The performance o f some c o r r e l a t i o n c o e f f i c i e n t s f o r a genera l b i v a r i a t e d i s t r i b u t i o n . B i o m e t r i c a , 1960, 47 , 307-323. F i s h e r , R. A. The f requency d i s t r i b u t i o n o f the va lues o f the c o r r e l a t i o n c o e f f i c i e n t i n samples f rom an i n d e f i n i t e l y l a r g e p o p u l a t i o n . B i o m e t r i k a , 1915, 10, 507-521. F r e c h e t , M. A note on s imple c o r r e l a t i o n . Mathematical Magazine, 1959, 3 2 , 265-268. G r u n f e l d , Y . , & G r i l i c h e s , Z. I s aggrega t ion n e c e s s a r i l y bad? Review  o f Economics and S t a t i s t i c s , 1960, 4 2 , 1-13. Halm, J . UBC FAN: Fac to r a n a l y s i s . Vancouver: The U n i v e r s i t y o f B r i t i s h Columbia Computing C e n t r e , 1971 (Mimeo). Hannan, M. T. Problems o f a g g r e g a t i o n . I n H. M. B l a l o c k , J r . ( E d . ) , Causal models i n the s o c i a l sc iences . Chicago: A l d i n e -A t h e r t o n , 1971. Harman, H. H. Modern f a c t o r a n a l y s i s . (2nd ed . r e v . ) Chicago: The U n i v e r s i t y o f Chicago Press , 1967. H a r r i s , C. D. & U l lman , E. L. The na ture o f c i t i e s . Annals o f the American Academy o f P o l i t i c a l and Soc ia l Sc ience , 1945, 242, 7 -17. Hawley, A. H . , & Duncan, 0 . D. Soc ia l area a n a l y s i s : A c r i t i c a l a p p r a i s a l . Land Economics, 1957, 33 , 337-345. Haynes, K. E. S p a t i a l change i n urban s t r u c t u r e : A l t e r n a t i v e approaches t o e c o l o g i c a l dynamics. Economic Geography, 1 9 7 1 , 4 7 , 324-335. Heath, H. A. An e m p i r i c a l s tudy o f c o r r e l a t i o n i n v o l v i n g a h a l f - n o r m a l d i s t r i b u t i o n . Psycho log ica l R e p o r t s , 1 9 6 1 , 9 , 85 -86 . H e r b e r t , D. T. Soc ia l area a n a l y s i s : A B r i t i s h s t u d y . Urban S t u d i e s , 1967, 4 , 41-60 . Horsty i iP. v Fac tor a n a l y s i s o f da ta m a t r i c e s . New York : H o l t , R inehar t and Wins ton , 1965. 150 Hoyt , H. The s t r u c t u r e and growth o f r e s i d e n t i a l neighborhoods i n American c i t i e s . Washington: U. S. Federal Housing A d m i n i s t r a t i o n , 1939. Hunter , A. A. F a c t o r i a l eco logy : A c r i t i q u e and some sugges t i ons . Demography, 1972, 9 , 107-117. Janson, C. G. Some problems o f e c o l o g i c a l f a c t o r a n a l y s i s . In M. Dogan, & S. Rokkan ( E d s . ) , Q u a n t i t a t i v e e c o l o g i c a l a n a l y s i s i n the  s o c i a l sc iences . Cambridge: MIT Press , 1969. K a i s e r , H. F. The varimax c r i t e r i o n f o r a n a l y t i c r o t a t i o n i n f a c t o r a n a l y s i s . Psychomet r ika , 1958, 23 , 187-200. Kenney, J . F . , & Keeping, E. S. Mathematics o f s t a t i s t i c s : Par t two. P r i n c e t o n , N. J . : Van Nos t rand , 1951 . Kowa lsk i , C. J . The performance o f some rough t e s t s f o r b i v a r i a t e n o r m a l i t y be fo re and a f t e r c o o r d i n a t e t r a n s f o r m a t i o n s t o n o r m a l i t y . Technomet r i cs , 1970, 1 2 , 517-544. K o w a l s k i , C. J . On the e f f e c t s o f n o n - n o r m a l i t y on the d i s t r i b u t i o n o f the sample product-moment c o r r e l a t i o n c o e f f i c i e n t . App l ied  S t a t i s t i c s , 1972, 2 1 , 1-12. K o w a l s k i , C. J . & T a r t e r , M. E. Coord ina te t r a n s f o r m a t i o n to n o r m a l i t y and power o f normal t e s t s f o r independence. B i o m e t r i k a , 1969, 56, 139-148. ~~ M a r d i a , K. V. Measures o f m u l t i v a r i a t e skewness and k u r t o s i s w i t h a p p l i c a t i o n s . B i o m e t r i k a , 1970, 5 7 , 519-530. Murd ie , R. A. F a c t o r i a l ecology o f M e t r o p o l i t a n T o r o n t o , 1951 - 1961 :  An essay on t h e s o c i a l geography o f t h e x i t y . ChiCagor" The ' Department o f Geography, The U n i v e r s i t y o f Chicago, Research Paper No. 116, 1969. N o r r i s , R. C , & H je lm , H. F. Non-no rma l i t y and produc t moment c o r r e l a t i o n . Jou rna l o f Exper imental E d u c a t i o n , 1 9 6 1 , 2 9 , 261-270. Ogburn, W. F. The f a m i l y and i t s f u n c t i o n s . Recent s o c i a l t rends i n the Un i ted S t a t e s . New York : M c G r a w - H i l l , 1953. V o l . 1 , 661-708. P a t i l l o , R. W. The West End o f Vancouver: "A s o c i a l p r o f i l e " . Vancouver, B. C : Un i ted Community S e r v i c e s , 1969. P a t t e r s o n , J . M. , & W h i t a k e r , R. A. UBC CGROUP - H i e r a r c h i c a l g roup ing  a n a l y s i s w i t h o p t i o n a l c o n t i g u i t y c o n s t r a i n t . Vancouver: The U n i v e r s i t y o f B r i t i s h Columbia Computing C e n t r e , 1973 (Mimeo). 151 Pearson, E. S. A f u r t h e r development o f t e s t s f o r n o r m a l i t y . B i o m e t r i k a , 1930, 2 2 , 239-249. Pearson, E. S. The a n a l y s i s o f va r i ance i n cases o f non-normal v a r i a t i o n . B i o m e t r i k a , 1931 , 23, 114. Pearson, E. S . , & H a r t l e y , H. 0. B iomet r i ka t a b l e s f o r s t a t i s t i c i a n s : Vol 1 . ( 3 r d e d . ) London: Cambridge U n i v e r s i t y P ress , 1966. Peucker, T. K., & Rase, W. D. A f a c t o r i a l ecology o f Greater Vancouver. Contemporary Geography: Western v i e w p o i n t s . B. C. Geographical S e r i e s , 1971 , 12 , 80-96 . Rees, P. H. F a c t o r i a l eco logy : An extended d e f i n i t i o n , s u r v e y , and c r i t i q u e o f the f i e l d . Economic Geography, 1 9 7 1 , 47 , 220-233. Rees, P. H. Problems o f c l a s s i f y i n g subareas w i t h i n c i t i e s . I n B. J . L. Ber ry ( E d . ) , C i t y c l a s s i f i c a t i o n handbook: Methods and  a p p l i c a t i o n s . New York : W i l e y - I n t e r s c i e n c e , 1972. . Robinson, A. H. Mapping the correspondence o f i s a r i t h m i c maps. I n B. J . L. B e r r y , & D. F. Marble ( E d s . ) , S p a t i a l a n a l y s i s : A  reader i n s t a t i s t i c a l geography. Englewood C l i f f s , N. J . : P r e n t i c e - H a l l , 1968. Robinson, W. S. Eco log ica l c o r r e l a t i o n s and the behav io r o f i n d i v i d u a l s . American S o c i o l o g i c a l Review, 1950, 15, 351-357. Robson, B. T . Urban a n a l y s i s : A s tudy o f c i t y s t r u c t u r e . Cambridge: Cambridge U n i v e r s i t y P ress , 1969. Rummel, R. J . A p p l i e d f a c t o r a n a l y s i s . Evanston: Nor thwestern U n i v e r s i t y Press., .- !970. Sal i n s , P. D. Household l o c a t i o n p a t t e r n s i n American m e t r o p o l i t a n a reas . Economic Geography, 1 9 7 1 , 4 7 , 230-248. Schmid, C. F . , & T a g a s h i r a , K. E c o l o g i c a l and demographic i n d i c e s : A methodo log ica l a n a l y s i s . Demography, 1964, J_, 194-211 . Shevky, E . , & B e l l , W. S o c i a ! area a n a l y s i s . S t a n f o r d S o c i o l o g i c a l S e r i e s , No. 1 . S t a n f o r d : S t a n f o r d U n i v e r s i t y P ress , 1955. Shevky, E . , B e l l , W., & W i l l i a m s , M. The s o c i a l areas o f Los Ange les : A n a l y s i s and t y p o l o g y . B e r k e l e y : U n i v e r s i t y o f C a l i f o r n i a Press , T9W. Sweetser, F. L. F a c t o r i a l eco logy : H e l s i n k i , 1960. Demography, 1965, 2, 372-385. • • , 152 Theodorson, 6 . A. S tud ies i n human eco logy . Evanston, I l l i n o i s : Row, Peterson and C o . , 1961. T h u r s t o n e , L. L. M u l t i p l e f a c t o r a n a l y s i s . Chicago: U n i v e r s i t y o f Chicago Press , 1947. Veldman, D . . J . F o r t r a n programming f o r the behav io ra l s c i e n c e s . New York : H o l t , R inehar t and Wins ton , 1967. Ward, J . H. J r . H i e r a r c h i c a l g roup ing t o o p t i m i z e an o b j e c t i v e f u n c t i o n . Journa l o f the American S t a t i s t i c a l A s s o c i a t i o n , 1963, 58 , 236-244. Weiss, S. R e s i d e n t i a l deve loper d e c i s i o n s . Chapel H i l l : I n s t i t u t e f o r Soc ia l Research, U n i v e r s i t y o f Nor th C a r o l i n a , 1966. W i r t h , L. Urbanism as a way o f l i f e . American Journa l o f S o c i o l o g y , 1938, 4 4 , 1-24. Y u l e , G. U . , & K e n d a l l , M. G. An i n t r o d u c t i o n t o t h e t h e o r y o f s t a t i s t i c s . London: Char les G r i f f i n , 1950. APPENDIX A VARIABLE DEFINITIONS 153 DEFINITION OF VARIABLES A l a r g e number o f the v a r i a b l e s are descr ibed here as percentages. When the v a r i a b l e va lues were c a l c u l a t e d , however, they were computed as r a t i o s and so d i f f e r f rom the e q u i v a l e n t percentage values by a f a c t o r o f 100.0 . Th is i n no way a f f e c t s skewness, k u r t o s i s , o r c o r r e l a t i o n c o e f f i c i e n t s f o r the d a t a . However the averages and s tandard d e v i a t i o n s f o r these percentage v a r i a b l e s r e p o r t e d i n Appendix Table B-l must be m u l t i p l i e d by 100 t o g i v e e q u i v a l e n t percentage f i g u r e s . T h i s d e f i n i t i o n o f v a r i a b l e s assumes a knowledge o f bas i c Census o f Canada te rms , which can be found i n Canada ( 1 9 6 3 ) , pp. 28-29. 1 POP'N DENSITY Popu la t i on per square m i l e , c a l c u l a t e d as descr ibed i n S e c t i o n 3 . 3 . 2 POP'N POTENTIAL " P o p u l a t i o n per m i l e " - see Sec t ion 3 . 3 . 3 RECENT P0PN INCR Percent inc rease i n p o p u l a t i o n f rom 1956 t o 1961. 4 TIME-CITY CENTER Minutes o f au tomobi le t r a v e l t ime f rom the es t imated c e n t e r o f g r a v i t y o f EA p o p u l a t i o n t o the i n t e r s e c t i o n s o f Georgia and G r a n v i l l e S t r e e t s (see Sec t i on 3 . 3 ) . 5 FERTILITY RATE Number o f c h i l d r e n aged 0 t o 4 years per female aged 20 t o 44 years 6 % 0 - 4 YEARS Percent o f t he t o t a l p o p u l a t i o n between 0 and 4 years o f age. 7 % 5 - 9 YEARS Percent o f the t o t a l p o p u l a t i o n between 5 and 9 years o f age. 8 % 10-14 YEARS Percent o f the t o t a l p o p u l a t i o n between 10 and 14 years o f age. 9 % 15-19 YEARS Percent o f the t o t a l p o p u l a t i o n between 15 and 19 years o f age. 10 % 20-24 YEARS Percent o f t he t o t a l p o p u l a t i o n between 20 and 24 years o f age. 11-35 25-29 YEARS Percent o f the t o t a l p o p u l a t i o n between 25 and 29 years o f age. 12 % 30-34 YEARS Percent o f the t o t a l p o p u l a t i o n between 30 and 34 years o f age. 13 % 35-39 YEARS Percent o f the t o t a l p o p u l a t i o n between 35 and 39 years o f age. 14 % 40-44 YEARS Percent o f the t o t a l p o p u l a t i o n between 40 and 44 years o f age. 15 % 45-49 YEARS Percent o f the t o t a l p o p u l a t i o n between 45 and 49 years o f age. 16 % 50-54 YEARS Percent o f the t o t a l p o p u l a t i o n between 50 and 54 years o f age. 17 % 55-59 YEARS Percent o f the t o t a l p o p u l a t i o n between 55 and 59 years o f age. 18 % 60-64 YEARS Percent o f the t o t a l p o p u l a t i o n between 60 and 64 years o f age. 19 % 65-69 YEARS Percent o f the t o t a l p o p u l a t i o n between 65 and 69 years o f age. 20 % 70-74 YEARS Percent o f the t o t a l p o p u l a t i o n between 70 and 74 years o f age. 21 % 75-79 YEARS ... Percent o f t he t o t a l p o p u l a t i o n between.-75 .and 79 . -years o f age. 22 % 80-84 YEARS Percent o f the t o t a l p o p u l a t i o n between 80 and 84 years o f age. 23 % 85 + YEARS Percent o f the t o t a l p o p u l a t i o n 85 years o f age and o v e r . 24 % SINGLE <15 YRS Percent o f the t o t a l p o p u l a t i o n which i s s i n g l e and l e s s than 15 years o f age. 25 %SNGL>15YRS.MALE Percent o f the t o t a l p o p u l a t i o n which i s s i n g l e , mai and 15 years o f age o r ove r . 26 %SNGL>15YRS.FEM. Percent o f the t o t a l p o p u l a t i o n which i s s i n g l e , f e m a l e , and 15 years o f age o r ove r . 155 27 % MARRIED Percent o f the t o t a l p o p u l a t i o n whose m a r i t a l s t a t u s i s m a r r i e d . 28 % WIDOWED Percent o f the t o t a l p o p u l a t i o n whose m a r i t a l s t a t u s i s widowed. 29 % DIVORCED Percent o f the t o t a l p o p u l a t i o n whose m a r i t a l s t a t u s i s d i v o r c e d . 30 BORN B. C. Percent o f t h e t o t a l p o p u l a t i o n born i n B r i t i s h Columbia. 31 BORN CANADA-NOBC Percent o f the t o t a l p o p u l a t i o n born i n Canada elsewhere than B. C. 32 ORIG CANADA Percent o f the t o t a l p o p u l a t i o n o f Canadian o r i g i n . 33 ORIG NATIVE POPN Percent o f the t o t a l p o p u l a t i o n band o r non-band n a t i v e Ind ian o r Eskimo. 34 ETHN USA Percent o f t h e t o t a l p o p u l a t i o n born i n o r o r i g i n a t i n g f rom the USA. 35 BORN UK+REP.IREL Percent o f the t o t a l p o p u l a t i o n born i n England, S c o t l a n d , Wales, Nor te rhn I r e l a n d , o r the Republ ic o f I r e l a n d . 36 ORIG UK+REP.IREL Percent o f the t o t a l p o p u l a t i o n o r i g i n a t i n g f rom the c o u n t r i e s s p e c i f i e d i n v a r i a b l e 35. 37 BORN AUSTL+COMWL Percent o f the t o t a l p o p u l a t i o n born i n A u s t r a l i a , South A f r i c a , the West I n d i e s , o r o t h e r Commonwealth c o u n t r i e s . 38 ETHN FRANCE Percent o f the t o t a l p o p u l a t i o n born i n o r o r i g i n a t i n g f rom France. 39 BORN ITALY Percent o f the t o t a l p o p u l a t i o n born i n I t a l y 40 ORIG ITALY Percent o f the t o t a l p o p u l a t i o n o r i g i n a t i n g f rom I t a l y . 41 BORN GERM+A+SWIT Percent o f the t o t a l p o p u l a t i o n born i n Germany, A u s t r i a , o r S w i t z e r l a n d . 42 ORIG GERM+A+SWIT Percent o f the t o t a l p o p u l a t i o n o r i g i n a t i n g f rom Germany, A u s t r i a , o r S w i t z e r l a n d . 43 BORN BELG+NETHLD Percent o f the t o t a l p o p u l a t i o n born i n Belgium o r t he Ne the r lands . 156 44 ORIG BELG+NETHLD Percent o f t o t a l p o p u l a t i o n o r i g i n a t i n g f rom Belgium o r the Ne the r l ands . 45 BORN SCANDINAVIA Percent o f the t o t a l p o p u l a t i o n born i n Denmark, Norway, Sweden, F i n l a n d o r I c e l a n d . 46 ORIG SCANDINAVIA Percent o f the t o t a l p o p u l a t i o n o r i g i n a t i n g from the c o u n t r i e s s p e c i f i e d i n v a r i a b l e 45. 47 BORN CENT EUROPE Percent o f the t o t a l p o p u l a t i o n born i n Czeckos lovak ia . Po land, Hungary, Roumania, o r Yugos lav ia . 48 ORIG CENT EUROPE Percent o f the t o t a l p o p u l a t i o n o r i g i n a t i n g f rom the c o u n t r i e s s p e c i f i e d i n v a r i a b l e 47. 49 ETHN GREECE Percent o f the t o t a l p o p u l a t i o n born i n o r o r i g i n a t i n g f rom Greece. 50 ETHN INDIA PAKIS Percent o f the t o t a l p o p u l a t i o n born i n o r o r i g i n a t i n g f rom I n d i a or P a k i s t a n . 51 BORN USSR Percent o f t he t o t a l p o p u l a t i o n born i n the U.S.S.R. 52 ORIG USSR Percent o f the t o t a l p o p u l a t i o n o f B y e l o r u s s i a n , E s t o n i a n , L a t v i a n , L i t h u a n i a n , Russ ian, o r Ukran ian o r i g i n s . 53 BORN CHINA Percent o f the t o t a l p o p u l a t i o n born i n China. 54 ORIG CHINA Percent o f the t o t a l p o p u l a t i o n o f Chinese o r i g i n . 55 ETHN JAPAN Percent o f the t o t a l p o p u l a t i o n born i n or o r i g i n a t i n g f rom Japan. 56 %0F IMGTS BEF'21 Percent o f the t o t a l immigrants t o Canada t h a t immigrated b e f o r e 1921 . 57 % OF IMGTS 21-40 Percent o f the t o t a l immigrants t o Canada t h a t immigrated between 1921 and 1940. 58 % OF IMGTS 41-50 Percent o f the t o t a l immigrants t o Canada t h a t immigrated between 1941 and 1950. 59 % OF IMGTS 51-55 Percent o f the t o t a l immigrants t o Canada t h a t immigrated between 1951 and 1955. 60 % OF IMGTS 56-59 Percent o f t he t o t a l immigrants t o Canada t h a t immigrated between 1956 and 1959. 61 % OF IMGTS 60-61 Percent o f the t o t a l immigrants t o Canada t h a t immigrated between I960 and 1961. 157 62 RELIG PROTESTANT Percent o f the t o t a l p o p u l a t i o n whose r e l i g i o n i s A n g l i c a n , B a p t i s t , L u t h e r a n , P r e s b y t e r i a n , U n i t a r i a n , o r Un i ted Church o f Canada 63 RELIG ROM CATHOL Percent o f the t o t a l p o p u l a t i o n whose r e l i g i o n i s Roman C a t h o l i c 64 RELIG JEWISH Percent o f the t o t a l p o p u l a t i o n whose r e l i g i o n i s Jewish 65 RELIG OTHERS Percent o f the t o t a l p o p u l a t i o n whose r e l i g i o n i s o t h e r than those s p e c i f i e d i n v a r i a b l e s 62 , 63 , o r 64. 66 LANG NOT ENG, FR Percent o f the t o t a l p o p u l a t i o n whose mother tongue i s n e i t h e r Eng l i sh nor French. 67 ED-M-NO-<GRADE 1 Percent o f the male p o p u l a t i o n who are no longer a t t e n d i n g school and who have completed less than grade 1 68 ED-M-NO-ELEMENTY Percent o f the male p o p u l a t i o n who are no longer a t t e n d i n g school and who have completed between grade 1 and grade 4 . 69 ED-M-NO-HIGH NC Percent o f the male p o p u l a t i o n who are no longer a t t e n d i n g school and who have at tended but not completed h igh s c h o o l . 70 ED-M-NO-HIGH FIN Percent o f the male p o p u l a t i o n who are no l onge r a t t e n d i n g school and who have completed h igh s c h o o l . 71 ED-M-NO-UNIV NC Percent o f the~male p o p u l a t i o n who are no l onge r a t t e n d i n g school and who have a t tended but not completed u n i v e r s i t y . 72 Ed-M-NO-UNIV DEG Percent o f the male p o p u l a t i o n who are no l onge r a t t e n d i n g school and who have completed a u n i v e r s i t y degree. 73 ED-M-AT-ELEM+PRE Percent o f the male p o p u l a t i o n c u r r e n t l y a t t e n d i n g e lementary o r p r e - s c h o o l . 74 ED-M-AT-HIGH SCH Percent o f the male p o p u l a t i o n c u r r e n t l y a t t e n d i n g h igh s c h o o l . 75 ED-M-AT-UNIV UGD Percent o f t h e male p o p u l a t i o n c u r r e n t l y a t t e n d i n g u n i v e r s i t y a t the undergraduate l e v e l . 158 76 ED-M-AT-UNIV GRD Percent o f the male p o p u l a t i o n c u r r e n t l y a t t e n d i n g u n i v e r s i t y a t the graduate l e v e l . 77 ED-F-NO-<GRADE 1 Percent o f the female p o p u l a t i o n who are no longer a t t e n d i n g school and who have completed less than grade 1 . 78 ED-F-NO-ELEMENTY Percent o f the female p o p u l a t i o n who are no longer a t t e n d i n g school and who have completed between grade 1 and grade 4 . 79 ED-F-NO-HIGH NC Percent o f t he female p o p u l a t i o n who are no longer a t t e n d i n g school and who have at tended bu t no t completed h igh s c h o o l . 80 ED-F-NO-HIGH FIN Percent o f the female p o p u l t i o n who are no longer a t t e n d i n g school and who have completed h igh school 81 ED-F-NO-UNIV NC Percent o f the female p o p u l a t i o n who are no longer a t t e n d i n g school and who have at tended but not completed u n i v e r s i t y . 82 ED-F-NO-UNIV DEG Percent o f the female p o p u l a t i o n who are no longer a t t e n d i n g school and who have completed a u n i v e r s i t y degree 83 ED-F-AT-ELEM+PRE Percent o f t he female p o p u l a t i o n c u r r e n t l y a t t e n d i n g e lementary o r p r e - s c h o o l . 84 ED-F-AT-HIGH SCH Percent o f t he female p o p u l a t i o n c u r r e n t l y a t t e n d i n g h igh s c h o o l . 85 ED-F-AT-UNIV UGD Percent o f the female p o p u l a t i o n c u r r e n t l y a t t e n d i n g u n i v e r s i t y a t the undergraduate l e v e l . 86 ED-F-AT-UNIV GRD Percent o f the female p o p u l a t i o n c u r r e n t l y a t t e n d i n g u n i v e r s i t y a t the graduate l e v e l . 87 MALE-FEMAL RATIO Rat io o f t o t a l male p o p u l a t i o n t o t o t a l female p o p u l a t i o n . 88 % o f POP "RURAL" Percent o f the t o t a l p o p u l a t i o n who are " r u r a l " . 89 AVG # PERSON/FAM Average number o f persons per f a m i l y . 90 %P0P IN FAMILIES Percent o f the p o p u l a t i o n who are i n a f a m i l y , where a f a m i l y c o n s i s t s o f a husband and w i f e w i t h o r w i t h o u t c h i l d r e n who have never m a r r i e d , o r a paren t w i t h one o r more c h i l d r e n never m a r r i e d , l i v i n g t o g e t h e r i n the same d w e l l i n g . 159 91 %FPOP UNMARRIED Percent o f the f a m i l y p o p u l a t i o n t h a t i s unmarr ied . 92 %FAM W W.E.HEADS Percent o f f a m i l i e s w i t h a wage earner head 93 AVG FAM EARN WEH Average f a m i l y earn ings o f f a m i l i e s w i t h a wage earner head ($100 . ) 94 %FAM-HEAD < 25 Y Percent o f f a m i l i e s w i t h head aged less than 25 y e a r s . 95 %FAM-HEAD 25-34 Percent o f f a m i l i e s w i t h head aged f rom 25 to 34 y e a r s . 96 %FAM-HEAD 35-44 Percent o f f a m i l i e s w i t h head aged f rom 35 t o 44 y e a r s . 97 %FAM-HEAD 45-54 Percent o f f a m i l i e s w i t h head aged f rom 45 t o 54 y e a r s . 98 %FAM-HEAD 55-64 Percent o f f a m i l i e s w i t h head aged f rom 55 t o 64 y e a r s . 99 %FAM-HEAD 65-69 Percent o f f a m i l i e s w i t h head aged f rom 65 t o 69 y e a r s . 100 %FAM-HEAD > 70 Percent o f f a m i l i e s w i t h head aged over 70 y e a r s . 101 %FAM-WEH <$2000 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing l e s s than $2,000 102 %FAM-WEH $2-2999 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing f rom $2,000 t o $2,999. 103 %FAM-WEH $3-3999 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing f rom $3,000 t o $3 ,999. 104 %FAM-WEH $4-4999 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing f rom $4,000 t o $4,999. 105 %FAM-WEH $5-5999 Percent o f f a m i l i e s w i t h a wage_earner head who i s ea rn ing f rom $5,000 to $5,999. 106 %FAM-WEH $6-6999 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing f rom $6,000 t o $6 ,999. 107 %FAM-WEH $7-9999 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing f rom $7,000 t o $9,999. 108 %FAM-WEH $>10000 Percent o f f a m i l i e s w i t h a wage earner head who i s ea rn ing over $10,000. 109 %FAM NO CHILDREN Percent o f f a m i l i e s w i t h no c h i l d r e n . 110 %FAM CHLD N0NE>6 Percent o f f a m i l i e s w i t h 1 o r more c h i l d r e n under 6 , bu t none over 6 years o f age. 160 111 %FAM CHLD 15-24 Percent o f f a m i l i e s w i t h 1 or more c h i l d r e n 15 t o 24 years o f age b u t none under 15 and none over 24 a t home. 112 %FAM CHLD ALL<15 Percent o f f a m i l i e s w i t h 1 o r more c h i l d r e n under 15 years o f age bu t none over 15 a t home. 113 %FAM HBD+WF WORK Percent o f f a m i l i e s w i t h both husband and w i f e i n the l abou r f o r c e . 114 %P0P IN HOUSHLDS Percent o f the t o t a l p o p u l a t i o n i n households, where a household c o n s i s t s o f a person o r group o f persons occupying one d w e l l i n g . 115 #0F PERSONS/HSLD Average number o f persons per household. 116 %HSLD 1 FAM HSLD Percen t o f households which are one f a m i l y households. 117 %HSLD>1 FAM HSLD Percent o f households which a re two or more f a m i l y households. 118 %HSLD NONFAM HLD Percent o f households which a re n o n - f a m i l y households, 119 %HSLD W LODGERS Percent o f households w i t h l o d g e r s . 120 #HPERSONS/DWELL Average number o f persons i n households per d w e l l i n g . 121 %DWEL OWNED,SNGL Percent o f t o t a l d w e l l i n g s which are owned, s i n g l e f a m i l y non- farm d w e l l i n g s . 122 %DWEL SNG DETACH Percent o f t o t a l d w e l l i n g s which are s i n g l e detached. 123 %DWEL DUPLEX Percent o f t o t a l d w e l l i n g s whch a re s i n g l e a t tached o r dup lex . 124 %DWEL APT OTHER Percent o f t o t a l d w e l l i n g s wh ich a re apartments o r f l a t s . 125 %DWEL RENTED Percent o f t o t a l d w e l l i n g s which are r e n t e d . 126 %DWEL RESID+BUSN Percent o f t o t a l d w e l l i n g s which are used f o r res idence and bus iness . 127 %DWEL OCUP 0-2YR Percent o f t o t a l d w e l l i n g s which are occupied by the p resen t occupant f o r 2 years o r l e s s . 128 %DWEL OCUP 3-5YR Percent o f t o t a l d w e l l i n g s which a re occupied by the p resent occupant f rom 3 t o 5 y e a r s . 129 %DWEL 0CUP6-10YR Percent o f t o t a l d w e l l i n g s which a re occupied by the p resen t occupant f rom 6 to 10 y e a r s . 161 130 %DWEL OCUP >10YR Percent o f t o t a l d w e l l i n g s which are occuped by the p resent occupant f o r more than 10 y e a r s . 131 %DWEL CONS <1920 Percent o f t o t a l d w e l l i n g s which a re c o n s t r u c t e d p r i o r to 1920. 132 %DWEL CONS 20-45 Percent o f t o t a l d w e l l i n g s which are cons t ruc ted between 1920 and 1945. 133 %DWEL CONS 46-59 Percent o f t o t a l d w e l l i n g s which are c o n s t r u c t e d between 1946 and 1959. 134 %DWEL CONS 60-61 Percent o f t o t a l d w e l l i n g s which are cons t ruc ted i n 1960 o r 1961. 135 %DWEL C0ND GOOD Percent o f t o t a l d w e l l i n g s i n good c o n d i t i o n . 136 %DWEL C0ND OK Percent o f t o t a l d w e l l i n g s i n need o f minor r e p a i r s . 137 %DWEL C0ND FAIR Percent o f t o t a l d w e l l i n g s i n need o f major r e p a i r s 138 %DWEL W SEWER Percent o f t o t a l d w e l l i n g s whose sewage d i sposa l i s w i t h a sewer. 139 %DWEL NO SEPT TA Percent o f t o t a l d w e l l i n g s whose sewage d i sposa l i s Other than w i t h a sewer or s e p t i c t a n k . 140 %DWEL OWN WATER Percent o f t o t a l d w e l l i n g s whose source o f wa te r i s p r i v a t e on p r o p e r t y . 141 AVG # R00MS/DWEL Average number o f rooms per d w e l l i n g . 142 AVG # PERS/R00M Average number o f persons per room. 143 AVG #BDR00M/DWEL Average number o f bedrooms per d w e l l i n g . 144 %DWEL OS <$1200 Percent o f owned, s i n g l e , non- farm d w e l l i n g s o f va lue less than $12,000. 145 %DWEL0S$13-17000 Percent o f owned, s i n g l e , non- farm d w e l l i n g s o f va lue between $13,000 and $17,000. 146 %DWEL0S$18-27000 Percent o f owned, s i n g l e , non- farm d w e l l i n g s o f va lue between $18,000 and $27,000. 147 %DWEL(0S)>$28000> Percent o f owned, s i n g l e , non- farm d w e l l i n g s o f va lue over $28,000. 148 %DWEL NO TV Percent o f t o t a l d w e l l i n g s w i t h no t e l e v i s i o n . 162 149 %DWEL 1 TV 150 %DWEL NO CARS 151 %DWEL 1 CAR 152 %DWEL >1 CARS 153 %DWEL 1ST MORTG 154 %DWEL > 1 MORTG 155 %DWEL NO MORTGAG 156 %DWEL BANK MORTG - 157 %DWEL PRIV MORTG 158 %DWEL W GARAGE 159 AVG# TENANT/DWEL 160 %POP TENANTS 161 AVG GROSS RENT 162 AVG # FAM/DWELL 163 %MALLF EMPLOYER 164 %MALLF OWN ACCNT 165 %MALLF WAGE EARN 166 %FEMLF EMPLOYER 167 %FEMLF OWN ACCNT Percent o f t o t a l d w e l l i n g s w i t h one t e l e v i s i o n . Percent o f t o t a l d w e l l i n g s w i t h no au tomob i les . Percent o f t o t a l d w e l l i n g s w i t h one au tomob i le . Percent o f t o t a l d w e l l i n g s w i t h more than one au tomob i le . Percent o f t o t a l d w e l l i n g s w i t h a f i r s t mortgage o n l y . Percent o f t o t a l d w e l l i n g s w i t h more than one mortgage. Percent o f t o t a l d w e l l i n g s w i t h no mortgage. Percent o f t o t a l d w e l l i n g s w i t h a mortgage he ld by a bank. Percen t o f t o t a l d w e l l i n g s w i t h a mortgage he ld by a p r i v a t e i n d i v i d u a l . Percent o f t o t a l d w e l l i n g s w i t h a garage. Average number o f tenan ts per d w e l l i n g . Percent o f the t o t a l p o p u l a t i o n which are non- farm t e n a n t s . Average gross r e n t . Average number o f f a m i l i e s per d w e l l i n g . Percent o f males i n t h e .experienced l a b o u r , f o r c e who are c lassed as employers. Percent o f males i n the exper ienced labour f o r c e who are c lassed as own account . Percent o f males i n the exper ienced labour f o r c e who are c lassed as wage e a r n e r s . Percent o f females i n the exper ienced labour f o r c e who are c lassed as employers. Percent o f females i n the exper ienced l abou r f o r c e who are c lassed as own account . 163 168 %FEMLF WAGE EARN Percent o f females i n the exper ienced labour f o r c e who are c lassed as wage earners , 169 %FEMLF UNPD FAMW Percent o f females i n the exper ienced labour f o r c e who are c lassed as unpaid f a m i l y worke rs . 170 %ECAMP0P EMPLOYD Percent o f - e c o n o m i c a l l y a c t i v e males who are employed. 171 %ECAFP0P EMPLOYD Percent o f economica l ly a c t i v e females who are employed. 172 IND FOOD&BEVERGS Percent o f the exper ienced labour f o r c e i n food and beverage manu fac tu r ing i n d u s t r i e s . 173 IND WOOD FINISH Percent o f the exper ienced labour f o r c e i n wood manufac tu r ing i n d u s t r i e s . 174 IND PAPER&ALLIED Percent o f the exper ienced l abou r f o r c e i n paper and a l l i e d manu fac tu r ing i n d u s t r i e s . 175 IND METAL FABRCN Percent o f the exper ienced labour f o r c e i n metal f a b r i c a t i n g i n d u s t r i e s except t r a n s p o r t a t i o n equipment and e l e c t r i c a l machinery. 176 IND CONSTRUCTION Percent o f the exper ienced l abou r f o r c e i n c o n s t r u c t i o n i n d u s t r i e s . 177 IND TRANSP+STORG Percent o f the exper ienced labour f o r c e i n t r a n s p o r t a t i o n and s to rage i n d u s t r i e s . 178 IND TELECOMMUNCN Percent o f the exper ienced labour f o r c e i n communication - r a d i o , t e l e v i s i o n , t e lephone , t e l e g r a p h - a n d - p o s t - o f f i c e . -179 IND TRADE WHOLSL Percent o f the exper ienced labour f o r c e i n wholesa le t r a d e i n d u s t r i e s . 180 IND TRADE-RETAIL Percent o f the exper ienced labour f o r c e i n r e t a i l t r ade i n d u s t r i e s . 181 IND FINANCE+INS Percent o f the exper ienced l abou r f o r c e i n f i n a n c e , insurance or r e a l e s t a t e . 182 IND EDUCATION Percent o f the exper ienced labour f o r c e i n educa t ion and r e l a t e d s e r v i c e i n d u s t r i e s . 183 IND HEALTH WELFR Percent o f the exper ienced l abou r f o r c e i n h e a l t h and w e l f a r e s e r v i c e i n d u s t r i e s . 164 184 IND HOTEL RESTNT Percent o f the exper ienced labour f o r c e i n h o t e l , r e s t a u r a n t , and t a v e r n s e r v i c e i n d u s t r i e s . 185 IND PUB ADMN DEF Percent o f the exper ienced labour f o r c e i n p u b l i c , a d m i n i s t r a t i o n and defence. 186 OCC MANAGERIAL Percent o f the exper ienced l a b o u r f o r c e i n manager ia l o c c u p a t i o n s . 187 OCC PROFL TECHNL Percent o f the exper ienced labour f o r c e i n p r o f e s s i o n a l and t e c h n i c a l o c c u p a t i o n s : 188 OCC CLERICAL Percent o f the exper ienced labour f o r c e i n c l e r i c a l o c c u p a t i o n s . 189 OCC SALES Percent o f the exper ienced labour f o r c e i n sa les o c c u p a t i o n s . 190 OCC FARMER Percent o f the exper ienced l abou r f o r c e i n f a r m i n g , s t o c k - r a i s i n g , and fa rm l a b o u r i n g o c c u p a t i o n s . 191 OCC LOGGER Percent o f the exper ienced labour f o r c e i n l o g g i n g and r e l a t e d o c c u p a t i o n s . 192 OCC FISH TRAP Percent o f the exper ienced labour f o r c e i n f i s h i n g , t r a p p i n g , and h u n t i n g o c c u p a t i o n s . APPENDIX B TESTS FOR NORMALITY 165 EXPLOR: Data e x p l o r a t i o n w i t h t e s t s f o r n o r m a l i t y Type o f Program: F0RTRAN IV SUBROUTINE subprogram Purpose: To c a l c u l a t e t he mean, s tandard d e v i a t i o n , minimum, maximum, and moment c o e f f i c i e n t s o f skewness and k u r t o s i s . These are p r i n t e d ou t a long w i t h a h is togram and the r e s u l t s o f the t e s t s f o r n o r m a l i t y based on skewness and k u r t o s i s . The program i s in tended f o r use as a f a s t p r e l i m i n a r y , check on the na tu re o f a u s e r ' s d a t a . How t o use : EXPL0R can be c a l l e d f rom any F0RTRAN program by i n c l u d i n g a s ta tement o f the f o r m : CALL EXPLOR (ARRAY, N, NAME) where ARRAY i s a one-d imensional a r r a y o f t ype REAL*8 c o n t a i n i n g the da ta v a l u e s . N i s t he t o t a l number o f o b s e r v a t i o n s i n ARRAY NAME i s a h o l e r i t h l i t e r a l o r v a r i a b l e ( s ) o f f rom 1 t o 43 c h a r a c t e r s used as i d e n t i f i c a t i o n f o r the d a t a . A l l c h a r a c t e r s a re p r i n t e d up t o t he f i r s t occurrence o f a semi -co lon ( " ; " ) , two consecu t i ve b l a n k s , o r any non-p r i n t a b l e c h a r a c t e r . The va lues o f t h e mean, s tandard d e v i a t i o n , minimum, maximum, skewness and k u r t o s i s a re a c c e s s i b l e f rom the c a l l i n g program i n a l a b e l e d common s to rage area c a l l e d PARAMS. I n c l u d i n g the f o l l o w i n g card i n the c a l l i n g program w i l l a l l o w access^. COMMON/ PARAMS/ AVG, SD, AMIN, AMAX, SKEW, AKURT Note t h a t a l l va lues are o f t ype REAL*8 and must be dec la red so. 2 3 4 Method: Let h, = Ex. ; h , = Fx . ; = E x . ; h„ = I x . 1 N c N N 4 T 1 Then m 2 = h £ - h\ = s ( x _ _ ^ 2 N 166 m 3 = h 3 - 3 h ] h 2 + 2h3} = z(* - x )3 N m 4 = n 4 " 4 h l h 3 + 6 h l2 h 2 - 3 h ]4 = E ( x . - x ) 4 N are the 2nd, t h i r d , and f o u r t h moments about the mean ( = h , ) The va r iance = m„ 1 "2 The s tandard d e v i a t i o n = / m 2 ; w i t h B e s s e l ' s c o r r e c t i o n = >4TI2 x N N-l The moment c o e f f i c i e n t o f skewness = g = m-177 m 2 The moment c o e f f i c i e n t o f k u r t o s i s = g 2 = m^ 2 m 2 When sampl ing f rom a p o p u l a t i o n w i t h a normal d i s t r i b u t i o n the expected va lue o f g.j i s 0 , and o f g 2 i s 3. The sampl ing d i s t r i b u t i o n s o f these s t a t i s t i c s are not s u f f i c i e n t l y c l o s e t o any s tandard d i s t r i b u t i o n s t o p e r m i t easy c a l c u l a t i o n o f p r o b a b i l i t i e s assoc ia ted w i t h observed values o f the s t a t i s t i c s . The approach used i n t h i s program was t o f i t f u n c t i o n s th rough pub l i shed t a b l e s o f c r i t i c a l va lues f o r v a r i o u s numbers o f data p o i n t s and t o i n c o r p o r a t e these f u n c t i o n s i n t o the program. Values c a l c u l a t e d f o r a g iven se t o f data are then compared w i t h the c r i t i c a l va lues assoc ia ted w i t h the g iven number o f data p o i n t s and a s t e r i s k s used t o i n d i c a t e the r e s u l t s . - No a s t e r i s k s i n d i c a t e n o - s i g n i f i c a n t d i f f e r e n c e between the observed values and those va lues f o r a normal d i s t r i b u t i o n ; two a s t e r i s k s , a s i g n i f i c a n t d i f f e r e n c e a t the 99% con f idence l e v e l ; and one a s t e r i s k , a p o s s i b l y s i g n i f i c a n t d i f f e r e n c e (between t h e 95 and 99% conf idence l e v e l s ) i n a 1 - t a i l e d t e s t . Note t h a t t h e l e a s t squares f u n c t i o n approx imat ion i s a c c u r a t e over t he range 20< N<100 ,000 f o r skewness and 40 < N <100,000 f o r k u r t o s i s . 167 SUBROUTINE EXPLOR (A, N, NAME) C PRINTS OUT THE MEAN, ST. DEVIATION, MINIMUM, MAXIMUM, C SKEWNESS, AND KURTOSIS OF THE REAL*8 ARRAY "A" CONTAINING "N" C DATA POINTS, ALONG WITH A HISTOGRAM. SIX SUCH SUMMARIES ARE C PRINTED PER PAGE. "NAME" IS A HOLER IT H LITERAL OR VARIABLE (S) C OF FROM 1 TO 43 CHARACTERS USED AS IDENTIFICATION FOR THE C DATA. ALL CHARACTERS ARE PRINTED DP TO THE FIRST OCCURRENCE C OF: A SEMI-COLON (";"), TWO CONSECUTIVE BLANKS (" " ) , OR ANY C INVALID CHARACTER. C BEST SUITED TO RELATIVELY LARGE ARRAYS OF DATA, SINCE THE C SIGNIFICANCE TESTS FOR SKEWNESS AND KURTOSIS ARE INAPPROPRIATE C BELOW ABOUT 40 DATA POINTS. C THE HISTOGRAM HAS 64 POSITIONS IF THERE ARE MORE THAN 600 C DATA POINTS, 32 IF N > 300, 16 IF N > 150, AND 8 IF N > 40. C NOTE THAT THE HISTOGRAM PRINTING RELIES UPON THE WAY EATA IS C STORED IN IBM'S /360 SERIES OF COMPUTERS. IF ANOTHER MACHINE C IS USED, EXTENSIVE MODIFICATIONS MAY BE NECESSARY. C C PROGRAMMER: J. M. PATTERSON, DEPARTMENT OF GEOGRAPHY, UBC. C IMPLICIT REAL+8 (A-H,0-Q,S-Z) DIMENSION A(1) ,B(64) ,CE(80) , RE (160) ,C1 (8) ,C2 (16) ,C4 (8) , 1 C5 (8) ,C6 (8) ,C7 (8) ,C8 (8) ,C9 (8) ,C10 (8) ,NCLAS (5) , DNAM (8) , 2 JSTART (8) , JFINIS (8) INTEGER*2 CI (320) ,IAST/***'/ LOGICAL* 1 CURVE (640) , AST/ **'/,IFIR ST/. FALSE . / , NA ME (4 3) , N AM (43) , 1 VALID (256) ,LCONP (4) COMMON / PARAMS / AVG, SD, AM IN, AMAX, SKEW, A KURT EQUIVALENCE (CURVE (1) ,CE (1) ,RE (1) ,CI (1) , C1 (1) ) , (CE(9) , C2 ( 1) ) , 1 (CE (25) ,C4(1)) , (CE(33) ,C5(1)) , (CE (41) ,C6 (1) ) , (CE (49) ,C7 (1)) , 2 (CE (57) ,C8 (1)) , (CE (65) ,C9 (1) ) , (CE (73) ,C10 (1) ) , (ENAM ( 1 ) , 3 NAM (1) ) , (LCOMP (1),ICOMP) DATA BLANK/' */,NCLAS/8,16,32,32,64/,RA,RB,RC,RAST 1 /• ',' * ** ',«****'/,NLASr/-1/,UNCER/' •/ 2 ,JSTART/74,90,107,122,193,209,226,240/, 3 JFINIS/80,97,111,126,201,217,233,249/ C START AT TOP OF PAGE IF THIS IS THE FIRST CALL IF (IFIRST) GO TO 140 WRITE(6,10) 10 FOR MAT (' ;') IFIRST=.TRUE. C SET DP CHECK FOR VALID CHARACTERS. DO 100 1=1,256 100 VALID(I)=•TRUE. VALID(65) = . FALSE. DO 120 1=1 ,8 JS = JSTART(I) JF=JFINIS(I) DO 120 J=JS,JF 120 VALID(J + 1)=.FALSE. C INITIALIZE NAME 140 DO 160 1=1 ,8 160 DNAM(I)= BL ANK C SCAN INPUT NAME FOR SEMI-COLON, 2 CONSECUTIVE BLANKS, OR AN C INVALID CHARACTER. NB = 0 ICOMP=0 168 DO 200 1=1,43 LCOMP (4) =NAME (I) IC=ICOMP + 1 IF (VALID (IC) ) GO TO 220 IF(IC. EQ.95) GO TO 220 NAM (I) =LCOMP (4) IF (IC. EQ.65) GO TO 180 NB=0 GO TO 200 180 NB=NB • 1 IF (NB. EQ. 2) GO TO 220 200 CONTINUE 220 IF (N.LE. 1) GO TO 660 C IF 11N" HASN'T CHANGED FROM THE LAST CALL, EON • T EUPLICATE C CALCULATIONS. IF (N.EQ.NLAST)GO TO 240 C DETERMINE # OF CLASSES FOR HISTOGRAM ICLIND=N/150 + 1 IF(ICLIND. GT.5) ICLIND=5 NCLASS=NCLAS(ICLIND) NBAR=8 IF (ICLIND.EQ.1) NBAR=4 FI=1. DO/DFLOAT (N) DCOR=DSQRT (DFLOAT (N)/DFLOAT (N - 1)) X A= DSQRT (FI) XB=FI XC=FI*FI XD=XC*FI C LEAST SQUARES CURVE APPROXIMATIONS TO SKEWNESS AND KURTOSIS C CONFIDENCE INTERVALS AHIGH=-.2998432D-4 + 5.6879D0*XA +.4816522D0*XB - 75.13115D0*XC • 385.4993D0*XD AL0W=-.5168405D-3+ 4.073526D0*XA - 1 . 346945D0*XB - 45.00911D0*XC + 356.9341D0*XE BHIGH=2.986854D0 + 12.53813D0*XA + 27.78154D0*XB - 1473.694D0*XC + 19249. 14 DO*XD BLOW =3.001417D0 - 1 1. 26386D0*XA + 35.66988D0*XB - 717.4646D0*XC + 14353.41DO *XD CHIGH=2.993379D0 + 8.7076 19D0*XA - 6.240714D0*XB - 288.21 31D0*XC + 724.9355 CO *XD CLOW =2.998436D0 - 7.847057D0*XA + 12.88377D0*XB + 93.07472D0*XC - 4206.590 DO*XD NLAST=N C INITIALIZE TO FIND SUMS OF POWERS ANC MIN AND MAX 240 X=0.D0 X2=0.D0 X3=0.D0 X4=0.D0 AMIN=1.D70 A MA X=-1.D7 0 C GET INFORMATION FROM DATA DO 260 1=1 ,N V=A(I) IF (V.LT.AMIN) AMIN=V IF (V. GT. AMAX) AMAX=V VT=V X=X + VT 169 VT=VT*V X2=X2 + vr VT=VT*V X3=X3 + vr 260 X4 = X4 + VT*V C CHECK FOR ALL DATA VALUES EQUAL IF (AMIN.EQ.AMAX)GC TO 640 C CALCULATE MEAN, S. D., SKEWNESS AND KURTOSIS AM=X*FI AVG=AM AM2=AM*AM AM3= AM2* AM A M4=A M3 *AM VAR=X2*FI - AM2 C CHECK FOR VARIANCE = 0 IF (VAR. LT. 1. D-70) GO TO 640 SD = DSQRT (VAR) VARI=1.DO/VAR SKEWN=((X3 - 3.D0*AM*X2)*FI + AM3 + AM3)*VARI/SD AKURT=((X4 - 4.D0*AM*X3 + 6.D0*AM2*X2)*FI - 3 . DO*AM4 ) *VARI 1 *VARI SD=SD*DCOR C CHECK WHETHER SKEWNESS AND KURTOSIS ARE SIGNIFICANTLY C DIFFERENT FROM VALUES EXPECTED FOR A NORMAL DISTRIBUTION. ASK=DABS (SKEWN) IF (ASK. GT. AHIGH) GO TO 280 IF (ASK.LT. ALOW) GO TO 300 RSK= RB GO TO 320 280 RSK=RC GO TO 320 30 0 RS K= RA 320 IF (AKURT.GT.BHIGH)GO TO 340 IF (AKURT.LT.BLOW) GO TO 340 IF(AKURT.LT.CHIGH.AND.AKURT.GT. CLOW)GO TO 360 RKU= RB GO TO 380 340 RKU=RC GO TO 380 360 RKU= RA C HISTOGRAM - INITIALIZE ARRAY TO STORE FREQUENCIES. 380 DO 400 I=1,NCLASS 400 B(I)=0.D0 C CALCULATE FREQUENCIES IN EACH OF "NCLASS" CLASSES COVERING C RANGE OF DATA. • CON= (DFLOAT (NCLASS) - 2.D-4)/(AMAX - AMIN) CONS=-AMIN*CON + 1.0001D0 DO 420 I=1#N J=IDINT (A(I)*CON + CONS) 420 B(J)=B(J) + 1.D0 C FIND MAXIMUM FREQUENCY BMAX=-10.D0 DO 440 1=1,NCLASS IF (B (I) . GT. BMAX) BMAX=B (I) 440 CONTINUE C INITIALIZE WRITING ARRAY DO 460 1=1,80 1 7 0 460 CE (I) =BLANK C SET ARRAY WITH APPROPRIATE ASTERISKS CON=9.9998D0/BMAX GO TO (520,520, 560, 560,480) ,ICLIND 480 DO 500 1=1 ,64 BI=B(I)*CON IF (BI.LT.1,D-60)GC TO 500 IND=I + 640 - 64*IDINT(BI + 1.0001D0) CURVE (IND) =AST 500 CONTINUE GO TO 600 520 DO 540 1=1,NCLASS BI=B(I)*CON IF (BI. LT. 1 . D-60) GO TO 540 IND=I + 160 - 16*IDINT(BI + 1.0001EO) RE (IND) =RAST 540 CONTINUE GO TO 600 560 DO 580 1=1 ,32 BI=B(I)*CON IF (BI.LT.1 .D-60)GO TO 580 IND=I + 320 - 32*IDINT(BI + 1.0001DO) CI (IND) =IAST 580 CONTINUE C WRITE OUT RESULTS 600 WRITE (6, 12) C1,NAM,C2 WRITE(6,14)AVG,C4,SD,C5 WRITE(6, 16) AM IN , C 6, AM AX, C 7 WRITE(6,18)SKEWN,RSK,C8 WRITE(6,20)AKURT,RKU,C9 WRITE (6 ,22) N,C10 WRITE(6,24) (UNDER, 1=1,NBAR) 12 FORMAT(46X, 8A8/3X, 43A1, 8A8/46X, 8A8) 14 FORM AT(• MEAN =•,F20.7,17X, 8A8/« S.D. =»,F20.7,17X, 1 8A8) 16 FORM AT (* MIN = •, F20. 7, 17X , 8A8/« MAX = • ,F2 0.7 , 17X, 1 8A8) 18 FORMAT(• SKEW =«,F20.7,A4,13X, 8A8) 20 FORMAT(' KURT =',F20.7,A4,13X, 8A8) 22 FORMAT (• N =»,I10,27X, 8A8) 24 FORMAT (••• ,45X,8A8) 620 RETURN 640 WRITE (6,26) A (1),NAM 26 FORMAT (' »/«0****** ALL DATA VALUES ARE EQUAL TO«,G16.8, 1 • FOR THE VARIABLE : • , 43 Al / • - V'O '/•O •) GO TO 620 660 WRITE (6 ,28) N,NAM 28 FORMAT (' «/»0****** N =',110,' FOR THE VARIABLE : «,I»3-A1/ 1 '-'/'O'/'O •) GO TO 620 END APPENDIX C VARIABLE CHARACTERISTICS AND TRANSORMATIONS 171 APPENDIX TABLE C-1: VARIABLE CHARACTERISTICS BEFORE NORMALIZATION NAM E 1 POP'N DENSITY 2 POP'N POTENTIAL 3 RECENT POPfl INCR 4 TIME-CITY CENTER 5 FERTILITY RATE 6 % 0 -• 4 YEARS 7 % 5 - 9 YEARS 8 % 10-14 YEARS 9 % 15-19 YEARS 10 % 20-24 YEARS 11 % 25-29 YEARS 12 % 30-34 YEARS 13 % 35-39 YEARS 14 % 40-44 YEARS 15 % 45-49 YEARS 16 % 50-54 YEARS 17 % 55-59 YEARS 18 % 60-64 YEARS 19 % 65-69 YEARS 20 % 70-74 YEARS 21 % 75-79 YEARS 22 % 80-84 YEARS 23 % 85 + YEARS 24 % SINGLE <15 YRS 25 95SNGL>15YRS.MALE 26 5?SNGL>15 YRS.FEM. 27 % MARRIED 28 % WIDOWED 29 % DIVORCED 30 BORN B. C. 31 BORN CAN ADA-NOBC 32 ORIG CANADA 3 3 ORIG NATIVE POPN 34 ETHN USA 35 BORN UK+REP.IREL 36 ORIG UK+REP.IREL 37 BORN AUSTL+COMWL 38 ETHN FRANCE 39 BORN ITALY 40 ORIG ITALY 41 BORN GERM +A+SWIT 42 ORIG GERM+A+SWIT 43 BORN BEL G+N ET HL D 44 ORIG BELG+NETHLD 45 BORN SCANDINAVIA 46 ORIG SCANDINAVIA 47 BORN CENT EUROPE 48 ORIG CENT EUROPE 49 ETHN GREECE 50 ETHN INDIA PAKIS 51 BORN USSR 52 ORIG USSR MEAN S.D. MIN. 1902.2 3590.6 47.398 193582 59699. 41730. 1. 1102 .33456 .13100 21.420 12.440 .00000 .28185 .11724 .00000 .09760 .04054 .00000 .08997 .03706 .00000 .08295 .03302 .00000 .06395 .02042 .00000 .05768 .02417 .00000 .06257 .02521 .00000 .06879 .02071 .00000 .07337 .02014 .00000 .07020 .01745 .00000 .06883 .01862 .01355 .05750 .0 1971 .00297 .04641 .01962 .00000 .04003 .02019 .00000 .03578 .02134 .00000 .03593 .02415 .00000 .02760 .01868 .00000 .01417 .01138 .00000 .00668 .00674 .00000 .27052 .09555 .00267 .09324 .05245 .01812 .07152 .04140 .00000 .49103 .04920 . 19136 .06340 .03875 .00000 .01030 .01051 .00000 .42326 . 10681 .07588 . 27199 .05969 .00000 .01130 .03124 .00000 .00551 .06252 .00000 .02711 .0 1426 .00000 . 14479 .05996 .00000 .62159 .14594 .00000 .00599 .00727 .00000 .03715 .03360 ^00000 .01321 .02955 .00000 .02533 .04206 .00000 .02534 .02034 .00000 .07634 .04700 .00000 .01162 .01190 .00000 .03116 .02181 .00000 .02085 .01578 .00000 .02808 .02133 .00000 .02086 .01744 .00000 .03786 .02404 .00000 .00416 .01011 .00000 .00529 .01960 .00000 .01196 .01411 .00000 .03865 .02390 .00000 MAX. SKEWNESS KURTOSIS 68767. 9.589** 135.1** 280797 -.5577** 2.359** 3.2737 2.227** 12.36** 67.000 1. 135** 4.304** 1.4000 .7328** 9.843** . 24206 . 1900** 3. 197 .20635 -.1754** 2.924 .25455 .5077** 6.426** .20829 .1373* 5.568** .27804 1.707** 11.89** .25000 .9557** 6.514** .17935 .4621** 4.110** . 15331 .6344** 4.293** . 15000 .4347** 4.245** .26667 1.156** 13.10** .20000 .7283** 6.470** . 15214 .6771 ** 4.553** . 15094 . 9388** 4.895** . 15447 1.532** 6.746** .21680 1.881** 8.978** .13953 1.533** 6.511** .07891 1.714** 6.803** .04545 2.222** 9.765** .55556 -.4541** 3.316* .61880 4.175** 29.72** .52309 2.892** 20.13** .80000 -.9146** 8.529** .33721 1.701** 7.558** .08025 2.274** 10.27** 1.0000 -. 1056 5.470** .68430 -.0446 8.620** .36818 5.879** 48.25** 1.0000 15.10** 232.3** . 13953 1. 737** 9.469** .40000 .7934** 4.193** 1.0000 -.9740** 5.079** .07592 3.394** 23.43** .55741 8.082** 102.8** .30503 4.307** 25.57** .40881 3.891** 21.51** .25000 2.415** 17.03** .46889 2.732** 17.29** .09438 2.530** 13.27** . 18631 1.753** 8.625** .15625 2.486** 15.15** .20000 1.896** 9.156** .10744 1.428** 5.612** .25000 1.648** 10.34** . 13580 5.486** 46.74** .52000 16.75** 400.4** . 14792 4.099** 29.92** .21627 1.440** 8.194** 172 N AM E MEAN S.D. MIN. MAX. SKEWNESS KURTOSIS 53 BORN CHINA .01403 .05779 .00000 . 90244 8.893** 95.46** 54 ORIG CHINA .02385 .07724 .00000 .98916 7.483 ** 67.43** 55 ETHN JAPAN .00912 .03370 .00000 .66397 12.50** 196.4** 56 %OF IMGTS BEF'21 .36827 .11631 .00000 .83333 .1642** 3.654** 57 % OF IMGTS 21-40 . 18684 .05832 .00000 .58333 .8102** 5.647** 58 % OF IMGTS 41-50 . 10191 .05987 .00000 1.0000 4. 170** 48.74** 59 % OF IMGTS 51-55 . 15500 .06778 .00000 .75000 .8717** 7.773** 60 % OF IMGTS 56-59 .16053 .07890 .00000 .47205 . 560 1** 3.366** 61 % OF IMGTS 60-61 .02665 .02806 .00000 .28571 2.694** 17.71** 62 RELIG PROTESTANT .69327 .12619 .00000 1.0000 -1.542** 7.219** 63 RELIG ROM CATHOL .16655 .09296 .00000 1.0000 3.268** 23.55** 64 RELIG JEWIS.H .01016 .02673 .00000 .34642 5.620** 45.75** 65 RELIG OTHERS . 13001 .08316 .00000 .89702 2.980** 18.70** 66 LANG NOT ENG, FR .01103 .03109 .00000 .52033 7.883** 90.06** 67 ED-M-•NO-<GRADE 1 . 12640 .05232 .00000 .34884 .2789** 3.482** 68 ED-M-- NO-ELE ME NT Y .20983 .11659 .00353 .75238 .9105** 4.712** 69 ED-M-•NO-HIGH NC .30927 .07627 .00526 .57529 -.3756** 4.036** 70 ED-M-•NO-HIGH FIN .05194 .05399 .00000 .44800 2.134** 9. 103** 71 ED-M --NO-UNIV NC .04018 .03028 .00000 .23944 1.470** 6.184** 72 ED- M-•NO-UNIV DEG .04008 .04936 .00000 .32157 2.006** 7.207** 73 ED-M-•AT- EL EM +.PRE . 15020 .05986 .00000 .38527 -.0512 4.017** 74 ED-M-•AT-HIGH SCH .05517 .02589 .00000 . 15331 .3638** 3.317* 75 ED-M-•AT-UNIV UGD .01302 .01523 .00000 .16258 3.330** 22.09** 76 ED-M- AT-UNIV GRD .00391 .01219 .00000 .28144 14.57** 284.6** 77 ED-F-•NO- <GR ADE 1 .12353 .05855 .00000 .46296 .6742** 4.725** 78 ED-F- NO-ELE ME NTY .18890 .10053 .00000 .78992 .6459** 3.978** 79 ED-F-•NO-HIGH NC .36980 .09056 .02747 .85714 -.2399** 4.763** 80 ED-F-NO-HIGH FIN .06567 .07465 .00000 .48062 2.058** 7.451** 81 ED-F-NO-UNIV NC .03659 .03121 .00000 .23656 1.432** 5.593** 82 ED-F-NO-UNIV DEG .01829 .02519 .00000 . 191 18 2.474** 11.00** 83 ED-F-AT-ELEM+PRE . 13943 .05904 .00000 .41713 .1630** 4.232** 84 ED-F-AT-HIGH SCH .05039 .02405 .00000 .20000 .5522** 4.360** 85 ED-F-AT-UNIV UGD .00608 .00805 .00000 .07602 2.445** 12.12** 86 ED-F-AT-UNIV GRD .00132 .00323 .00000 .03125 4.078** 25.26** 87 MALE-FEMAL RATIO 1.0228 .52404 .28450 11.111 11.88** 183.8** 88 % OF POP "RURAL" .00927 .04875 .00000 .99739 11.10** 173.2** 89 AVG # PERSON/FAM 3.3460 .46328 2.0238 6.0541 .1010 5.215** 90 3SPOP IN FAMILIES .84245 .13834 .08642 .98858 -2.325** 9.295** 91 %FPOP 1 UNMARRIED .39830 .08925 .02353 .69643 -1.210** 5.553** 92 5CFAM W W .E.HEADS .6781 1 .1 1244 .00694 1.0000 -.9559** 4.883** 93 AVG FAM EARN WEH 5266.5 1348.7 1533.3 12296. 1.373** 6.507** 94 *FAM-HEAD < 25 Y .03688 .02659 .00000 .25000 1.441** 8.304**' 95 %FAM- HEAD 25-34 .19901 .07914 .00000 .52880 .4699** 3.548** 96 %FAM- HEAD 35-44 .24117 .08430 .00000 .67442 1.185** 6.730** 97 %FAM- HEAD 45-54 .22284 .05921 .03896 .66667 .8654** 5.907** 98 *FAM HEAD 55-64 .14014 .05016 .00000 .35135 .2123** 3.243* 99 SFAM-HEAD 65-69 .05109 .03155 .00000 .50000 3.109** 37.16** 100 5EFAM-HEAD > 70 .10886 .06549 .00000 .48333 1.566** 7.619** 10 1 55FAM-W EH <$2000 .09979 .07537 .00000 .56250 2.049** 9.600** 102 95FAM-WEH $2-2999 .09471 .06518 .00000 .50000 1.718** 8.235** 103 3SFAM-WEH $3-3999 .17898 .08147 .00000 .58763 .1203* 3.409** 104 5SFAM-WEH $4-4999 .23082 .08831 .00000 .60000 -.3563** 3.127 105 %FAM- WEH $5-5999 .15313 .06530 .00000 .50000 .0902 3.279* 106 55FAM-WEH $6-6999 .09262 .05759 .00000 .33654 .6290** 3. 138 173 NAME MEAN S. D. MIN. MAX. 107 5SFAM-WEH $7-9999 .08046 .08810 .00000 1.0000 108 %FAM- WEH $>10000 .04092 .09442 .00000 .83019 109 XFAM NO CHILDREN .37733 .14652 .02721 .95238 1 10 %FAM CHLD NONE>6 . 14681 .06255 .00000 .41667 111 XFAM CHLD 15-24 .11356 .04126 .00000 .23741 112 %FAM CHLD ALL<15 .12077 .06157 .00000 .43636 113 XFAM HBD+WF WORK .20650 .06864 .00000 .51592 114 %POP IN HOUSHLDS .96446 .09305 .04354 1.0000 115 #OF PERSONS/HSLD 3.3197 .60876 1.2739 6.3333 116 %HSLD 1 FAM HSLD .78946 .14401 . 14844 .99315 117 XHSLD>1 FAM HSLD .03127 .02938 .00000 .31429 118 %HSLD NON FAM HLD . 17927 .14339 . 0 0 0 0 0 .84375 119 XHSLD W LODGERS .08619 .07405 .00000 .62857 120 # HPERS ONS/DW ELL 3.3407 .67648 1.1815 6.6842 121 XDWEL OWNED,SNGL .63989 .29525 .00000 1.0000 122 % DW EL SNG DETACH .75207 .29731 .00000 1.0000 123 %DWEL DUPLEX .09304 .12645 .00000 .92063 124 % DW EL APT OTHER .15250 .25450 . 0 0 0 0 0 1.0000 125 ^DWEL RENTED .30712 .27596 .00000 1.0000 126 % DWEL RESID+BUSN .03450 .09460 .00000 1.0000 127 35DWEL OCUP 0-2YR .33894 .15789 .00000 1.0000 128 55 DWEL OCUP 3-5YR .21312 .09574 .00000 1.0000 129 ftDWEL 0CUP6-1OYR .19117 . 10636 .00000 .77273 130 % DWEL OCUP >10YR .25676 .14479 .00000 1.0000 131 35DWEL CONS <1920 .18615 .23994 .00000 1.0000 132 % DWEL CONS 20-45 .36321 .24471 .00000 1.0000 133 %D WEL CONS 46-59 .42675 .28740 .00000 1.0000 134 %DWEL CONS 60-61 .02390 .04817 .00000 .42500 135 5EDWEL COND GOOD .77123 .21213 .00000 1.0000 136 % DW EL COND OK .18529 .16743 .00000 1.0000 137 £DWEL COND FAIR .04348 .09468 .00000 1.0000 138 % DW EL W SEWER .71903 .41309 .00000 1.0000 139 XD WEL NO SEPT TA .01098 .05748 . 0 0 0 0 0 1.0000 140 % DWEL OWN WATER .02582 .11719 .00000 1.0000 141 AVG # ROOMS/DWEL 5.0491 1.0015 1.7822 9.8203 142 AVG # PERS/ROOM .66986 .12504 .33333 2.5000 143 AVG #BDROOM/DWEL 2.3788 .56452 .31023 4.4821 144 % DWEL OS <$12000 .39673 .29148 .00000 1.0000 145 ?SDWELOS$13-17000 .31894 .20686 .00000 1.0000 146 %DWELOS$18-27000 . 19187 .22106 .00000 1.0000 147 SDWEL (OS)>$28000 .05608 .15826 .00000 1.0000 148 5CDWEL NO TV . 13662 .1 1976 .00000 1.0000 149 %D WEL 1 TV .81432 .12149 .00000 1.0000 150 %DWEL NO CARS .30042 .19978 .00000 1.0000 151 SDWEL 1 CAR .58186 .15918 .00000 1.0000 152 5&DWEL >1 CARS .11772 .12240 .00000 .83333 153 «DWEL 1ST MORTG .28207 .20107 .00000 1.0000 154 % DWEL > 1 MORTG .05686 .06220 .00000 .34043 155 *DWEL NO MORTGAG .30096 .17753 .00000 1.0000 156 %DWEL BANK MORTG .20601 .19835 .00000 1.0000 157 SDWEL PRIV MORTG .09337 .07791 . 0 0 0 0 0 .39024 158 %DWEL W GARAGE .36248 .22929 .00000 1 . 0 0 0 0 159 AVG# TENANT/DWEL .30572 .27659 .00000 1.0000 160 %POP TENANTS .11007 .13162 .00000 .83520 SKEWNESS KURTOSIS 2 . 3 3 6 * * 3 . 9 7 1 * * 1 . 0 3 7 * * . 4 4 0 7 * * . 2 5 8 8 * * . 4 7 0 6 * * . 2 9 6 8 * * 5 . 9 3 2 * * . 2 9 1 5 * * 1 . 4 2 7 * * 2 . 8 7 6 * * 1 . 6 3 3 * * 2 . 6 2 8 * * . 0 6 8 5 . 8 4 2 3 * * 1 . 2 2 3 * * 2 . 6 5 2 * * 1 . 8 7 9 * * 1 . 0 8 0 * * 6 . 2 1 7 * * . 6 9 5 2 * * . 9 8 5 4 * * . 9 6 7 3 * * . 3 8 3 4 * * 1 . 7 8 5 * * . 4 9 7 2 * * . 1 4 8 7 * 3 . 3 1 6 * * 1 . 2 8 3 * * 1 . 3 4 5 * * 5 . 0 4 6 * * . 9 8 9 8 * * 9 . 8 7 6 * * 6 . 8 6 0 * * . 5 3 8 9 * * 3 . 2 5 6 * * . 3 0 0 2 * * . 1 4 6 7 * . 3 5 1 5 * * 1 . 4 6 3 * * 3 . 8 2 6 * * 2 . 7 1 5 * * 2 . 0 7 7 * * . 8 3 0 8 * * . 7 7 8 5 * * 9 6 5 * * . 4 8 0 8 * * 1 . 2 8 3 * * . 2 6 6 5 * * 1 8 4 * * . 8 6 4 3 * * . 3 6 0 0 * * 1 . 0 8 3 * * 2 . 1 0 2 * * 1 1 1 4 . 0 6 * * 2 1 . 6 1 * * 4 . 6 5 9 * * 3 . 5 5 6 * * 2 . 9 1 0 3 . 6 7 1 * * 4 . 2 1 0 * * 4 6 . 8 5 * * 4 . 9 1 1 * * 4 . 8 5 0 * * 1 8 . 1 9 * * 5 . 4 6 5 * * 1 3 . 5 1 * * 4 . 5 7 1 * * 2 . 4 9 0 * * 3 . 1 9 9 1 2 . 4 9 * * 5 . 4 8 9 * * 3 . 0 1 7 5 0 . 4 3 * * 3 . 3 9 0 * * 7 . 2 5 7 * * 5 . 1 2 2 * * 3 . 3 1 2 * 5 . 6 0 4 * * 2 . 4 9 6 * * 1 . 9 1 4 * * 1 7 . 6 8 * * 4 . 3 4 4 * * 5 . 0 5 2 * * 3 9 . 0 9 * * 2 . 1 2 6 * * 1 2 7 . 5 * * 5 1 . 8 9 * * 4 . 9 4 1 * * 4 2 . 4 7 * * 4 . 3 7 0 * * 1 . 8 9 8 * * 2 . 9 8 6 4 . 6 9 5 * * 1 8 . 6 3 * * 1 5 . 0 5 * * 1 1 . 2 6 * * 3 . 5 9 5 * * 3 . 9 7 3 * * 8 . 2 1 4 * * 2 . 6 8 3 * * 4 . 5 4 1 * * 2 . 9 9 8 4 . 0 9 9 * * 3 . 4 7 2 * * 2 . 6 2 0 * * 3 . 0 1 7 7 . 6 2 7 * * 174 NAME MEAN S.D. MIN. MAX. SKEWNESS KURTOSIS 161 AVG GROSS RENT 83.363 35.543 .00000 299.00 .4174** 5.973** 162 AVG # FAM/DWELL .86780 .17120 .20588 3.5000 2.427** 51. 16** 163 %MALLF EMPLOYER .07544 .06458 .00000 .48148 2.560** 11.72** 164 XMALLF OWN ACCNT .05514 .03572 .00000 .44167 3.701** 32.35** 165 XMALLF WAGE EARN .86819 .07704 .45679 1.0000 -1.901** 7.871** 166 5EFEMLF EMPLOYER .01915 .02963 .00000 .50000 7.942** 116.7** 167 %FEMLF OWN ACCNT .04028 .04399 .00000 .50000 3.262** 22.54** 168 5SFEMLF WAGE EARN .92571 .06597 .00000 1.0000 -4. 106** 40.93** 169 %FEMLF UNPD FAMW .01405 .02456 .00000 .33333 4.189** 37.46** 170 55ECAMPOP EMPLOYD .94240 .05695 .52632 1.0000 -2.877** 15.89** 171 %ECAFPOP EMPLOYD .96012 .05013 .00000 °1.0000 -7.113** 116.1** 172 IND FOOD SBEVERGS .04213 .03014 .00000 .42424 4.722** 50.17** 173 IND WOOD FINISH .05892 .05273 .00000 .69841 3.328** 26.61** 174 IND PAPE R6ALLIED .01375 .01304 .00000 . 12030 2.062** 11.04** 175 IND METAL FABRCN .02391 .01490 .00000 . 11538 .8185** 4.564** 176 IND CONSTRUCTION .05696 .02906 .00000 .21333 .4539** 3.377** 177 IND TRANSP+STORG .08821 .03716 .00000 .55882 3.468** 38.52** 178 IND TELECOMMUNCN .02811 .0 1509 .00000 .09714 .5446** 3.774** 179 IND TRADE-W HOLSL .07824 .03057 .00000 .24155 .5834** 4.699** 180 IND TRADE-RETAIL . 11704 .03994 .00000 .50000 .5628** 9.870** 181 IND FINANCE+INS .05623 .03450 .00000 .23353 .9897** 4.103** 182 IND EDUCATION .04608 .04188 .00000 .42400 3.286** 20.93** 183 IND HEALTH WELFR .05725 .04388 .00000 .73663 4.648** 54.46** 184 IND HOTEL RESTNT .03792 .04237 .00000 .50000 4.487** 33.29** 185 IND PUB ADMN DEF .06953 .05090 .00000 .92889 10.90** 176.7** 186 OCC MANAGERIAL . 10670 .07489 .00000 .51542 1.693** 6.244** 187 o c c PROFL TECHNL . 10601 .08006 .00000 .68984 1.468** 6.672** 188 OCC CLERICAL .16225 .05984 .00000 .45251 .4780** 4. 192** 189 o c c SALES .08714 .03499 .00000 .22826 .4155** 3.778** 190 o c c FARMER .00900 .04161 .00000 .58235 9.374** 106.9** 191 o c c LOGGER .00588 .01418 .00000 .35000 15.09** 322.5** 192 o c c FISH TRAP .00711 .02174 .00000 .40559 11.88** 176.4** 175 APPENDIX TABLE C-2 : NORMALIZATION AND RESULTING CHARACTERISTICS 'ARIABLE TRANS- CONSTANT CONSTANT SKEWNESS KURTOSIS NUMBER FORM 1 2 1 ARCTAN (X) .00109 .73389 .0723 2.828 2 ARCSIN (X) 3.210 -6 .09851 -.0450 2. 109 ** 3 ARCTAN (X) 2.6901 -2.587 -.0212 2.893 4 LOG (X) .09646 1.0000 .0423 3.067 5 ARCTAN (X) 2.9852 -.7814 -.0908 2.798 6 LOG (X) 1.8500 1.0000 -.0090 3.052 7 ARCSIN (X) 1.8887 .30514 -.0485 2.963 8 ARCTAN (X) 15.748 -1.321 -.0977 2.910 9 ARCTAN (X) 19.584 -1.267 -.0632 2.893 10 ARCTAN (X) 20.272 -.6892 -.1325 * 2.736 * 11 ARCTAN (X) 11.107 -.3150 .0543 2.868 12 ARCTA N (X) 13.522 -.7302 .0242 2.908 13 ARCTAN (X) 19.865 -1.258 .0775 2.850 14 ARCTAN (X) 22.919 -1.489 . 1100 2.802 15 ARCTAN (X) 17.456 -1.101 .0836 2.912 16 ARCTA N (X) 17.762 -.9212 .0946 2.736 * 17 LOG(X) 16.308 1.0000 -.0008 3.31 1 * 18 LOG (X) 29.605 1.0000 .0076 3.119 19 ARCTAN (X) 21.090 .14541 .0419 2.859 20 LOG (X) 115.93 1.0000 -.0753 3. 350 21 LOG(X) 107.08 1.0000 -.0259 3.020 22 LOG (X) 329.50 1.0000 -.0732 2.902 23 LOG(X) 734.49 1.0000 -.1121 2.691 4 4 24 ARCSIN (X) .31397 .76507 -. 1503 * 3.288 4 25 ARCTAN (X) 16.110 -.6020 .0563 3.067 26 ARCTAN (X) 16.427 -.1748 .0929 2.866 27 ARCTAN (X) 11.178 -5.679 -.0473 3.209 28 LOG (X) 77.414 1.0000 .0809 3.220 29 LOG(X) 666.05 1.0000 -.0934 2.792 30 ARCTAN (X) 3.9321 -1.964 .0348 3. 143 31 ARCTAN (X) 8.3764 -2.278 -.0923 3.052 32 LOG (X) 3.201 +11 1.0000 I .2945 4 4 1.112 4 4 33 LOG(X) 1.599 +11 1.0000 .8098 4 4 1 .680 4 4 34 ARCTAN (X) 29.445 -. 0484 .0784 2.941 35 ARCTAN (X) 4.3359 -. 1028 .0847 3.03 3 36 EXP(X) 1.7473 1.0000 -.0095 3.060 37 LOG(X) 687.81 1.0000 .0793 2.395 4 4 38 ARCTAN (X) 26.782 .15507 -. 1063 3.234 39 LOG(X) 4399.7 1.0000 .01 36 1.990 4 4 40 LOG (X) 891.61 1.0000 .0178 3. 156 41 LOG(X) 184.37 1.0000 -.0538 3.015 42 ARCTAN (X) 11.915 .09044 -.0055 2.972 43 LOG(X) 302.61 1.0000 .0587 2.572 4 4 44 LOG (X) 73.376 1.0000 .1133 2.000 45 LOG(X) 145.79 1.0000 .0039 3.230 46 LOG (X) 145.22 1.0000 -.0723 3.051 47 LOG(X) 171.97 1.0000 -.0604 2.422 4 4 48 LOG (X) 45.762 1.0000 -.0026 3.079 49 LOG(X) 9.891 +11 1.0000 .3602 4 4 1.144 4 4 50 LOG(X) 5.103 +11 1.0000 -.0976 1.034 4 4 51 LOG(X) 478. 30 1.0000 -.0411 2.834 52 LOG (X) 37.664 1.0000 -.0399 3. 119 VARIABLE TRANS- CONSTANT NUMBER FORM 1 53 LOG (X) 6921.6 54 LOG (X) 2848.4 55 LOG (X) 45101. 56 ARCTAN (X) 2.4074 57 ARCTAN (X) 4.4581 58 ARCTAN (X) 9.3536 59 ARCTAN (X) 4.4264 60 SQRT (X) 26.616 61 LOG(X) 105.05 62 EXP(X) 2.9321 63 ARCTAN (X) 6.9926 64 LOG (X) 1.361 +6 65 ARCTAN (X) 8.4174 66 LOG (X) 4503.6 67 SQRT (X) 9. 1736 68 LOG (X) 4.0140 69 ARCTAN (X) 5.6380 70 LOG (X) 277.83 71 LOG(X) 165. 12 72 LOG (X) 486.25 73 ARCTAN (X) 7.5177 74 ARCSIN(X**.25) 3.0896 75 LOG(X) 321. 83 76 LOG (X) 8.203 +16 77 LOG(X) 6.1482 78 LOG(X) 2.4419 79 ARCTAN (X) 5.5210 80 LOG (X) 301.39 81 LOG(X) 128.16 82 LOG (X) 516.06 83 ARCTAN (X) 4.7426 84 LOG (X) 10.781 85 LOG(X) 869.89 86 LOG(X) 3.093 +12 87 ARCTAN (X) 5.9156 88 LOG (X) 2.051 +11 89 ARCTAN (X) .90659 90 EXP (X) 9.1078 91 ARCTAN (X) 5.4899 92 ARCTAN (X) 3.5575 93 ARCTAN (X) .00044 94 LOG (X) 43.868 95 LOG(X) 3.1589 96 ARCTAN (X) 6.5243 97 ARCTAN (X) 5.0666 98 LOG (X) 1.5950 99 ARCTAN (X) 12.361 100 ARCTAN (X) 6.4137 101 LOG(X) 30.515 102 LOG (X) 22.093 103 ARCTAN (X) 2.4550 104 ARCSIN (X) .05662 105 ARCTAN (X) 1.9907 106 ARCSIN(X**.25) 2.2572 176 CONSTANT SKEWNESS KURTOSIS 2 1.0000 .0613 2.127 ** 1.0000 .1118 1.886 ** 1.0000 .0815 1.278 ** -.8806 . 1242 * 2.970 -.3830 -.0212 3.221 -.3532 .0202 2.750 * -.5361 . 1193 * 2.731 * 1.0000 -.0269 2.915 1.0000 .0287 2.302 ** 1.0000 -.0475 3. 173 -.2647 -.0980 2.979 1.0000 .0394 1.143 ** .30566 -.0697 3.185 1.0000 . 1078 1.956 ** 1.0000 -.1007 3.216 .98582 .0958 2.883 - 1.894 .0316 2.851 1.0000 -.0764 2.904 1.0000 -.1407 * 2.740 * 1.0000 -.0944 2.437 ** - 1.179 -.1125 2.798 .14954 .0255 3.201 1.0000 .1089 2.660 ** 1.0000 . 1569 * 1.029 ** 1.0000 -.0916 3.220 1.0000 . 1600 * 2.731 * -2.142 -.0349 2.906 1.0000 -.1049 2.829 1.0000 -.0495 2.435 ** 1.0000 . 1007 2.084 ** -.6712 -.0113 3.213 1.0000 -.0766 3.321 * 1.0000 .1004 1.861 ** 1.0000 1. 220 ** 2.494 ** -5.651 -.0546 2.846 1.0000 1.692 ** 3.901 ** -3.108 -.0424 3.055 .2129 1 .0950 2.273 ** -2.587 -.1078 3.218 -2.912 .0397 2.943 -1.993 .0978 2.996 1.0000 -. 1345 * 2.754 * 1.0000 -.0107 3.011 -1.273 -.0398 2.887 -.6791 -.0396 3.058 1.0000 -.0003 3.166 .1184 2 -.0672 2.834 -.0982 .0843 2.851 1.0000 .0889 3.217 1.0000 .0556 3. 102 -.4894 .0753 2.922 .94693 -.1465 3.442 ** -.3348 .0753 3.009 .06094 .0471 2.803 177 VARIABLE TRANS- CONSTANT CONSTANT SKEWNESS KURTOSIS NUMBER FORM 1 2 107 LOG (X) 90.809 1.0000 -.0541 2.449 ** 108 LOG (X) 5295.5 1.0000 .0816 1.481 ** 109 ARCTA N (X) 2.9348 -.5574 -.0641 3.086 110 LOG (X) 3.6254 1.0000 -.0256 3.005 111 ARCSIN(X**. 25) 2.0358 .16882 .0585 3.025 112 LOG (X) 3.4108 1.0000 -.0080 3.034 113 ARCTA N (X) 4.8075 -.8427 -.0760 3.021 114 ARCTA N (X) 19.344 -19.66 -1.080 ** 3.020 115 ARCTAN (X) .82134 -2.952 -.0183 3.083 116 SXP(X) 5.2081 .22692 . 1076 2.436 ** 117 LOG(X) 126.49 1.0000 -.0369 3.024 118 LOG (X) 41.844 1.0000 .0475 3.020 119 LOG(X) 89.803 1.0000 -.0356 3.332 * 120 ARCTAN (X) .59129 -2. 125 .0424 3. 129 121 ARCSIN (X) .11108 .88892 -.1464 * 2.264 ** 122 ARCSIN (X) .08067 .91933 -.4585 ** 2.092 .** 123 LOG(X) 63.264 1.0000 .1116 1 .958 ** 124 LOG(X) 825. 15 1.0000 . 1592 * 1.383 ** 125 ARCSIN(X**. 25) .94426 1.000 -8 -.0309 3.446 ** 126 LOG(X) 1.057 +11 1.0000 .3713 ** 1. 149 ** 127 LOG(X) 2.7868 1.0000 .0882 2.878 128 ARCTA N (X) 3.3946 -. 4535 .0862 2.912 129 LOG(X) 4.7010 1.0000 .0047 3.283 * 130 ARCSIN(X**. 25) .70445 .14778 .0917 3.029 131 ARCSIN(X**. 25) .99383 1.000 -8 . 1353 * 2.759 * 132 ARCSIN (X**. 25) .97363 .02637 .0531 2.808 133 LOG(X/(1 - X)) .90466 .04894 -.0110 2.546 ** 134 LOG (X) 2.076 +11 1.0000 .6296 ** 1.403 -** 135 EXP(X) 3.3470 1.0000 -.1062 1.858 *• 136 LOG (X) 13.140 1.0000 .0681 2. 191 ** 137 LOG (X) 1.056 +16 1.0000 .1545 * 1.029 ** 138 ARCSI N (X) .07118 .92882 -.7642 ** 1.784 ** 139 LOG(X) 1.740 +11 1.0000 2.478 ** 7.160 ** 140 LOG(X) 8.533 +10 1.0000 1. 353 ** 2.850 141 ARCTAN (X) .41936 - 1.967 - .0024 3. 194 142 ARCTAN (X) 3. 1990 -1.943 .0308 3.046 143 ARCTAN (X) .92114 -2.341 -.0094 2.964 144 . LOG (X/ (1 - X)) .96062 .02889 . 1235 * 3. 195 145 ARCSIN(X**. 25) .77349 .1 1326 -.0104 2.837 146 LOG (X) 20.848 1.0000 . 1425 * 1.877 ** 147 LOG(X) 6.319 +10 1.0000 1.021 ** 2.062 ** 148 LOG (X) 27.066 1.0000 -.0574 3.340 * 149 EXP(X) 3.8521 1.0000 .0439 2.845 150 ARCSIN (X**. 2 5) .45049 .01467 -.1509 * 2.778 * 151 EXP(X) 1.3004 1.0000 -.0786 3.301 * 152 LOG (X) 30.230 1.0000 -.0227 2.306 ** 153 ARCSIN(X**. 25) .90538 .09462 .1490 * 2.599 ** 154 LOG (X) 36.174 1.0000 . 1551 * 1.777 ** 155 ARCSIN(X**. 25) .73229 .13386 ^.0356 2.860 156 LOG (X) 10.083 1.0000 .0869 1.974 ** 157 LOG(X) 10.782 1.0000 .0707 2.119 ** 158 ARCSIN(X**. 25) .941 14 .05886 .0214 2.864 159 ARCSIN(X**. 25) .94859 1.000 -8 -.0449 3.435 ** 160 LOG (X) 205.14 1.0000 -. 1596 * 2.849 VARIABLE TRANS- CONSTANT NUMBER FORM 1 161 ARCTAN (X) .01829 162 ARCTAN (X) 7.0095 163 ARCTAN (X) 17.034 164 ARCTAN (X) 12.738 165 EXP(X) 7. 1716 166 LOG(X) 180.45 167 LOG(X) 68.197 168 EXP (X) 9.0953 169 LOG(X) 4.072 +16 170 EXP (X) 14.135 171 EXP(X) 18.352 172 ARCTAN (X) 17.255 173 LOG(X) 75.143 174 LOG (X) 143.88 175 LOG(X) 27.645 176 LOG (X) 6.1945 177 ARCTAN (X) 12.245 178 LOG (X) 18.066 179 ARCTAN (X) 13.084 180 ARCTA N (X) 7.5105 181 LOG(X) 31.882 182 ARCTA N (X) 20.178 183 ARCTAN (X) 15.953 184 . ARCTAN (X) 306.79 185 ARCTAN (X) 15.323 186 LOG (X) 53.412 187 LOG(X) 57.461 188 ARCTA N (X) 6.6849 189 ARCTAN (X) 8.5751 190 LOG (X) 2.403 +11 191 LOG(X) 1537.7 192 LOG (X) 2414.7 178 CONSTANT SKEWNESS KURTOSIS 2 -1.495 -.0012 3.078 -6.683 .0485 2.895 .81494 -.0224 2.934 .09760 .0890 3. 132 -2.276 .0386 3.298 * 1.0000 . 1328 * 2.044 *» 1.0000 .0905 2.798 1.0000 -.0951 2.635 ** 1.0000 .2348 ** 1.058 ** -6.440 -.0730 2. 193 ** 1.0000 -.0627 1.913 ** -. 1269 -.0190 2.776 * 1.0000 .0766 3.110 1.0000 .0550 2.545 ** 1.0000 .0567 2.815 1.0000 .0795 2.776 • -.8302 .0362 3.084 1.0000 -.0280 2.982 -.8237 -.0399 2.866 -.8670 -.0711 3.165 1.0000 -.0358 2.803 .42028 .0522 2.719 * -.1133 .0188 2.886 13.367 . 1168 * 2.980 -.6154 -.0653 3.035 1.0000 . 1337 * 3.246 * 1.0000 -.1368 * 2.669 ** -. 8846 -.0529 2.844 -.5972 .0585 2.829 1.0000 .7034 ** 1.517 ** 1.0000 .0535 1.764 *• 1.0000 . 1614 * 1.766 ** APPENDIX D PROBABILITY IN AN F-DISTRIBUTION 179 FUNCTION PROBF (F,FNUM,FDEN) C ** NOTE ** ALL VALUES ARE DOUBLE PRECISION FLOATING C POINT NUMBERS. RETURNS THE PROBABILITY OF ACHIEVING A C VALUE AS HIGH AS "F" IN AN F-DISTRIBUTION WITH "FNUM" C DEGREES OF FREEDOM IN THE NUMERATOR AND "FEEN" IN THE C DENOMINATOR. THE CALCULATION IS DONE BY NUMERICALLY C INTEGRATING THE F-DISTRIBUTION FUNCTION USING M-POINT C GAUSS-CHEBYSHEV QUADRATURE FOR FDEN < 60 OR FNUM < 10. C OTHERWISE THE STANDARD APPROXIMATION IS US EC, WITH C EXECUTION TIME APPROXIMATELY 0.002 SECONDS PER EVALUATION C ON THE IBM 360/67. FOR INTEGRATION, IF FDEN > 4, B IS C SET TO 50 AND EXECUTION TIME = 0.05 SECONDS; FOR FDEN = 3, C M = 100, T = 0.09 SEC; FDEN = 2, M = 200, T = 0.2 SEC; C AND FDEN = 1, M = 500, T =0.5 SECONDS. RESULTS DIFFER C FROM TRUE VALUES BY NOT MORE THAN 6 IN THE FIFTH SIGNIFICANT C FIGURE OVER THE WHOLE RANGE OF FNUM AND FDEN VALUES. C REFERENCES: CARNAHAN, B. , LUTHER, H. A. , & WILKES, J. 0. C "APPLIED NUMERICAL METHODS". NEW YORK: WILEY, 1969, C PP. 115-121. C ABRAMOWITZ, M. A., 6 STEGUN, I. A. C "HANDBOOK OF MATHEMATICAL FUNCTIONS". NEW YORK: DOVER, C 1965, P. 947. C PROGRAMMER : MIKE PATTERSON, DEPARTMENT OE GEOGRAPHY, C UNIVERSITY OF BRITISH COLUMBIA, FEBRUARY, 1972. IMPLICIT REAL*8 (A-H,0-Z) DIMENSION IC(4) DATA PI/3.141592653589793/,IC/500,200,100,50/ PROBF=0.D0 IF (FNUM*FDEN*F.EQ.O.DO)RETURN IF{FDEN.LT.60.DO)GO TO 100 IF(FNUM.LT.IO.DO)GO TO 100 GO TO 200 100 IND=MIN0 (4 ,IDINT (FDEN + 0.999D0)) M=IC (IND) DN=F NU M DD= FDEN A=DN*0.5D0 B=DD*0. 5D0 C= (DN - 2. DO) *0.5DO AB= A + B CON=DLGAMA (AB) + A*DLOG (DN) + B*DLCG (CD) - CLGAMA (A) 1 - DLGAMA (B) AF=DABS(F) AMI= 1. DO/DFLOAT (2*M) ZIS=-PI*AMI ZINC=-2.D0*ZIS XC=AF*0.5D0 SUM=0. DO DO 180 J=1 ,M ZIS=ZIS + ZINC ZI=DCOS(ZIS) XI=ZI*XC + XC 180 SUM=SUM + DEXP(CON + C*DLOG(XI) - AB*CLOG(CC * DN*XI)) 1 *DSQRT (1. DO - ZI*ZI) PROBF= AF*PI*SUM*AMI RETURN 200 FDI=2.DO/9.DO FNI = FDI/FN UM FDI= FDI/FDEN FS=F** (1 .DO/3 .DO) XN=(1.D0 - FDI) *FS + FN I - 1.D0 XN = XN/(DSQRT (FDI*FS**2 + FNI) ) PROBF=0.5D0 + 0.5D0*DERF(XN/DSQRT(2. DO) ) RETURN END APPENDIX E FACTOR LOADINGS MATRICES 181 TABLE E-l FACTOR LOADINGS MATRIX - NORMALIZE! LATA FACTOR VARIABLE 1 2 3 4 5 6 7 1 POP • M I DENSITY -.4536 — . 1867 -.3784 .3 065 -.1564 • 5039 .0387 2 POP ' N I POTENTIAL -.3749 -.2 126 -.1032 .5019 .0350 • 5554 .0713 3 RECENT POPN OCR . 4250 . 1544 .0720 . 0623 -.1536 * 2403 .0 528 4 TIME-•CITY CENTER .4530 -. 0731 .2273 -.4225 .0698 * 53 04 .0973 5 FERTILITY RATE . 6097 -. 2801 . 2238 -.0235 -.1674 • 2715 -.0240 6 % 0 -• 4 YEARS .7639 -.2409 . 1 534 -.0092 -.2717 • 1943 .0776 7 % 5 - 9 YEARS . 8627 -. 07 37 . 2265 -. 159C -.0233 • 1399 .0605 8 1 10- 14 YEARS .7317 .0030 .2556 -. 163 1 . 3674 • 0580 .0 1 34 9 % 15-19 YEARS . 2871 . 1076 . 3008 -.06 10 .7521 • 0279 .0185 10 * 20-24 YEARS -.2558 -. 1508 -.2040 .2882 .151 1 • 02 18 -.0331 11 % 25-29 YEARS -.0221 -. 2307 -.3159 . 2209 -.4191 • 0330 .0620 12 % 30-34 YEARS . 3593 -. 0823 -. 1 602 . 1200 -.5244 m 01 14 .1451 13 % 3 5-•39 YEARS . 5894 . 0605 -.0944 -.0081 -.1819 0768 .1954 14 51 40-44 YEARS .2999 . 1899 .0522 -.0875 .1999 — ' • 193 1 .1424 15 % 45-•49 YEARS -.2648 . 3368 . 1 128 . 0278 .4704 -. 2 196 - .0444 16 % 50-54 YEARS -. 5966 . 1993 .0742 . 1428 . 3446 -. 2028 -. 1 38 5 17 % 5 5-59 YEARS -.7812 .0421 .0342 . 1773 .0937 • 05 04 -.1874 18 S 60-64 YEARS -.8512 .0059 -.0133 . 0919 .0340 • 0169 -.1323 19 % 65-69 YEARS -.8758 .0079 -.0100 . 092C -.0729 • 0025 - .0602 20 3 70-74 YEARS -.8834 .0165 -.0376 .0177 -.0394 0032 -.0 162 21 % 75-79 YEARS -.8758 -.0077 -.0602 -.0296 -.0234 0010 .0709 22 % 80-84 YEARS -.7529 .0401 -.0 521 -.0624 -.0066 • 0122 .0590 23 % 85 + YEARS . -.6055 . 1098 -.0041 -. 04 96 .0120 0180 .0577 24 K SINGLE : <15 YRS .91 82 - . 1367 . 1957 -. 1371 -.0212 • 1432 .0518 25 %SNGL>15YRS.MALE 5344 -. 1989 -.1586 .2814 .2483 • 1059 -.2943 26 $SNGL>15YRS.FEM. -.5888 .3867 -.0885 . 0935 . 3060 181 1 -.0088 27 % MARRIED -.1471 . 0492 . 1927 -.0226 -.0282 * 0089 .2531 28 % WIDOWED -.8782 . 1018 -. 1148 . 0054 .0304 104 2 .0611 29 % DIVORCED -. 4967 -. 0575 -. 2995 . 0848 -.0315 • 152 1 -.0133 30 BORN B- c. .7627 . 0388 . 3374 -. 1760 .1412 1 196 . 1 143 31 BORN CANADA-NCBC -. 1245 . 2432 -.2886 -.2623 -.1156 m 0466 .0168 3 2 ORIG CANADA .0324 . 0729 -.0107 -. 055 1 -.0418 • 017 1 .1371 33 ORIG NATIVE PCPN .0401 -.2513 -.0996 . 0768 -.08 60 1464 - . 1 321 34 ETHN USA -.3606 .3892 -.0104 -. 1572 .0019 m 0609 -. 1707 35 BORN UK+REP.IREL -.6788 . 2775 -.0715 -.3 04 8 .0022 * 1455 .2699 36 ORIG UK+REP.IREL -.0362 .6769 .0957 -.5110 .0868 • 0832 .20 58 37 BORN AUSTL+COMWL -.2689 . 3996 -.1513 -.0 101 -. 1349 """" • 0877 .0474 38 ETHN FRANCE .0574 - .2765 -.2159 •-.0228 -.0579 • 1468 -.0433 39 BORN ITALY -.1235 -.4 296 .0246 .4526 .0038 • 17 12 -.0098 40 ORIG ITALY .02 83 -.3 927 .0351 .4804 .0046 • 1792 .0685 4 1 BORN GERM+A+SWIT -.2786 -. 18 13 -.1491 .2 140 -.1020 0692 - .0238 42 ORIG GERM+A+SWIT .0192 - .2947 -.0699 . 1056 -.0595 0725 . 1 279 43 BORN BE LG + NETHLD .01 58 -. 18 10 -.0506 -.C13 3 .0443 42 16 .1136 44 ORIG BELG+NETHLD .1809 -.1882 -.0000 0630 .0702 m 4917 . 1672 45 BORN SCANDINAVIA -.2369 -.4208 -.0905 . 1 125 -.1171 1 136 - .0251 46 ORIG SCANDINAVIA -.1096 - .3 160 -.0704 .4974 -.0430 m 0125 -.0087 47 BORN CENT EUROPE -. 3252 -. 2578 -. 1056 . 5475 -.0232 0959 -.1 128 48 ORIG CENT EUROPE -.0902 -.3628 -.0941 .4844 -.0421 0663 -.0455 4 9 ETHN GREECE -.1135 . 1589 -.0853 . 1253 -.0108 • 0826 - .0177 50 ETHN INDIA PARIS -.0497 . 1848 -.0596 . 0420 -.0448 • 0456 .0 161 51 BORN USSR -.2916 -. 1896 -.0631 .4663 -.0027 006 2 .0 343 52 ORIG USSR .0070 -.4701 -.0865 . 3696 -.0385 • 0679 .0885 182 VARIABLE 1 2 53 BORN CHINA -.1616 -.C914 54 . ORIG CHINA -.0863 -.2206 55 ETHN JAPAN -.0810 -.0597 56 %0? IMGTS BEF«21 -.6270 . 08 14 57 % OF IMGTS 21-40 .1000 1380 58 % OF IMGTS 4 1-50 . 5270 . 2514 59 % OF IMGTS 51-55 .3861 -.04 17 60 % OF IMGTS 56-59 .0874 -. 122 1 6 1 % OF IMGTS 60-6 1 -.0773 . 1774 62 RELIG I PROTESTANT .1014 .5407 63 RELIG I ROM CATHOL -.0139 -.5651 64 RELIG I JEWISH -. 1963 .4657 65 RELIG ! OTHERS -.1284 -.3874 66 LANG NOT ENG r FR -.1143 -.3284 67 ED-M-NO-<GRA DF 1 . 6881 -.3030 68 ED-M-NO-ELEMENTY -.3679 -.7525 69 ED-M-NO-HIGH NC -.2330 -. 1005 70 ED-M-NO-HIGH FIN -.2454 .4379 7 1 ED-M-NO-UNIV NC -.2356 .7429 72 ED-M-NO-UNIV DEG -.0915 .8783 7 3 ED-M-AT-ELEM+ PRE .7540 . 0564 74 EC-M- AT—HIGH SCH .2565 .3263 75 ED-M-AT-UNIV UGD -.0943 .6 340 76 ED-M-AT-UNIV GRD -.0412 .4295 77 ED-F-NO—<GRADF 1 . 6584 -.3931 78 ED-F-NO-ELEMENTY -.3955 -.7142 79 ED-F-NO-HIGH NC -.2326 . 125 1 80 ED- F-NO-HIGH FIN -.2497 .4912 81 ED-F-NO-UNIV NC -. 2075 . 7880 82 ED-F-NO-UNIV DEG -.1717 .8112 83 ED-F-AT—E LEM+ ERE . 80 70 -.07 1 1 84 ED-F-AT-HIGH SCH .3340 . 2528 85 ED-F-AT-UNIV UGD -.0944 .5626 86 ED-F-AT-UNIV GRD -.1098 .4555 87 MALE-FEMAL RATIO . 3445 -.4674 8 8 % OF POP "RURAL" .2106 -.0895 89 AVG # PERS ON/FAM .8755 -.0802 90 *POP IN FAMILIES .8067 . 0523 91 5EFPOP UNMARRIED . 8833 -. 1268 92 3FAM W W. E. HEADS .6376 -.2034 93 AVG FAM EARN WEH .1947 .7989 94 3FAM-HEAD < 25 Y -.0042 -.4360 95 % FAM-HEAD 25-34 . 4086 -.2925 96 % F AM-HEAD 35-44 .8106 . 0403 97 % FAM-HEAD 45-54 .0748 . 2920 98 X FAM HEAD 55-64 -.6741 . 0445 99 % FAM— HEAD 65-69 -.7133 .0005 1C0 % FAM-HEAD > 70 -.8416 . 0554 101 *FAM-WEH <$2000 -.4032 -.4095 102 % FAM-WEH $2-2999 -.4042 -.4866 103 % FAM-WEH $3-3999 -.2267 -.6426 104 *FAM-WEH $4-4S99 .1506 -.54 05 105 %FAM- WEH $5-5999 . 31 10 .0112 106 % FAM-WEH $6-6999 .2807 .5249 FACTCR 3 4 5 6 7 -.1127 . 6351 -.0067 -.0885 -.0912 -.0787 . 6973 -.0009 -.0896 -.0364 -.0187 .4 34 1 .0182 . 0199 .0037 . 2009 -.2688 .1112 . 0580 .1899 .1333 . 0048 .1541 . 2030 .0831 .0 630 -.0499 .1107 -.0039 .0518 -.0575 .2383 -.1820 .0185 .0 120 -.2563 . 2745 -.1315 -. 1026 -.1086 -.2306 . 2053 -.0623 -.0664 -.2284 .1562 -. 553 1 .0518 -.0167 .2477 -. 1435 .33 13 -.0465 -. 1226 -.1639 -.0804 . 1906 .0069 -. 1277 -.0505 -. 1321 .456 8 -.0529 . 2347 .0337 -.0553 .5322 .0205 . 03 14 -.0691 . 1 566 . 019C -.2623 . 2090 .0761 -.1026 .2079 -.0946 . 0195 -.1319 -. 1058 -.0773 -.0475 -.0292 .3956 -.0944 -.05 12 -.0202 .0172 .0465 -.0761 -.1370 -.0828 -.0782 .0208 -.0007 -.1305 -.0034 -.0544 -.0154 . 2367 -. 1547 .2313 . 0630 .0810 .2915 -.0620 .5594 -.0076 .1527 .0871 . 0520 .2820 -. 1 152 .0388 .0017 . 0719 .1167 -. 0300 .0706 . 1 179 . 1374 -.2548 .2089 -.0210 -.0652 . 2503 -.0714 -. 055 1 -.0975 -.0582 -. 1481 -.0328 -.0652 .3751 -.0728 -.0896 -.0097 . 0470 .0432 -.0409 -. 1631 .0180 . 0035 -.0167 .0113 1222 .0157 -. 04 75 -.0979 . 1786 -. 1223 . 1612 . 0634 -.0174 .3090 -.0366 .5604 . 06 10 .0800 . 1470 . 0 193 .2821 -.0571 .0233 -.0100 . 0300 .0838 -.0382 -.0623 .0826 . 0870 -.0797 . 1382 -.2041 .0616 -.0474 -.0636 . 66 14 -.0334 . 2273 -.0866 .1120 . 1696 -.0676 . 2729 -. 1710 .0155 . 1 102 . 1 30 2 . 1783 -.0863 .1290 . 1409 -.0690 -.0972 -. 1381 -.0851 -.2335 .3704 . 2 433 -.1891 .1436 -.0938 .0936 -.2766 . 0890 -.0238 . 0938 . 1057 -. 2058 .1307 -.5166 . 1004 .0960 .0399 -.0795 -.0025 . 0074 . 1775 . 2492 -.0134 .6568 -. 18C0 -.0618 .1793 . 1554 .21 10 . 0030 -.2057 . 1202 . 0913 .0254 . 0773 -.0635 .0628 -.0609 .0283 . 0456 .0583 -.2782 . 2297 -.0733 . 0233 -.1593 -.2587 .2605 -.0543 -. 023 1 -.1328 -.1798 . 1576 .0389 . 0654 -.0371 .0043 -. 1368 .0613 . 128 1 .4144 . 1432 -.2209 -.0557 . 0791 .5896 .1826 -.2447 -.0162 -. 0048 .3704 183 V A R I A E L E 1 2 1 0 7 % F A M - W E H $ 7 - 9 9 9 9 . 1 8 5 7 . 7 7 5 4 1 0 8 S F A M - W E H $ > 1 C C 0 0 . 0 9 9 1 . 7 2 6 3 1 0 9 % F A M N O C H I L D R E N - . 9 3 0 8 . 0 4 5 8 1 1 0 >? F A M C H L D N O N E > 6 . 5 1 7 2 - . 2 6 1 4 1 1 1 3 S F A M C H L D 1 5 - 2 4 - . 2 0 7 2 . 2 0 7 7 1 1 2 9cF A M C H L D A L L < 1 5 . 8 2 1 0 - . 1 4 4 5 1 1 3 % F A M H B D + W F W O R K - . 1 0 5 4 - . 0 3 3 4 1 1 4 X P O P I N H O U S H L D S . 3 4 0 6 . 0 2 4 3 1 1 5 # O F P E R S O N S / H S L D . 7 8 3 5 - . 0 6 6 6 1 1 6 3 K S L D 1 F A M H S L D . 7 9 8 3 . 1 7 8 4 1 1 7 % H S L D > 1 F A M . H S L D . 0 2 1 6 - . 2 6 7 6 1 1 8 2 H S L D N O N F A M H L D - . 8 1 1 1 - . 1 2 3 5 1 1 9 55 H S L D W L O D G E R S - . 4 9 1 0 - . 2 0 4 1 1 2 0 # H P E R S O N S / D W E L L . 7 2 0 0 - . 0 4 7 9 1 2 1 5 J D W E L O W N E D , S N G L . 3 9 8 1 . 2 1 5 1 1 2 2 5? DW E L S N G D E T A C H . 5 1 2 1 . 1 1 3 4 1 2 3 % D W E L D U P L E X - . 2 1 4 1 - . 2 4 3 2 1 2 4 X D W E L A P T O T H E R - . 4 7 3 9 - . 0 8 7 6 1 2 5 % D W E L R E N T E D - . 3 5 3 4 - . 1 8 7 9 1 2 6 $ DW E L R E S I D + B U S N - . 1 7 3 3 - . 1 2 1 0 1 2 7 * D W E L O C U P 0 - 2 Y R - . 0 5 1 7 - . 0 0 4 2 1 2 8 X D H ' E l O C U P 3 - 5 Y P . 2 8 6 2 . 0 5 8 1 1 2 9 55 D W E L O C U P 6 - 1 0 Y R . 1 6 1 9 . 0 3 5 4 1 3 0 ?f D W E L O C U P > 1 0 Y B - . 2 6 6 9 - . 1 1 0 6 1 3 1 9 S D W E L C O N S < 1 9 2 0 - . 4 3 2 5 - . 3 4 0 9 1 3 2 X D W E L C O N S 2 0 - 4 5 - . 3 3 3 6 . 0 5 7 1 1 3 3 5SDWEL C O N S 4 6 - 5 9 . 5 3 5 2 . 1 7 7 3 1 3 4 3SDWEL C O N S 6 0 - 6 1 . 2 4 1 8 . 0 8 8 7 1 3 5 9 o D W E L C O N D G O O D . 1 2 9 0 . 4 0 4 2 1 3 6 ? D W E L C O N D O K - . 1 2 6 4 - . 3 5 6 9 1 3 7 3 S D W E L C O N D F A I R . 0 1 5 7 - . 3 3 4 5 1 3 8 % DW E L W S E W E R - . 4 3 4 9 - . 0 4 6 0 1 3 9 % D W E L N O S E P T T A . 1 3 4 6 - . 1 9 1 2 1 4 0 % D W E L O W N W A T E R - . 0 1 9 5 - . 0 5 6 1 1 4 1 A V G # R O O M S / D W E L . 3 8 0 8 . 4 8 6 0 1 4 2 A V G i P E R S / R O C M . 4 7 0 9 - . 6 3 0 8 1 4 3 A V G # B D R O O M / D W E L . 4 6 0 1 . 3 4 2 7 1 4 4 H DW E L O S < $ 1 2 C 0 0 . 0 7 4 4 - . 7 3 3 9 1 4 5 % D W E L O S $ 1 3 - 1 7 0 0 0 . 1 3 7 0 . 0 0 2 7 1 4 6 * E W E L O S $ 1 8 - 2 7 0 0 0 . 0 6 6 7 . 6 2 3 4 1 4 7 % D W E L ( O S ) > $ 2 8 C O 0 . 1 2 8 5 . 5 1 1 3 1 4 8 S D W E L N O T V - . 4 4 6 9 - . 0 7 9 4 1 4 9 5 S D W E L 1 T V . 3 3 6 5 - . 1 8 6 2 1 5 0 5? D W E L N O C A R S - . 5 6 7 9 - . 5 0 0 8 1 5 1 % D W E L 1 C A R . 4 4 6 1 . 0 7 7 1 1 5 2 ? D W E L > 1 C A R S . 3 4 6 4 . 5 7 2 0 1 5 3 9 5 D W E L 1 S T M O R T G . 5 9 1 8 . 2 6 5 8 1 5 4 % DW E L > 1 M O R T G . 3 6 5 4 . 1 0 3 8 1 5 5 5 S D W E L N O M O R T G A G - . 0 8 0 5 - . 1 4 8 2 1 5 6 9 J D W E L E A N K M O R T G . 4 8 2 6 . 3 8 9 8 1 5 7 % D W E L P R I V M O R T G . 1 1 8 2 - . 2 4 2 3 1 5 8 X D W E L W G A R A G E . 0 4 2 1 . 1 7 6 6 1 5 9 A V G # ' T E N A N T / D W E L 3 5 2 0 - . 1 8 7 2 1 6 0 3SPOP ' T E N A N T S - . 4 2 7 7 - . 2 0 2 5 FACTOR 3 4 5 6 7 . 2 0 5 3 - . 2 0 4 9 . 0 5 0 8 * 0 2 1 7 . 1 3 0 5 . 1 3 2 8 - . 1 5 7 3 . 0 4 4 7 • 0 2 4 1 - . 0 5 9 0 - . 1 3 0 5 . 0 4 5 5 - . 0 7 1 5 0 4 2 4 - . 0 1 6 9 - . 0 3 0 4 . 1 5 9 7 - . 3 7 0 2 • 1 1 1 2 . 0 8 6 5 . 3 0 1 1 . 1 5 5 0 . 6 0 0 8 • 1 2 6 9 . 0 4 7 9 . 1 5 9 4 - . 0 7 6 3 - . 1 2 8 4 • 1 4 0 4 . 0 3 5 9 - . 3 9 8 6 . 1 1 6 5 - . 0 3 4 3 2 6 2 2 . 3 8 6 2 . 1 5 7 7 . c c o o . 1 8 8 6 0 0 5 0 . 1 7 9 8 . 3 4 1 7 . 0 5 4 7 . 0 3 8 8 • 0 1 7 4 - . 0 8 9 0 . 2 9 7 3 - . 1 5 3 0 . 0 4 4 7 • 0 6 3 4 . 1 1 0 0 . 3 2 2 9 . 3 7 2 5 . 0 9 7 2 • 1 9 5 1 . 0 4 2 8 - . 3 5 7 0 . 0 7 5 3 - . 0 4 9 5 • 0 0 2 0 - . 1 0 5 8 . 0 2 2 3 . 3 9 2 6 . 0 4 1 6 2 0 8 5 . 0 0 1 4 . 3 3 8 3 . 0 5 3 0 . 0 3 0 7 • 0 1 2 3 - . 0 9 0 4 . 8 3 4 7 - . 1 0 3 4 . 0 6 1 7 • 0 0 8 4 . 0 7 4 0 . 7 1 4 9 - . 1 5 6 3 . 1 2 2 7 m 1 2 6 7 . 0 3 0 4 - . 4 0 1 5 . 1 9 1 7 - . 0 2 4 5 m 0 7 2 8 . 1 0 2 5 - . 6 0 4 7 . 1 2 6 7 - . 0 5 4 2 -. 1 3 8 2 - . 0 1 8 1 - . 8 4 9 2 . 0 7 3 6 - . 0 8 8 0 - . 0 5 16 - . 0 4 0 3 - . 2 4 2 3 . 1 3 0 6 - . 0 2 6 6 • 0 9 8 8 - . 0 1 2 8 - . 6 7 5 5 . C 4 7 7 - , 2 4 7 4 • 1 3 4 5 - . 0 8 6 5 . 0 8 0 3 - . 0 7 3 8 - . 3 9 8 6 • 0 1 2 0 . 0 2 9 3 . 2 7 1 0 - . 0 4 8 8 . 1 3 3 5 - . 0 8 4 2 . 0 5 9 8 . 5 0 4 1 . 0 4 3 4 . 4 8 9 3 • 0 2 2 9 . 1 0 9 0 - . 2 1 0 3 . 2 1 3 1 . 0 0 2 4 • 1 3 6 4 - . 0 9 5 0 . 3 3 6 5 - . 0 0 0 9 . 4 0 4 6 0 4 8 0 . 0 8 2 8 - . 0 3 3 3 - . 2 1 0 6 - . 2 1 7 4 m 1 7 8 1 , 1 5 6 4 - . 0 7 0 4 - . 1 0 7 8 - . 1 8 5 0 m 3 8 7 0 . 0 4 0 5 . 0 8 3 3 - . 0 6 1 3 - . 0 0 0 5 m 0 8 8 0 . 1 3 4 8 - . 0 5 9 8 . 0 5 8 7 . 0 1 9 8 m 0 4 3 9 - . 0 8 7 6 - . 0 6 9 1 . 0 6 5 0 . 0 0 2 7 2 2 2 9 - . 0 0 3 0 - . 0 6 4 1 . 3 2 6 7 . 1 6 0 1 5 6 5 8 . 0 6 6 0 . 0 1 6 7 - . 1 3 5 3 - . 0 0 0 7 • 4 7 0 6 - . 2 1 6 9 - . 0 2 4 1 . C O 1 5 - . 0 6 8 9 1 9 3 9 - . 0 9 2 7 . 5 7 8 6 . 0 3 5 3 . 1 8 6 4 • 0 4 1 8 - . 0 6 8 6 - . 1 6 4 3 . 0 5 0 4 - . 1 1 8 8 • 0 4 0 5 . 0 4 0 5 . 5 6 7 1 . 0 1 9 4 . 1 5 2 2 • 0 2 0 1 - . 0 8 9 6 . 0 9 0 9 - . 0 1 5 9 . 0 8 7 0 • 0 5 9 3 . 0 0 5 0 . 1 2 8 4 . 0 6 6 6 . 0 0 5 3 • 0 2 1 4 . 6 1 3 6 . 1 2 7 3 - . 0 3 8 4 . 0 3 8 6 • 0 8 7 6 . 0 8 6 4 . 1 1 9 0 - . 0 6 1 3 . 0 3 8 1 m 1 9 4 4 - . 2 3 3 6 - . 2 8 9 7 . 0 5 8 3 - . 0 4 2 5 0 7 2 9 - . 1 7 7 1 . 1 4 2 0 - . 1 3 6 8 . 0 1 2 6 • 0 1 9 6 . 3 0 8 1 - . 3 0 8 9 . 1 9 2 7 - . 0 4 9 3 1 2 3 7 - . 0 5 2 3 . 1 6 1 1 - . 2 5 0 3 - . 0 0 4 5 m 1 3 5 2 . 3 9 5 4 . 3 4 2 5 - . 0 8 7 0 . 1 4 6 4 • 1 4 4 4 - . 0 4 8 0 . 5 1 0 7 - . 0 7 2 4 - . 0 5 7 1 • 0 5 6 6 . 2 1 4 1 . 3 8 1 4 - . 0 1 3 0 - . 0 9 1 8 • 1 5 1 6 . 33 3 6 . 7 3 7 2 - . 0 5 1 1 . 3 4 1 2 m 1 1 8 0 - . 0 2 3 9 . 5 0 0 2 - . 0 2 3 0 - . 0 4 6 6 m 0 9 1 7 . 2 3 5 4 . 5 0 4 1 - . 0 7 1 5 . 1 1 1 3 • 0 9 5 8 . 2 6 9 3 . 7 6 6 2 . 0 4 1 1 . 3 6 9 7 • 0 0 2 4 . 1 5 9 9 - . 8 4 7 2 . 0 7 1 9 - . 0 8 7 3 * 0 7 5 1 - . 0 2 9 0 - . 8 1 5 2 . 0 6 4 8 - . 0 6 2 3 • 0 4 2 0 . 0 0 3 3 184 VARIABLE 1 2 16 1 AVG GROSS RENT .0456 .3928 162 AVG # FAM/DWELL . 6246 -. 0 15 2 16 3 3MALLF EMPLOYES .0061 .6843 164 %MALLF OWN ACCNT -.2262 . 074 1 165 2MALLF WAGE EARN .1124 -.5926 166 %FEMLF EMPLOYER -.0910 .2363 167 5?FEMLF OWN ACCNT -.2,304 -.0392 168 SSFEMLF WAGE EARN . 1477 -.05 14 169 5?EEMLF UNPD FAMW . 1026 .0356 170 •SECAMPOP EMPLCYD . 2419 .4 304 171 KECAFPOP EMPLCYD -.0458 . 1588 172 IND FOODS BEV ERGS . 1296 -.4344 173 IND WOOE FINISH .2024 -.5242 174 IND PAPERS ALLIED . 1488 . 04 38 175 IND METAL FABRCN . 1952 -.2779 176 IND CONSTRUCTION . 1643 -.5177 177 IND TR ANSP+STC RG .0694 -.3528 178 IND TELECOMMU NCN .0450 . 1868 179 IND TRADE-WHOLSL .1643 .3740 180 IND TRADE-RETAIL .0895 . 0648 181 IND FINANCE+INS -.2252 .7095 182 IND EDUCATION -. 1055 .7 130 183 IND HEALTH WELER -.2382 .2243 184 IND HOTEL RESTNT -. 30 4 6 -.4045 185 IND PUB ADMN DEF .0904 . 1608 186 OCC MANAGERIAL .0506 .7859 187 OCC PROFL TECHHL -.0655 .8816 188 OCC CLERICAL -.3257 .3 17 1 189 OCC SALES .1412 .525 1 190 OCC FARMER . 1021 -. 174 1 191 OCC LOGGER .0510 -.3788 192 OCC FISH TRAP .0916 -.2353 FACTOR 3 4 c. 6 7 -.0351 • 0287 -.0778 -. 1794 .2435 . 4254 m 0968 -.0376 -.0540 .0076 .1965 m 1 154 . 1 404 . 289 1 -.0656 -.0460 m 1909 -.0049 .4852 -.0158 -.1413 • 1975 -.1213 -.4500 .1267 .0157 0793 .0175 . 0832 -.0584 -.0756 • 0829 -.0 354 . 0659 -.0614 .0236 — m 0275 .0133 -. 15 19 .1602 -.0067 m 0712 .0763 . 16 15 .0722 .2157 ~~ m 1597 .0698 . 06 19 .0333 .0970 - . 0853 .0448 -.0507 .0085 . 1492 m 3 166 .0997 . 1138 .1209 .0679 - . 0112 .0830 .2065 -.0 1 54 .0 2 30 • 0 130 .0472 . 028 1 -.0032 .1410 • 1058 .0182 -.0685 .2973 .0843 1028 -.1300 . 2067 .0738 .0790 • 127 1 -.1012 -.0228 .1348 .0534 • 0101 .0055 -. 092 1 . 3658 .0494 m 0756 -.1101 -.1190 .2153 .0731 — m 0544 .1014 -. 18 15 .5566 -.0706 m 0538 -.0192 -. 1512 .1380 . 1254 m 16 12 .0940 . 0954 . 1 191 -.1498 m 0914 .1169 -. 0206 . 1 200 -.2457 m 32 15 -.1227 -.2118 -.1 1 53 .0144 m 3328 .1377 . 0974 .3190 . 1373 m C7 13 .0311 . 0446 -.0086 .0414 m 196 0 .0132 -.0203 .0778 -. 1596 •9 03 1 1 .0294 -.2646 .49 59 .0520 m 2099 -.0386 -.1891 .40 31 .0353 m 0 6 93 -.0930 . 4 85 9 -.0643 -.0304 m C2 07 -.0655 . 057 1 -.0708 .0719 m 2377 -.0070 . 2669 .1008 185 TABLE E-2 REORDERED FACTOR LOADINGS MATRIX NORMALIZED DATA VARIABLE 1 FACTOR 3 4 24 % SINGLE <15 YRS .9182 91 %FPQP UNMARRIED .8833 89 AVG # PERSON/FAM .8755 7 % 5 - 9 YEARS .8627 112 5JFAM CHLD ALL<15 .8210 96 %FAM-HEAD 35-44 .8106 83 ED-F-AT-ELEM+PRE .8070 90 %POP IN FAMILIES .8067 116 ToHSLD 1 FAM HSLD .7983 115 #OF PERSONS/HSLD .7835 6 % 0 - 4 YEARS .7639 30 BORN B. C. .7627 73 ED-M-AT-ELEM+PRE .7540 8 % 10-14 YEARS .7 317 120 #HPERSONS/DWELL .7200 67 ED-M-NO—<GRADE 1 .6881 77 ED-F—NO-<GR ADE 1 .6584 92 %FAM W W. E. HEADS .6376 162 AVG # FAM/DWELL .6246 5 FERTILITY RATE .6097 153 %DWEL 1ST MORTG .5918 13 % 35-39 YEARS .5894 133 9SDWEL CONS 46-59 .5352 58 % OF IMGTS 41-50 .5270 110 %FAM CHLD NONE>6 .5172 151 %DWEL 1 CAR .4461 3 RECENT POPN INCR .4250 131 %DWEL CONS C1920 -.4325 119 SSHSLD W LODGERS -.4910 29 % DIVORCED -.4967 25 9SSNGL>1 5YRS. MALE -.5344 150 %DWEL NO CARS -.5679 26 %SNGL>15YRS.FEM. -.5888 16 % 50-54 YEARS -.5966 23 % 85 + YEARS -.6055 56 %OF IMGTS 3EF'21 -.6270 98 %FAM HEAD 55-64 -.6741 35 BORN UK+REP.IREL -.6788 99 /SFAM-HEAD 65-69 -.7133 22 % 80-84 YEARS -.7529 17 % 55-59 YEARS -.7812 118 %HSLD NONFAM HLD -.8111 100 %FAM— HEAD > 70 -.8416 18 % 60-64 YEARS -.8512 21 % 75-79 YEARS -.8758 19 % 65-69 YEARS -.8758 28 % WIDOWED -.8782 20 % 70-74 YEARS -.8834 109 %FAM NO CHILDREN -.9308 4254 5 107 -.5008 187 OCC PROFL TECH NL 72 ED-M-NO-UNIV DEG .8816 . 8783 186 VARIABLE FACTOR 4 82 ED-F-NO-UNIV DEG .8112 93 AVG FAM EARN WEH .7989 81 ED-F-NO-UNIV NC .7880 186 OCC MANAGERIAL .7859 107 %FAM—WEH $7-9999 .7754 71 ED-M-NO-UNIV NC .7429 108 %FAM—WEH $>10000 . 7263 182 IND EDUCATION .7130 181 IND FINANCE+INS .7095 163 SMALLF EMPLOYER .6843 36 ORIG UK+REP.IREL .6769 75 ED-M-AT-UNIV UGD .6340 146 %DWELOS$18-27000 .6234 152 5SDWEL >1 CARS . 5720 85 ED-F-AT-UNIV UGD .5626 189 OCC SALES .525 1 106 XFAM-WEH $6-6999 .5249 147 %DWEL (OS) >$28000 . 51 13 64 RELIG JEWISH .4657 86 ED-F-AT-UNIV GRD .4555 76 ED-M-AT-UNIV GRD .4295 37 BORN AUSTL+COMWL .3996 161 AVG GROSS RENT .3928 34 ETHN USA .3892 179 IND TRADE-WHOLSL .3740 50 ETHN INDIA PARIS .1848 33 ORIG NATIVE POPN -.2513 177 IND TRANSP+STORG -.3528 191 OCC LOGGER -.3788 184 IND HOTEL REST NT -. 4045 101 5EFAM-WEH <$2000 -.4032 -.4095 45 BORN SCANDINAVIA -.4208 172 IND FOOD5BEVERGS -.4344 87 MALE-FEMAL RATIO -.4674 52 ORIG USSR -.4701 102 %FAM—WEH $2-2999 -.4042 -.4866 176 IND CONSTRUCTION -.5177 173 IND WOOD FINISH -.5242 104 XFAM-WEH $4-4999 -.5405 63 RELIG ROM CATHOL -.5651 165 95MALLF WAGE EARN -. 5926 142 AVG # PERS/ROOM .4709 -.6308 103 %FAM—WEH $3-3999 -.6426 78 ED-F-NO-ELEMENTY -.7142 144 5EDWEL OS <$12000 -. 7339 68 ED-M-NO-ELEMENTY -.7525 -.5110 ,4031 .4 144 4500 121 SDWEL OWNED,SNGL 158 %DWEL W GARAGE 155 %DW EL NO MORTG AG 122 %DWEL SNG DETACH 14 1 AVG # ROOMS/DWEL 143 AVG #BDR03M/DWEL 157 5EDWEL PRIV MORTG 5121 4601 4860 ,8 347 ,7662 ,7372 ,7 149 ,5786 5671 ,5041 187 V A R I A B L E FACTOR 4 130 5? DWEL 156 %DW EL 154 55DWEL 126 %DWEL 113 %FAM 123 %DWEL 124 %DW EL 127 %DWEL 160 %POP 159 AVG# 125 5&DWEL OCUP >10YR BANK MORTG > 1 MORTG RESID+BUSN HBD+WF WORK DUPLEX APT OTHER OCUP 0-2YR TENANTS TEN ANT/DWEL RENTED 54 ORIG CHINA 53 BORN CHINA 47 BORN CENT EUROPE 66 LANG NOT ENG, FR 46 ORIG SCANDINAVIA 48 ORIG CENT EUROPE 40 ORIG ITALX 51 BORN USSR 65 RELIG OTHERS 39 BORN ITALY 55 ETHN JAPAN 117 %HSLD>1 FAM HSLD 185 IND PUB ADMN DEF 62 RELIG PROTESTANT 9 % 15-19 YEARS 97 %FAM-HEAD 45-54 111 %FAM CHLD 15-24 84 ED-F-AT-HIGH SCH 74 ED-M-AT-HIGH SCH 15 % 45-49 YEARS 132 XD.WEL CONS 20-45 128 J&DWEL OCUP 3-5YR 95 %FAM-HEAD 25-34 12 % 30-34 YEARS 88 % OF POP "RURAL" 4 TIME-CITY CENTER 44 ORIG BELG+NETHLD 190 OCC FARMER 164 %MALLF OWN ACCNT 139 %DWEL NO SEPT TA 134 %DWEL CONS 60-6 1 192 OCC FISH TRAP 140 SDWEL OWN WATER 1 POP'N DENSITY 2 POP'N POTENTIAL 138 %DWEL W SEWER 145 5£DWELOS$13-1 7000 105 XFAM-WEH $5-5999 180 IND TRADE-RETAIL 4826 4739 4277 .5041 .5002 .3814 . 2423 .3986 ,4015 .6047 ,6755 .8152 ,8472 ,8492 5 .4893 -.4296 . 54 07 ,4086 .4530 4536 4349 6973 6351 54 75 5322 4974 4 844 4 804 4 663 4568 4526 4 341 3 725 3328 5531 .7521 .6568 .6008 .5604 .5594 .4704 .4046 .3986 .5166 .5244 -.4225 . 501 9 6614 5304 4917 4859 4852 4706 3870 2669 1939 5039 5554 5658 ,6 136 ,5896 ,5566 188 VARIABLE FACTOR 4 188 OCC CLERICAL 178 IND TELECDMMUNCN 175 IND METAL FABRCN 32 ORIG CANADA 14 % 40-44 YEARS 61 % OF IMGTS 60-61 11 % 25-29 YEARS 94 %FAM-HEAD < 25 Y 10 % 20-24 YEARS 168 5?FEMLF WAGE EARN 169 %FEMLF UNPD FAMW 166 %FEMLF EMPLOYER 167 %FEMLF OWN ACCNT 27 % MARRIED 114 %POP IN HDUSHLDS 49 ETHN GREECE 60 % OF IMGTS 56-59 43 BORN BELG+NETHLD 59 % OF IMGTS 51-55 42 ORIG GERM+A+SWIT 41 BORN GERM+A+SWIT 70 ED-M-NO-HIGH FIN 80 ED-F-NO-HIGH FIN 57 % OF IMGTS 21-40 38 ETHN FRANCE 174 IND PAPERS ALLIED 183 IND HEALTH WELFR 31 BORN CAN AD A-NOBC 136 %DWEL COND OK 137 %DWEL COND FAIR 135 3SDWEL COND GOOD 79 ED-F-NO-HIGH NC 69 ED-M-NO-HIGH NC 129 %DWEL OCUP6-10YR 14 8 %DWEL NO TV 149 %DWEL 1 TV 170 %ECAMPOP EMPLOYD 171 X1CAFPOP EMPLOYD .4959 .3658 . 2973 . 137 1 -.4 19 1 -.4360 4216 4379 4912 .4042 -.4469 4304 189 APPENDIX TABLE E-3: FACTOR LOADINGS MATRIX - ORIGINAL DATA FACTOR VARIABLE 1 2 3 4 5 6 7 1 POP' N DENSITY .3269 -.0779 -. 2455 . 3862 - . 0 5 1 1 _ .123 5 - . 0 7 2 0 2 POP' N POTENTIAL . 3466 - . 1 7 4 6 -.1319 .1538 .0488 . 2092 .0464 3 RECENT POPN INCR .3457 .0987 .0114 -.0069 .1186 - . 1903 . 0 2 3 2 4 TIME -CITY CENTER .3618 -.1325 . 2221 - . 1725 -. 1313 - . 1676 - . 0 3 5 8 5 FERTILITY RATE .6359 - . 2 1 2 7 .2060 . 1 4 7 9 .1161 - .2122 -.0807 6 % 0 - 4 YEARS .7573 -.2218 .1216 -.0144 . 2840 - . 3293 .0162 7 % 5 - 9 YEARS . 8841 - . 0 1 5 7 . 1722 - . 1 3 0 3 -.1255 - . 1527 - . 0 1 7 7 8 % 10 -14 YEARS .7869 .041 1 . 1337 -. 1002 -.3505 . 2287 -.0629 9 % 15 -19 YEARS . 3751 . 1186 .2930 -. 1660 -.0727 .6708 -.0123 10 % 20 -24 YEARS .2422 -. 1999 -. 2211 .0055 .6061 . 2590 -.0130 11 % 25 -29 YEARS -.0851 -.2857 -.2892 .0721 .6882. - .2732 .0882 12 % 30 -34 YEARS .2893 -. 1060 -. 1784 -.0992 . 4337 - .5048 . 1992 13 % 35 -39 YEARS .5662 .0765 -.2076 -.2099 -.1647 -.2817 .1664 14 f 40 -44 YEARS .3012 .2014 -.0831 -.2225 -.3910 . 1241 . 1133 15 % 45 -49 YEARS -.2555 .3572 .0446 -.1491 -.2243 .4941 .0069 16 Jt 50 -54 YEARS .5697 .2286 . 0697 .0590 -.0755 . 4317 - . 0 2 5 8 17 % 55 -59 YEARS -.7482 .0846 . 0 1 8 7 .1381 -.0873 .1315 -.1125 18 •% 60 -64 YEARS -.8129 .0487 .0192 . 2306 -. 1438 .0326 -.0704 19 % 65 -69 YEARS -.8321 .0154 -.0027 .2624 -.1554 - . 1323 - . 0 8 0 8 20 % 70--74 YEARS -.8240 . 0 2 8 5 -.0339 . 2460 -. 2042 -. 1286 - . 0 6 1 1 21 % 75 -79 YEARS -.8306 -.0201 -.0509 . 1 199 -.2041 -.1045 -.0491 22 K 80--84 YEARS -.6995 .0114 -.0629 .0767 - . 1719 - . 0726 -.0637 23 % 85 + YEARS -.5162 .0578 -.0264 -.0124 -.04 02 .0108 -.0336 24 5? SINGLE <1 5 YRS .9364 -.0860 . 1645 -. 0914 -.0494 -. 1199 -.0218 25 %SNGL>15YRS.MALE -.4121 -. 1264 -.1491 .3700 -.0448 .0738 -.1412 26 5?SNGL>15YRS.FEM. -.5867 .2898 -.2144 -. 1578 . 1800 . 1978 .0350 27 % MARRIED -. 1150 .0208 . 2452 -.0214 .0358 .0029 .1876 23 % WIDOWED -.8269 .0669 -. 1853 -.0628 -.0777 - .0076 -.0104 29 % DIVORCED -.5538 -.0738 -.3726 -.0639 .0818 -.0435 -.0782 30 BORN B. C. .7776 .0496 . 2826 -. 1924 -.0865 . 0982 .0719 31 BORN CANADA-NOBC -. 1706 . 1411 -.3099 -.4175 -.1348 - .0992 - . 0 1 8 9 32 ORIG CANADA -.0229 -.0269 .0166 -.0461 -.0338 -. 0565 -.0258 33 ORIG NATIVE POPN . 1 106 -.0818 .0311 -.0280 .0496 - .0108 -.1138 34 ETHN USA -.3522 .3879 -.0062 -. 1572 -.0902 - . 0604 -.2593 35 BORN UK+REP.IREL - .7187 .1264 -.0146 -.3054 -.0351 -.0105 . 1158 36 ORIG UK+REP.IREL -.0761 .5232 .0691 -. 4799 -. 1521 . 0529 . 1688 37 BORN AUSTL+COMWL -.3505 . 2854 -. 1962 -.0297 .2859 -.1087 -.0416 38 ETHN FRANCE .0812 -. 1979 -.1407 -.0683 .0365 -.0510 -.0723 39 BORN ITALY - .0058 -.2447 .0386 .1757 .1128 .0225 -.0871 40 ORIG ITALY .0238 -.2579 .0634 . 1511 .0961 .0264 -.0669 41 BORN GERM+A+SWIT -. 2490 -.1987 -.1417 -.0505 .3139 - .0339 -.0913 42 ORIG GERM+A+SWIT .0142 -.3015 .0123 -.0786 . 1641 -.0371 -.0455 43 BORN BELG+N ETHLD .0626 -.2110 . 0 3 2 5 -.0861 .1295 - . 0 0 4 6 -.0187 44 ORIG BELG+NETHLD .2092 -.2517 . 0666 -.1265 .0707 -.0508 -.0157 45 BORN SCANDINAVIA -.1932 -.3798 -.0727 . 0779 .1175 -.0555 -.1670 46 ORIG SCANDINAVIA -.1003 -.2923 -.0453 .0833 . 1199 .0165 -.0449 47 BORN CENT EUROPE -.2700 -.2121 -.1073 . 1032 .0982 .0124 -.0404 48 ORIG CENT EUROPE -.0822 -.3163 -. 0851 .0000 .0790 - .0304 -.0290 49 ETHN GREECE -.2214 .0599 -.0942 -.0166 .0730 .0214 .0621 50 ETHN INDIA PARIS .0290 -.0732 -.0276 .0927 . 1189 .0276 -.0823 51 BORN USSR -. 1787 -.1527 -.0099 .1352 -.0056 . 0 4 3 2 -.0095 52 ORIG USSR .0421 -.4396 - . 0 6 4 6 .0260 -.0077 -.0327 -.0425 190 FACTOR VARIABLE 1 2 3 4 5 6 7 53 BORN CHINA -.0868 -.0541 -. 0545 . 9558 .0194 -.04.00 -.0070 54 ORIG CHINA -.0667 -.0683 -.0590 .9405 .0247 -.0178 -.0164 55 ETHN JAPAN .0124 -.0769 -.0339 . 1291 .0073 .0421 -.0246 56 %OF IMGTS BEF'21 .5886 .0508 . 2350 .0044 -.4639 . 1295 .0618 57 % OF IMGTS 21-40 . 1249 -. 1172 .0764 -.1119 -.4417 .0380 -.0037 58 % OF IMGTS 41-50 .5121 . 1878 -.0853 -.0783 -.2248 . 0411 -.0205 59 % OF IMGTS 51-55 .3582 -.0296 -.0450 . 0585 .4557 -.2009 . 1061 60 % OF IMGTS 56-59 .0749 -.1539 -.2261 .0656 . 6285 -.0464 -.0689 6 1 % OF IMGTS 60-61 .0615 . 1727 -.2215 .0656 .4033 -.0363 -.1576 62 RELIG PROTESTANT .0428 . 3841 . 1239 -.4877 -. 1228 .0179 .1520 63 RELIG ROM CATHOL .0495 -.3984 -.0856 .0118 .1493 -.0212 -.1411 64 RELIG JEWISH .1572 .4715 -.0501 -.0055 -. 1306 . 1467 -.0468 65 RELIG OTHERS .0694 -.2916 -.0747 .7381 .0637 -.0519 -.0587 66 LANG NOT ENG, FR •.0638 -. 1304 -.0353 . 8477 .0681 -.0273 -.0366 67 ED-M-NO-<GRADE 1 .6869 -.2657 . 1339 .0908 .2314 -.3191 .0147 68 ED-M-NO-ELE MENTY .3020 -.6140 -.0882 . 3941 -. 1004 -.0600 -.2254 69 ED-M-NO-HIGH NC .2785 -.2399 -.0737 -.2967 .1142 .0082 .3375 70 ED-M-NO-HIGH FIN .3237 .2806 -.0886 -.1360 . 1037 . 0079 .0761 71 ED-M-NO-UNIV NC . 3409 .6885 -. 1356 -.1664 .0813 -.0786 .0528 72 ED-M-NO-UNIV DEG .1032 .8921 .0188 -.0786 -.0036 . 0384 -. 1736 73 ED-M-AT-ELEM+PRE . 8030 .0823 . 1482 -.1205 -.2872 .1035 -.0103 74 ED-M-AT-HIGH SCH .2849 .3005 . 3055 -.1487 -. 1072 . 5062 . 1014 75 ED-M-AT-UNIV UGD -.0997 . 5442 .0614 -.0411 .0727 .2529 .0019 76 ED-M-AT-UNIV GRD .0453 .2184 -.0709 -.0114 . 1464 -.0129 .0371 77 ED-F-NO-<GR ADE 1 .6350 -.2862 .0710 .4263 .1601 -.3063 -.0219 78 ED-F-NO-ELE MENTY -.3165 -.6184 -.0572 . 2490 -.0793 -.0325 -.2091 79 ED-F-NO-HIGH NC -.2851 -.0061 -.0601 -.3307 .0800 .0320 . 3228 80 ED-F-NO-HIGH FIN -.3304 .3196 -.0754 -. 1304 .0948 .0014 .0271 81 ED-F-NO-UNIV NC -.2738 .7369 -.0692 -. 1290 .0806 .0049 .0029 82 ED-F-NO-UNIV DEG -.1443 .7818 .0383 -.0709 .0530 .0669 -.2068 83 ED-F-AT-ELEM+PRE . 8442 .0148 . 0869 -.0431 -.2726 .0268 -.0737 84 ED-F- AT-HIG.H SCH .3481 .2581 . 3042 -.0692 -. 1589 . 4996 .0605 85 ED-F-AT-UNIV UGD -.0779 .5701 . 1745 -.0292 .0116 .3052 -.0622 86 ED-F-AT-UNIV GRD -.0926 . 3851 -.0008 -.0 201 .0732 . 0871 -.1364 87 MALE-FEMAL RATIO -.0392 -.1156 .0170 .5768 -.1609 -.1139 -.0281 88 % OF POP '"RURAL" .0895 -.0382 .0025 .0165 -.0368 .0236 -.0219 89 AVG # PERSON/FAM .9103 -.0088 . 1705 . 0498 -.1296 .0139 -.0897 90 3POP IN FAMILIES .7115 .0172 . 3506 -.2858 -.0314 . 1061 .0764 91 %FPOP UNMARRIED .9153 -.0725 .2199 .0625 -.0679 .0919 -.0760 92 #FAM W W.E.HEADS .5604 -.2873 -.1352 -.1321 . 1863 -.0313 .3639 93 AVG FAM EARN WEH .1601 .8399 .1812 -.2222 -.1128 .1223 -.0823 94 3FAM-HEAD < 25 Y .0134 -.4753 -.2568 -.0136 . 3852 .0786 -.0567 95 %FAM-HEAD 25-34 . 3693 -.3130 -.1907 . 1202 .5713 -.4312 . 1103 96 XFAM-HEAD 35-44 .8021 .0754 -. 1500 -. 1596 -.2613 -. 1335 .0866 97 %FAM-HEAD 45-54 . 1057 . 3907 .1726 -.0888 -.2363 .6193 -.0531 98 XFAM HEAD 55-64 -.6213 .1203 . 2045 . 1447 -.0628 . 2549 -. 1044 99 %FAM-HEAD 65-69 -.6348 .0185 . 1779 . 1765 -.0989 -.0288 -.0487 100 5JFAM-HEAD > 70 -.7959 .0204 . 1275 -.0516 -.2053 -.0889 -.0702 101 55FAM-WEH <$2000 -.3328 -.3017 -.2622 .4327 .1375 -.0395 -.1695 1 02 SFAM-WEH $2-2999 -.3354 -.3768 -.2159 . 4085 .1128 .0261 -. 1183 103 %FAM-WEH $3-3999 -. 1980 -.6784 -.1181 -.0079 .1834 .0430 -.1421 104 9EFAM- WEH $4-4999 .1670 -.6792 . 1007 -. 2621 .0566 -.0065 .1248 105 %FAM-WEH $5-5999 .2855 -. 1302 . 1730 -.2757 -.1025 -. 1028 .4608 106 XFAM-WEH $6-6999 .2521 .4690 . 1 176 -. 20 29 -. 1716 -.0827 . 4574 191 VARIABLE 1 2 107 %FAM- WEH $7-9999 .1570 .7946 108 *FAM-WEH $>10000 .0865 .7974 1 09 %FAM NO CHILDREN -.9451 .0243 1 10 5 FAM CHLD NONE>6 .4948 -.2694 1 11 % FAM CHLD 15-24 -.1316 . 1997 112 SFAM CHLD ALL<15 .8309 -.0527 1 13 %FAM HBD+WF WORK -.1755 -. 1715 1 14 9JPOP IN HOUSHLDS .3035 .0318 1 15 #OF PERSONS/HSLD . 8288 .0203 116 XHSLD 1 FAM HSLD .7570 .1145 1 17 %HSLD>1 FAM HSLD .0363 -. 1888 118 *HSLD NONFAM HLD -.7679 -.0763 1 19 %HSLD W LODGERS -.3533 -. 1402 120 #HPER SONS/D WELL .7731 .0424 121 % DW EL OWN ED,SNGL .4801 . 1791 122 *DWEL SNG DETACH .5887 .0757 123 % DW EL DUPLEX -. 1861 -.1626 124 *DWEL APT OTHER -.5927 -.0074 125 %DWEL RENTED -.4574 -. 1584 1 26 SDWEL RESID+BUSN -.2396 -.1067 127 %DW EL OCUP 0-2YR -.1620 -.0807 128 *DWEL OCUP 3-5YR .2474 .1016 129 %DWEL OCU P6-10YR . 2314 . 1187 130 9«D WEL OCUP >10YR -.1610 -.0700 131 %DWEL CONS <1920 -.3094 -.2567 132 SDWEL CONS 20-45 -.2837 .0465 133 % DW EL CONS 46-59 . 4787 .1597 134 JED WEL CONS 60-61 .1268 .0894 1 35 %DWEL COND GOOD .0690 .3302 136 %D WEL COND OK -.0637 -.3073 137 XDWEL COND FAIR -.0419 -. 1950 138 5JDWEL W SEWER -.3924 -.0738 1 39 %DW EL NO SEPT TA .0426 -.0757 140 3DWEL OWN WATER -.0036. -.0427 1 4 1 AVG # ROOMS/DWEL .4074 .6016 142 AVG # PERS/ROOM .4303 -.5169 143 AVG # BDROOM/DW EL . 5319 .4019 144 XD WEL OS <$12000 .1207 -.7393 145 %DWELOS$13- 17000 . 1313 -.1121 146 *DWELOS$18-27000 -.0226 .6193 147 % DW EL (OS) >$28000 -.0161 .5913 148 ?D WEL NO TV -.4269 -.0930 149 %DW EL 1 TV . 3418 -. 1584 150 ftDWEL NO CARS -.5275 -.4353 151 % DW EL 1 CAR .4177 -.0045 152 *DWEL >1 CARS .3185 .7191 153 % DW EL 1ST MORTG .5953 . 3206 1 54 XDWEL > 1 MORTG .3427 .0969 155 %DWEL NO MORTGAG .0036 -.1008 156 XDWEL BANK MORTG .4908 .5042 157 %DWEL PRIV MORTG .1550 -.3180 158 ?D WEL W GARAGE .0892 .2122 1 59 AVG# TENANT/DWEL -.4582 -. 1584 1.6 0 *POP TENANTS -.5966 -.0674 FACTOR 3 4 5 6 7 . 1123 -.0998 -. 1603 .0151 .1374 .1212 -.0033 -.0766 . 1369 -.4675 -.1614 -.0591 .0047 .0767 -.0069 -.0503 . 1585 . 4901 .3212 .0687 . 3075 -.0420 .0060 .6214 .0230 .0566 .0075 -.0908 . 2685 -.0061 -.4013 -. 1039 .2327 .0614 .3515 .0956 -.4992 .0966, . 1504 .0470 .3171 .1859 -.0519 .0342 -.0207 .4108 -. 2052 -.0970 .0582 . 1146 . 2361 .4225 .1957 . 1217 -.0053 -. 4614 . 1191 .0569 . 0837 -.1143 -.0020 .4755 .2059 .0904 .0296 . 2906 . 1708 -.0365 .0126 -.0421 . 7799 -. 1358 -.1371 .0016 .0853 . 7036 -. 10 16 -. 1830 .0357 .0414 -.3272 . 1302 .1965 .0091 .0244 -. 6577 .0557 .1171 -.0392 -.0605 -.7978 . 1338 .1216 -.0084 -.0958 -. 2063 . 3629 -.1101 -.0232 -.0691 -.6333 -.0636 .3204 - . 1423 -.0275 .0152 .0100 -.0300 -. 4339 .0185 . 1398 .0408 -.3796 .0239 -.0373 .5835 .0287 -.0322 . 4575 .0576 -.2716 .3769 .1119 .0955 -.1782 . 4527 -.1262 .1111 . 3485 .0321 -.1357 -.1923 -.2054 -.3553 . 1187 -. 1393 -.0889 . 1015 -. 1266 .0160 .0582 -.2042 -.0510 - .0297 .1241 -.0169 . 10 16 .0823 .0401 -. 1183 -. 1017 .2774 -.0299 -.0013 -.0644 -.0748 .0910 -.0103 . 2714 .0355 -.0262 .0114 -.0310 .0272 -.0880 .0161 .0006 -.0558 - . 0338 -.0681 .5183 .0375 -.0245 . 1753 -.0607 -. 1886 . 1719 -.0237 -. 1412 -.0736 .5635 .0994 -.0511 .1300 -.0367 . 1556 .0463 .0633 .0297 -.2825 . 1488 -.0248 .0635 .0373 .5748 -.0617 -.0797 -.0918 . 0032 . 2090 .0110 .0335 .0521 .0819 -.4699 -.2958 . 4288 .0165 - .0590 -. 1429 . 1946 -.4237 -.0046 -.0033 . 2260 -.3181 . 2932 .0330 - .0239 -.2077 .2280 -.3253 -.0248 - .0591 .4219 . 2227 -.0552 -.0163 . 1185 -.2261 . 3678 -. 1 120 -.1806 -. 1726 .1695 . 2677 -. 1052 .0162 -. 1397 .2456 . 7826 -.0620 -.02 74 .2522 -.1423 . 2473 -.0647 -.1655 -. 1578 . 2264 .5175 -. 1422 .0148 - .0102 -.0213 .7501 -.0992 -. 1896 . 2975 .0394 -.7979 .1324 .1209 -.0085 -.0945 -.7303 .0287 .0689 -. 0693 -.0897 192 VARIABLE 1 2 161 AVG GROSS RENT .0090 .3154 162 AVG # FAM/DWELL .5605 .0022 163 %MALLF EMPLOYER .0253 .7596 164 $MALLF O W N ACCNT -.1689 .0381 165 %MALLF WAGE EARN .0572 -.6526 166 KFEMLF EMPLOYER -.0315 . 1902 167 %FEMLF OWN ACCNT -. 1915 -.0455 168 2FEMLF WAGE EARN .1060 -.0662 169 %FEMLF UNPD FAMW .0812 .0263 170 XECAMPOP EMPLOYD .2534 .3234 171 %ECAFPOP EMPLOYD -.0323 . 1356 172 IND FOOD6BEVERGS .1473 -.3036 173 IND WOOD FINISH .2177 -.3901 1 74 IND PAPER £A LLIED .1452 .0347 175 IND METAL FABRCN .2071 -.3147 176 IND CONSTRUCTION .1975 -.5107 177 IND TRANSP+STORG .0822 - .2996 178 IND TELECOM MUNCN .0191 .0812 179 IND TRADE-W HOLSL . 1406 . 3801 180 IND TRADE-RETAIL .0859 -.0519 181 IND FINANCE+INS -.3268 .6465 182 IND EDUCATION -.0948 .5400 183 IND HEALTH WELFR -.2380 .0737 184 IND HOTEL RESTNT -.2780 -.2150 185 IND PUB A D M N DEF . 1500 .0076 1 86 OCC MANAGERIAL .0753 .8567 187 OCC PRO FL T ECHNL -.1021 .8177 188 OCC CLERICAL -.4213 .1280 189 OCC S ALES . 1061 . 4371 1 90 OCC FARMER .0700 -.0496 19 1 OCC LOGGER .0421 -. 1719 1 92 OCC FISH TRAP .0664 -.0947 FACTOR 3 4 5 6 7 .0216 -.1158 .0736 -.0545 .1316 .4071 . 1795 -.0802 .0207 .0535 .1650 .0119 -.1183 . 1515 -.3342 .0034 . 1073 -.0563 -.0361 -.0376 -.1391 -.0646 . 1254 -.1135 . 2960 .0208 -.0 202 -.0279 -.0076 -.0855 -.0278 .2467 .0077 -.0418 -.0701 -.0114 -. 1605 -.0093 .0469 .0917 .0177 .0828 -.0463 -.0655 -.0503 .2154 -.2186 .0103 .0448 . 1257 .0695 -.0783 -.04 66 .0331 .0491 . 1040 . 1122 -.0307 . 0520 -.0234 .0646 .0791 -.0228 .0022 -.1157 .0190 -.0420 -.0028 . 0610 -.0609 . 1426 -.1185 .0027 -.0039 .0609 . 1018 -.0318 .0203 -. 1643 -.0991 .0774 -.0695 -.0618 -.0990 -.0572 .0376 -. 1983 -.0384 .0127 . 2549 -.0233 -.0858 -.0128 -.0779 .1944 .0708 -.2171 -. 1821 . 1011 .3532 -.1295 -.2084 .0767 -.0026 .1217 . 1195 -.0934 .0130 .0547 .0338 -.1604 -.1446 .0945 .1216 . 1001 -. 1823 . 6466 .0518 -. 1010 -. 1122 -.0736 -. 1478 -.0716 .0106 .0239 . 1041 -.0328 -. 1738 .0537 -. 1231 .0162 -.1314 .0357 .0189 .0717 -.1753 -.3228 . 1259 .0858 . 3922 .0053 -.1985 -.1136 -.0476 .3955 .0038 .0871 -.0231 .0352 -.0558 -.0255 . 0769 -.0060 -.0247 -.0545 .0204 .0 133 .0085 .0344 -.0010 193 APPENDIX TABLE E-4 : REORDERED FACTOR LOADINGS MATRIX - ORIGINAL DATA FACTOR VARIABLE 1 2 3 4 5 6 7 24 % SINGLE <15 YRS .9364 91 2FPOP UNMARRIED .9153 89 AVG # PERSON/F AM .9103 7 X 5 - 9 YEARS .8841 83 ED-F-AT-ELEM+PRE .8442 112 SFAM CHLD ALL<15 .8309 115 #OF PERSONS/HSLD .8288 73 ED-M-AT-ELEM+ PRE .8030 96 % FAM-HEAD 35-44 . 8021 8 f 10-14 YEARS .7869 30 BORN B. C. .7776 120 #HPERSONS/DWELL .7731 6 % 0 - 4 YEARS .7573 116 SHSLD 1 FAM HSLD .7570 90 %POP IN FAMILIES .7115 67 ED-M-NO-<GRADE 1 .6869 5 FERTILITY RATE .6359 77 ED-F-NO-<GRADE 1 .6350 153 %DW EL 1ST MORTG .5953 13 f 35-39 YEARS .5662 162 AVG # FAM/DWELL .5605 92 %TkM W W.E.HEADS .5604 58 % OF IMGTS 41-50 .5121 110 XFAM CHLD NONE>6 .4948 133 %DWEL CONS 46-59 .4787 154 %DWEL > 1 MORTG .3427 37 BORN AUSTL+COMWL -.3505 138 <?D WEL W SEWER -.3924 188 OCC CLERICAL -.4213 150 XDWEL NO CARS -.5275 29 % DIVORCED -.5538 16 % 50-54 YEARS -.5697 26 %SNGL>15YRS.FEM. -.5867 56 SOF IMGTS BEF«21 -.5886 98 %FAM HEAD 55-64 -.6213 99 /{FAM-HEAD 65-69 -.6348 22 % 80-84 YEARS -.6995 35 BORN UK+REP.IREL -.7187 17 % 55-59 YEARS -.74 82 118 ?HSLD NONFAM HLD -.7679 100 3SFAM-HEAD > 70 -.7959 18 % 60-64 YEARS -.8129 20 % 70-74 YEARS -.8240 28 % WIDOWED -.8269 21 % 75-79 YEARS -.8306 19 % 65-69 YEARS -.8321 109 %FAM NO CHILDREN -.9451 4108 4263 4071 . 4901 -.4353 . 4317 -.4639 -.4614 72 ED-M-NO-UNIV DEG 186 OCC MANAGERIAL 93 AVG FAM EARN WEH .8921 .8567 .8399 194 V A R I A B L E F A C T O R 4 1 8 7 O C C P R O F L T E C H N L 1 0 8 5 & F A M - W E H $ > 1 0 0 0 0 1 0 7 2 F A M - W E H $ 7 - 9 9 9 9 8 2 E D - F - N O - U N I V D E G 1 6 3 X M A L L F E M P L O Y E R 8 1 E D - F - N O - U N I V N C 1 5 2 X D W E L > 1 C A R S 7 1 E D - M - N O - U N I V N C 1 8 1 I N D F I N A N C E + I N S 1 4 6 % D W E L O S $ 1 8 - 2 7 0 0 0 1 4 1 A V G # R O O M S / D W E L 1 4 7 % D W E L ( O S ) > $ 2 8 0 0 0 8 5 E D - F - A T - U N I V U G D 3 6 O R I G U K + R E P . I R E L 1 5 6 * D W E L B A N K M O R T G 6 4 R E L I G J E W I S H 1 0 6 S F A M - W E H $ 6 - 6 9 9 9 1 8 9 O C C S A L E S 3 4 E T H N U S A 1 7 9 I N D T R A D E - W H O L S L 1 6 1 A V G G R O S S R E N T 1 7 7 I N D T R A N S P + S T O R G 1 7 5 I N D M E T A L F A B R C N 4 5 B O R N S C A N D I N A V I A 5 2 O R I G U S S R 9 4 % F A M - H E A D < 2 5 Y 1 7 6 I N D C O N S T R U C T I O N 1 4 2 A V G # P E R S / R O O M 6 8 E D - M - N O - E L E M E N T Y 7 8 E D - F - N O - E L E M E N T Y 1 6 5 * M A L L F W A G E E A R N 1 0 3 % F A M - W E H $ 3 - 3 9 9 9 1 0 4 2 F A M - W E H $ 4 - 4 9 9 9 1 4 4 % DW E L O S < $ 1 2 0 0 0 . 8 1 7 7 . 7 9 7 4 . 7 9 4 6 . 7 8 1 8 . 7 5 9 6 . 7 3 6 9 . 7 1 9 1 . 6 8 8 5 . 6 4 6 5 . 6 1 9 3 . 4 0 7 4 . 6 0 1 6 . 5 9 1 3 . 5 7 0 1 . 5 2 3 2 . 4 9 0 8 . 5 0 4 2 . 4 7 1 5 . 4 6 9 0 . 4 3 7 1 . 3 8 7 9 . 3 8 0 1 . 3 1 5 4 - . 2 9 9 6 - . 3 1 4 7 - . 3 7 9 8 - . 4 3 9 6 - . 4 7 5 3 - . 5 1 0 7 . 4 3 0 3 - . 5 1 6 9 - . 6 1 4 0 - . 6 1 8 4 - . 6 5 2 6 - . 6 7 8 4 - . 6 7 9 2 - . 7 3 9 3 - . 4 6 7 5 . 5 1 8 3 - . 4 6 9 9 - . 4 7 9 9 . 4 5 7 4 1 5 5 %DWEL N O M O R T G A G 1 2 1 S D W E L O W N E D , S N G L . 4 8 0 1 1 5 8 % D W E L W G A R A G E 1 2 2 S D W E L S N G D E T A C H . 5 8 8 7 1 3 0 5 S D W E L O C U P > 1 0 Y R 1 4 3 A V G # B D R O O M / D W E L . 5 3 1 9 1 5 7 % D W E L P R I V M O R T G 1 3 2 3DWEL C O N S 2 0 - 4 5 1 2 3 % D W E L D U P L E X 1 1 3 ? ? F A M H B D + W F W O R K 1 2 7 %' DW E L O C U P 0 - 2 Y R 1 2 4 5 I D W E L A P T O T H E R - . 5 9 2 7 1 6 0 % P O P T E N A N T S - . 5 9 6 6 1 2 5 2 D W E L R E N T E D - . 4 5 7 4 1 5 9 A V G # T E N A N T / D W E L - . 4 5 8 2 5 3 B O R N C H I N A 5 4 O R I G C H I N A 6 6 L A N G N O T E N G , F R . 4 0 1 9 . 7 8 2 6 . 7 7 9 9 . 7 5 0 1 . 7 0 3 6 . 5 8 3 5 . 5 6 3 5 . 5 1 7 5 . 4 5 2 7 - . 3 2 7 2 - . 4 0 1 3 - . 6 3 3 3 - . 6 5 7 7 - . 7 3 0 3 - . 7 9 7 8 - . 7 9 7 9 4 5 7 5 . 9 5 5 8 . 9 4 0 5 . 8 4 7 7 195 FACTOR VARIABLE 1 2 3 4 65 RELIG OTHERS 184 IND HOTEL RESTNT 87 MA LE-FE MAL RATIO 119 %HSLD W LODGERS 1 0 1 KFAM-WEH < $ 2 0 0 0 148 % DW EL NO TV 102 3FAM-WEH $2-2999 1 POP'N DENSITY 131 $D WEL CONS <1920 126 % DW EL RESID + BUSN 31 BORN CANADA-NOBC 149 %DW EL 1 TV 62 RELIG PROTESTANT 114 %POP IN HODSHLDS -.4269 .7381 .6466 . 5768 .4755 . 4327 .4288 . 4085 . 3862 . 3769 . 3629 -.4175 .4237 •. 4877 .4992 11 % 25-29 YEARS 60 f OF IMGTS 56-59 10 % 20-24 YEARS 95 9SFAM-HEAD 25-34 59 % OF IMGTS 51-55 61 S OF IMGTS 60-61 129 %DWEL OCUP6-10YR 14 % 40-44 YEARS 57 % OF IMGTS 21-40 9 % 15-19 YEARS 111 XFAM CHLD 15-24 97 %FAM-HEAD 45-54 74 ED-M-AT-HIGH SCH 84 ED-F-AT-HIGH SCH 15 % 45-49 YEARS 128 %DW EL OCUP 3-5YR 12 % 30-34 YEARS -.4312 . 6 8 8 2 . 6285 . 6 0 6 1 . 5713 . 4 5 5 7 . 4033 . 3 7 9 6 . 3 9 1 0 . 4 4 1 7 .6708 .6214 .6193 . 5062 .4996 . 4941 -.4339 .4337 -.5048 1 45 5?DWELOS$13-17000 105 %FhM-WEH $5-5999 151 *DWEL 1 CAR 180 IND TRADE-RETAIL 178 IND TELECOMMUNCN .4177 51 BORN USSR 42 ORIG GERM+A+SWIT 47 BORN CENT EUROPE 41 BORN GERM+A+SWIT 48 ORIG CENT EUROPE 2 POP'N POTENTIAL . 5748 .4608 .4219 .3532 . 2549 190 OCC FARMER 88 % OF POP "RURAL" 43 BORN BELG+N ETHLD 44 ORIG BELG+NETHLD 4 TIME-CITY CENTER 140 SDWEL OWN WATER 196 FACTOR VARIABLE 1 2 3 4 5 6 7 40 ORIG ITALY 39 BORN ITALY 63 RELIG ROM CATHOL 46 ORIG SCANDINAVIA 25 %SNGL>15YRS.MALE -.4121 49 ETHN GREECE 170 XECAMPOP EMPLOYD 27 % MARRIED 33 ORIG NATIVE POPN 19 1 OCC LOGGER 139 %DWEL NO SEPT TA 192 OCC FISH TRAP 55 ETHN JAPAN 164 %MALLF OWN ACCNT 172 IND FOOD8BE VERGS 76 ED-M-AT-UNIV GRD 86 ED-F-AT-UNIV GRD 75 ED-M-AT-UNIV UGD .5442 182 IND EDUCATION .5400 38 ETHN FRANCE 173 IND WOOD FINISH 183 IND HEALTH WELFR 174 IND PAPER6ALLIED 80 ED-F-NO-HIGH FIN 70 ED-M-NO-HIGH FIN 69 ED-M-NO-HIGH NC 79 ED-F-NO-HIGH NC 167 3FEMLF OWN ACCNT 117 %HSLD>1 FAM HSLD .4225 1 68 9SFEMLF WAGE EARN 137 %DWEL COND FAIR 171 *ECAFPOP EMPLOYD 135 %DWEL COND GOOD 136 %DWEL COND CK 134 3DWEL CONS 60-61 3 RECENT POPN INCR 50 ETHN INDIA PARIS 166 3SFEMLF EMPLOYER 23 S 85 + YEARS -.5162 169 %FEMLF UNPD FAMW VARIABLE 1 185 IND PUB A DM N DEF 32 ORIG CANADA FACTOR 2 3 4 

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