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A Statistical analysis of math 100 grades related to B.C. high school factors McKeeman, Cheryl Ann 1978

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A STATISTICAL ANALYSIS OF MATH 100 GRADES RELATED TO B.C. HIGH SCHOOL FACTORS "by CHERYL ANN McKEEMAN B.Sc, Concordia U n i v e r s i t y , 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF 'MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES The I n s t i t u t e of Applied Mathematics and S t a t i s t i c s We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1978 (c) Cheryl Ann McKeeman , 1978 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of H^fljtlul^vUaJ^ The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 i i . ABSTRACT In t h i s thesis we consider the records of students who graduated from B r i t i s h Cloumbia high schools, and who registered at U.B.C. for the f i r s t time i n the 1977 f a l l term, including f i r s t - y e a r calculus (Math 100) as one of t h e i r courses. Using canonical cor-r e l a t i o n techniques, we scale the Math 100 grades and the grade 12 mathematics grades. Other methods of multidimensional s c a l i n g are also considered. Analyses of variance are c a r r i e d out to examine re l a t i o n s h i p s between grade achieved i n Math 100 and several-school f a c t o r s , namely type of semesterization, Math 12 textbook, and geometry background for the 1976-77 high school graduating c l a s s . Then grade 12 mathematics grade i s introduced as a covariate, and the above' analyses repeated. F i n a l l y the methods of correspondence analysis are described and an example of t h e i r a p p l i c a t i o n presented. TABLE OF CONTENTS Chapter Page LIST OF TABLES v LIST OF FIGURES v i i ACKNOWLEDGMENTS v i i i INTRODUCTION 1 I THE DATA 4 I I ON METHODS OF SCALING 15 I I - l . Introduction 15 II-2. Canonical Correlations 17 I1-3. Canonical Correlations on Subgroups 32 .11-4. Alternate Method Number Two 37 II-5. Alternate Method Number Three 39 I I - 6. Conclusions 43 I I I DATA ANALYSIS 46 I I I - l . Introduction ' - 46 III-2 . I n i t i a l Analysis 47 II I - 3 . Analysis of Variance 49 III- 4 . Analysis of Covariance 58 III - 5 . Semesterization '63 III-6. Textbook 70 III- 7 . Geometry 75 ( I I I - 8 . Discussion 80 III - 9 . Interactions 83 111-10. Sections 91 i i i CORRESPONDENCE ANALYSIS " IV-1. I n t r o d u c t i o n IV^2. Geometric Representation IV-3. A l t e r n a t e Representation : Incidence Data IV-4. General Rules f o r I n t e r p r e t a t i o n IV-5. A p p l i c a t i o n IV-6. Conclusions CONCLUSIONS BIBLIOGRAPHY APPENDIX LIST OF TABLES Number I Page I I I I I IV V . VI V I I V I I I IX X XI X I I X I I I XIV D i s t r i b u t i o n of Math 12 Grades f o r Students Who Dropped Out of Math 100 and f o r Those Who Did Not D i s t r i b u t i o n of Math 12 Grades f o r Students who F a i l e d Math 100 and For Students Who Dropped Out of Math 100 Sample of,Data F i l e Semester Types and Hours of Math 12 Taught f o r 54 Large Schools Textbook Data Geometry Data Test i n g Canonical C o r r e l a t i o n s v i a B a r t l e t t ' s S t a t i s t i c Canonical C o r r e l a t i o n Scores and Standardized Scores f o r the O r i g i n a l Count.Data O r i g i n a l Data Di v i d e d i n t o Subgroups and Assigned Scores .Comparison of Math 12 Scores Obtained by the D i f f e r e n t Methods A n a l y s i s of Variance : MODEL A A n a l y s i s of Variance - MODEL A Semesterization E f f e c t s A n a l y s i s of Variance - MODEL A Textbook E f f e c t s A n a l y s i s of Variance - MODEL A Geometry E f f e c t s , 6 8 11 12 12 24 26 33 45 51 53 55 56 V I Number XV XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI XXVII XXVIII XXIX Analysis of Covariance -Semesterization E f f e c t s Semesterization ANOCOVA Results Analysis of Covariance -Textbook E f f e c t s Textbook ANOCOVA Results Analysis of Covariance -Geometry E f f e c t s Geometry ANOCOVA Results Assignment of Students by School Type for Analysis, of Covariance with Interactions Analysis of Covariance MODEL E -Interactions ANOCOVA Data Two-Way Information D i s t r i b u t i o n of MAth 100 Grades for Honours and Nonhonours Students Data Grouping Used for Correspondence Analysis Table of Eigenvalues, Eigenvectors and Percentages of Variance Explained Table of Information on Math 100 . Categories Information on Math 12 Page. 65 • 67 71 72 76 77 87 . 89 .90-91 104 105 106 108 LIST OF FIGURES Frequency polygon of Number of Students Who Received Each.Math 100 Grade Plot of the Grades with respect to Their Assigned Scores P l o t of the Regrouped Grades as They Relate to Their Newly Assigned Scores Normal D i s t r i b u t i o n Divided into r Intervals on (-» , +°°) Plot of Average Math 12 Scores vs Average MAth 100 Scores for a l l 49 Schools Average Math 12 Score vs Average Math 100 Score for Schools I d e n t i -f i e d by Textbook Type Regression l i n e s and Average Scores Under Geometry Treatments P l o t of the Categories of Math 100 Grades and of the Math 12 Grades w.r.t. the Scores Assigned v i a Correspondence Analysis V l l ACKNOWLEDGMENTS I would l i k e to thank Dr. James V. Zidek for suggesting the. topic of t h i s t h e s i s , and for h i s guidence throughout the study. I am also indebted to Dr. Stanley W. Nash for h i s comments, help and encour-agement, and for his time given so generously to reading t h i s thesis i n i t s many forms. Dr. George W. Bluman's assistence i n c o l l e c t i n g the data was invaluable. I am very g r a t e f u l for h i s h e l p f u l suggestions made during the wri t i n g of the th e s i s , and for the many hours that he spent i n discussion with me. Professor M. Greenacre of the Department of S t a t i s t i c s , UNISA, supplied the documentation and ca r r i e d out the computing required for the correspondence analysis of Chapter IV. His help i s greatly appreciated. The f i n a n c i a l support of the National Research Council of Canada and of the Department of Mathematics of the Univ e r s i t y of B r i t i s h Columbia,is g r a t e f u l l y acknowledged. 1 INTRODUCTION Since 1975 George Bluman, [5], [6], [7], [8] and [9], has compiled data on student performance i n the f i r s t year calculus course at the Unive r s i t y of B r i t i s h Columbia (U.B.C.). In p a r t i c u l a r he considered graduates from B r i t i s h Columbia (B.C.) high schools. Dr. Bluman reported on factors such as high school geographical l o c a t i o n , high school semes-t e r i z a t i o n type, and students' grade 12 mathematics grades i n order to study the asso c i a t i o n between Math 12 and Math 100 grades. During the 1976-77 school year, B.C. high schools were at various stages of a change-over i n t h e i r mathematics programs. Some schools were teaching a course c a l l e d Algebra 12; others were teaching Math 12. Throughout t h i s thesis we r e f e r to a student's grade 12 mathematics course as Math 12. Some schools had offered a grade 10 geometry course u n t i l 1975. Thus students from these schools had some background i n geometry. Some schools have only grades 11 and 12 - t h e i r students can come from any one of several "feeder" schools. The feeder schools for a given high school need not have a l l offered a grade 10 geometry course. Thus some students i n f i r s t year calculus at U.B.C. had had no geometry, some students had had geometry, and we do not know about the others. There were also differences between schools v i s - a - v i s the textbook used i n Math 12. Most schools used the book by D o l c i a n i . A few used "Using Advanced. Algebra" (U.A.A.) , and many schools used both texts. One school used a book by Keedy and B i t t i n g e r ; another used "Precalculus Mathematics" (PCM) . 2 We define a."large" school as one from which at l e a s t ten students "with f u l l information" registered f o r a f i r s t year calculus course at U.B.C. i n September, 1977. (A student with f u l l information i s one for whom we have a Math 12 grade, and who did not drop out of the calculus course. ) With respect to semesterization, we can c l a s s i f y large schools as one of the following : 1. UNSEMESTERED. In an unsemestered school, Math 12 i s a f u l l - y e a r course taught, f o r example, for 5 hours i n every 7 days.. 2. SINGLE-SEMESTERED. Math 12. i s taught usually f o r llA hours per day for one (approximately f i v e month) semester. There i s no other mathematics course a v a i l a b l e during the months when Math 12 i s not taught. 3. DOUBLE-SEMESTERED. Math 12 i s a two-term f u l l - y e a r course, taught for one hour per school day. 4. MIXED. This category r e f e r s to schools which are unusual. For example, some sections of Math 12 could be single-semestered while others i n the same school were unsemestered. 5. INDEPENDENT. These are pri v a t e schools. We s h a l l examine l a t e r how the choice of semesterization, textbook and geometry a f f e c t s student performance i n t h e i r calculus course. Geo-graphical locations are not considered here. In the U.B.C. academic year 1977-78, f i r s t - y e a r calculus was divided into two parts : Math 100 and, contingent on passing Math 100, 3 Math 101. Most f i r s t - y e a r students who take Math 100 do so i n the F a l l term, and then take Math 101 i n the Winter term. Graduates from B.C. high schools were normally not admitted to Math 100 unless they had a grade of 'C or better i n Math 12. (Possible grades from Math 12 were A, B, C+, C, Pass, and F a i l . ) There were 39' sections of Math 100 i n the 1977 f a l l term. Three of these sections, c a l l e d "Honours Sections", were designed f or students who had demonstrated a b i l i t y i n mathematics, through a high grade i n the Scholarship Exam or Basic Math Test. Students from these sections had the same curriculum as, but harder problems than, students from other sections. Students from the honours sections wrote the same f i n a l exam at Christmas as the students from the other sections. In t h i s thesis we consider the students who registered i n a f i r s t - y e a r Math 100 course at U.B.C. i n the 1977 f a l l term. In Chapter I we describe the data and explain some of the ways i n which the data were edited. Then i n Chapter II we examine methods of s c a l i n g the Math 100 and Math 12 grades. Using a set of numerical scores determined by the methods of Chapter I I , we proceed with the data analysis i n Chapter I I I . In that chapter we. examine how school semesterization type, textbook • used i n Math 12, and amount of geometry studied a f f e c t students' Math 100 grades. We also i n v e s t i g a t e the i n t e r a c t i o n of these three v a r i a b l e s . As w e l l , an analysis i s c a r r i e d out to determine whether students i n honours sections had s i g n i f i c a n t l y higher grades i n Math 100 than students i n nonhonours sections. In Chapter IV we describe correspondence a n a l y s i s , and apply t h i s method to our data. 4 CHAPTER I THE DATA Almost 1700 of the students who registered i n the Mathematics 100 course at U.B.C. i n the 1977 f a l l term had graduated from a B.C. high school, and were attending U.B.C. for the f i r s t year. These students represented more than 150 high schools, of which 59 sent ten or more students who completed ( i.e.passed or f a i l e d , but did not drop out of) the Math 100 course and who had a grade of at le a s t C i n Math 12. The Math 12-grades that we s h a l l use i n t h i s . t h e s i s are not the grades a c t u a l l y received by the students. In the spring of each year, the Registrar's o f f i c e receives from the high schools estimates of t h e i r students' f i n a l grades. The Math 12 grades a c t u a l l y attained are not entered on the students' records. Thus.the grades used throughout t h i s thesis are-the estimated Math 12,grades. The Registrar's o f f i c e provided us with the following data for each B.C. high school graduate who registered i n Math 100 i n the 1977 f a l l term : 1. the high school from which the student graduated; 2. the l a s t i n s t i t u t i o n attended by the student; 3. the mark (A,B,C+,C,Pass or Unknown) that was sub-mitted as an estimate of the student's Math 12 grade; 4. the section of Math 100 i n which the student was regi s t e r e d ; 5. the mark out of 75 obtained i n Math 100; and 6. the corresponding letter-grade achieved i n Math 100 (under 38 = F; 38 to 48 = P; 49 to 59 = 2nd c l a s s ; over 59 = 1st c l a s s ) . For the purposes of t h i s study we are interested only i n those students who.graduated from a B.C. high school i n 1977; more s p e c i f i c a l l y we are interested i n those who took Math 12 i n the 1976-77 academic year. Unfortunately the year of graduation from high school was not av a i l a b l e . We discarded from the study any students for whom 1. and 2. above did not match because i t seemed l i k e l y that any such students had spent time at another post-secondary i n s t i t u t i o n before entering U.B.C. However as mentioned above, the only students for whom we have informa-t i o n are f i r s t - y e a r students who had graduated from a B.C. high school. Thus we do not consider students who are taking Math 100 for the second or t h i r d time at U.B.C, nor do we consider students who resumed t h e i r studies at U.B.C'. a f t e r taking some time o f f , nor do we consider students i n t h e i r second or t h i r d year at U.B.C. who are taking Math 100 for the f i r s t time. We were supplied with neither r e g i s t r a t i o n number nor year of high school graduation f o r any student, hence we cannot i d e n t i f y students who took time o f f between high school and U.B.C. However current work by Dr. George Bluman indicates that at most 3% of the 1670 students considered here take time o ff between leaving high school and f i r s t entering u n i v e r s i t y . Seventeen students had no Math 12 grade recorded. These students were not used i n the study. Twenty students who had received a grade of 'P' (Pass) i n Math 12 were admitted c o n d i t i o n a l l y to Math 100. These students' records were also deleted since normally a student i s not admitted to Math 100 unless he had a grade of 'C or better i n Math 12. The one student whose grade was recorded as 'C-' was also not used. We turn now to the students who did not f i n i s h .Math 100,. i . e . of MATH 12 GRADE A B C+ . C Passed or F a i l e d Math 100 517 546 324 137 Dropped out of Math 100 . 10 '40 33 25 TABLE I : D i s t r i b u t i o n of the Math 12 Grades f o r Students Who Dropped Out of Math 100 and for Those Who Did Not . MATH 12 GRADE A . B C+ C Fa i l e d Math 100 25'.' 134 130 75 Dropped Out Of Math 100 10 40 33 25 TABLE II : D i s t r i b u t i o n of Math 12 Grades for Students Who F a i l e d Math 100 and for Students Who Dropped Out Of Math 100 the students whose Math 12 grade was at least a 'C, we consider those whose Math 100 grade was 'N' . We reason as follows : i f the Math 12 grades of those students who dropped out of Math 100 come from the same population as the grades of those students who did not drop out, then we could consider replacing 'N' by some mark. For example, i f an 'N' student has an 'A' from Math 12, we could assign as h i s mark the average Math 100 grade achieved by a l l 'A' students who did not drop the course. In Table I we compare the d i s t r i b u t i o n of Math 12 grades for students who completed ( i . e . did not drop out of ) Math 100 with the 2 d i s t r i b u t i o n f o r students who dropped out of Math 100. A Pearson's x ~ 2 test applied to Table I y i e l d s x = 43.49 with three degrees of freedom, which i s highly s i g n i f i c a n t . Table II compares the d i s t r i b u t i o n of Math 12 grades for students who f a i l e d Math 100 with the d i s t r i b u t i o n of Math 12 grades for students 2 who withdrew from Math 100. A Pearson's x _ test applied to Table II 2 y i e l d s x = 1.54 with 3 degrees of freedom,which i s highly n o n s i g n i f i c a n t . It appears, then, that (for example) we could have replaced the 'N' grade attained i n Math 100 by students who received a Math 12 grade of 'A' by, the average Math 100 grade out of 75 obtained by 'A' students who f a i l e d Math 100. In t h i s thesis we decided to exclude the records of students who withdrew from Math 100. This d e c i s i o n i s discussed further i n the Conclusions (Chapter V). ' Table I I I shows a t y p i c a l page i n i t s o r i g i n a l condition from our data f i l e . Each l i n e gives the information for a d i f f e r e n t student. The columns designated 'a' represent the student's section of Math 100. C o l -umns 'b' give the mark out of 75 received i n Math 100, column 'c' the c e. 1 2 3 4 5 6 7 14 15 18 19 24 1 075 015 57365 1 075038015 P 57365 1 . 075065015 1 57365 1 075038015 P 57365 1 075033 F 57365 1 075065015 1 57365 1 075046015 P 59095 1 075025 F 59095 1 075050015 2 59095 1 075060015 1 60155 1 075046015 P 60155 1 075050015 2 60155 1 075047015 P 60155 1 075062015 1 61185 1 075 N 61245 1 075043015 P. 61245 1 075040015 P 61245 1 075045015 P 61245 1 075047015 P 61245 1 075055015 2 61245 1 075054015 2 61245 1 075032 F 61245 1 075055015 2 61265 1 075039015 P 61485 1 075038015 P 61496 1 075050015 2 61496 PRINCE GEORGE SR SEC 570MATH12 PRINCE GEORGE SEC PRINCE GEORGE SEC PRINCE GEORGE SEC PRINCE GEORGE SEC PRINCE GEORGE SEC SOUTH PEACE SEC SOUTH PEACE SEN SEC SOUTH PEACE SR SEC NORTH PEACE SR SEC NORTH PEACE SEC NORTH PEACE SEC NORTH PEACE SR SEC VICTORIA SEC OAK BAY SEN SEC OAK BAY SEC OAK BAY SEC OAK BAY SEC OAK BAY SEC OAK BAY SEC OAK BAY SEC OAK BAY SR SEC SPECTRUM SEC ESQUIMALT SEC CAMOSUN COLLEGE . MT DOUGLAS SEC 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 570MATH12 C+ 570MATH12 A 570MATH12 B 570MATH12 C+ 570MATH12 B 570MATH12 A 570MATH12 A 570MATH12 C+ 570MATH12 C 570MATH12 A 570MATH12 B 570MATH12 C+ 570MATH12 C B A B B A B TABLE III : Reproduction of Part of the Or i g i n a l Data F i l e corresponding " l e t t e r " grade, columns Vd' the high school code, columns 'e' the name of the l a s t i n s t i t u t i o n attended, and columns ' f the Math 12 grade. A blank i n columns 'b' or 'c' means that the student dropped out of the course. Some Math 12 grades are missing. Notice the student ( l i n e 25) who went to Camosun College at some time between;leaving high school and entering U.B.C. The other set of data required for our study consists of informa-t i o n on the schools themselves. In.order.to examine the r o l e s of semes-t e r i z a t i o n , textbook and geometry i n determining the Math 100 grade, we r e s t r i c t our attention to the 59 large schools mentioned e a r l i e r . (Recall that a large school i s one which had ten or more students with f u l l data sets taking Math 100 i n the 1977 f a l l term.) The Mathematics Departments' heads at these schools were interviewed by telephone to determine : 1. whether t h e i r school was semestered i n 1976-77 and, i f so, since when; 2. the time organization for Math 12 i n 1976-77, for example, whether Math 12 was taught for 1 hour per school day for two semesters, or 5 hours every 7 days for one school year, etc. 3. the text used i n Math 12 i n 1976-77,as w e l l as any supplementary material; 4. whether or not the students who took Math 12 i n 1976-77 had had a geometry course i n grade 10 ; 5. the approximate number of students i n grade 12 i n 1976-77 . 1 0 6. the approximate number of students i n Math 12 i n 1976-77 ; and 7. the approximate number of sections of Math 12 i n 1976-77 i n t h e i r school. We also determined i n each case whether changes were planned for 1978-79, and whether there were differences between the current 1977-78 school year and the 1976-77 year. One department head could not be reached, three department heads declined to answer any of our questions, and one school had burned down since l a s t year. That l e f t 54 schools and 1112 students. Omit-ti n g private and mixed schools we obtain 49 schools and 1035 students. Tables IV, V.and VI summarize some of the data c o l l e c t e d . With respect to textbook, categories 2. through 7. were combined, so that three textbook groupings were obtained : 1. U.A.A. only , 2. D o l c i a n i only, and 3. Combinations and Other. The geometry categories were also combined to form three groupings : 1. no geometry 10 offered, 2. geometry 10 offered, and 3. do not know. Figure 1 i s a frequency polygon for a l l 1670 students, excluding those who dropped out of Math 100. We see there the number of students who received each of the possible marks i n Math 100. Very few students get marks of 37, 48 or 59 since these marks are a l l just below the lowest mark i n the next grade-category. Thus, for example, a student would get a high second class of 58 or a low f i r s t class of 60, but not a borderline grade of 59 out of 75. This explains the peaks at 38, 49 and 60. 11 NUMBER OF SCHOOLS NUMBER OF HOURS 9 5 hours i n 7 days 4. 5 hours ' i n 8 days 1 5 hours i n 9 days . 5 3 hours i n 5 days 1 hours i n 10 days TOTAL : 20 Unsemestered Schools 14 1^25 hours per day 2 . .1 hour per day 1 3 100-minute periods i n 4 days 2 80 minutes per day TOTAL : 19 Single Semestered Schools 10 1 hour per day TOTAL : 10 Double Semestered Schools ALSO : 2 Private Schools, 3 Mixed'Schools TABLE IV : Semester Types and Hours of Math 12 Taught for 54 Large Schools 12 TEXT COMBINATION NUMBER OF SCHOOLS '. 1.. U.A.A. only 3 f 2-U.A.A. with D o l c i a n i 4 U.A.A. with D o l c i a n i and PCM 1 4. Keedy and B i t t i n g e r L "Corr\\po V 5. Some sections U.A.A., some 2 sections D o l c i a n i 6. Do l c i a n i with U.A.A. 3 1 7. Dol c i a n i with calculus sup- .2 plement 8. Dol c i a n i only 38 TABLE V : Textbook Data GEOMETRY -BACKGROUND - NUMBER OF SCHOOLS• 1. No Geometry 10 Offered 28 2: Offered Geometry 10 19 "Hoi \ m 3. Some Students had Geometry 10, Some Did Not 5 • C " 5 ure 4. Do No t Know 2 TABLE VI : Geometry Data 13 O o .Ifl \r>. Figure 1 Frequency polygon of the number of students who received each Math 100 grade (out of 75), with the corres-ponding letter-grades indicated, for a l l 1670 students, except those 116 who dropped the course. The horizon t a l axis i s the Math 100 grade out o 75. The v e r t i c a l axis i s the number students. - ' 15 CHAPTER II ON METHODS OF SCALING GRADES .11-1. Introduction For each student, we have a Math 100 mark out of 75, and a Math 12 grade (A, B, C+ or C). The Math 12 grades are simply l a b e l s that have an order. We want more than j u s t an ordering : we want to be able to place these grades on a scale so as to examine the r e l a t i v e d ifferences between the various grades. S i m i l a r l y , we want .to see how the Math 100 marks compare when, subject to some c r i t e r i o n , we scale them. Thus each mark and grade w i l l have associated with i t a new numerical score. We can examine these scores to see i f they reveal any patterns i n the data. Later, we w i l l use the scores to perform analyses of covariance. In t h i s chapter we s h a l l f i r s t l y describe methods of determining scores to assign to the categories of two v a r i a b l e s . We s h a l l then apply these methods to our data. F i n a l l y we s h a l l choose one of the r e s u l t i n g numerical scales, and use these scores to proceed with our data a n a l y s i s . For t h i s chapter, l e t A and B be va r i a b l e s with, f and c clas s e s , r e s p e c t i v e l y . The empirical j o i n t d i s t r i b u t i o n i n a sample of s i z e n i s given by the following contingency table, where n „ i s the number of observations that f a l l into the i t h cla s s of v a r i a b l e A and the c r i t h class of v a r i a b l e B, where n. = ) n., , n , = I n... and : - / - i > x n r c n = T 7 n. . . i= l j = l 16 VARIABLE B 1 2 c . Total Scores 1 n l l l c "•' n l . 2 n21 n 2 2 n 2 c n2. X2 . r n r l n r 2 . . n rc n r . r T o tal n . l n.2 n . c n Scores y l y 2 y c w PQ <! H < > Let x^ be the score to be assigned to the i t h category of v a r i a b l e A. Let y^ . be the score to be assigned to the j th category of v a r i a b l e B. We s h a l l assume that the categories are ordered, so that X , < X „ < . . . < X 1 — 2 — — r and y1 i y 2 i We now need to define a c r i t e r i o n by which to choose these scores, 17 II-2. Canonical Correlations One way to f i n d the scores i s to choose those values which maxi-mize the product moment c o r r e l a t i o n . The product moment c o r r e l a t i o n i s p , given by r c I I n,, (x - x) (y, - y) i = l 3=1. 13 1 P = i = i x- 1 l n . (y - y). 2 3=1 ° J Since p i s invariant under a r b i t r a r y a f f i n e transformations of the data we may, without loss of generality, r e s t r i c t both the x's and the y's to have mean zero and variance one, i . e . x r • >_ _ I" n x /n = 0 = I n y / n = y 1=1 1 j = l "J 3 and r 2 ° 2 var x = 7 n. x. / (ri-1) = 1 = j n". y, /(n-1) = var y i-1 1 1 1 j-1 ' J J We seek to maximize p subject to these conditions, where.now we can write r c ( I I - D P = I." I n ± j x. Y j / (n-1) i = l j = l Introducing the Lagrange m u l t i p l i e r s \i , ^2 > ^3 a n ( ^ 1^4 > t n e 18 .Lagrangian I i s given ([1]') by r c L = 7 y n. . x. . y . / (n-i = l j = l 1 J 1 J 1) A, / n. x. /n 1 ^ l . l i = l j = i J .3 I n. / (n-1) - 1 l . l r - x4 i n , y / (n-1) - 1 j = l ° J Setting 3L/9y_. = 0 , we have (H-2) r c y }" n., x. y,. L L ik l Jk i = l k=l 1=1 n. . x, 13 i (n-1) - A? n .•/ n 2 A 4 n . y. / (n-1) = 0 . -3 3 Summing over j , we obtain r c y J n., x. y, L L i k l •'k i = l k=l r c i = l j-1 2 AL I n . y / (n-1) j = l ° 3 = 0 Therefore., A 2 = 0. Mu l t i p l y (II-2) by y and sum over j r c ~\ 2 I [ . ^ x. y. / (n-1) 2 i = l j = l 2 A, I n , y, / (n-1) = 0 . 3=1 ° J J Therefore, A^ =? p Substituting these values of X2 and back into equation (II-2) and using ( I I - l ) we have (II-3) y. = I . n.. x / (n , p ) i = l Square (II-3) and multiply the r e s u l t by n to obtain ' • J n . y. = • 1 J I n.. x. ( n . p ' ) . • J Sum the l a s t expression over j and rearrange the terms i n the r e s u l t so that (n-4) p 1 n-1 j = l J n.. x. *-• T I T i = l n . since I n . yf / (n-1) = 1 3=1 •3 3 If the x. were known, the maximum c o r r e l a t i o n could be calculate using (II-4) , and then the y scores determined by (II-3). Observe that the x. would f i r s t be standardized to have mean zero and variance 1 one, i . e . they would be. replaced by ( x^ - x ) /:& ' , where r x = )" n. x. / n i= l X- 1 and 0 r \ = 'I n i . ( x i " x ) 1 ( n " i ; ) • i= l ' If the x. were not known, then we would next maximize L with 1 respect to the x_^  . The f i r s t term i n the Lagrangian L i s p , which can also be written as i n (II-4) . Substituting (II-4) i n L and s e t t i n g E)L / 8x_^  = 0 we obtain c r (II-5) _2_ , I n I n . x / n - X x n / n 1 j = l 1 J k=l k J k , : 1 a -n-1 J - 2 A 3 n. x. / (n-1) = 0 . Summing over i , we f i n d that Aj. = 0. As i n the case of A l, , we now f i n d that A 3 = p •''. Following Anderberg [1], we now determine, i n the following sequence of equivalent conditions, a form of (II-5) s u i t a b l e for c a l c u l a t i n g the x^ : c r _ J _ I n I n K I n - 2..p . n x / ( n - 1 ) = 0 . 1 j = l 1 J k=l , J k ° 1 ' 1 n-1 J X/ . y 1 I n.. I n . x V n k . \ /""A 1 ~ T A - k-i ^ k\vJA%fe p n. x. / n-1 n-1 \ 1. c r I V'2-I I n k i • n i . i /v* j = l k=l , • . >2 x k p 2 Y v f x ± = 0 n-1 r c V / T T n v , V2 ' r .n.. n, . - 2 (x.. n . 1 ) (II-6) I (x n> ) I LI k.i = p 1 1 . / k=l K K- j = l , ,VZ ( n i . n k . } n . . i -2 Thus we see that the f e a s i b l e values of p are eigenvalues of the matrix BB' where B i s the r by c matrix with ( i , j ) t h element n.. / (n. n .) i j i - • J We can write (IT-6) i n matrix notation as , -2 BB'u p u , where u i s an r x 1 eigenvector with elements x^ Vn7 . Thus to f i n d the x_^ , we form the matrix BB' (or B'B i f c < r ) and f i n d the nonzero eigenvalues. There w i l l be up to min(r,c) such eigenvalues, de-pending on the rank of B. The larges t eigenvalue w i l l be 1 , with-corresponding eigenvector proportional to u^ — ( , /n^ , . . . , >/n~ To see that (l,u^) i s indeed an eigenpair for BB', observe that B = 11 >/ru n 1. .1 "21  / n 2 . n . l r l . / n n . r. .1 n 12 >/ru n 1. ".2 '22 / n2. n.2 r2 ~n~ r. 1 c / n, ~n 1. . c n 2c J n„ n 2. .c rc n n r. . c and so that we can write BB'.u^ as n. . j = l • i = l /n, - n ~ v/n~ 1. -3 c n . Y LL j = l / — — J v n n . r . .3 r n, , .1 i = l ; v n j That i s , BB'u, c n n j = 1 ^ n . j c n . n . j = l 7 J v n n . r . . j Thus Now P Q ^ = 1 i s obtained by assigning the same score, x^ = y = 1 (say to each category - i . e . by collapsing each variable's c l a s s i f i c a t i o n into a single class. The remaining nonzero eigenvalues are the canonical - 2 correlations. The largest i s the maximum squared.correlation, , between our two variables. We then calculate the eigenvector corres-~ 2 ponding to whose elements have mean zero and variance one. This gives us one set of scores - (namely, the x. i f r < c and the y_. other-wise) . Then we can find the other set by using (II-3) or c (II-7) x = I n y / (n p. ) . j=l . Corresponding to the kth nonzero eigenvalue, we can calculate a kth set of scores with mean zero and variance one. In this way we obtain m = min ( r - l , c - l ) sets of scores. Kendall and Stuart [17] describe a test by B a r t l e t t to test whether, assuming that k canonical correlations are nonzero, the remaining ones are zero. The test s t a t i s t i c i s k 1 -j mi n ( r - l , c - l ) „ - { n - l - k - ( r + c + 1 ) + 1 : 2 V log TT ( 1 - p. ) , i= l i 1  2 2 j=k+l which i s approximately x 2 with [(r-k)(c-k)] degrees of freedom. Thus once we have calculated the eigenvalues, we may test to see which of them are not s i g n i f i c a n t l y d i f f e r e n t from zero, and calculate the scores based on these nonzero canonical correlations. We can apply t h i s method of canonical correlations to our data. Let variable A represent the Math'100 grade out of 75. There are 71 categories, since the grades 0, 1 , 7, 37 and 59 were not assigned to any students. Variable B represents the Math 1 2 grade, which i s divided intofour categories : A, B, C+ and C. The canonical correlatipn NULL HYPOTHESIS VALUE OF BARTLETT'S STATISTIC DEGREES OF FREEDOM x29 5 ( d . f . ) DECISION Assume nonzero and test and p^ equal zero X 2 = 16.7-59 270 308.46 Accept at a=0.05 Assume and p^ nonzero, and .test P 3 = 0 X2 = 64.66 138 166.41 Accept at a=0.05 TABLE VII : Testing Canonical Correlations Via B a r t l e t t ' s S t a t i s t i c 25 method i s applied to the 71 x 4 matrix N = {n..}, n.. being the number of students who f e l l into the i t h category of v a r i a b l e A.and the j t h - 2 category of v a r i a b l e B. The eigenvalues obtained are = 1.0 , P x 2= 0.3619433 , p 2 2 = 0.0672655 and p 3 2 = 0.0421717. Table VII 2 ' shows the r e s u l t of using B a r t l e t t ' s test to see whether p 2 and 2 p ^ are zero. From the r e s u l t s of t h i s table, we conclude that p 2 & p ^ are zero. Table VIII shows the count data, together with the transformed - 2 " 2 (to mean zero and variance one) scores corresponding to p ^ and p 2 . - 2 The scores based on p^. are l a b e l l e d "CCl Scores" ; those based on „ 2 P 2 are l a b e l l e d "CC2 Scores" . Table VIII also contains the "Standard-ized Scores" ; these we discuss l a t e r . The Math 100 "CCl Scores" do not preserve the monotone-increasing . property of the Math 100 grades; that i s , i f we project the scores onto an axis and l a b e l each with the corresponding Math 100 grade, then the order 2,3,4,...75 w i l l not be preserved. Furthermore, we see that the grades with proportional counts get the same "CCl Score" - f o r example, Math 100 grades of 3, 5 and 48 a l l get a score of -0.381. We started with a 71 x 4 matrix, which had rank at most 4, so that there are at most four nonzero eigenvalues. Since the nonzero eigenvalues of the transpose of a matrix are equal to the eigenvalues of the o r i g i n a l matrix, we work with the 4 x 71 matrix and f i n d i t s eigenvalues. For each eigenvalue, we c a l c u l a t e the scores corresponding to each of the four Math 12 grades. The Math 100 grades are expressed as l i n e a r combinations of these Math 12 scores. That i s , i f y^, y 2» y^ and y^ are the Math 12 scores corres-ponding to the eigenvalue p , then the t-th Math 100 score xfc i s given by ^ 1 ^ n , y. / n x_ = , L t j ' j t. t - i = l J J MATH 10G GRADE A 2 0 3 0 4 0 5 0 6 0 8 0 9 1 10 0 11 0 12 1 13 0 14 0 15 1 16 0 17 0 18 0 19 0 20 0 21 2 22 0 23 1 24 2 25 1 26 2 27 1 28 1 29 0 30 4 MATH 12 GRADES B C+ C 0 0 1 1 0 0 0 1 1 1 0 0 0 1 0 0 1 0 1 0 0 0 2 1 1 1 2 3 5 1 1 3 0 2 1 2 6 8 3 1 2 2 1 3 1 2 1 2 5 3 1 8 5 4 4 3 4 0 3 2 4 10 3 6 6 6 8 7 4 13 10 7 3 6 3 4 1 3 2 2 0 18 17 10 TOTAL CC1 1 -2.366 1 -0.381 2 -2.064 1 -0.381 1 -1.762 1 -1.762 2 0.876 3 -1.964 4 -1.719 10 -1.019 4 -1.417 5 -1.452 18 -1.186 5 -1.728 5 -1.607 5 -1.452 9 -1.062 17 -1.255 13 -0.924 5 -2.004 18 -1.340 20 -1.140 20 -1.140 32 -1.090 13 -1.283 9 -0.917 4 -1.072 49 -1.060 STANDARDIZED CC2 SCORES 8.320 -2.955 -4.461 -2.886 4.555 -2.817 -4.461 -2.748 0.790 -2.679 0.790 -2.540 -1.225 -2.471 -3.300 -2.402 3.242 -2.333 0.090 -2.263 -0.523 -2.194 1.702 -2.125 0.363 -2.056 2.752 -1.987 1.246 -1.918 1.702 -1.848 -1.290 -1.779 0.091 -1.710 1.679 -1.641 3.802 -1.572 0.946 -1.502 1.596 -1.433 0.257 -1.364 0.380 -1.295 1.410 -1.226 1.102 -1.157 -1.835 -1.087 0.498 -1.018 TABLE VIII : Canonical Correlation Scores and Standardized Scores for the O r i g i n a l Data „ . ™ T nnn M A ™ 1 2 GRADES MATH 100 GRADE A B C+ C TOTAL CCl CC2 STANDARDIZED SCORES 31 0 6 4 1 11 -1.064 -1.390 -0.949 32 1 9 9 4 23 -1.158 0.098 -0.880 33 0 11 7 2 20 -1.063 -1.345 -0.811 34 4 8 1 3 16 -0.211 -0.118 -0.741 35 . 3 5 6 2 16 -0.676 0.319 -0.672 36 0 0 1 0 1 -1.762 0.790 -0.603 38 18 50 52 28 148 • -0.936 0.589 -0.465 39 8 20 14 1 43 -0.409 -1.250 -0.396 40 12 13 14 5 44 -0.360 0.427 -0.326 41 8 21 12 5 46 -0.520 -0.576 -0.257 42 13 15 10 3 41 -0.066 -0.193 -0.188 43 6 6 8 1 21 -0.283 -0.003 -0.119 44 2 8 6 1 17 -0.690 -1.094 -0.050 45 19 25 14 4 62 -0.050 -0.467 0.020 46 13 18 4 1 36 0.318 -1.185 0.089 47 3 9 . 3 2 17. -0.415 -0.888 0.158 48 0 2 0 0 2 -0.381 -4.461 0.227 49 22 50 15 4 91 . -0.088 -1.469 0.296 50 18 22 12 . 1 53 0.123 -0.833 0.365 51 10 8 5 0 23 0.412 -0.505 0.435 52 10 13 4 1 28 0.249 -0.943 0.504 53 19 19 5 0 43 0.570 -0.991 0.573 54 11 10 3 0 24 0.599 -0.838 0.642 55 17 6 0 1 24 1.318 0.656 0.711 56 18 14 2 2 36 0.689 -0.223 0.781 57 28 15 3 0 46 1.060 -0.179 0.850 58 13 7 1 0 21 1.110 -0.204 0.919 60 41 22 4 2 69 0.976 0.060 1.057 TABLE VIII continued MATH 100 MATH 12 GRADES GRADE A B C+ C TOTAL • CCl CC2 STANDARDI2 SCORES 61 • 25 3 1 0 29 1.740 1.300 1.126 62 20 7 0 0 27 1.482 0.333 1.196 63 15 10 0 0 25 1.128 -0.578 1.265 64 22 5 0 0 27 1.668 0.813 1.334 54 - 20 5 • 0 o - 25 1.631 0.717 1.403 66 7 1 0 0 8 1.820 1.202 1.47.2 67 9 1 0 0 10 1.883 1.364 1.541 68 19 2 1 0 22 1.728 1.367 1.611 69 6 1 0 0 7 1.775 1.087 1.680 70 7 1 0 0 8 1.820 1.202 1.749 71 11 3 1 . 0 15 1.371 0.635 1.818 72 8 0 0 0 8 2.134 2.011 1.887 73 4 0 0 0 4 2.134 2.011 1.957 74 8 0 0 0 8 2.134 2.011 2.026 75 2 0 0 0 2 2.134 2.011 2.095 TOTAL 517 546 324 137 1524 CCl 1.284 -0.229 -1.060 -1.424 * 2 P = o.: 36194 CC2 0.522 -1.157 0.205 2.158 p = O.l 36727 STANDARDIZED 1.260 -0.181 -1.054 -1.540 p 2 = 0.31393 SCORES TABLE VIII concluded 29 From t h i s equation we see that i f we had two grades s and t such that ng_. = c nt_. f o r j = 1 to 4 and for some p o s i t i v e integer c, then n = c n and n . / n = n . / n , so that x ..and x would be s. t. sj s. t j t. s t equal. Since with t h i s data we are expressing seventy-one scores as l i n e a r combinations o f . j u s t four others, random v a r i a t i o n leads almost i n e v i t a b l y to proportional c e l l s of the type described, and hence to the anomaly that d i f f e r e n t Math 100 grades are assigned the same score. Figure 2 shows how the scores correspond [11]. The h o r i z o n t a l axis i s the score with respect to the f i r s t canonical c o r r e l a t i o n ; the v e r t i c a l axis i s the score with respect to the second canonical c o r r e l -a t i o n . Each l e t t e r e d point describes a Math 12 grade. Each numbered point represents the corresponding Math 100 grade out of 75. The graph emphasizes how l i t t l e the scores describe the increasing nature of the grades. We should also comment on the Math 12 scores. We see that the CCl-score for a grade of A i s the only non-negative Math 12 CCl-score. The CCl-scores for C and C+ are close, i n d i c a t i n g that t h i s scoring system does not d i f f e r e n t i a t e well between C and C+ students. In summary, we have determined the scoring scheme that maximizes the product moment c o r r e l a t i o n between our two v a r i a b l e s . However the scores obtained do not maintain the natural ordering of the v a r i a b l e categories. C-2.V,S3) ••5.0 -f • 4.0 + 3.0} 2 1 10 Ho 30 111 oZ 0 .0 . j •o z o z , < |.o-0 . A Z 0 0 UJ D.O-f -i.o4 © •2.0 2.4-£3 2& 36 3 0 IS ^> 1 5 _ ^ 3 5 2 0 5 2 . a 4o 1 3 34 15 48 t7 55 fes t2. 4 A to 1.3 4v5 si 41 &3 SO 44 31 4-7 39 52 5 * 53 41 -2.Q -I.O O O l.O F IR .ST C-/\hiOH\C^\L_ ' 5 C O & E 2.0 Figure 2 Plot of the grades with respect to t h e i r assigned scores. The h o r i z o n t a l axis i s the score corresponding to the f i r s t canonical c o r r e l a t i o n . The v e r t i c a l axis i s the score corresponding to the second canonical c o r r e l a t i o n . Both the Math 100 and the Math 12 grades are portrayed. 32 II-3. Canonical Correlations on Subgroups In the preceding section we had two v a r i a b l e s , one with many cate-gories, the other with few. Some categories contained many observations, while others had only one or two members. With such low counts, the p r o b a b i l i t y of having duplicate or proportional categories (with respect to the count data) i s high. We now seek scores which preserve the natural ordering of the grades. Our f i r s t step i s to reduce the number of Math 100 categories by combining adjacent categories. This w i l l also help to reduce the large differences i n category s i z e , and to eliminate proportional categories. We then apply the canonical c o r r e l a t i o n analysis of section II-2 to the regrouped data. Table IX shows the twenty-eight subgroups formed by grouping the Math 100 categories. It also indicates the grades (out of 75) combined to form each new category. The Math 100 standings (1st c l a s s , - 2 2nd c l a s s , etc.) are i d e n t i f i e d . The eigenvalues obtained are =1.0, ' * 2 2 2 = 0.34930 , p 2 =0.04357 and p 3 = 0.01176, which means that the maximum c o r r e l a t i o n squared i s 0.34930, as compared with 0.36194 for the preceding a n a l y s i s . Consider the scores themselves, presented.in Table IX as "CCl-Scores" and "CC2-Scores". The Math 12 scores are again i n the correct order, and are almost i d e n t i c a l to the scores obtained i n the preceding section. The CCl-scores for the new Math 100 categories seem to preserve the order s l i g h t l y better but, as Figure 3 shows, we s t i l l do not have a completely monotone scale. MATH 12 GRADES CEGORY CATEGORY A B C+ C TOTAL CCl CC2 OP1 OP2 JMBER NAME •SCORES SCORES SCORES SCORES 1 : 2-15 3 16 23 11 53 -1.268 1.120 -2.221 2.974 2 16-22 2 21 20 16 59 -1.325 1.201 -1.623 1.240 3 23-25 4 18 23 13 58 -1.218 1.198 -1.335 0.599 4 26-29 4 22 19 13 58 -1.124 0.683 -1.131 0.219 5 30 4 18 17 10 49 -1.078 0.625 -0.977 -0.026 6 31-33 1 26 20 7 54 -1.125 -0.866 -0.848 -0.203 7 34-36 7 13 8 5 33 -0.493 0.117 -0.749 -0.323 8 38 18 50 52 28 148 -0.951 0.760 -0.566 -0.506 9 39 8 20 14 1 43 -0,418 -1.448 -0.387 -0.635 10 40 12 13 14 .5 44 -0.363 0.602 -0.311 -0.676 11 41 8 21. 12 5 46 -0.532 -0.719 -0.234 -0.708 12 42 13 15 10 3 41 -0.067. -0.207 -0.160 -0.730 13 43-44 8 14 14 2 38 -0.471 -0.475 -0.094 -0.744 14 45 19 25 14 4 62 -0.053 -0.563 -0.012 -0.751 15 46-48 16 29 7 3 55 0.061 -1.580 • 0.085 -0.746 16 49 22 50 15 4 '9 1 -0.097 -1.869 0.207 -0.721 17 50 18 22 12 .1 53 0.124 -0.985 0.331 -0.672 18 51-52 20 21 9 1 51 0.326 -0.911 0.423 -0.620 19 53 19 19 5 0 43 0.576 -1.250 0.510 -0.561 20 54-55 28 16 3 1 48 0.974 -0.161 0.597 -0.488 21 56 18 14 2 2 36 0.699 -0.369 0.682 -0.408 22 57-58 41 22 4 0 67 1.094 -0.265 0.7 94 -0.284 TABLE IX : O r i g i n a l Data i s Divided Into Subgroups, and Scores are Assigned u> MATH 12 GRADES CATEGORY CATEGORY A B C+ C TOTA] NUMBER NAME 23 60 41 22 4 2 69 24 61-62 45 10 1 0 56 25 63-64 37 15 0 0 52 26 65-67 36 7 0 0 43 27 68-70 32 4 1 0 37 28 71-75 33 3 1 0 37 TOTAL 517 546 324 137 1524 CCl SCORES 1.290 -0.244 -1.046 -1.420 CC2 SCORES -0.522 1.205 -0.352 -2.001 OP1 SCORES 1.169 -0.052 -0.951 -1.954-OP2 SCORES 0.724 -1.003 -0.389 2.186 TABLE IX concluded CCl CC2 . OP1 OP2 SCORES SCORES SCORES SCORES 0.993 0 .020 0.959 -0.069 1.648 1 .009 1.137 0.210 1.433 0 .114 1.326 0.559 1.759 1 .154 1.544 1.025 1.795 1 .584 1.810 1.693 1.865 1 .808 2.360 3.413 35 2.5 2.0 28 1.5 27 LU 0 0 vf) j U Z z < p z o o UJ if) 2 3 u> o.s o.o -o.s 8 /o 13 \Z 14 ".•2t 24 23 zo 2Z El 25 -i.o 18 ® 17 •I.S' 15 -20 -\.5 -\.o -0.5 lb oo o.s i.o \. F I R S T C A N O N V C f \ L Figure 3 Plot of the regrouped data with respect to t h e i r newly assigned canonical scores. The h o r i z o n t a l axis i s the score corres-ponding to the f i r s t canonical c o r r e l a t i o n . The v e r t i c a l axis i s the score correspondin to the second canonical c o r r e l a t i o n . H-4. A l t e r n a t e Method Number Two R e c a l l from s e c t i o n I I - 2 that i f one set of scores i s known, then we can c a l c u l a t e the other set of scores using equation (II-3) (or equation (II-7) ). One d i f f i c u l t y w i t h these data i s that the l a r g e number of p o s s i b l e Math 100 grades makes i t d i f f i c u l t to determine a s c o r i n g scheme' that maintains the monotonicity d i s p l a y e d by the grades. So l e t us assume that the a c t u a l number-grades themselves serve adequately as scores. A l l we need to do then i s to c a l c u l a t e the s t a n -dardized Math 100 scores x. , i = 1 to 71 , and then to determine the l Math 12 scores y^ , j = 1 to 4 , using 71 ^ y. I n. . x. / (p n .) , 2 ±t± i J 1 -3 where . 2 / By c a r r y i n g out these- c a l c u l a t i o n s , we o b t a i n the "Standardized Scores" shown i n Table V I I I . The Math 12 "Standardized Scores" are very sim-- 2 i l a r to the "CCl-Scores". p^  i n t h i s case i s 0.31393, as compared to 0.36194 which i s the maximum p o s s i b l e c o r r e l a t i o n between the two v a r i a b l e s , based on the matri x r e p r e s e n t a t i o n of Table V I I I . Let us compare the "Standardized" and "CCl" Math 100 scores. They d i f f e r markedly i n the 41 to 50 range, but are s i m i l a r elsewhere. * The "Standardized Scores" x_ a re, of course, i n the c o r r e c t order t s i n c e they were c a l c u l a t e d as x* = ( x - 44.71785 ) / 14.45490 , . where x f c i s the a c t u a l Math 100 grade out of 75, and x t = 0, 1,...,75, x^ r 0, 1, 7, 37, 59. ( R e c a l l that the x^ are c a l c u l a t e d u s i n g n-1 4 I 3=1 71 i = l n. ,x. 13 i 38 (x - x)/4 and that x and -6 are computed using the 1524 students with t X X f u l l information and "allowable" (C or better) Math 12 grade.) II-5. Alternate Method Number Three This i s the l a s t method of assigning scores that we consider. It i s suggested by Kendall and Stuart [16], page 573. (See also Barnett and Lewis[3].) In th i s case we assume an underlying normal d i s t r i b u t i o n . Consider . v a r i a b l e A which, we r e c a l l , has r categories. Suppose that the underlying normal d i s t r i b u t i o n i s divided into r i n t e r v a l s , with boundary points given by -oo = u < u.. < u„ < . . . < u = +».. o 1 2 r If p_^  = n^'/n i s the proportion observed i n the i t h i n t e r v a l , then we l e t t. be the mean value of the i t h i n t e r v a l , i . e . -z 2 • ze dz V27 t. 1 l - l •u. e~" " dz J . U i - 1 f * -z 2/2 /2TT~ V Since we know the p^ we can then c a l c u l a t e the u^ (see Figure 4) using the Biometrika Tables for S t a t i s t i c i a n s , [19]. Notice that - u 2 J2 1. -u 2/2 e l - l : e 1 /2TT' /2TT the d i f f e r e n c e of the standard normal d i s t r i b u t i o n at u^_^ and at u^ . Then we can compute the . Our.scores are determined by standardizing the t ^ so that they have mean zero and variance one. The process i s repeated f o r v a r i a b l e B. We car r i e d out these computations for the 28 X k matrix of data 40 Figure 4 Normal d i s t r i b u t i o n , divided into r i n t e r v a l s on (-«, +•») .; The propor-t i o n observed i n the i t h i n t e r v a l i s p^ , the i t h i n t e r v a l being given by 42 given i n Table IX , fi n d i n g scores x. , i = 1 to 28 and y , j = 1 to 4. These scores are l a b e l l e d "OPl-Score.s" i n Table IX. The Math 12 "OPl-Scores" are not too d i f f e r e n t from the "CCl Scores". The process of c a l c u l a t i n g the Math 100 "OPl, Scores" ensures that they are i n the correct order. The scores for groups 5 to 22 (corresponding to Math 100 marks of 30 to 58 out of 75) are close together, while the scores are -2 more spread out i n the t a i l s , p i s 0.30750. Thus i n t h i s method we assume that we are sampling from a normal d i s t r i b u t i o n . We assign to each category a score that depends on the proportion observed i n that category. II-6. Conclusions We sought a set of scores to use l a t e r i n the analysis of covariance of our data. Of the sets of scores we considered, the only ones that preserved the order of both the Math 100 and the Math 12 grades were the methods of sections.11—4 and .11-5. The method of section II-2 maximizes the c o r r e l a t i o n between the "2 two v a r i a b l e s . The maximum c o r r e l a t i o n squared obtained i s p = 0.36194. However the corresponding Math 100 scores do not have the desired monotone property. In section II-3 we regroup the data and f i n d scores corresponding to the maximum c o r r e l a t i o n for these revised data. The maximum c o r r e l a t i o n squared i s 0.34930, and the scores obtained do not preserve the natural order of the categories. In section II-5 we force the marginal d i s t r i b u t i o n s to be as nearly normal as possibl e , and assign scores corresponding to the proportion observed i n each category. The correlation-squared of the two va r i a b l e s using the scores obtained -2 ' • i n t h i s way i s p = 0.30750. The scores we s h a l l use are those of section II-4. Since the Math 100 marks 0 through 75 are meaningful (as opposed to simply being l a b e l s ) , i t i s not unreasonable to adopt them as a set of scores. The l o s s i n c o r r e l a t i o n using t h i s method versus using the method of section II—2 i s not too large ( /0.31393 = 0.56029 . versus /0.36194 = 0.60161 ), and the Math 12 grade-scores obtained are not very d i f f e r e n t from those found by maximizing the product-moment c o r r e l a t i o n . One reason for the d i f f i c u l t y i n obtaining scores which preserve the inherent order of the grades i s that we have "too many" Math 100 categories expressed i n terms of j u s t a few Math 12 grades. Even expressing twenty-eight scores as l i n e a r combinations of four.others did not help to preserve the ordering. Perhaps other regroupings of the Math 100 grades should be examined, to determine an optimal (in some sense) regrouping. It does seem, however, that with only four Math 12 grades we should not have too many more than four Math 100 grade-categories . As a f i n a l comment, we compare the Math 12 scores found by each method. Table X shows the approximate Math 100 grade corresponding to each Math 12 score. To form t h i s table, we take the assigned score for the Math 12 grade, look for (approximately) the same score i n the l i s t of Math 100 scores , and then look at the corresponding Math 100 grade out of 75. Notice that i n three cases out of four, B corresponds (approximately) to a Math 100 mark of 44-. out. of 75. I t turns out that the median Math 100 mark obtained by considering a l l students except dropouts i s 44.969. The mean for those students i s 44.534. METHOD A-SCORE APPROX MATH 100 GRADE/75 B-SCORE APPROX MATH 100 GRADE/75 C+-SC0RE APPROX MATH 100 GRADE 7 7 5 C-SCORE APPROX MATH 100 GRADE /75 Method of Canoni-c a l Correlations on 71X4 Matrix 1.284 63.5 -0.229 44 -1.060 33 -1.424 10 As above, on 28 X 4 Matrix 1.290 60 -0.244 44 -1.046 31 -1.420 0 Method of Sec-tion II-4 : Assume Math 100 Marks Correct 1.260 63 -0.181 42 -1.054 29.5 -1.540 22.5 Method of Sec-tion II-5 : Im-posing a Normal Metric-1.169 61.5 -0.052 44 -0.951 30 -1.954 16 TABLE X : Comparison of Math 12 Grade-Scores Obtained by the Different Methods CHAPTER III DATA ANALYSIS I I I - l . Introduction Now that we have selected a method of assigning numerical scores to our data, we can proceed with our analyses i n order to answer the questions of i n t e r e s t . We f i r s t l y want to know i f the students from the forty-nine schools perform equally well i n Math 100 at U.B.C. If not, we w i l l want to examine the e f f e c t s of semesterization type, textbook used and geometry background on students' Math 100 grades. Another question of i n t e r e s t i s whether the students from the honours sections had, on the average, higher grades i n Math 100 than students from the nonhonours sections. In t h i s chapter we use analysis of variance and analysis of covariance techniques.to examine the above questions. 47 III-2. I n i t i a l Analysis Our f i r s t question i s whether the i n d i v i d u a l schools are d i f f e r e n t with respect to the average of t h e i r students' Math 100 grades. Here we have our p schools, with n. students from the i t h school. Let n = / n. be the t o t a l number of students considered. Let v.. be the i = l J Math 100 score of the j t h student from the i t h school. The scores we use are those determined i n section II-4. The p populations of Math 100 grades are assumed to be normally d i s t r i b u t e d with equal variances. The observations are assumed to be independent. Our analysis of variance table i s SOURCE OF VARIATION D.'F. SUMS OF SQUARES MEAN SQUARES Among Treatments ': P-1 S b 2 = J.ni. ^1 ~ 1=1 s b 2 = S b 2 / (p-1) Within Treatments n-p s 2 = I I 1 (y.. - y.) 2 V = S o 2 1 ( n - p ) Total n-1 1=1 j = l where y = \ y../n. , y = \ I y,,/n = 1 n y /ri 1 3=1 3 i = l 3=1 1 J i = l 1 1 To test the hypothesis that the p populations have a common mean, we look at 2 2 F = s^ / S Q r^, F(p-1,n-p) and r e j e c t H i f F i s too b i g . For each of the p=49 "large " schools 48 we can average •the Math 100 scores of the students who went to U^B.C. from that school. In order to test whether these averages are the same for a l l forty-nine schools, we obtain the following analysis of variance table SOURCE OF SUMS OF MEAN VARIATION D.F. SQUARES SQUARES F-RATIO F-PROB Among Schools 48 82.44 1.7175 1.8114 0.0008 Error 986 934.90 0.9482 Total 1034 1017.34 We reject the n u l l hypothesis of equal school means, even at l e v e l of significance a=0.001. Now that we are confident that there are differences i n the school averages, we want to see i f these differences are caused by some underlying variable. This leads us to investigate the effects of semesterization type, Math 12 textbook used, and geometry 10 background on student grades i n Math 100. I I I - 3 . A n a l y s i s of Variance In t h i s s e c t i o n , we w i l l consider one treatment ( e i t h e r semes-t e r i z a t i o n , textbook or geometry) at a time. Each treatment has three l e v e l s : f o r s e m e s t e r i z a t i o n , we consider unsemestered, single-sem-estered and double-semestered schools only ; f o r textbook we have the c l a s s i f i c a t i o n s U.A.A.-only, D o l c i a n i - o n l y , and Combinations/Others ; i n the case of geometry schools e i t h e r o f f e r e d geometry 10, d i d not o f f e r geometry 10, or we do not know. There are b_^  schools nested w i t h i n the i t h treatment l e v e l , w i t h n.. students from the i t h school i n treatment l e v e l i . Each student has a Math 100 score and a Math 12 score recorded. We assume that the treatments and schools are f i x e d e f f e c t s . "We want to see whether there are d i f f e r e n c e s among schools w i t h i n each treatment l e v e l , and whether there are d i f f e r e n c e s among the treatment means. The a n a l y s i s of va r i a n c e model can be w r i t t e n as ([20]) y. = u + a. + B,.. . + e,. < N l (MODEL A) ^ l j k l ( i ) j ( i j ) k where u i s the mean Math 100 score f o r a l l three treatment l e v e l i s the part due to the i t h treatment l e v e l , i = l to p ; 8 /.N . i s the part due to the p a r t i c u l a r school used as the (i)j j t h school i n treatment l e v e l i , j = l to b^ ; e,.. S l i s the c o n t r i b u t i o n from the p a r t i c u l a r student (ij)k chosen, k=l to n.. : and y.., i s the student's Math 100 score.-" • ijk The e ^ j y ^ a r e assumed to be independently normally d i s t r i b u t e d w i t h 2 the same vari a n c e a . The hypotheses we want to t e s t are H ( 2 ) : B , . s , = 0 Vj and Vi , o (i)j 50 and •: a = .. . = a C=0) o 1 p (here, p = 3 treatment l e v e l s ) . The analysis of variance table i s given i n Table XI. There, b. • l n. = T n.. , b. i s the number of schools nested within the i t h treatment l e v e l , n = ) n. , y.,= / y..,/n , i-1 l ! l j k=l l j k « b. n. . b . n. . i i j _ p I I J y i . . = * y i j k I n i . ' a n d y...= J. I I y i j k / n ' 3=1 k=l J i = l j = l k=l J Furthermore, b. n. . f I I ( y i 1 k " y ) 2 = ' f n (y - y ' ) 2 i = l j = l k=l 1 J K i = l 1* 1 • b. b. n. . . + ! l *±*<Ju-h ) 2 + ? ^ ( yiik-yii ) 2 i = l j = l 1 J 1 J X " i = l j = l k=l 1 J t C 1 J gives the between treatments, between.subgroups (schools) within treatments, and r e s i d u a l within subgroups sums of squares. In Table XI i f the F-Ratio s A 2 / (p-1) So2 / ( n - k ) i = l i s s i g n i f i c a n t at l e v e l of s i g n i f i c a n c e a then we r e j e c t the hypothesis of no differences between treatment means. In other words, from o MODEL A we can write the mean Math.100 score, c a l l i t u\- , of the i t h treatment l e v e l as u. = u + a. . Rejection of H^' means that the u. i l o t 1 are not equal. If the second F-Ratio i n Table XI i s s i g n i f i c a n t a t . l e v e l of s i g n i f i c a n c e SOURCE OF VARIATION D.F. SUMS OF SQUARES EXPECTED MEAN SQUARES F-RATIO Between Treatments p-1 1=1 2 , 1 ( 2 a H > n. a. P-1 1=1 1 - 1 s 2 / (p-D S* / (n- I b ) 1=1 Between Schools (Within Treatments) i=l p b i 9 V P n 1 « + i r — I I n i i • • f c b . - D 1 - 1 j = 1 i = i 1 SBW < W1" S 2 / (n- _ b ) 1=1 Between Students Within Schools (Error) n- )b. i = l \ p b i n i j 3=1 2 • a • .• Total n-1 p b i n i 3 9 i = l j=l k=l TABLE XI : Analysis of Variance ' i j k H i ( i j ) k (MODEL A) SOURCE OF VARIATION D.F. SUMS OF SQUARES MEAN SQUARES F-RATIO F-PROB Between Semester Types 2 6.262 3.131 3.3020 0.0363 Between Schools (Within Semester Types) 46 76.177 1.656 1.7465 0.0018 Between Students (Within Schools) 986 934.900 0.948 Total 1034 1017.339 TABLE XII : Analysis of Variance - MODEL A, SEMESTERIZATION EFFECTS 54 (2) a , then we reject H . We conclude, therefore, that the b. schools o ' 1 within the i t h treatment l e v e l have different:average Math 100 scores, for i = 1, 2, 3. Consider f i r s t l y the sit u a t i o n where the semesterization types are the treatment l e v e l s . There are 20 unsemestered schools, 18 single-semestered schools and 11 double —semestered schools. The analysis of variance table i s shown i n Table XII . We accept at l e v e l of significance a=0.01 , and reject at a=0.05, the n u l l hypothesis of no differences between semester types. We reject at cx=0.01 the hypo-thesis of no differences among schools (within semesterization types). Thus the schools within any given semesterization type vary with respect to how the i r students did i n Math 100.- However even taking this into account, students from different types of semesterization backgrounds do not, on the average, perform equally well i n Math 100. When the treatments are textbook types, the analysis of variance table i s as i n Table X I I I . We consider the same 49 schools as we used to test-semesterization effects. We find here that at l e v e l of significance a=0.05, we reject both the n u l l hypothesis of no textbook effects and the n u l l hypothesis of no differences between schools (within textbook types) . Thus the three'(one for each textbook type) mean Math 100 scores (average of the scores for a l l the students within each textbook type) are s i g n i f i c a n t l y d i f f e r e n t . Table XIV gives the analysis of variance table for testing geometry effects. We reject at a=0.05 the n u l l hypothesis that i f we calculate the average Math 100 scores for students from each school, then within each geometry-background type these averages are di f f e r e n t . However with much greater than 95% confidence we conclude SOURCE OF VARIATION D.F. SUMS OF SQUARES MEAN SQUARES F-RATIO F-PR0B Between Textbook 2 15.492 7.746 8.1696 0.0004 Types Between Schools (Within Text) 46 66.947 1.455 1.5349 0,0137 Error 986 934.900 0.948 Total 1034 •1017.339" TABLE XIII : Analysis of Variance Textbook E f f e c t s - MODEL A, SOURCE OF VARIATION D.F.. SUMS OF SQUARES MEAN SQUARES F-RATIO F-PROB Between Treatments - 2 3.302 1.6509 1.7412 0.1735 . Between Schools (within Treatments) 46 79.137 1.7204 . 1.8144 0.0009 • Error 986 934.900 0.9482 Total 1034 1017.339 TABLE XIV : Analysis of Variance - MODEL A, Geometry Effects Cn ON • 57 that there i s no s i g n i f i c a n t geometry e f f e c t , i . e . students from each of the geometry-background categories do not, on the average, perform s i g n i f i c a n t l y d i f f e r e n t l y i n Math 100. 58 III-4. Analysis of Covariance The analyses of variance c a r r i e d out i n the l a s t section i n d i c a t e that based on grades achieved i n Math 100, there are differences among students from schools of d i f f e r e n t semesterization types, there are differences among students from schools which used d i f f e r e n t textbooks to teach Math 12 i n the 1976-77 school year, and there are no s i g n i f i c a n t differences among students with d i f f e r e n t geometry 10 backgrounds. Now there are some schools f o r which few students with low (say, C or C+) Math 12 grades were admitted to Math 100, while .there are other schools which had proportionately higher numbers of such students taking Math 100- at U.B.C. We w i l l use analysis of covariance techniques to eliminate the e f f e c t of students having d i f f e r e n t Math 12 grades, and then we w i l l examine any diff e r e n c e s between the treatment l e v e l s , f o r each of our three treatments. The general model ([12]') f o r our analysis of covariance i s y. = y + a. + . + y. (x... -x ) + e ... (MODEL B) • i j k i d ) j i i j k ... ( i j ) k where u, a., e ... s, and y... are as before for MODEL A ; i (1)3 ( i j ) k ' l j k x... i s the student's Math 12 grade-score ; l j k b.. n.. P i i j x = 1 I I x i i k 7 n 5 a n d •*• i = l j = l k=l 1 J K i s the regression c o e f f i c i e n t f o r the i t h treatment l e v e l , i = 1 to p. The computer package used here was UBC ANOVAR [14] which, l i k e most analysis of covariance packages encountered, assumes a common regression c o e f f i c i e n t , that i s y. = y + a. + B + Y ( x . .. -x ) + e,., N. (MODEL C) . l j k l (1)3 13k ... ( i j ) k Before proceeding, we want to ensure that t h i s assumption of equal regression l i n e slopes i s tenable. In order to describe.how UBC ANOVAR works, l e t the treatments be denoted by A and the schools by B. Then B(A) s i g n i f i e s that B i s nested within A.. Considering one treatment type (semesterization, textbook or geometry) at a time, l e t there be n „ students i n the j t h school under treatment l e v e l i . Let the students from t h i s school have Math 100 scores y . a n d Math 12 scores x . , k = 1 to n.. . We f i r s t calculat xjk 13 k 13 a regression c o e f f i c i e n t yAA for each school, Y i j n. . . yi3 ^ x i j k y i j k n „ 1 x i i k l k = l 1 J t c . ' n.. ^ ^ y i ik k=l 1 J k . n. . ^ x i i k " k=l 1 J 1 C n. . 13 ^ X i i k k=l 1 J R , ( i j ) 3XY 3 ( i 3 ) XX The variance of y.. i s 13 Var = Var I 3 u k=l i j k X i j . ^ y i j k Z n. . 13 _ 2 2 ^ • ( x i j k - X i j . } ° ( i j ) . k=l 'xx which i s estimated by the standard error squared, > where 60 The c o e f f i c i e n t s y.. are averaged over B to obtain 3=1 b. 1 I 3=1 .(ij) XX b. 3=1 ; 1 J  S ( l j ) bXY . ( i j ) XX :l XX 3=1 3=1 X j=l ^ with variance Var ( Y ± ) b. c ( i 3 ) Var b . n. . i _i3 j=iJi ^k^ij^^ijk-yij.) XX 3=1 which i s estimated by 6Z. • x The are averaged over A to obtain Y ' = 1 19. / &<L. x=l I 1 7 < i=l , that is 61 (III-l) 3 B i / b Notice from equation ( I I I - l ) that y ±s the same for each of our three treatment types (semesterization, textbook and geometry) , since we simply sum the appropriate sums of squares over a l l schools. Denote the standard error of y by 42. . The error term used to test the equality '.'of the y's i s the sum of squares about the regression l i n e s for the schools, summed over a l l schools. The numerator of the F - s t a t i s t i c used for t e s t i n g H : y_ = Y'2. "'Yg ( = Y) i s the weighted sums of squares of the c o e f f i c i e n t s y^ about t h e i r weighted mean y . The numerator for te s t i n g H : Y--i = ••• ~ y. i_ ( = Y. ) , i = 1,2,3- i s the "sum over A of the l l i , b . l l weighted sum over B of the squares of the c o e f f i c i e n t s y^ about t h e i r weighted mean over B " [14] . An informal test for slope equality can be ca r r i e d out where we do not i d e n t i f y the i n d i v i d u a l schools. That i s , we compare only the students from each treatment l e v e l . The analysis of covariance model used i s y.. = u + a. + y.Xx..-x ) + e.. (MODEL D) where y i s the o v e r a l l mean Math 100 e f f e c t ; a_£ i s the part due to the i t h l e v e l of the treatment ; i s the regression c o e f f i c i e n t for the i t h treatment l e v e l x.. i s the Math 12 score for the j t h student i n treatment i j l e v e l i , and y.. i s the Math 100 score : i j 3 m i x i = l j = l *1 62 m i s the number of students i n treatment l e v e l i ; and 1 ' 3 •• n = 7 m. i s the t o t a l number of students. i-1 1 A regression c o e f f i c i e n t i s calculated f or each l e v e l , m. J x..y.. .1=1 1 J 1 J m. x. Lj=i_ m. v 1 2 L x- • j = l m. I with standard error . The c o e f f i c i e n t s are then averaged to obtain with standard error 62.. We can then' informally decide "whether the regression c o e f f i c i e n t s are s u f f i c i e n t l y s i m i l a r to warrant proceeding" [10]. Once we are convinced that the assumption of equal slopes i s tenable, we continue with the analysis of covariance using MODEL C and examine, for each treatment type, the hypothesis H (2) there are no s i g n i f i c a n t differences between the average Math 100 scores of students from d i f f e r e n t schools (within each treatment l e v e l ) , and -H^' •* there are no s i g n i f i c a n t differences among the average Math 100 scores of students from the three treatment l e v e l s . The next three sections describe the r e s u l t s of applying the above analyses to our data. Section III-5 i s quite det a i l e d ; sections III-6 and III-7 follow the same sequence of steps and thus are not so d e t a i l e d . The r e s u l t s are discussed i n section III-8. 63 III-5. Semesterization As i n section III-3, we consider the 49 large schools, c l a s s i f i e d as unsemestered (20 schools, 452 students) , single-semestered (18 schools, 304 students ) or double-semestered (11 schools, 279 students). A t o t a l of 1035 students i s considered. We want to perform an^analysis of covariance on these data, using Math 100 score as the dependent v a r i a b l e and Math 12 score as the covariate, to i nvestigate the hypotheses and H^2^ described i n the preceding section. Our f i r s t step i s to convince ourselves that the assumption of equal within-semesterization-type regression l i n e slopes i s indeed tenable. We f i r s t carry out an informal slope t e s t where we do not d i s t i n -guish between schools. We have,the three semesterization l e v e l s , with students within each l e v e l . We perform an analysis of covariance and obtain the regression c o e f f i c i e n t s , weighted average and standard errors given below. There do not seem to be large differences between the re-gression c o e f f i c i e n t s . Treatment Level Regression C o e f f i c i e n t Standard Error Weighted Average 1. Unsemestered 0.558 0.039 0.578 2. Single-Semestered 0.585 0.046 3. Double-Semes tered 0.600 0.049 Suppose that we do group the students by school. Then within eadh semesterization type we can test whether the regression l i n e s for the scores of students from the schools within that semesterization type are p a r a l l e l . Secondly, we can test whether the within-semesterization-l e v e l regression l i n e s are p a r a l l e l . To test the f i r s t conjecture we c a l c u l a t e F - (28.91415D/46 = 1 - Q 2 2 3 > (576.107536)/937 which i s not s i g n i f i c a n t , even at l e v e l of s i g n i f i c a n c e ct=0.1O. The numerator sum of squares i s the sum over the semesterization l e v e l s of the weighted sum over the schools of the c o e f f i c i e n t s for each school about t h e i r weighted mean. To test whether the within-semesterization-level regression l i n e s are p a r a l l e l , we compute F - (Q-342518)/2 = 0 > 2 7 8 5 ^ (576.107536)7937 . which i s not s i g n i f i c a n t , even at level.a=0.10 . The numerator sum of squares here i s the weighted sum of squares of the c o e f f i c i e n t s about t h e i r weighted mean. Thus we f e e l confident that the assumption of equal slopes across treatment l e v e l s i s tenable, and we proceed with the analysis of covariance using MODEL C. The analysis of covariance table i s reproduced i n Table XV. To test the equality of the average Math 100 scores f or students from the schools of each semesterization type, we look at the F-Ratio F=2.34'57 with (46,985) degress of freedom. This i s s i g n i f i c a n t even at l e v e l of s i g n i f i c a n c e a=0.0001. To test we see that o F=14.8356 with (2,985) degrees of freedom, which i s s i g n i f i c a n t even at l e v e l of s i g n i f i c a n c e a=0.0001. Thus we conclude that there are s i g n i f i c a n t d ifferences among the average Math 100 scores of students SOURCE OF VARIATION D.F. SUMS OF SQUARES MEAN SQUARES. F-RATIO F-PROB Between Treatments. 2 18.235 9.1177 14.8356 0.0000 Between Schools (Within Treatments) 46 66.315 1.4416 2,3457 0.0000 Error 985 605.364 0.6146 TABLE XV : Analysis of Covariance - MODEL C, Semesterization E f f e c t s 66 from each of the three semesterization types, even when we take into account the d i f f e r e n t Math 12 grades. The regression c o e f f i c i e n t s and t h e i r standard errors are given below, together with the'equation of the regression l i n e for each treatment. Figure 5 i s a p l o t of the average Math 12 scores versus the average Math 100 scores for students from each school, with the schools i d e n t i f i e d by semester type. The s o l i d l i n e s are the regression l i n e s with slopes > i = 1J2,3 as i n Table XVI ..The weighted average of the regression c o e f f i c i e n t s i s y = 0.591. ' These r e s u l t s are discussed i n section III-8. SEMESTERIZATION TYPE REGRESSION COEFFICIENT y STANDARD ERROR se. REGRESSION LINE EQUATION y = y± (x - x ±) +'y± 1. Unsemestered 0.585 0.039 y = 0.585 x + 0.054 2. Single Semestered 0.571 0.046 y = 0.571 x -.0.086 3. Double Semestered 0.620 0.049 y = 0.620 x + 0.270 TABLE XVI : Semesterization ANOCOVA Results Figure 5 Plot of the average Math 12 scores versus the average Math 100 scores f or a l l 49 schools. Unsemestered schools are denoted by c i r c l e s , and t h e i r regression l i n e i s l a b e l l e d 'U'. Single semestered schools are indicated by t r i a n g l e s ; t h e i r regression l i n e has the l a b e l 'SS'. Double semestered schools are shown as squares; t h e i r regression l i n e has 'DS' as a l a b e l . The X's denote the treatment means. 70 III-6. Textbook In t h i s section, we present the r e s u l t s of the analysis of covariance when the treatments are the d i f f e r e n t textbook categories. The f i r s t category i s 'U.A.A.-only',made up of 3 schools and 82 students. The second category i s ' D o l c i a n i - o n l y 1 , comprised of 40 schools or 801 students. Level three i s 'Combinations/Others' , which incorporates 6 schools (152 students). If we ignore school d i s t i n c t i o n and perform an analysis of covariance on the three groups of students, we f i n d regression coef-f i c i e n t s 0.610, 0.540, and 0.657(with standard errors 0.088, 0.029, and 0.069 respectively) which seem f a i r l y close. We. proceed to test for equality of slopes when schools are i d e n t i -f i e d . The y „ are computed, then averaged to obtain the y^. F - s t a t i s t i c s are computed to compare the c o e f f i c i e n t s , amd are found to be norisignifleant even at l e v e l of s i g n i f i c a n c e a=0.10. Thus we can assume that the slopes i n the d i f f e r e n t treatment l e v e l s are equal. The analysis of covariance table appears i n Table XVII. Figure 6 shows the average scores f o r each school. The schools are i d e n t i f i e d by textbook type. The l i n e s are the regression l i n e s for each treatment l e v e l , with equations given irt Table XVIII. SOURCE OF VARIATION D.F. SUMS OF SQUARES r MEAN SQUARES F-RATIO F-PROB Between Treatments 2 9.4343 4.7171 7.6753 0.0000 Between Schools (Within Treatments) 46 74.5863 . 1.6214 2.6383 0.0000 Between' Students Within • Schools (Error) 985 605.3641 0.6146 TABLE XVII : Analysis of Covariance - MODEL C, Textbook E f f e c t s TEXTBOOK TYPE REGRESSION . COEFFICIENT y • STANDARD ERROR se. l REGRESSION LINE EQUATION y = Y i(x-x ±) + y± 1. U.A.A. Only , 0.612 0.088 y = 0.612 x - 0.251 2. Dolci a n i Only 0.583 0.029 y = 0.583 x + 0.103 3. Combo / Others 0.623 0.069 y =0.623 x - 0.072 TABLE XVIII : Textbook ANOCOVA Results 73 Figure 6 Average Math 12 scores versus average Math 100 scores for schools i d e n t i f i e d by textbook type. U.A.A.-only schools are denoted by squares; t h e i r regression l i n e i s marked by a 'U'. Dolciani-only schools aire c i r c l e s , with t h e i r regression l i n e l a b e l l e d 'D'. The other schools are t r i a n g l e s , with t h e i r regression l i n e l a b e l l e d '0'. The X's are the average scores for the textbook l e v e l s . 75 III-7. Geometry There were 27 schools that did not o f f e r a grade 10 geometry course i n the 1974-75 school year, 15 schools that did o f f e r a geometry 10 course, and 7 schools that either d i d not know or that had feeder schools so that some students did have geometry 10 and some did not. These categories accounted for 600, 305 and 130 students r e s p e c t i v e l y . We f i r s t l y do not d i f f e r e n t i a t e between schools and perform an analysis of covariance using MODEL D on the mathematics scores of the students within each geometry type. The three regression c o e f f i c i e n t s computed are 0.545, 0.554 and 0.685. We proceed to the f u l l a nalysis of covariance, now i d e n t i f y i n g schools nested within geometry type. The regression c o e f f i c i e n t s given i n Table XX are tested and found to be equal at l e v e l of s i g n i f i c a n c e a=0.20 . Then using MODEL C we perform an analysis of covariance assuming equal slopes. . The r e s u l t i n g table i s shown i n Table XIX. Figure 7 shows the regression l i n e s for each of the three treatment l e v e l s . The regression c o e f f i c i e n t s and l i n e equations aire shown i n Table XX. The weighted average of the regression c o e f f i c i e n t s i s y = 0.591. SOURCE OF VARIATION D.F. SUMS OF SQUARES MEAN SQUARES F-RATIO F-PROB Between Treatments 2 1.8448 0.9224 1.5009 0.2217 Between Schools (Within Treatments) 46 82.3319 1.7898 2.9123 0.0000 Between Students Within Schools (Error) 985 605.3641 i 0.6146 TABLE XIX : Analysis of Covariance - MODEL C, Geometry E f f e c t s GEOMETRY TYPE REGRESSION COEFFICIENT y STANDARD ERROR se. l REGRESSION LINE_EQUATION y = Y i(x-x j.) + y± 1. No Geometry 10 Offered . 0.565. 0.034 y = 0.565 x + 0.088 2. Geometry 10 Was Offered 0.593 0.045 y = 0.593 x + 0.080 3. Do No t Know 0.711 0.074 y = 0.711 x - 0.032 TABLE XX : Geometry ANOCOVA Results 79 Figure 7 . Regression l i n e s and average scores under geometry treatments. "No Geometry" schools are c i r c l e s , with regression l i n e l a b e l l e d 'N'. "Yes Geometry" schools are squares with regression l i n e l a b e l l e d 'Y'. Other schools are t r i a n g l e s , with regression l i n e l a b e l l e d '0'. X's are treatment l e v e l means. 80 III-8. Discussion The students used i n the analyses of the preceding three sections were the 1035 students with f u l l data sets from t h e 4 9 large schools. R e c a l l that when we computed the Math 12 scores, we transformed them to have mean zero. S i m i l a r l y the o r i g i n a l Math 100 grades out of 75 were standardized to have mean zero. In order to compute those scores we used a l l 1524 students who had both a Math 12 grade and a (nonzero) Math 100 grade recorded. Now the 1035 students used i n the analyses of the preceding three sections had an average Math 100 score of 0.0592, corresponding to the grade 45.57 out of 75. Thus the average Math 100 grade for these students i s higher than the average grade f or the 1524 students. On the other hand, t h e i r average Math 12 grade-score was -0.0176, which i s les s than the average (for a l l 1524 students) of 0.0 . Consider the r e s u l t s of section III-5 on semesterization. In the analysis of covariance on the scores of students from schools nested within semesterization types, we f i n d that even at l e v e l of s i g n i f i c a n c e (2) a=0.0001 we r e j e c t H q . This means that within a given semesterization type, the schools are d i f f e r e n t with respect to the average Math 100 scores of t h e i r students, even a f t e r taking into account the students' Math 12 grade-scores. We also r e j e c t at l e v e l of s i g n i f i c a n c e a=0.0001. Thus the average Math 100 scores of students from the three semes-t e r i z a t i o n types are s i g n i f i c a n t l y d i f f e r e n t . These two s i t u a t i o n s are i l l u s t r a t e d i n Figure 5. Consider f o r example the students from unsemestered schools. The average Math 100 and'.Math 12 scores are plotted i n Figure, 5, with the averages for un-semestered schools i d e n t i f i e d by c i r c l e s . Notice how spread out these c i r c l e s are with respect to both Math 100 and Math 12 average scores. 81 S i m i l a r l y the single-semestered and double-semestered schools vary widely with respect to the averages of t h e i r students' scores. Fur-thermore, when we compare the regression l i n e s of the three semester-i z a t i o n types we see that for a given average Math 12. score, the average Math 100 score of students from double-semestered schools seems to be higher than the average Math 100 score of students from single-semes-tered schools. We turn now to the textbook e f f e c t s , keeping i n mind the large differences i n numbers of students i n the d i f f e r e n t textbook categories. (2) At l e v e l of s i g n i f i c a n c e a=0.0001 we r e j e c t H q . This we i n t e r p r e t as follows : consider textbook type i . Compute the average Math 100 score and the average Math 12 score of the students f o r each school of te x t -book type i . Then even i f two schools have the same average Math 12 score, they may have s i g n i f i c a n t l y d i f f e r e n t Math 100 averages. We also r e j e c t H^"^ at l e v e l of s i g n i f i c a n c e a=0.0001 for the tex t -book e f f e c t s . This means that average Math 100 scores of students from the three textbook categories are s i g n i f i c a n t l y d i f f e r e n t . If we look at Figure 6, we see that for the same average Math 12 score, schools that used only the "Using Advance Algebra" text to teach Math 12 i n 1976-77 seem to have a lower average Math 100 score than schools that used D o l c i a n i only, or combinations of texts. We should remember that of the 1035 students that we have considered here, only 82 came from schools that used only U.A.A. Another question we.could have considered would have been to compare schools that used U.A.A. to schools that did not (thus we would have one category made up of Text Combinations l . , 2 . , 3 . , 5 . and 6. from Table V, and another category comprising the other Combinations). The categories that; we did use compare 82 schools that used only U.A.A. to schools that used only D o l c i a n i , and to schools that used combinations of texts. F i n a l l y we consider the geometry e f f e c t s . R e c a l l from Table XIX (2) that i n the analysis of covariance we rejected at l e v e l of s i g n i f -icance a=0.0001 , thereby concluding that even when we take into account the d i f f e r e n t Math 12 scores, the average Math 100 scores f o r schools w i t h i n geometry-background types are s i g n i f i c a n t l y d i f f e r e n t . From Figure 7 we see the v a r i a b i l i t y with respect to both average Math 12 and average Math 100 scores for the schools within geometry-background category. In t h i s case, we accepted , even at l e v e l of s i g n i f i c a n c e a=0.20. That i s , we could not f i n d any s i g n i f i c a n t d ifferences among the average Math 100 scores of the students from the three geometry-back-ground categories. In Figure 7 the regression l i n e s seem nearly i d e n t i c a l , from which we conclude that on the average, students from the three geometry-background categories perform equally w e l l i n Math 100 when they have the same Math 12 score. , III-9. Interactions. The computer package UBC GENLIN {13] i s used to perform a three-way crossed c l a s s i f i c a t i o n with i n t e r a c t i o n analysis of covariance on our data. The three factors are semesterization, textbook and geometry-background, each at three l e v e l s . The covariate i s Math 12 grade-score, the dependent (yield) v a r i a b l e i s Math LOO score. In th i s a n a l y s i s , schools are not distinguished; rather, students from schools belonging to the same semesterization, textbook and geometry categories are lumped together. Table XXI shows the data organization used. The model we consider includes a l l two-way i n t e r a c t i o n s , as well as the three-way i n t e r a c t i o n . We can test f o r the s i g n i f i c a n c e of each of the i n t e r a c t i o n s , and for the equality of slopes i n c e r t a i n c e l l s . Since t h i s design i s unbalanced with some empty c e l l s , the degress of freedom w i l l have to.be adjusted accordingly. As we s h a l l see, there are so many empty c e l l s that we,cannot test the three-way i n t e r a c t i o n , nor can we test f o r the equality of slopes of the regression l i n e s i n the f i f t e e n nonempty c e l l s defined by the three-way crossed design. The model we applied i s ([2]) y i j k l = U + a i + 6 j + Y k + ( a S ) i j + ( a ^ i k + ( 6 Y ) j k + (aBy) + 6(x.. -x ) + e . . f c l . (MODEL E) i j k i j k l . . . l j k l where u i s the o v e r a l l mean Math 100. score ; i s the e f f e c t of the i t h semesterization l e v e l , i = 1,2,3 ; • 3^  i s the e f f e c t of the j t h textbook l e v e l , j = 1,2,3 ; i s the e f f e c t of the kth geometry l e v e l , k =1,2,3 ; (a3)^^ represents the i n t e r a c t i o n e f f e c t of the i t h semes-t e r i z a t i o n l e v e l with the j t h textbook l e v e l ; 84 (aBY) . i s the three-v?ay i n t e r a c t i o n ; i j K • -x. , i s the Math 12 score of the 1th student from the l j k l ( i , j,k)th c e l l ; 6 i s the regression c o e f f i c i e n t ; e.. i n i s the error term ; and i j k l y..,.. i s the Math 100 score of the 1th student i n the i j k l (i,j,k)th c e l l . Each term i n the model i s tested using an F-test : the denominator sum of. squares i s the re s i d u a l sum of squares. The numerator sum of squares i s computed as the dif f e r e n c e of the sum of squares about two f i t t e d regression models : model 1 excludes the term being tested, while model 2 i s model 1 plus the term being tested. We need to f i n d a method by which to determine which terms to i n -clude i n these regression models. One method i s to include a l l terms contained i n the test term, and.to. exclude a l l terms that contain the test term. The maximal method i s of t h i s s o r t , i n c l u d i n g as many terms as poss-i b l e . For example, to test whether the (ag) 'are equal, we l e t model 1 be yijkl" - " + °i + W (aY)ik + (BY)jk + ^ ijkr*...) + e ijkl , and we l e t model 2 be y i j k l m v + a i + 3 j + Y k + ( a Y ) i k + ( 6 Y ) j k + ^ijkr*,./ • + ( a 6 ) i j + e i j k l The d i f f e r e n c e i n regression sums of squares for these two models i s then a t t r i b u t e d to the term being tested. [13]. We have seen then that the models used to produce the test sums of NUMBER OF SEMESTERIZATION TEXTBOOK USED GEOMETRY 10 ? STUDENTS Unsemestered U.A.A. only Not Offered 69 Offered 0 Other 0 Do l c i a n i only Not Offered 155 Offered 157 Other 54 Other . Not Offered 0 Offered 17 Other 0 . Single Semes- U.A.A. only Not Offered- 13 tered Offered 0 Other 0 Do l c i a n i only Not Offered 155 Offered 28 Other 28 Other Not Offered 10 Offered 22 Other 48 Double U.A.A. only Not Offered 0 Semestered Offered 0 Other 0 Do l c i a n i only Not Offered 143 Offered 81 Other 0 Other Not Offered 55 Offered 0 Other 0 TOTAL : 1035 TABLE XXI : Assignment of Students by School Type for Analysis of Covariance With Interactions 86 squares are the maximal ones, omit ting .only those terms that contain the term to be tested. Some properties of the maximal method are : 1. When a term i s judged s i g n i f i c a n t , there i s nothing else i n the f u l l model that could remove that s i g n i f i c a n c e . 2. If a term i s judged not s i g n i f i c a n t , i t could be because there i s another term i n the model that could explain away those differences that have been found, or i t could be that there are no di f f e r e n c e s . Thus an ANOVA table with no s i g n i f i c a n t e f f e c t s does not ne c e s s a r i l y mean there are no differences among c e l l means; i t could simply be due to the i n a b i l i t y to place the blame on any sing l e e f f e c t . The conservativeness of the method leads to no blame being placed. [13], 3. In te s t i n g an e f f e c t , nothing judged s i g n i f i c a n t i s l e f t out of the model for that test (in f a c t , nothing i s l e f t out). Regression c o e f f i c i e n t s are calculated for each c e l l . Slope tests are provided to aid i n deciding whether the assumption of a common slope. 6 i s tenable. These slope tests appear i n the lower h a l f of the analysis of covariance table i n Table XXII. The l i n e with Source of Va r i a t i o n given as "term"*Ml2 tests the equality of slopes within the c e l l s defined by "term". If we look a t the top of the table, the i n t e r -action between semesterization type and geometry background (Sem*Geom) i s highly s i g n i f i c a n t , and the Text*Geom and Sem*Text i n t e r a c t i o n s are s i g n i f i c a n t at a=0.05. The large number of empty c e l l s i n the design makes i t impossible to measure the three-way i n t e r a c t i o n . A s i g n i f i c a n t semesterization-geometry i n t e r a c t i o n i s interpreted as follows : the e f f e c t of semesterization on Math 100 grade i s not the 87 SOURCE OF SUMS OF MEAN VARIATION D.F. SQUARES SQUARES F-RATIO F-PROB' Sem 2 14.853 7.43 11.75 0.00001 Text 2 14.225 7.11 11.25 0.00001 Geom 2 2.393 1.20 1.89 0.15115 Sem*Text 3 6.452 2.15 3.40 0.01725 Sem*Geom 3 12.670 .4.22 6.68 0.00018 Text*Geom 2 3.956 1.98 3.13 0.04417 ML 2 1 336.70 336.70 532.71 0.00000 Sem*M12 2 1.46 0.73 1.16 0.31546 Text*M12 2 1.37 0.69 1.09 0.33815 Geom*Ml2 2 2.50 1.25 • 1.98 0.13867 ' Sem*Text*Ml2 • 3 1.79 0.60 0.94 0.41931 •Sem*Geom*MI2 3 2.30 0.77 1.21 0.30398 Text*Geom*M12 2 2.49 1.24 1.97 0.14053 Residual 1005 635.23 0.63 TOTAL 1034 1017.30 TABLE XXII : Analysis of Covariance -MODEL E, Interactions same for a l l l e v e l s of geometry-background; s i m i l a r l y , the e f f e c t of geometry-background on Math 100 grade i s not the same for a l l l e v e l s of semesterization, even a f t e r taking into account the d i f f e r e n t Math 12 grades. In the same way, we can in t e r p r e t the textbook-geometry and semesterization -geometry i n t e r a c t i o n s . For the Sem*Geom design, Table XXIII shows the observed mean Math 100 score, the corresponding Math 100 grade out of 75, the ob-served mean Math 12 score, and the number of students i n each c e l l . In Table XXIV the same information i s presented for the other two-way crossed designs. The i n t e r p r e t a t i o n of these r e s u l t s i s made d i f f i c u l t f i r s t l y because i n t e r a c t i o n i s present, and secondly because there are some empty c e l l s . Geometry background i n t e r a c t s strongly with semesteriza^-t i o n (which i s i t s e l f highly s i g n i f i c a n t ) and yet the geometry main e f f e c t i s s t i l l judged i n s i g n i f i c a n t . We f i n d that the main e f f e c t s semesterization and textbook are s t i l l both highly s i g n i f i c a n t . 89 GEOMETRY 1 2 3 0.035 0.045 0.104 45.2 •' • 45.4 46.2 1 -0.155 0.151 0.052 224 174 54 0.044 0.008 -0.227 45.4- 44.8 41.4 2 0.224 0.072 -0.176 178 50 76 0.158 0.220 47.0 47.9 3 -0.077 -0.337 0 198 81 TABLE XXIII : Two-Way Data from Interaction Analysis of Covariance Design - sem*geom. layout TEXT SEMESTERIZATION 1 2 3 -0.327 40.0 -0,179 69 0.095 46.1 0.009 366 0.532 52.4 0.183 17 GEOMETRY 1 2 -0.268 40.8 0.259 13 0.040 45.3 0.205 211 -0.172 42.2 -0.207 80 0.221 47.9 -0.135 224 -0.006 44.6 -0.221 55 -0.318 40.1 -0.110 0 0 82 0.153 -0.090 -0.012 46.9 43.4 44.5 0.017 0.008 0.079 453 266 82 0.056 0.051 -0.223 45.5 • 45.5 41.5 -0.139 -0.011 -0.356 65 39 48 TABLE XXIV : Two-Way Interaction Data for Text by Sem and Text by Geom Designs. In .each c e l l we have (from top to bottom) the observed average Math 100 score f o r the c e l l , the corresponding Math 100 grade out of 75, the observed average Math 12 score, and the number of students o 111-10. ' Sections In the 1977 f a l l term at U.B>C., there were 39 sections of Hath 100. Of these, three were l a b e l l e d "Honours Sections", and were designed for students who had demonstrated a b i l i t y i n mathematics -through achieving a high mark i n the Scholarship Exam or i n the Basic Math Test. These.students had the same curriculum as students i n the other sections, but worked on harder problems. Students i n the honours sections wrote the same f i n a l exam as students from the non-honours sections. 1st Class 2nd Class Pass F a i l T o t a l Honours 39 10 1 0 50 Nonhonours 254 379 476 364 1473 To t a l 1523 ' TABLE XXV : D i s t r i b u t i o n of Math 100 Grades for Honours and Nonhonours Students Table XXV shows how the students from nonhonours sections did i n Math 100 as compared withstudents from nonhonours sections, f o r the students with f u l l information. There are only 1523 students there because the records for one student had him i n a nonexistent section. 2 X i s 119.94 with 3 degrees of freedom, so there, i s a highly s i g n i f i -cant a s s o c i a t i o n between type of section and Math 100 grade attained. In f a c t , the data of Table XXV suggest that the honours students did much better i n Math 100 than the students i n nonhonours sections d i d . If instead of looking at Math 100 l e t t e r grades we look at the actual marks out of 75, we f i n d that the 1473 nonhonours students have an average score of -0.0463 (44.05 out of 75) with standard deviation - ' 92 0.9804, and the students from honours sections have an average score of 3.4260 (64.24 out of 75) with a standard deviation of 0.5343. Let the pooled estimate of the variance be 4 2 = (49)(0.5343) 2.+ (1472)(0.9804) 2 /(1521) 2 that i s 4 = 0.9303. Let y be the mean Math 100 score for the students H from honours sections; l e t u be the mean score for nonhonours stu-dents. Then a 95% confidence i n t e r v a l for the i r difference i s [3.4260 - (-0.0462) ] - (1.960) /(0.9303)(1/1473 + 1/50) that i s , 3.2003 < u - t i X T H <• 3.7441 . H NH Thus we see that students from honours sections had a significantly-higher average Math 100 grade that students from nonhonours sections. 93 CHAPTER IV CORRESPONDENCE ANALYSIS IV-1. Introduction In t h i s chapter, l e t N = ( n ^ } be an r x c contingency table, where n „ i s the number of i n d i v i d u a l s i n category i of v a r i a b l e A who are also i n category j of v a r i a b l e B, i = 1 to r , j = 1 to c. As before, c r r c The i j u t r c l e t n = _ n , n = _ n and n = _ I n 1 ' j = l 1 3 ° i = l 1 3 i = l j = l 3 correspondence analysis of J.P. Benzecri i s discussed, and an example of i t s a p p l i c a t i o n presented. Much of the material i n t h i s chapter follows the book by Lebart, Morineau and Tabart [18], and the paper by H i l l [15]. 94 IV-2. Geometric Representation c Suppose f i r s t l y that we are i n R , where the table N i s represented by r points. If the coordinates of these points are given by the raw data, i . e . by the n_^ _. , then there w i l l be some (highly populated) categories of v a r i a b l e A that w i l l have large-valued coordinates. S i m i l a r l y , there w i l l be some categories that have few observations, so that the corresponding points w i l l have a l l coordinates near zero. Now i t i s not an aim of our analysis to explain the d i f f e r i n g abundancies of the various categories. Thus rather than looking at the raw data, we consider the " p r o f i l e s " of v a r i a b l e A,namely the pro-portions of each category of v a r i a b l e B i n each category of v a r i a b l e A. This i s l i k e the co n d i t i o n a l p r o b a b i l i t y of belonging to category j of v a r i a b l e B, given membership i n category i of v a r i a b l e A. S i m i l a r l y , i n R we would consider the p r o f i l e s of v a r i a b l e B, which are the estimates of the p r o b a b i l i t y of being i n the i t h category of v a r i a b l e A, given membership i n the j t h category of v a r i a b l e B, i = l to r , j = l to c. Let the j t h component of the i t h vector (point) i n R° be n _ j / n _ > i = l to r points. In R , the i t h component of the j t h vector would be c n../n , . Notice that the table of new coordinates i n R i s not the r transpose of the new table i n R . We define the r e l a t i v e frequencies by f . . = n../n , with c r f. = J f.'. and f = y f . . . Then the c o n d i t i o n a l j = l J J i = l — 1 frequencies are given by f . . / f . (=n /n ) and f , . / f . (=n /n ) i j l • ' i j 1 • • i j .J i j .j for a l l i and j . In the geometric representation, we examine the data to see i f we can 95 f i n d c l u s t e r s of " s i m i l a r " or "close" points, or trends i n the data. We then see i f the existence of these trends or groups can be ex-plained by some underlying v a r i a b l e , be i t continuous(such as i n the case of some environmental or s o c i o l o g i c a l v a r i a b l e s ) , or d i s c r e t e ( i n which case we would have d i s t i n c t groups). We have chosen the p r o f i l e s (or c o n d i t i o n a l frequencies) to represent our data. We now need to define what we mean by the distance, between points. Let the distance between points (categories) i and i ' of v a r i a b l e A be d Z ( i , i ' ) ^ l~— 3=1. O c 1 ff.. f . , . \ 2 _ i l _ - i l l f. f., 1. 1 S i m i l a r l y ^ the distance between categories j and j ' of v a r i a b l e B i s . V i f f . . f . . , We choose t h i s distance because i t s a t i s f i e s the "equivalent d i s t r i -butions" property that i f we combine two categories of v a r i a b l e A (respectively, B) that have the same variable-B (respectively, -A) p r o f i l e s , then the distance between variable-B (respectively, -A) points i s unchanged. In other words, we do not lose much information by combining c e r t a i n c l a s s e s , nor do we gain much information by subdividing homogeneous clas s e s . c F i n a l l y we define the mass of the i t h point i n R to be c f-t (= 1 n ./n) . S i m i l a r l y we define the mass of the j t h point i n 1 ' j = l 1 3 r R to be f . (= j n../n ) . •3 • -. i'J J 1=1 J . Let F = {f..} be the r x c matrix of r e l a t i v e frequencies, l e t i j = diag {•£ } be the r x r diagonal matrix with i t h diagonal element f. , and l e t D = diag {f .} be the corresponding c x c matrix. The 1. c ,j c -1 coordinates of the r points i n R are given by the rows of D_ F. Let u be a vector such that U'D C = 1 • The r-vector v- of projections of the r points on the axis u can be written as v=D "ST) \ I . R e c a l l r c that we are looking at distances between p r o f i l e s , so as to gr a p h i c a l l y represent differences between p r o f i l e s . That i s , we want to maximize the weighted sums of squares of the proje c t i o n s , namely v'D^v , that i s U'D_ ^F'D_ > subject to U ' D C "^u ='1 . Introducing Lagrange m u l t i p l i e r s , the Lagrangian L i s L = u'D - 1F'D - 1FD _ 1 u - X(u*D - 1 u - 1 ) . c r c c Setting dL /3u equal to zero, we obtain D _ 1F'D - 1FD - 1 u - AD - 1 u = 0 . c r c c Premultiplying t h i s l a s t equation by D^ we see that u w i l l be an eigenvector of S , where S=F'D^ "^FD ""V , corresponding to the. largest eigenvalue A. The vector u i s c a l l e d the f i r s t f a c t o r i a l a x i s . The vector y=Dc \ i . , an eigenvector of S 1, i s c a l l e d the f i r s t f a c t o r . The projections on u are the components of the vector D_ ^FD_ "^ u , i . e . of the vector D "^Fy . Note that S={s,, ) where r j k r f f r i.1 i k s., _ J . J' i = l f. f ' I . ,k r For the corresponding analysis i n R , the f a c t o r i a l axis v i s an eigenvector of ;S^=FD TF'D^ "" , corresponding to the largest eigenvalue. Now from above, Su=Au can be written as F'D^ "^FD^  ^ u=Au . Premul t i p l y i n g t h i s expression by FI>c ^ , we obtain FD ^F'D _ 1 ( F D _ 1 u ) = A (FD - 1 u ) . c r c c Thus S and. S^have the same n o n t r i v i a l eigenvalues, and the components of each eigenvector f o r S are proportional to the weighted average.of the components of an eigenvector f o r . In order to s a t i s f y v'D^ ^v=l , we have that v=FDc ^ U / / A . . When projected on v, the c points have coordinates given by the components of "^ F'D^  \ r . x=D^ _ ^v i s the factor corresponding to the eigenvalue A. Notice that x=D _ 1v=D _ 1FD _ 1u//A = D _ 1Fy//A ; s i m i l a r l y y = D _ 1 F ' X / / A . That i r r c r 1 J J c the coordinates on a f a c t o r i a l axis i n one space are l i n e a r combinations of the components of the factor of : the other space, corresponding to the same eigenvalue. If we write x'=(x^, ... , x^) and .y'=(y^»••.,y c) then we obtain - i c f. . 1 V 3-J x- = — L — 1 / A r-i f • J (IV-1) 1 R 1 v i j y = .) — x, . /A i i i f 1 We represent a s o l u t i o n to the correspondence analysis of matrix F = ' t f 1 j } ' b v t h e t r i p l e (f\ , x , y ) IV-3. Alternate Representation : Incidence Data Consider again the r x e contingency table N= {n..}, with r c 1-' n= .T, y n.. .We can form the (r+c) x n incidence matrix A={a., } 3>1 , -. i j k l 3=1 associated with N as follows J.et a, ., = 1 i f the 1th i n d i v i d u a l has k l a t t r i b u t e k, and l e t a. . = 0 otherwise. Since each i n d i v i d u a l was k l o r i g i n a l l y described i n N by the two a t t r i b u t e s i t possessed (namely, the v a r i a b l e A category and the v a r i a b l e B category ) we see that each column i n A w i l l have exactly two l ' s and (r+c-2) zeroes. Correspondence analysis can be applied to A as a method of multidimensional s c a l i n g i n which we scale the rows and columns of A simultaneously. It turns out that correspondence analysis y i e l d s the same s o l u t i o n when applied to A as when applied to N. Let the scores assigned to the f i r s t r rows of A be x,, ... , x , -and the scores assigned to the 1 r l a s t c rows be y . In the s o l u t i o n , each column of A c o r r e s -ponding to an occurence of the a t t r i b u t e combination ( i , j ) w i l l have the same score, c a l l i t $ . I t follows from equation (IV-1) that i f p' i s an eigenvalue of A , then (IV-2) p'<(,.. = (x. +.y.) I 2 98 (IV-3) p . ^ . - J n *±/n±m 3=1 r (IV-4) p*y. = 1 xi. A. ./n , 3 ^ i j 13 - j Substituting (IV-2) i n (IV-3) we obtain c P ' X i = ^ n i i ( x i + y i ) / ( 2 n i p , ) ' 3=1 ' 2 c ^ that i s , . (2p' - l ) x . = c. _ n..y.In. ,2 (IV-5) S i m i l a r l y , (2p' - l ) y . = _ . n..x./n . Now (IV-5) i s a s o l u t i o n of the correspondence analysis on N. Thus the analysis of the contingency table N can be regarded as equivalent to the multidimensional s c a l i n g described by the analysis on the incidence matrix A . . The following chart i s a sample incidence matrix, for an r x c data matrix on n i n d i v i d u a l s . The scores x^,...,x_,y^,...,y_ are assigned to the a t t r i b u t e s . The score cj)^ _. i s assigned to an i n d i v i d u a l i f he f e l l into category i of v a r i a b l e A and into category j of v a r i a b l e B. COLUMN (INDIVIDUAL) NUMBER SCORES ROW NUMBER 1 2 3 4 k n X l 1 1 0 1 1 0 0 •' X2 2 0 1 .0 0 ' 0 1' • * • • • • • x. 1 i . 0 0 0 0 ... 1 0 • • • * X r r 0 o .; 0 0 0 ... 0 y l r+1 0 0 1 0 0 1 >2 r+2 1 0 0 1 0 ... 0 • • * • . ; - • r+j 0 0 o 0 1 ... 0 • • • V r+c 0 l 0 0 .0 0 ' • SCORES • l 2 * 2 c • l l . * 1 2 ... . • •2.1. 101 IV-4. General Rules for Interpretation Lebart, Morineau and Tabard [18J define two series of numbers, the absolute contributions and the r e l a t i v e c ontributions, corresponding to each f a c t o r i a l a x i s. The absolute c o n t r i b u t i o n of the i t h point i n R to the ath f a c t o r i a l axis i s defined to be CTR ( i ) = f. x } , a 1. a i where x . i s the score assigned to the i t h point with respect to the O i l . r ath a x i s . The above-mentioned authors use t h e i r geometric approach to correspondence analysis to show that the absolute c o n t r i b u t i o n shows the " r o l e played by a v a r i a b l e i n the variance explained by a f a c t o r " . c The r e l a t i v e c o n t r i b u t i o n of the i t h point i n R to the ath axis i s defined by , 0 COR ( i ) = (/A x . ) 2 a a ax f where A^ i s the eigenvalue corresponding to which the ath f a c t o r i a l axis was formed. Thus the r e l a t i v e contributions describe the "portion of the variance of a v a r i a b l e explained by a factor " . Thus, reason, Lebart,Morineau and Tabard, i f CTR^(i) i s low and COR^Cl) i s high, then point i does not enter g r e a t l y into the construc-t i o n of the ath axis ; however the ath axis explains a good deal of the variance of the i t h point. If CTR ( i ) i s high and COR ( i ) i s low, then a a -the i t h point contributes strongly to the ath a x i s . In R we can describe s i m i l a r quantities CTR (j) and COR ( i ) for • - a a the c points i n that space. If we consider correspondence analysis from the point of view of incidence data, then the axes that we f i n d f a l l into three categories 102 NODAL AXES. A nodum i s a set of rows and columns that go together to form a group. A nodal axis i s one with respect to which the data f a l l into separate groups. SERIAL AXES. With respect to s e r i a l axes, the data are spread out along the axes, not clustered into groups. An axis of s e r i a t i o n suggests an underlying trend rather than a c o l l e c t i o n of separate groups. POLYNOMIAL AXES. An axis i s said to be a polynomial axis i f i t i s an approximate polynomial i n one of the other axes. 103 IV-5. A p p l i c a t i o n . R e c a l l from Chapter II our methods of scoring, and the d i f f i -c u l t i e s we encountered i n obtaining a scoring system that preserved the order of the Math 100 grades. These problems are p a r t i a l l y caused by the large number of Math 100 grade categories r e l a t i v e to the number of Math 12 grade categories. We use correspondence anal-y s i s here to see i f there i s some other feature of the data that also contributes to the inconsistencies i n the scoring system. For example, we w i l l see i f there i s some underlying v a r i a b l e describing trends i n or c l u s t e r i n g of the grades. ' The way to use correspondence analysis to answer these questions i s as follows. Consider equations (IV-1) of t h i s chapter, and also equations (II-3) and (II-7) of Chapter I I . These equations are c i d e n t i c a l , with p 2 = X , since f . . / f . i s simply (n../n)/( Jn../n) i j !• IJ j = _ i J which i s equal to n../n. ; s i m i l a r l y f . ./f . = n../n ; . We s h a l l see that the scores obtained with correspondence analysis are, except for a scalar multiple, the same as the canonical scores we obtained e a r l i e r . Thus we can use the d i f f e r e n t point of view of correspondence analysis to see i f any c h a r a c t e r i s t i c s of the data are highlighted. To i l l u s t r a t e , we apply correspondence analysis to the data of Table XXVI.' The computer program used was written by N. Tabet of the Laboratoire de S t a t i s t i q u e Mathematique de Professeur J.-P. Benzecri, of the U n i v e r s i t e de P a r i s . A copy of the program appears i n Appendix I. Table XXVII gives the eigenvalues and eigenvectors found. Tables 104 MATH 100 MATH 12 CLASS CLASS NUMBER • NAME A B C4- C TOTAL 0-1 0 0 ' 0 0 0 1 2-15 3 16 23 11. 53 . 2 16-22 2 21 20 16 59 3 23-25 4 18 23 13 58 4 26-29 4 22 19 13 58 5 30 4 -18 17 10 - 49 6 31-33 1 26 20 7 54 7 34-36 7 13 8 5 33 37 0 0 0 0 0 8 38 18 50 52 28 148 9 39 8 20 14 l< 43 . 10 40 12 13 14 5 44 11 41 8 21 12 5 46 12 42 13 15 10 3 41 13 43-44 8 14 • 14 2 38 14 45 19 25 14 4 62 15 46-48 16 29 7. 3 . 55 16 49 22 50 15 4 91 17 50 . 18 22 12 1 53 18 51-52 20 21 9 1 51 19 53 19 19 5 0 43 20 54-55 28 16 3 \. 1 48 21 - 56 18 14 2 2 36 22 57-58 41 22 4 0 67 59 0 0 0 0 0 23 60 41 22 4 2 69 24 61-62 45 10 1 0 56 25 63-64 37 15 0 0 52 26 65-67 36 7 0 0 43 27 68-70 32 4 1 0 37 28 71-75 33 3 1 0 37 TOTAL 517 ' -546 324 . 137 1524 TABLE XXVI : Data Grouping Used for Correspon-dence Analysis EIGENVALUE : 1.000000 0.349299 0.04357 0.01176 EIGEN-• VECTOR -0.29983 -0.46109 -0.59856 -0.58244 0.42587 0.48248 . 0.14611 -0.75133 -0.60007 -0.16256 0.72179 -0.30416 PERCENT :. 86.325% 10.768% 2.907% TABLE XXVII : Table of Eigenvalues, Eigen-vectors and Percentages of Variance Explained 106 CLASS NUMBER CLASS NAME QLT MASS INR 1//F COR1 CTRL 2//F COR2 CTR2 : . i 2-15 981 35 54 -750 894 56 233 87 44 2 16-22 945 39 68 -783 857 68 250 88 56 3 23-25 1000 38 55 -720 892 56 249 108 55 4 26-29 964 38 45 -665 922 48 142 42 18 5 30 994 32 34 -637 954 37 129 40 13 6 31-33 990 35 42 -665 922 45 -181 68 27 7 34-36 876 22 5 -291 870 5 23 6 0 8 38 1000 97 82 -562 92.6 88 158 74 56 9 39 724 28 15 -247 290 5 -302 434 59 10 40 748 29 6 -215 557 4 125 191 10 11 41 992 30 9 -314 808 . 9 -150 184 16 12 '42 367 27 1 -39 167 0 -43 200 1 13 43-44 489 25 11 -279 .434 6 -99 " 55 6 14 45 879 41 2 -31 57 0 -117 822 13 15 46-48 817 36 12 35 10 0 -330 807 90 16 49 947 60 24 -57 20 1 -390 927 209 17 50 721 35 6 72 81 1 -206 640 34 . 18 51-52 949 33 6 192 481 4 -190 .468' 28 19 53 1000 28 13 339 630 9 -261 370 44 20 54-55 990 31 26 575 987 30 -34 3 1 21 56 849 24 12 412 820 12 -77 29 3 . 22 57-58 999 44 46 646 992 53 -55 7 3 23 60 985 45 39 586 . 985 45 3 0 0 24 61-62 1000 37 90 973 955 100 210 45 37 25 63-64 992 34 61 846 991 70 23 1 . 0 26 65-67 1000 28 80 1039 949 87 240 51..' 38 27 68-70 995' 24 74 1060 907 78 33.0 88 61 28 71-75 993 24 82 1101 889 84 377 104 79 TABLE XXVIII : Table of Information on Math 100 Categories 107 XXVIII and-XXIX present the information provided by the program. . In Tables XXVIII and XXIX, the column C0R1 gives the r e l a t i v e c o n t r i b u t i o n of each point to the f i r s t a x is, while C0R2 gives the r e l a t i v e contributions to the second a x i s . The column headed QLT gives the " q u a l i t y of representation" of each point, where QLT = CORl + C0R2. The column MASS gives the mass f. or f . , times 1000, of the cor-responding point. The column INR l i s t s the variance (or i n e r t i a ) f or each point, with the variances scaled to have sum 1. The coordinates (x 1000) i n the f i r s t canonical v a r i a b l e are given i n the column 1//F ; i . e . these are the numbers x./Zx.. and y.//x~. ; s i m i l a r l y for 2//F. i 1 J 1 F i n a l l y , the absolute c o n t r i b u t i o n . (scaled) of each point to the i n e r t i a of the axis of the f i r s t ( r e s pectively, second) canonical v a r i a b l e i s shown i n the column headed CTR1 (respectively, CTR2). Figure 8 i s a reproduction of the p l o t produced by the correspond-ence analysis program. The h o r i z o n t a l axis represents the coordinate i n the f i r s t canonical v a r i a b l e , while the v e r t i c a l axis gives the scores 2//F. Notice that we have plotted the Math 100 scores and the Math 12 scores with respect to the same axes. As we saw i n Chapter I I , the Math 100 categories do not maintain t h e i r inherent order under t h i s scoring scheme. The c i r c l e d Math 100 class.numbers have QLT < 0.900 . The squared points have QLT =1.000 from Table XXVIII. When QLT =1.0000 t h i s means.that a l l of the variance of that point i s represented by these f i r s t two axes. If QLT < 0.900 then a f a i r l y large part of the variance of the point has not been explained by the f i r s t two axes. R e c a l l from Table XXVII that together the f i r s t two axes explain a l m o 6 t .97% of the t o t a l variance. CLASS NUMBER CLASS NAME QLT MASS INR 1//F COR1 CTR1 2//F. C0R2 CTR2 1 A 1000 339 497 761 980 565 108 20 93 2 B 963 358 77 -144 238 21 -252 725 521 3 C+ 930 213 219 -618 917 233 73 13 26 4 C 948 90 206 -839 760 181 417 188 360 TABLE XXIX : Information on Math 12 109 H i l l [15] has shown that the. roughly horseshoe shape of the graph i s to be expected, i . e . that the second f a c t o r i a l axis i s often a quadratic axis i n the f i r s t . There do riot appear to be any separate groups, except perhaps for the points {24,26,27,28} and. {1,2,3,4,5,8} , and possibly {20,21,22,23}. Consider f i r s t the Math 12 scores. With respect to the f i r s t axis <j>^  ,. C and C+ are quite close, B i s close to them, and A i s away o f f by i t s e l f . With respect to the second axis ^ , C+ and A are very close, while C and B are f a r from C+ & A, and from each other. When we look at <f>^  and <j> together, the parabolic shape of the graph i s emphasized, with C and C+ i n one t a i l , A i n the other and B hear the vertex. With respect to <f>^  we can not r e a d i l y d i s t i n g u i s h between C and C+ students. As w e l l , t h i s scoring system does not get much information from the B students (in that the score assigned to B students i s near zero). The near-zero score assigned to B can be p a r t i a l l y explained by looking at Table XXVI. Notice that i n the Math 100 categories num-bered 1 through 25 , there are always between 13 and 50 , and usually between 13 and 30, students with a B i n Math 12. It seems that the number of people receiving B grades does not help us very much to d i f f e r e n t i a t e between Math 100 categories. In other words, we f e e l that the Math 12 B grades are ambiguous i n that there are as many B students who get high Math 100 grades as those who get low Math 100 grades. Also we f i n d that C and C+ students are not e a s i l y d i s t i n g u i s h a b l e with respect to t h i s scoring scheme. Perhaps some sort of test should be administered to C+ and B students before they enter Math 100 i n order to more accurately assess t h e i r mathematical a b i l i t i e s . Points 20 to 23 as a group seem to be " i n the right place" - i . e . 110 CM or S r Cvi ra In rf> o IT) f o--9 I l l Figure 8 Plot of the scores f o r the categories of Math 100 grades, and of the Math 12 scores. The h o r i z o n t a l axis i s the score cb^  assigned with respect to the f i r s t canonical v a r i a b l e . The v e r t i c a l axis i s the score $2 corresponding to the second canonical v a r i a b l e . between the scores for group 19 and group 24 - however as i n d i v i d u a l s t h e i r order i s quite wild. A l l four points have QLT >0.848.. However i n each case, both the r e l a t i v e contribution and the absolute contribu-t i o n to the second axis are very small : never more than 3% and usually l e s s than 1%. The point 25 also has low contributions to the second a x i s . Point 25 seems to be out of sequence, e s p e c i a l l y i n com-parison to points 24 and 26, both of which have perfect q u a l i t y (QLT = 1.000) . If we r e f e r back to Table XXVI , "we see that the four points 20 to 23 represent Math 100 marks of 54 to 60, included i n which i s the mark 59 , which i s the borderline 1st Class - 2nd Class mark. The whole area described by points 9 through 17 i s also confused almost none of these points have high q u a l i t y (QLT <0.900 here; QLT < 0.500 sometimes ) . The mark 48.- borderline between Pass and 2nd Class - i s included i n c l a s s number 15. We see that a l l these points (except point 13) have r e l a t i v e l y high C0R2 values, and cor-respondingly lower C0R1 values. Thus the second ( v e r t i c a l ) axis explains a f a i r amount of the variance of these points; the h o r i z o n t a l axis explains l i t t l e of the variance of these points. Consider points 1,2 and 3. They are very s i m i l a r with respect to 1#F, C0R1, CTR1, 2//F, C0R2, CTR2 and even i n the o r i g i n a l count data of Table XXVI . Point 3 has perfect q u a l i t y . Point 7 i s out of order, at l e a s t with respect to cb^ . A c t u a l l y , point 7 has r e l a t i v e l y low q u a l i t y , with very small absolute contribu-tions to both axes. Point 6 seems out of place with respect to cf^ • Notice that these points are near the Math 100 Fail-Pass borderline mark of 37 out of 75. We combined the o r i g i n a l seventy-one Math 100 grade categories so that each Math 100 category would have enough students that the t h e o r e t i c a l c e l l frequencies would not be too small i f we wanted to 2 use x • "• One consequence of t h i s regrouping i s that the numbers of students i n the new grade categories are l e s s e r r a t i c than were the numbers i n the o r i g i n a l 71 Math 100 categories. One thing that seems to be showing up i n t h i s correspondence analysis i s i r r e g u l a r i t i e s near the Math 100 grade (1st Class, 2nd Class, Pass and F a i l ) borderlines. Points 14 to 18 contributed strongly to the second axis, but not too much to the f i r s t a x i s , leading to t h e i r order with respect to cj>^  being i n c o r r e c t . Thus i t appears that i t i s the extreme points - i . e . the high Math 100 grades and the low Math 100 grades - that are most important i n constructing the f i r s t a x is, while the middling grades -represented by points 7 to 23 - have low absolute contributions to t h i s axis. Notice that what distinguishes the extreme classes from the others i s larger numbers of students who received either A or C/C+ i n Math 12. IV-6. Conclusions One phenomenon which may be causing nonmonotone Math 100 scores i s that we express the scores f o r 28 Math 100 grade-categories i n terms of only 4 Math 12 scores. The correspondence analysis of t h i s chapter indicates another reason f o r nonmonotonicity, namely the e f f e c t s of having four Math 100 grades ( F a i l , Pass, 2nd Class and 1st Class) . There could also be an e f f e c t from not assigning c e r t a i n marks i n Math 100 - i n p a r t i c u l a r the between-grade marks of 37, 48 and 59. The seemingly uniform d i s t r i b u t i o n of the Math 12 B grades with respect to the Math 100 grades could also be causing problems, since we end up having l i t t l e information from those 546 B students, at l e a s t with respect to t h i s subdivision of the Math 100 marks. A B grade i s , of course, an "average" grade i n comparison to A and C grades which are "extreme" grades. R e c a l l the invariance property of the distance function used i n correspondence a n a l y s i s . If the Math 100 marks, that we combined to form a given cl a s s were the same with regards to Math 12.- p r o f i l e , then our r e s u l t s hold for t h e s i t u a t i o n where we do not combine any grade i . e . where we consider a l l 71 grade-categories. CHAPTER V CONCLUSIONS In t h i s thesis we have considered the records of students who r e g i s -tered at U.B.C. for the f i r s t time i n the 1977 f a l l term, including Math 100 as one of t h e i r courses. As f a r as possible we only considered those students who had taken a grade 12 mathematics course i n a B r i t i s h Columbia high school during the 1976-77 school year. In Chapter I we concluded that students who withdrew from Math 100 were (with respect to t h e i r Math 12 grades) l i k e students who f a i l e d Math 100. Below we compare the proportion of students who received each Math 12 grade for withdrawals and f a i l u r e s . Math 12 Grade •'. A B c+ C F a i l e d Math 100 6.9% 36,9% . 35.7% 20.6% Withdrew from Math 100. 9.3% 37.0% . 30.6% . 23.1% Math 100 teachers often f e e l that most students who withdraw from the course were having trouble with i t . . If students were dropping the course for other reasons,, then one might expect twenty-five percent each of' A, B, C+ and C students to-withdraw. However we see from the above table that t h i s i s not the case. On the other hand, the i n c l u s i o n of students who withdrew leads to problems i n assigning a Math 100 score. The method of assigning to 'A' students who withdrew the average Math 100 grade attained by 'A' students who f a i l e d i s p l a u s i b l e . However the small number of students involved leads to these scores-being based on a small number of students. 116 Furthermore, assignment of scores i n t h i s way would lead to problems i n the s i g n i f i c a n c e l e v e l s of our s t a t i s t i c a l t e s t s , because the assumption of the s t a t i s t i c a l independence of the students' Math 100 grades would be v i o l a t e d . We might well have included i n the study those students who withdrew from Math 100. However to be conservative, and i n order not to lose s i g n i f i c a n c e , we deleted the records of the 116 (representing 7% of the t o t a l ) students who dropped out of Math 100. Barnett and Lewis [3] also deleted withdrawals, which they f e e l "may well i n f l a t e the c o r r e l a t i o n due to the wider spectrum of talent that would r e s u l t from t h e i r i n c l u s i o n " (page 224). It i s i n t e r e s t i n g to see that although we did not include students who withdrew, our conclusions on the e f f e c t s of semesterization and textbook are the same as the conclusions drawn by Drs. Bluman and Hoechsmann [9], who did include students who withdrew. (However i n that study, scores were not assigned to the Math 100 or Math 12 grades. Instead a "Passed Math 100 / F a i l e d Math 100 " c r i t e r i o n was used.) We also deleted the records of students for whom no Math 12 grade was a v a i l a b l e . The average grade i n Math 100 for students whose records were included i n the study was almost 45 out of 75 (a passing grade i s 38 or higher). We have defined a " l a r g e " school to be one: with at least ten students s a t i s f y i n g the c r i t e r i a j u s t described. Representatives of these large schools.were contacted, and the schools were c l a s s i f i e d according to semesterization type, textbook used to teach Math 12, and grade 10 background i n geometry for t h e i r 1977 graduating c l a s s . P r i v a t e schools were discarded because i t has been suggested [4] that the performance of students from p r i v a t e schools i s not representative of that of s t u d e n t s from the en t i r e school population. There were then three semesterization categories : nnsemestered, single-semestered, and double-semestered. There were several textbooks used i n Math 12 classes throughout the various B.C. high schools, and these texts were grouped into three categories : Using Advanced Algebra only, D o l c i a n i only, and Combinations/Others. Some schools offered a geometry course that t h e i r 1977 graduates took i n grade 10 ; some schools.did not o f f e r such a course ; and other schools either do not have a grade 10 or else the representatives contacted at the schools did not. know i f the students had taken such a course. Thus schools were c l a s s i f i e d into one of three categories : 118 did not o f f e r geometry 10, did o f f e r geometry 10, or not sure. We selected from the large schools those student records which contained both a Math 12 grade and a Math 100 mark. Then the students were c l a s s i f i e d according to these two grades, the section of Math 100 i n which they r e g i s t e r e d , and the high school that they had attended, while each school was c l a s s i f i e d according to semesterization typej textbook used, and geometry background. The students were not i d e n t i f i e d by name, student number, age, sex, f a c u l t y at U.B.G., or any other v a r i a b l e s . However they were a l l at U.B.C. for the f i r s t year. The schools were not further i d e n t i f i e d by geographical region, average cla s s s i z e , or many other v a r i a b l e s . We next proceeded to scale the Math 12 l e t t e r grades ("accept-able" grades are A, B, C+ and C). We sought numerical scores for these grades i n order to see how they were related to each other. We simultaneously scaled the Math 100 number grades out of 75. Several methods of s c a l i n g were considered, f i r s t on the actual.number grades, and l a t e r on categories of grades formed by combining adjacent grades. We found that the method of canonical c o r r e l a t i o n s f a i l e d to preserve the natural ordering of the Math 100 grades. This i s probably due i n part to the f a c t that the number of categories we used i s much larger than the number of Math 12 grade categories. A correspondence analysis indicated that t h i s could also be a r e s u l t of the underlying Math 100 letter-grade categories, of which there are four, namely F a i l , Pass, 2nd Class and 1st Class. Not only did we have trouble maintaining the order of the Math 100 grade categories, we also encountered d i f f i c u l t i e s with the Math 12 grade scores. Now A and C are "extreme" grades, while B and C+ are . 1 1 9 "average" grades. Examination of the data for each school indicated that some schools do not l i k e to assign A's, and some schools assign few C's. Thus some B students are r e a l l y i n a sense A students, and some C+ are r e a l l y C students. Perhaps some sort of test should be administered to B and C+ students before they enter Math 100 i n order to better i d e n t i f y them as A, B, C+ or C students. (This v a r i a b i l i t y of the grades i s r e a l l y not that s u r p r i s i n g . Consider any four-grade system with A,B,C+ and C. Then a. good A student i s always an A student since there i s no higher grade. However a good B student may be an A student, and a poor B student may be a C+ student. Hence there i s inherently more v a r i a t i o n i n the middle two grades than i n the two boundary grades.) Ultimately we determined scores for the Math 100 grades by accepting these 71 (the grades 0,1,7,37 and 59 were assigned to no students) grades as being adequate as scores, then standardizing them to have mean zero and variance one. We then determined Math 12 letter-grade scores, which were computed i n order to maximize the product moment c o r r e l a t i o n . With th i s set of scores, we then proceeded to analyses of variance and c o v a r i -ance i n order to examine differences between schools with respect to the average scores obtained by t h e i r students in" Math 100 at U.B.C. We f i r s t l y performed an analysis of variance to test the equality of the average Math 100 scores of students from each of the 49 large schools that we used. Without considering Math 12 grades, we found that these school averages are indeed s i g n i f i c a n t l y d i f f e r e n t . In our next analyses of variance, we concluded that there are s i g n i f i c a n t differences between average performance i n Math 100 of students from schools of each of the three semesterization types. We concluded that there are s i g n i f i c a n t differences i n the average Math 100 scores of students from each of the 120 three textbook categories. F i n a l l y we found that the average Math 100 scores of students from each of the three geometry-background categories are not s i g n i f i c a n t l y d i f f e r e n t . Next we sought to remove any e f f e c t s caused by the d i f f e r e n t Math 12 grades that students had. It i s true that the average Math 12 score from some schools i s much lower than the average score for other schools. That i s , some schools send proportionately more Math 12 'C' students than other schools do. Since 'C' students cannot, i n general, be expected to perform as Well i n Math 100 as students who received an 'A' grade i n Math 12, we want to avoid "penalizing" schools whose students had, on the average, lower Math 12 scores. We c a r r i e d out three analyses of covariance, with students' Math 12 score as covariate. In the f i r s t analysis we concluded that there are s i g n i f i c a n t differences i n the average performance of students from the three semesterization types. We pl o t t e d the average Math 12 versus aver-age Math 100 scores for the students of each school. Schools were iden-t i f i e d by semesterization type, and regression l i n e s drawn (Figure 5).. It appears from that graph that of. students with a Math 12 grade of, for example, 'A', those from single-semestered schools do the worst i n terms of t h e i r Math 100 grades. In the second analysis of covariance, we concluded that there are s i g n i f i c a n t differences i n the average Math 100 scores of students from each of the three "textbook used i n Math 12" categories. However these categories were somewhat a r b i t r a r i l y determined, and there are very few students i n the "U.A.A. only" category, while over 75% of the students considered are i n the "Dolciani only" group, Nevertheless Figure 6 indicates that of students with a Math 12 grade of, for example, 'A', 121 those from schools that used Using Advanced Algebra as the only text i n Math 12 did the worst i n terms of t h e i r Math 100 grades. This r e s u l t i s corroborated by that of G.Bluman and K.Hoechsmann [9] who found that of s i x schools that used p r i m a r i l y U.A.A., one was "average" with respect to Math 100 performance, and f i v e were "below average". In the t h i r d analysis of covariance we considered the e f f e c t s of Geometry 10 background of the students. We concluded that there were no s i g n i f i c a n t differences i n the average Math 100 scores of students from the three "geometry-background" categories. This conclusion i s i l l u s -trated i n Figure 7 where the regression l i n e s are very s i m i l a r . [The old Geometry 10 course emphasized Euclidean (deductive) geometry which one might not expect to be d i r e c t l y relevant to Math 100 performance. The emphasis on geometry problems d i f f e r s s i g n i f i c a n t l y among the various Math 11 and Math 12 textbooks.] We also performed an analysis of covariance to see i f there were s i g n i f -icant i n t e r a c t i o n s between the three "treatments" of semesterization-type, Math 12 textbook used, and Geometry 10 background. For t h i s anal-y s i s we did not d i s t i n g u i s h among schools, u n i t i n g instead students from schools that were the same with respect to the three treatments. We found a l l two-way in t e r a c t i o n s to be s i g n i f i c a n t . For example, we conclude that the e f f e c t on Math 100 grade of geometry background varies with the textbook used i n Math 12, and the e f f e c t on Math 100 grade of the textbook used i s not the same for a l l l e v e l s of geometry background. Interpretation of the r e s u l t s of this analysis i s d i f f i c u l t because there are d i f f e r e n t numbers of students i n the twenty-seven c e l l s formed by the three-way crossed c l a s s i f i c a t i o n . In conclusion we sUggest that further studies be c a r r i e d out, including 122 withdrawals. As well i t would be i n t e r e s t i n g to do an analysis using only the four Math 100 grade categories (1st Class, 2nd Class, Pass and F a i l ) together with the four Math 12 grades. F i n a l l y a study of the se c t i o n / i n s t r u c t o r e f f e c t s might be c a r r i e d out for completeness. [Dr. G. Bluman did a feedback analysis (unpublished study) f o r the 1974-75 graduating class which used adjusted school grades for students i n each section. I t showed that marks did d i f f e r s i g n i f i c a n t l y from section to section. Taking into account section e f f e c t s , he adjusted the-Math 100 grades f o r each student. Using these adjusted Math 100 grades, he showed that the average Math 100 performance of students from large schools was not appreciably af f e c t e d by s e c t i o n a l d i f f e r e n c e s . This i s not s u r p r i s i n g : students from schools with ten or more students spread themselves among the approximately 40 sections, so that section e f f e c t s seemed to cancel out.] Nonetheless a d e t a i l e d study of section e f f e c t s should be undertaken. Other variables that could be examined are section s i z e i n Math 100, follow-up performance i n higher mathematics courses, geographical e f f e c t s , school s i z e , and fa c u l t y registered i n at U.B.C. -123 BIBLIOGRAPHY [1] Anderberg, M.R. (1973), Cluster Analysis for Applications, Academic Press.. [2] Bancroft, T.A. (1968), Topics i n Intermediate S t a t i s t i c a l Methods, Volume 1, Iowa State U n i v e r s i t y Press. [3] Barnett, V.D. and Lewis, T. (1963), "A Study of the Relation Between G.C.E. and Degree Results", Journal of the Royal S t a t i s t i c a l Society, Series A, 126, pages 187-225. [4] Bloom, B.S. and Peters, F.R. (1961), The Use of Academic P r e d i c t i o n Scales for Counselling and Selecting College Entrants, Free Press of Glencoe. [5] Bluman, G.W. (1977), "A school-by-school study c o r r e l a t i n g Math 12 grades and performance i n Math .100 for the 1976 classes of various B.C. high schools", Unpublished Report. [6] (1977), "Study of the.Effect of Semesterization i n Secondary Schools on Performance i n F i r s t Year Calculus Courses at U.B.C. (1976 High School C l a s s ) " , Unpublished Report. [7] (1978), "A school-by-school study c o r r e l a t i n g Math 12 grades and performance i n Math 100 for the 1977 classes of various B.C.' high schools", Unpublished report. [8] ' and Falk, C. (1976), "A school-by-school study c o r r e l a t i n g the Math 12 grades and performance i n Math 100 for the 1974 and 1975 classes of various B.C. high schools", Unpublished Report. ' ' " • ' ' . - ^ 124 [9] Bluman,G. and Hoechsmann, K. (1978), "Performance i n Math 100 at U.B.C. ( f a l l of 1977) for graduates from schools of various types and areas", Unpublished Report. [10] Dunn,O.J. and Clark,V.A. (1974), Applied S t a t i s t i c s :. Analysis  of Variance and Regression, John Wiley -Sons. [11] Falkenhagen, E.R. and Nash, S.W. (1977), "Multivariate C l a s s i f i -cation i n Provenance Research : A Comparison of Two S t a t i s t i c a l Techniques", submitted for p u b l i c a t i o n to Silvae Genetica. [12] Federer, W.T. (1957), "Variance and Covariance Analysis f or Unbal-anced C l a s s i f i c a t i o n s " , Biometrics ,13_, pages 333-362. [13] Greig, M. and B j e r r i n g , J . (1977), "UBC GENLIN - A General Least Squares Analysis of Variance Program", Computing Center, The Un i v e r s i t y of B r i t i s h Columbia. [14] Greig,M. and O s t e r l i n , D. (1977), "UBC ANOVAR : Analysis of Variance and Covariance ", Computing Center, The U n i v e r s i t y of B r i t i s h Columbia. [15] H i l l , M.O. (1974), "Correspondence Analysis : A Neglected M u l t i -v a r i a t e Method", Applied S t a t i s t i c s , 23_, No. 3, pages 340-354. [16] Kendall,M. and Stuart, A. (1967), The Advanced theory of S t a t i s t i c s ,  Volume 2 Inference and Relationship, Second E d i t i o n , Charles G r i f f i n & Company, l i m i t e d . [17] (1976), The Advanced Theory of S t a t i s t i c s , Volume 3 Design and Analysis, and Time-Series., Third E d i t i o n , Hafner Press. [18] Lebart, L. ,Morineau,A. and Tabard,N. (1977), Techniques de l a  Description S t a t i s t i q u e , Methodes et Logiciels- pour 1'Analyse  des Grands Tableaux, Dunod. [19] Peasrson, E.S. and Hartley,H.O. (1976), Biometrika Tables for  S t a t i s t i c i a n s , Volume 2, Cambridge U n i v e r s i t y Press. [20] Scheffe, H. (1959), The Analysis of Variance, John Wiley & Sons,Inc. 126 APPENDIX This appendix contains a copy of the computer program used for the computations of the correspondence analysis of Chapter IV. The author of t h i s program i s N, Tabet, of the Un i v e r s i t e de Paris VI. Professor M. Greenacre of UNISA did the actual computing, and provided the docu-mentation. _. CQRRESPCNQE.NCS.. ANA.'-YS.IS PROGRAM BY N . T* 3 E T LABORATQlRe OE STAT IS T I O LE DE PROF. J . - P . AENZECR1 UHIVdfiSITE Ot PARIS VI _' . PA. HIS S£ i. FRAM.Z _ o MATHS GRADES 12 VERSOS 100. 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E Q . 0 ) GD TO 19 R E A 0 I 5 . 2 1 ( NT < J 1 • J - 1 • N J ) . 2' FORMAT I 3C I 1 > NT EMP = 0 I 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 12 J i l . N J 12 N T E M P = N T E M P * N T « J 1 ' IF 1 N T E M P . E 0 . N J 2 ) GO TO 14 P R I N T 13 0 0 2 6 0 0 2 7 0 0 2 6 13 F O R M A T ! ' NUMBER. GF S U P P L E M E N T A R Y V A R I A B L E S I N C O R R E C T L Y I N D I C A T E D • CN T E M P L A T E C A R O M IE RR= 1" , RETURN 0 0 2 9 0 0 3 0 0 0 3 1 0 0 3 2 14 C O N T I N U E J l =0 J 2=0 03 1 5 J=l•NJ 0 0 3 3 0 0 3 4 0 0 3 5 0 0 3 6 IF ( N T ( J ) . E Q . l ) GO TO 16 J l = J U 1 _ N O M S I J l » = N O M j l J ) GO T • 1 5 • 0 0 3 7 0 0 3 8 0 0 3 9 0 0 4 0 16 J 2 = J 2 +1 NO MSCNJ1rJ2) =NQMJC J 1 15 C O N T I N U E 0 0 4 1 0 0 4 2 0 0 4 3 0 0 4 4 17 NOMJI J ) =NOMS1J» 19 C O N T I N U E IF C I & H . f . E . O ) R t " A O I 5 . 2 > ( € KGR I I . J ) . 1 = 1 . 1 0 » . J » 1 . 8 > IF 1 IFnO.fU.CI B r ' l t 1 1 . 1 1 f FNTI 11 • 1 201 0 0 4 5 0 0 4 6 0 0 4 7 IF < L E C . N E . S I REWIND L EC 00 22 2 1=1 .N I " , IF 1 I F G R . E O . C ) GO TO 4 I RF i n II Ff.FNC = QO 9 . F RR = 7fl 7 1 NtlX . 1 « U l . J = l . NJ1 i - 1 Co . 0 0 4 9 0 0 5 0 C GO TO 4 2 41 C O N T I N U E 0 0 5 1 c USUAL DATA RE-AO R E A D ! L E C . F M T . G N D = 9 0 9 . E R R i 7 0 7 1 NOM. (X 1 J » . J = I . N J ) c F O R T R A N I V l-G'.L.EV.EI 21. _1_£.CT-Ufij- _OAX£ - - * 7 f i 1 26 --R-A6E 0 00 2 -0 0 5 2 . QQ53... 0 0 5 * 0 0 5 5 0 0 5 6 0 0 5 7 0 0 5 a 0 0 5 9 0 0 6 0 _ 0 0 6 1 _ 0 0 6 2 0 0 6 3 0 0 6 * 0 0 6 5 . 0 0 6 6 0 0 6 7 0 0 6 8 0 0 7 0 0 0 7 1 0 0 7 2 .-007 3._ 0 0 7 * 0 0 7 5 0 0 7 6 _J3Q77 0 0 7 8 0 0 7 9 0 0 8 0 _Q03 1 _ 0 0 8 2 0 0 8 3 0 0 8 * -.003 5 4 2 C O N T I N U E I F i M r . F n . n i G-I Tn A< J l =0 J2 =0 00 * 5 ' J = 1 • N J ..IF. I N JL ! JO jL£ i J l = J 1 * I XS I J 1 ) = X I J } GO 13 * 5 4 6 J 2 =J 2• 1  XS I N J 1 • J 2 J = X I J ) 4 5 C O N T I N U E CO 4 7 J = 1 . N J * 7 X«.J l = *SJ.Jl 4 9 CO N'T I NUE PI 1 1 =0 . CO 60 J = l . N J P J I J ) =>PJI J ) +XI J ) 6 0 .OR. I • G T . N i l ) GO TO, 7 9 7 0. JUL I F I I . I • r .N . l l 1 IF I I . L E . N I 1 ) CO NT.I NUE IF ( P I I I . L E . G DP 7 0 J = 1 . N J 1 . : — 00 70 J J = l . J A l J , J J ) = A I J . J J ) * X I J ) » X I J J » / P I I I AK T= AKT * P I I I NO , •«R*T- l l f f l . K f l M . P l t I. I X l . l l ..1=1 . N J I GO TO 2 2 2 7 0 7 P R I N T 7 0 8 . 1 . NO M 7 0 6 F O R M » T I • ERROR I N D I V I D U A L lE-RR5-LEHli±-l __ 22 2 C O N T I N U E GO TO 8 0 9 0 9 N I = 1 - 1 PR INT 9 10 . NI I 4 . 2 X . A * ) 0 0 8 6 0 0 8 7 0 0 6 6 0 0 8 9 __ 0 0 9 0 0 0 9 I 0 0 ) 2 _0-Q.f 3 0 0 9 * 0 0 9 5 0 0 9 6 . 0 0 9 7 0 0 9 8 0 0 9 9 0 100 o.iai__ 0 13 2 0 10 3 0 1 0 * 0 105 9 1 0 _&.Q_. I N D I V I D U A L S IN DATA C E C K " ) FO RM AT I • • . I 4 I E R R = l E R R » l RETURN J.f-_( L££..N£_i-5-l REWIND L E C :  00 2 5 0 J = l . N J l P J J=PJ IJ> IF I P J J . L t . O . ) GO TO 250 P.] 2« ' i .1.1= l . J . - _ P J J J = P J ( J J ) IF ( P J J J . L E . U . I GQ TQ £ 4 5 A l J . J J ) = A I J . J J ) / S Q R T I P J J » P J J J ) A l JJi J).=.AI-J.-JJ1_ . _ C O N T I N U E C O N T [ N U E R E W . I N C - I E C ' n i O ; T I I I I M ; 2 * 5 2 5 0 I F : i l T • OUTPUT *T A Lo 3 0 0 29 9 (-RINT 3 C C . - T . I T R c ---F 3 PM AT( • I • V 2 G A 4 / / ) f » I M 29V F O R M A T ! J l X . ' C A T i M A T R I X * . / . 3 1 X . » = = === = ====•// ) .N — 1 ~ H • 3 3 -42 ^ X -• Z - j O , - - N J .1 ID O — 0\ V ( < — O 1*1 •* ' 11 11 11 — a. H - O J M C C O n -) a, u. t- < Z X — a a o a. u. -> 3 • o — </> -1 • it • T X — S -> -a. a ^ \ i v xd so - q I t- *!-•<•-ac z x z, x z II — a — tr — ocr a a n a U.Q. U.Q.U.D. o N — o C l t»! >or- d » o — r\i <*t <t urd o o r t o - - - - - - 1 f- 10 o> o — in 1*1 — — — OJ IM(\J <M -> X -> * n • T O • — T — - O X • X 7- CM a • z * « **t x o — ft — • \-\- < Z T — a ao a u. c*l O d i '1°" 3 o -io = 2 1 -— Hw a. H _ I • •Io •* +l< • o • t- > s w — >-z I- -C T < -z 1--aq crzH a au. a a a iu— u. u * —o — in rr (*! O a z z z - a * Z H • u a u z oJcr <J a ul r \ j p j f \ | t M r \ | r * t ( * ; f * I o o 1*1 * m o 1*1 f*t /** f*t 0 0 0 1 _QQ£L2 _I_V..-&_L£.SUJ_ Ff tTTHR --D-A-Ti^-*—7-8-1-26- 1 1 / 3 7 / V 4 - P A 6 E 0 0 0 1 -5 U 5 R 0 U T 1 N E • N J l ) F A C T O R * N O M J . P J . V A L . V A R J . P • F I « V E C • F J S » A K . T . N R . p I I U 1 1 , F T 1 I . r'rm I I K I H . U I B I I M I l . P I U • N J F 2 t N J . N F . F A C T R O O 1 F A C T R O O 1 }} . F A C T P n D ? 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 b 0 0 1 0 0 0 1 1 001 2 . 001 3 0 0 1 4 001 5 001 6 1 V E C C N J I .NR> ,FJSCNJF2) C 0 M H O N / 1 M / H I i N J J . M l . N J l I i N 1 2 . H J 2 . N F F CQ MMUN/10/ L E C . J E C . IRWD. I B F . ICR QQ JQ-1 M r . 2 l t i 8 . : : VA LC M ) =S0RT I VAL I M ) » 00 3 C 0 J= l t N J 1 V E C C J . M - I ) = 0 . IF I P.II J 1 .1 F . U . 1 n u i i i V E C C J . M - l ) = V £ C C J . M ) « > V A L C M ) » S U R T « A K T / P J C J > > 3 0 0 C O N T I N U E 30 I C O N T I N U E _DD. 3 J 25._J=-LxHJ . : ^ 3 2 5 V A R J C J ) =0 . IF C N J 2 . t Q . O ) GOTO 7 DO 8 I » l . N J F 2 H F l<=.f H z f l . 001 e 0 0 1 9 0 0 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 6 0 0 2 7 0 0 2 8 - 0 0 2 9 0 0 3 0 0 0 3 1 0 0 3 2 -QQ3 3 0 0 3 4 0 0 3 5 0 0 3 6 0 0 3 7 0 0 38 0 0 39 0 0 4 0 _Q0.*-1_ 0 0 4 2 0 0 4 3 0 0 4 4 0 0 4 5 0 0 * 6 0 0 4 7 0 0 4 8 . . 0 0 4 9 -0 0 5 0 0 0 5 1 0 0 5 2 0 0 5 3 0 0 5 * 0 0 5 5 0 0 5 6 00 3 1 5 I = 1 . N I RE AD f I E C . E N D ' 7 0 5 ) VA R I I I = C . DO 2Q J=_L»-NJ N S . P I I I I.P PP=P I I I I * V A L ( M » l ) F l M = 0 . i r C P P . L E . O . ) G O T O nn -".nf. 1 = 1 . N . i l 3 0 8 3 0 6 3 0 8 20 1 - 3Q3 F 1 M = F I M + V E C C J . M ) » P ( J ) F l M = F I M / P P I F C N J 2 . E O . O ) GOTO 3 0 5 I P X L ' & J L s N - I _ L > _ GOTO 3 0 5  DO 201 J = l . N J 2 M J = ( J - I ) « N F » H F J S C M J > = F J S < H J ) * P « J * N J 1 ) » F I M f ! 1 M ) = F I H N S t V A R I 1 I.P I I I I »F I 3 1 5 wR IT E t I 8 F ) 7 0 5 RE• I NO I EC I F < N J 2 . E 0 . 0 ) GOTO 211 DO 2 0 2 _J = 1 tNJ -2 _ PP =P J ( J * N J 1 ) DO 2 0 2 M=2.NR M J = C J - I ) *N F • M— 1 UF_iP-E .xJi J I U J _UO.XD zm F A C T R O O 3 F A C T R O O * F A C T R 0 0 3 _ E A C X R 0 Q 6 -F A C T R 0 0 7 F A C T R 0 0 8 F A C T R 0 0 9 F A C T PO1n F A C T R O 1 1 F A C T R O 1 2 F A C T R O 1 3 _ £ A C T . R O l - A _ F A C T R O 1 3 F A C T R O 1 6 F A C T R O 1 7 F^rran\ft F A C T R O I 9 F A C T RO 20 F A C T RO 21 A.CTRQ-22_ F A C T R 0 2 3 F A C T R 0 2 4 F A C T RO 2 5 F A C T R O 31 F A C T RO 3 2 F A C T R O 3 3 F i r T o n t A F A C T R 0 3 5 F A C T R O 3 6 F A C T R O 3 7 _FACXB0_38_ F A C T R 0 3 9 F A C T RO 4 0 F A C T R O A 1 r i i - T P n A ? F A C T R 0 4 3 F A C T RO44 F A C T R O 4 5 _ F A C T R 0 4 6_ F A C T R04 7 F A C T R 0 4 8 F A C T R 3 4 9 F ^ T o n s n 20 2, 21 t .50 I ' F J S I M J ) = F J S I M J l / P P / V A L C M l C O N T I N U E . w R I T EC 1 a F1 = 1 H U H J I J ).. V A R J C. J ).. P.J I J 1-. «-* & C C J . M ).. M = 1 . N F ). I F ( N J 2 . E O . O ) - 1 . 3 T 0 7 0 1 0 0 5 0 0 J = I • N J 2 J K J = ( J - I I ' N F F A C T RO 51 F A C T R 0 5 2 F A C T R O 5 3 - F A C T R O 5 *_ F A C T RO 5 3 F A C T R 0 5 6 F A C T R 0 5 7 . F O R T R A N . I V ( i . . . L £ Y E I -21 : . ! EAOXSia '. DA.TJI_=^Za.l26 ^_I . l>L3-7/5*—: . r- P A 6 E OO0 2 0 0 5 7 5 0 0 w R I T E C I B F ) N O M J ( J * N J 1 ) .V A R J ( J * N J 1 1 . P J < J * N J 1 ) . ( F J S I J K J * L ) • L = 1 . N F ) F A C T R 0 5 8 0 0 5 8 701 REWIND I 8 F F A C T R 0 5 9 3H5.Sc Rr TIIHN • : : : : F A C T B 0 6 0 0 0 6 0 END F A C T R 0 6 1 O O O P O O O D O O O O O O O D O O O O O O o o o o o o o c o o o o o o o b o o c o o o o r^u — p ceo ^ joui A OJ N o .c cr - J a in I ! 1 : • . • ! o o o o o o o l o o o o b o o olo OO O O O O OD o o E o o o o c o o b o o o o o o o o o o o o c o o o o o C ^ u w u N f v . N r v ' N ^ w i \ ) r u N ) ' - r - * - " " , " " " " " " " * " * " 0 0 (j> u ) f o « - : o £.cc -jjoiui * u* w *" O o o -J rj> ui * ui p*<o cc - o II > <_ ~ II •- r • r L " 7 z o 01 o n / . o i » b r r o o - c o c D r r o r i x I - II — r II — II f- II z *-ui* ui u> o *- r A - J o — c cn • p r M O r o • ' i o > o i Ul * Li O op — II • L. II i/l * II r c - r r x (- r — s • — . > r . 2 - r z » r ~ - r * *- ' ? > r -r e m ui 01 t/i v> 01 m -« •< -c < •< •< < Z X X E X X X o o o o o o o S S Z 33.2 33 33 • I u« mmgiui u>ui -j o> u» k> u> w >-9.ei > •n -- z z • • z m x c -» N) II - O 7 J X X z II II -o z • . i o z • a - o c : — —: r t • c — JO || • L O H O C : - ml • : II c - L. ui n» II — :n » - r - r i f - - I i- c • • u>» x L - C Z -l/> PI c I — II 1  U> 01 t c.ctx X X ft — V • *• E u 1  u c L r • r • r — ui m ui ui u> ui ui - •<-<•<-<•<•< X X X K X X X OO O D O O O 33 33 33 33 33 33 33 ca -J b»m *u> — — lui oo> o •--n (coco ii-;x i — — MI ru II ui • • Jul o o o — m _ n r > m r r »~ Ul Cx Ul > c -X — II II . • —me • (_ O I <- — II • • II H - zt>- r r ci» — • II r l — u> II c J> X r x — > K» 01 r o 33 'f -I uir c -z ll > — I o o o o o b o o o o o o o o o o oo o o o - i a <* * OJ ro " b o. — x o r x pc i>o>-> i X - x n x r - K x ii r- r n kio » > p • o X X - • • > — m x!u o - r • r * X "» X •c X Ul Ul Ul 01 1 01 01 ui Ul ui Ul ui [•<-<<«-<-<-''•-.*-<-< x x i x i x r O O O D O O O 3333335033333310 • • * u> C XI : U> I i >-'o.o S> -J o m OJ ui Li OJ UI UI W L J M — pioai^i In moi ui j k •<-<-< x i i r 11 C O o p o O o X 33 33 33 33 33 33 I /I 01 I Ll WM 10 n 01 Ul U) LA Ul U> Ul <<-<-< •<•<<< : x x xx x x x I D O O D O O D 3 33 33 30 !D 33 33 33 j N W r o k j N M " -» 0 > > Ul kl •* O 10 O X • •• » ? — X ff\ > Ll DJ o » r » o r r r II w to-il ~ »- r • •' r D X z r m m r 0 2 Z O 33 U 01 aix x » m.p c • !•»> H > *H CD -J II II >l 01 z r M *- o • PI 33 b a PI c to H Z Ln oi in Ui ui oi oi oi K •<-<•« K •<•< •< X X X E X X X o o o o o o o |D 33 33 33 p 33 33 33 bUM — piO0B>4b.OI*ul|i'»-• < PI > s z c m K oi c tO 33 u> o H E5 r pi m • r — z ro >• EC oioioioijnoi « < • < - < • < • < •< •< o o o o o o o fa IS in 33 D 33 33 33 33 33 (0 33 to-F O R T R A N IV.<_. L E V E l 2 . 1 . _SJf.MQ_a -DATE- __'.fll__6- P A f i F 0 0 0 ? 0 0 5 8 0 0 5 9 _-__Q6J__ 0 0 6 1 0 0 6 2 0 0 6 3 0 0 6 4 . 0 0 6 5 0 0 6 6 0 0 6 7 0 0 6 . 6 . 0 0 6 9 0 0 7 0 0 0 7 1 0 0 7 2 0 0 7 3 0 0 7 4 0 0 7 5 :0j_26_ 00 1 60 I =LL1 AC l . L _ . ) = G . O l ^ n At i i .1 l a n . n — 170 AC L L . L L ) = 1 . 180 NU=N N U M l = N - t SH F T =1_. : K l =K0 2 1 0 00 2 2 0 N N L = l . N U M l NL = N U M 1 - N N L • 1 I F * A B S C E C N O ) . 2 2 0 C O N T I N U E GO TO 2 4 0 _2.3Q._eC NL)_=_Q.«.Q N L = N L * 1 . E . S U M ) GOTO 230 I F C N L . N E . N U ) GO TO 2 4 0 IFCNUM1 . E Q . l ) GOTO 4 9 9 NU = NUH 1 _ 0 0 7 7 0 0 7 8 0 0 7 9 0 0 8 0 0 0 8 1 0 0 8 2 0 0 8 3 -0Q9 4 . NU M1 =NU-1 GO TO .210 2 4 0 E C N U ) = E C N U ) * l . JF C tl tiU ) J.6.I.JLI T.LE-RJ—BXJURN-S U M = C O C N U M I ) - D C N U > ) / 2 . S U H l = S Q R T C S U M » S U M » - E « N U M I ) « c C N U M l > > IF I S U M . L T . O . ) S U M l = - r S U M l K?=HI N U I - H N U H I > »F INUM1 I /C ' . I H t S U H l 1. 0 0 3 5 0 0 8 6 0 0 d 7 . ooae, 0 0 8 9 0 0 9 0 0 0 9 1 _0_ 0 0 9 3 0 0 9 4 0 0 9 5 0 0 9 6 . 0 0 9 7 0 0 9 8 0 0 9 9 _Q.1IL0_ [ F * S H F T . N E . I . ) G O T O 2 7 0 IFCABSCIC2- IC1 I . L T . 0 . 5 » A B S C I C 2 » > <>° T O 2 6 0 K l = K2 _K=_KO r- — GO TO 3 00 T • TO RETURN SI N C O S Q>= EC N O A S S I G N 310 A S S I G N 500 _5.aQ._ee =_____. S.CJ?.I : QU •= A B S I 0 > I F I O Q . G T . P P ) GO TO N O R M = P P « S Q R T C 1 . 0 f.n T n z?Q 5 1 0 • C O O / P P ) • C Q Q / P P >) 0 10 1 0 10 2 0 1 0 3 0.10 4 0 1 0 3 0 106 0 1 0 7 _a.iaa_ 0 1 0 9 0 110 0 111 0 1 1 2 0 1 1 3 0 114 0 115 5 1 0 IF C O O . E O . O . 0 ) GO TO 5 3 0 NORM = Q O » S O R T C l . O • C P P / 0 0 > « I P P / O Q ) ) 5 2 0 C = P / N C R M _ _ S = Q/MQRM GO TO R E T U R N . C 3 1 0 . 3 4 0 . 3 6 0 ) 5 3 0 C = 1 . 0 3 = 0 . 0 S r M O R 5 8 SYMQR 5 9 SYMOR 6 1 SYMQR 6 2 SYMQR 6 3 _ s r . _ o B - A - -SYMOR 6 5 SYMQR 6 6 SYMQR 6 7 SYMQR 6 9 SYMQR 7 0 SYMQR 7 1 __SYJ4QH_7_2_ SYMOR 7 3 SYMOR 7 * SYMQR 7 3 SYMQR 7 7 SYMOR 7 8 SYMQR 7 9 _S.Y.MQR—8 0_ SYMQR 8 1 SYMOR 8 2 SYMQR 8 3 _____Ml)B ft A SYMQR 8 3 SY <QR 8 6 SYMQR 8 7 _S-YMQR Rfl SYMQR 8 9 SYMQR 9 0 SYMQR 9 1 S v u n p QP SYMQR. 9 3 SYMOR 9 * SYMQR 9 3 ___YMQR_96_ SYMQR 9 7 SYMQR 9 8 SYMQR 9 9 S Y M Q R l 0 1 S Y M Q R 1 0 2 S Y M Q R l 0 3 _ S Y H Q R 1 0 4 -SYMQR1OS S Y M Q R 1 0 6 S Y M Q R l 0 7 _______LQR1 Oft . GO T J R E T U R N . C 3 1 0 . 3 4 0 . 3 6 0 ) 3 10 00 3dO l=N_. . f«UMI DO 3<;0 J = l • N TEMP = C » A C J..I ) t S » AC J . 1 *.i ) — AC J . I » l ) = - S » A C J . I ) * C » A C J . 1 • ! ) 320 AC J . I ) = TEMP OC l ) = C » 0 C I ) * C * r 2 . » C « E C I ) * S « p C I * l ) ) » S S Y M Q R l 0 9 S Y M Q R l 1 O S Y M Q R l 1 1 . .SYMQRl 1.2-S Y M 0 R 1 1 3 S Y M Q R l 1 4 S Y M Q R l 1 3 U> ^4 F O R T R A N . I V . & . L E l / E L . . . 2 L -S-Y-MU-H-L OAT E _ = _ Z _ 4 2 6 - 1 J. •37/34- - R A G E 0 0 0 3 -0 1 1 6 0 117 -J3. l l . t i  01 1 9 0 120 0 1 2 1 0 1 2 2 _ 0 1 2 3 0 124 0 125 - Q 1 2 6 0 127 0 128 0 1 2 9 0 1 3 0 0 1 3 1 0 132 0 1 3 3 0 1 3 4 01 I + 1 ) = - S « £ I I>• C » D I I•1 ) E ( I ) = - S « K IF 1 I . F Q . M J M 1 ) G-1 T P ^Hfl 1F{ A B S I S 1 . G T . A B S I O ) GO TO 3 5 0 R = _ / C C( 1 * 1 > = - S » E I l ) « » D ( I t l ) _P__0( I t 1, 1 r-K E l I* I ) = C * E t, I • ! 1 0 = EI I • 1 ) A S S I G N 3 » 0 T u R E T U R N _____] s i u rns . i s u m ;  S Y M Q R l 1 6 S Y M Q R l 1 7 S Y M Q R l I 8 S Y M Q R l 1 9 S Y M Q R l 2 0 S Y M Q R l 2 1 - _ Y - R Q R L _ 2 _ S Y M Q R 1 2 3 S Y M Q R l 2 4 S Y M Q R l 2 3 3 4 0 3 5 0 £ 1 I ) = R » N 0 R M GO TO 3 8 C P= C» E l I ) +S»0 ( .Q _ S » E . t - _ . _ U • 1 I E l 1*1 > = C » E < I+1) 01 I • I ) = C » P / S *K A S S I G N 360 TCI R E T U R N 0.0. TO S I N C D S . 1 5 0 0 1 S Y M Q R l 2 7 S Y M Q R l 2 8 S Y M Q R l 2 9 - S Y M Q R l 3 0 S Y M Q R l 3 1 S Y M Q R l 3 2 S Y M Q R l 3 3 tiYunni u 0 I 35 0 1 3 6 01 37 0 1 3 8 . 0 1 39 0 140 0 141 _____ 3 6 0 E l I ) = N0RM 3 8 0 C O N T I N U E T E M P = C « E I N U M 1 ) + S » 0 I N U ) .D-LN-U 1 = - S « £ I N U M l 1 + C » D INU1 £ 1 N U M 1 ) - T E M P GO TO 2 1 0 4 9 9 00 4 0 0 L = l . N A Y M A X - - 1 . S Y M Q R l 3 5 S Y M Q R l 3 6 S Y M Q R l 3 7 -_-__QR_3__. S Y M Q R l 3 9 . S Y M Q R l 4 0 S V M Q R 1 4 1 - S - - Q B 1 A 2 0 1 4 3 0 144 0 1 4 5 . 0 -146. 0 1 4 7 0 148 O 1 » 9 0 I 5 CL. 0 151 0 1 5 2 0 15 3 . 0 1 5 4-0 153 0 156 0 157 j__5.a_ 00 4 1 0 J = l . N A IF I D ! J ) . _ £ . V M A X ) V M A X - D I J 1 _J__=J ' GOTO 4 1 0 4 1 0 -fin C O N T I N U E IK I D * J J V J L I D - Y M A X n'' .i.n - - i . S Y M Q R l 4 3 S Y M Q R l 4 4 S Y M Q R l 4 5 - S Y M Q R U L f i -S Y M Q R l 4 7 S Y M Q R l 4 8 S Y M Q R l 4 9 S Y M Q B l 5 0 TA 8 E T S = — V A L I 11 00 42 2 I - l i N A T A B E T S = T A 3 E T S » V A L ( 1 > -0 _____ 1 L - l . N A . L_ 4 2 1 01 L ) = A I I t L ) 00 4 2 2 J « l * N R J J = I K l J ) S Y M Q R l 3 1 S Y M Q R l 5 2 .SYMQRl 5 3 -SY-MQR1-54-S Y M Q R l 5 3 S Y M Q R l 5 6 S Y M Q R l 3 7 S Y M Q R l 5 8 0 159 0 160 0 16 1 0 1 6 2 0 1 6 3 0 164 0 165 - Q 1 6 6.. 0 167 0 166 0 llQ 0 1 70 0 17 1 0 172 0 17 3 IF I I Q . E Q . 0 1 P R I N T 4 3 3 . I J DO 4'3 0 1=1 -*3Q..PR I .M.__1. 7 3 8 C O N T I N U E PR INT 6 6 4 . V A H 1 ) C U M T A B = 0 . C_ A 12 1=2 iNA R E T U R N J = l i 8 ) . I V A L I J ) . J = l . N R l NA J 1 . •_ = 1 . N P 1  S Y M Q R l 6 0 S Y M Q R l 6 1 - S Y . M Q R : 6 2 -S Y M Q R l S 3 S Y M Q R l 6 4 S Y M Q R l 6 5 S Y M Q R l 6 f t ii 3 2 4 3 1 4 3 2 P O u R C T =VAL I I ) • I 0 0 . / T A b E T S C 0 C T 4 8 =CUMTA3 t P O U R C T t S i = V A L l I I t 6 u / w » L I 2 1 * l . 5 IE 1 I I = t l K.I.L.H _ . _. P R I N T 4 3 2 . I . 1E M I . VAL « I > . P G U R C T . C UMTA 6. I E P S . K K = F O R M A T ! IH O B J . • . I 3 . • / • .61 ZX.F10 . 6 . 2 X . • / • ) » F 3 RM A T I ' / • • 2 I I 3 . ' / .-• ) . F 1 O . a . 2 1 • / ' . F 7 . 3 ) . ' / • 1 • I S3 ) . 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