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Additivity of component regression equations when the underlying model is linear 1983
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Title | Additivity of component regression equations when the underlying model is linear |
Creator |
Chiyenda, Simeon Sandaramu |
Publisher | University of British Columbia |
Date Created | 2010-05-02 |
Date Issued | 2010-05-02 |
Date | 1983 |
Description | This thesis is concerned with the theory of fitting models of the form y = Xβ + ε, where some distributional assumptions are made on ε. More specifically, suppose that y[sub=j] = Zβ[sub=j] + ε [sub=j] is a model for a component j (j = 1, 2, ..., k) and that one is interested in estimation and interference theory relating to y[sub=T] = Σ [sup=k; sub=j=1] y[sub=j] = Xβ[sub=T] + ε[sub=T]. The theory of estimation and inference relating to the fitting of y[sub=T] is considered within the general framework of general linear model theory. The consequence of independence and dependence of the y[sub=j] (j = 1, 2, ..., k) for estimation and inference is investigated. It is shown that under the assumption of independence of the y[sub=j], the parameter vector of the total equation can easily be obtained by adding corresponding components of the estimates for the parameters of the component models. Under dependence, however, this additivity property seems to break down. Inference theory under dependence is much less tractable than under independence and depends critically, of course, upon whether y[sub=T] is normal or not. Finally, the theory of additivity is extended to classificatory models encountered in designed experiments. It is shown, however, that additivity does not hold in general in nonlinear models. The problem of additivity does not require new computing subroutines for estimation and inference in general in those cases where it works. |
Subject |
Linear Models (Statistics) Regression Analysis Estimation Theory |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | Eng |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project |
Date Available | 2010-05-02 |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0075140 |
Degree |
Doctor of Philosophy - PhD |
Program |
Forestry |
Affiliation |
Forestry, Faculty of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/24279 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0075140/source |
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ADDITIVITY OF COMPONENT REGRESSION EQUATIONS WHEN THE UNDERLYING MODEL IS LINEAR by SIMEON SANDARAMU CHIYENDA .S.F., The University of B r i t i s h Columbia, 1974 M.S., Iowa State University, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Forestry We accept t h i s t hesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1983 © Simeon Sandaramu Chiyenda, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 DE-6 (3/81) ABSTRACT This thesis i s concerned with the theory of f i t t i n g models of the form y = X$ + e, where some distributional assumptions are made on e. More specifically, suppose that y = X(5\ + is a model for a component j (j = 1, 2, ..., k) and that one is interested in estimation and inter- im ence theory relating to y T = Z j = 1 y = X 3 T + e T- The theory of estimation and inference relating to the f i t t i n g of y T is considered within the general framework of general linear model theory. The consequence of independence and dependence of the y (j = 1, 2, ..., k) for estimation and inference is investigated. It is shown that under the assumption of independence of the y^, the parameter vector of the total equation can easily be obtained by adding corresponding components of the estimates for the parameters of the component models. Under dependence, however, this additivity property seems to break down. Inference theory under dependence is much less tractable than under inde- pendence and depends c r i t i c a l l y , of course, upon whether y^ is normal or not. Finally, the theory of additivity i s extended to classificatory models encountered in designed experiments. It is shown, however, that additivity does not hold in general in nonlinear models. The problem of additivity does not require new computing subroutines for estimation and inference in general in those cases where i t works. i i TABLE OF CONTENTS ABSTRACT i i LIST OF TABLES v LIST OF FIGURES v ACKNOWLEDGEMENT v l DEDICATION v i i i Chapter I INTRODUCTION 1 II PRELIMINARIES, NOTATION AND PROBLEM DEFINITION 4 2.1 Preliminaries and Notation 4 2.2 Problem Definition 6 III LITERATURE REVIEW 7 IV ADDITIVITY IN THE CASE r. = p 13 3 4.1 Inferences for Total Model when e ^ N(<j>, Io?) 15 4.2 Inferences for Total Model when E j N(<j>, Vo?) 19 V ADDITIVITY WHEN r. < p WITH r. < p FOR SOME j 23 3 3 5.1 Estimation when e. ̂ N(<j>, Io?) 24 5.2 Inference when ^ N((j>, I°j) 29 5.3 Estimation and Inference when ^ N(((>, Vo?) .... 32 VI A GENERALIZATION OF THE ADDITIVITY PROBLEM 33 VII OTHER ASPECTS OF THE ADDITIVITY PROBLEM 35 7.1 The Case y., y (j * I) Dependent 36 3 i 7.1.1 Estimation for Total Model under Dependence 39 i i i 7.1.2 Inference for Total Model when yit y2» • • •» a r e J M V N 41 7.1.3 Inference for Total Model when yi> Y 2 » •••> y. are not JMVN 43 K. 7.2 The Case of X Random or Measured with Error .. 44 7.2.1 The Case of X Random 44 7.2.2 The Case when X is Measured with Error 46 7.3 Other General Complements 48 VIII SOME EXTENSIONS OF THE THEORY 53 8.1 Extension to Classificatory Models 54 8.2 Extension to Nonlinear Models 56 IX COMPUTATIONAL CONSIDERATIONS 60 X SOME ILLUSTRATIVE EXAMPLES 67 10.1 Assessing Multivariate Normality of y^ 67 10.2 Example 1 72 10.3 Example 2 75 10.4 Example 3 76 XI CONCLUSIONS AND REMARKS 78 REFERENCES 80 APPENDIX I 86 APPENDIX II 90 APPENDIX III 92 iv 7.1.2 Inference for Total Model when yi» yi> • • • » y^ are JMVN 41 7.1.3 Inference for Total Model when Yi> Y2> •••> y^ are not JMVN 43 7.2 The Case of X Random or Measured with Error .. 44 7.2.1 The Case of X Random 44 7.2.2 The Case when X is Measured with Error 46 7.3 Other General Complements 48 VII SOME EXTENSIONS OF THE THEORY 53 8.1 Extension to Classificatory Models 54 8.2 Extension to Nonlinear Models 56 IX COMPUTATIONAL CONSIDERATIONS 60 X SOME ILLUSTRATIVE EXAMPLES 67 10.1 Assessing Multivariate Normality of 67 10.2 Example 1 72 10.3 Example 2 75 10.4 Example 3 76 XI CONCLUSIONS AND REMARKS 78 REFERENCES 80 APPENDIX I 86 APPENDIX II 90 APPENDIX III 92 iv LIST OF TABLES Table 1 Results of Koziol's (1982) test for multivariate normality on three data sets 69 2 Results of Koziol's test for bivariate normality 70 3 Comparison of predicted values and residuals of unrestricted total equation with those of total conditioned equation 94 LIST OF FIGURES Figure 1 Scatter of Residuals from Total Unrestricted Equation and from Total Conditioned Equat ion 95 v ACKNOWLEDGEMENT I would like to express my indebtedness to Dr. Antal Kozak, my supervisor, for his advice, encouragement, and for a degree of under- standing of a rare order. If i t were not for his encouragement and unstinting support, this thesis would hardly have reached this stage. My most sincere gratitude is due to other members of my research commit- tee, namely Drs. Michael Schulzer of the Department of Mathematics, J. H. G. Smith, D. D. Munro, and D. H. Williams, a l l of the Faculty of Forestry, for providing much-needed guidance and constructive criticism at various stages of this study. I am especially grateful to Dr. Schulzer who gave freely of his time to discuss various aspects of my research. I would like to extend my appreciation to Dr. Stanley Nash, now Professor Emeritus in the Department of Mathematics at the University of British Columbia,for kindly reviewing an earlier draft of this thesis. My two-year association with Dr. James Zidek of the Department of Mathematics at this university, prior to his sabbatical leave at Imperial College (London), was a truly rewarding experience and I would like to express my admiration for a person so selfless and so endowed with academic vision. I would like to record my deepest gratitude to the Faculty of For- estry for financial support during the period of my studies and to the Food and Agriculture Organization for a generous fellowship. My grati- tude also goes to Professor Lewis K. Mughogho for always having confidence in me and to Professors Chimphamba and Edje for supporting me whenever I v i needed support. Finally, I express much gratitude and love to my wife Jane and daughters Lucy and Margaret for putting up with a husband and daddy who was hardly home during the preparation of this thesis. I am indebted to Mrs. Nina Thurston for a typing job truly well done. There are many others that I should mention, including some very special people. To these and others, I would like to say "I have not forgotten about you." v i i DEDICATION I dedicate this thesis to the loving memory of my mother Katherine who passed away in February 1982 while I studied at the University of British Columbia v i i i CHAPTER I 1.0 INTRODUCTION The main objective of this thesis is to formalize and extend or generalize results obtained by Kozak (1970) concerning conditions that ensure that predicted values calculated from component regression equa- tions add up to those obtained from a corresponding total equation. Kozak (1970) derived his results within the context of forest biomass prediction using component biomass equations and a corresponding total biomass equa- tion. He cites examples in other areas of forestry and forest research where such a problem arises. A broader view is adopted in this thesis with regard to areas of application of the 'additivity' problem. Since biomass analysis is of interest to scientists in various other disciplines of applied biology than forestry, a formalization and generalization of the additivity problem and i t s related s t a t i s t i c a l theory w i l l be of value to a large number of scientists, including those in agriculture and ecology. To f i x ideas with regard to the additivity problem as perceived in this thesis, suppose as in Kozak (1970), that for some tree species weight of bole (Yi), weight of bark on the bole (Y2), weight of crown ( Y 3 = branches and foliage), and total weight ( Y = E ? , Y.) can each be modelled as some function (in the linear regression sense) of diameter at breast height (X). In this setting, one refers to the equations 1 2 expressing each of Y i , Y2, and Y3 as functions of X as component equations and that giving Y^ as a function of X as the total equation. More generally, one can envisage k components of an organism or system characterized by measurable attributes Yi, Y2, Y and their K. sum Y = Y. where Yi, Y 2, Y are each related to a common K+l j — 1 j k+1 set of p independent variables Xi, X2, X̂ according to a multiple linear regression model. Using the case (with k = 3) described in the preceding paragraph, Kozak (1970) states conditions under which one need only f i t the component equations, the total equation being completely determined by adding coefficients of corresponding independent variables in the component equations. Kozak's (1970) results pertain to the situa- tion where each component equation contains a l l independent variables under consideration. In the sequel, our objective is essentially four-fold. F i r s t , i t is intended to demonstrate, in a rather simple way, why Kozak's (1970) conditions hold and to derive explicit expressions for statistics of interest for the total equation from those of the component equations. Secondly, i t w i l l be shown that additivity can be assured even when d i f f e r - ent terms are retained in the component equations (that i s , when r^ S p with st r i c t inequality for at least one j , where r^ is the number of independent variables in component equation j and p is the number in the total equation). This w i l l be achieved by appropriately correcting the total equation in order to take into account the conditioning that forces some independent variables not to appear in some component equa- tions. Thirdly, i t w i l l be shown that the case considered by Kozak (1970), where each component equation has the same number of independent variables as the total equation (that i s , r. = p) , can be derived as a 3 s p e c i a l case of a g e n e r a l i z a t i o n of the c o n d i t i o n i n g p r i n c i p l e mentioned above. E s t ima t i on and i n fe rence theory w i l l be developed f o r the above o b j e c t i v e s based p r i m a r i l y on the assumption that Y. and Y are indepen- J *• dent f o r each j * £ ( j , £ = 1, 2, . . . , k) and an app rop r i a te d i s t r i b u - t i o n a l assumption on the e r r o r term corresponding to model j . Because of i t s re levance when con s i de r i n g many b i o l o g i c a l phenomena, theory r e l a t i n g to the case where Y. , Y ( j * £) are dependent w i l l be con - J * s i de red . F i n a l l y , some examples w i l l be worked out to i l l u s t r a t e the a p p l i c a t i o n of the theory . I t i s important to emphasize here that the main o b j e c t i v e of t h i s d i s s e r t a t i o n i s to i n v e s t i g a t e the a d d i t i v i t y problem and i t s r e l a t e d s t a t i s t i c a l theory w i t h i n a f a i r l y genera l framework. There fo re , the use of c e r t a i n models i n c o r p o r a t i n g and, p robab ly , e xc l ud i ng p a r t i c u l a r independent v a r i a b l e s i n the examples and elsewhere i n t h i s t h e s i s should not be construed as suggest ing that the equat ions are best i n a p r e d i c t i v e sense. More s p e c i f i c a l l y , a l though the subject of the t h e s i s has a d i r e c t bea r i ng upon biomass p r e d i c t i o n problems, i t i s not the o b j e c t i v e here to a r r i v e at a best biomass equat ion i n any p a r t i c u l a r sense. On the other hand, the v iew i s taken that the dete rminat ion of best equat ions f o r p r e d i c t i o n purposes i s best l e f t to p a r t i c u l a r a p p l i c a t i o n s of the theory to be presented here. CHAPTER II 2.0 PRELIMINARIES, NOTATION AND PROBLEM DEFINITION We begin with some definitions and rules of convention regarding notation to be adopted in the sequel. 2.1 Preliminaries and Notation Since the development in this thesis w i l l be concerned with linear regression models, we f i r s t seek to identify this class of models precisely. Accordingly, we define a regression model following Gallant (1971). Let X C Rm, n C R? and n and p be positive integers such that n > p. The elements of X and Q w i l l be denoted by x and g, respectively. Further- oo , oo more, we shall let l e t ) t _ ^ he a sequence of random variables, ' L x t J t _ ^ a sequence from X, f(x, g) a real-valued function with argument (x, g) and go to be a point in Q. Definition 1. A regression model is defined here to be the sequence of random variables {yt}^_-^ given by y t = f ( x t , go) + e t- We emphasize that we owe the basic idea behind this definition to Gallant (1971). Our definition is not as rigorous as Gallant's (1971) but i t wi l l suffice for our purposes. Note that f is a mapping of points from the product space X x fi into the real line R1 (that i s , f : X x fi ->• R1) . Now, suppose we denote the set of a l l possible regression models generated according to definition 1 by R* and define a set of regression 4 models of the form P y t = <t»o(xt) + Z B j * j ( x t > + e t where <K : X R 1 , i = 0, 1, p. Again we owe t h i s f o rmu la t i on to G a l l a n t (1971). Note that i n the above, f ( x t , So) i n d e f i n i t i o n 1 has been rep laced by <t>o ( x t ) + E ? = 1 ^ ^ ( X j - ) - G a l l a n t (1971) d e s i g - nates the c l a s s of r eg re s s i on models whose members are s p e c i f i e d accord ing to the l a s t equat ion by L, which i s the c l a s s of l i n e a r r e g re s s i on models. Thus we have a second d e f i n i t i o n . D e f i n i t i o n 2. A r eg re s s i on model r * i s c a l l e d a l i n e a r r e g re s s i on model i f r * E L, where L i s as de f ined above. Note, f o r completeness, that L C R*. Th i s t h e s i s w i l l be concerned w i t h the theory of e s t ima t i on and i n fe rence f o r members of L under c e r t a i n c o n d i t i o n s . As i s w e l l known, any member of L can be w r i t t e n i n mat r i x form. With regard to n o t a t i o n , a mat r i x r e p r e s e n t a t i o n w i l l be adopted throughout most of the development here . Th is has an obvious a e s t h e t i c appeal bu t , more impo r t an t l y , leads to b r e v i t y and a r a t h e r compact p r e - s e n t a t i o n of r e s u l t s which would otherwise be cumbersome us ing o rd i na r y s c a l a r a r i t h m e t i c . A c co rd i n g l y , l e t y^ denote an n x 1 vec to r of r e a l i - z a t i on s of an observable random v a r i a b l e corresponding to the j * " * 1 com- ponent, X an n x (p + 1) mat r i x de f i ned so that X = (X 0 IXi I . . . IX ) = (tfIXiI...I X ), where X 0 = # i s an n x 1 vec to r w i t h I I p P components i d e n t i c a l l y equal to u n i t y and X^ ( i = 1, 2, . . . , p) i s an n x 1 vec to r c o n s i s t i n g of r e a l i z a t i o n s of the independent v a r i a b l e X^, 8j i s a (p + 1) x 1 parameter vec to r and i s the corresponding n x 1 vec to r of e r r o r s or d i s t u rbance s . 6 For any matrix A, say, we s h a l l w r i t e A*" to denote the transpose of A and A ^ to denote the inverse of A, provided that the inverse e x i s t s . -1 t Where A does not e x i s t , we s h a l l have occasion to use A , the ge n e r a l i z e d inverse of A, to o b t a i n more general r e s u l t s . In any case, new n o t a t i o n may be introduced i n the discourse as the need a r i s e s but i n every case our n o t a t i o n w i l l be c o n s i s t e n t w i t h that used i n standard t e x t s i n l i n e a r algebra (e.g., Noble & D a n i e l , 1977; S e a r l e , 1966; Stewart, 1973; Strang, 1976). 2.2 Problem D e f i n i t i o n With the above conventions and n o t a t i o n , we s h a l l be concerned, i n t h i s d i s s e r t a t i o n , w i t h models of the form y^ = Xgj + £j ( j = 1, 2, k) (2.2.1) and y x = £ j = 1 y- = x e T + e T (2.2.2) where (2.2.1) gives us k component models and (2.2.2) the corresponding t o t a l model. We s h a l l focus i n t e r e s t i n the sequel on c h a r a c t e r i z i n g estimators of 3rr, and on propounding a theory of inference r e l a t i n g to the t o t a l model under a number of assumptions concerning the behavior of the E j ( j = 1, 2, k ) . The theory to be presented here w i l l be based on well-known general l i n e a r model theory. Note that s p e c i f i - c a t i o n of the behavior of the leads, i n general, to s p e c i f i c a t i o n of the behavior of try.. In the next chapter, a review i s made of some of the work i n the l i t e r a t u r e r e l a t i n g to the a d d i t i v i t y problem before proceeding to propose a u n i f i e d theory of e s t i m a t i o n and inference f o r t h i s problem i n succeeding chapters. CHAPTER III 3.0 LITERATURE REVIEW Theoretical and applied biologists have traditionally viewed biomass as a useful index for assessing the productivity of various flora and fauna with respect to designated environments or ecosystems (Ovington, 1962). This index has also been used for cataloguing, in the form of inventories, the quantities of biological matter available at a given time in a given environment. A thorough reading of the literature on biomass and related studies indicates development in two main directions. The early part of the literature indicates that scientists essentially sought ways of quantita- tively describing biomass production and productivity of various biolog- ica l organisms. This approach was especially dominant in ecological studies for many years. Quite often, systematic sampling schemes (e.g., line transects) were used to obtain data which were subsequently summarized to give crude estimates of biomass. In many cases, these estimates were reported by component of the organism or system under consideration. -In general, l i t t l e or no s t a t i s t i c a l information, such as measures of precision, accompanied the summarizations. In any case, the very nature of the sampling schemes upon which the estimates were based militated against a meaningful s t a t i s t i c a l interpretation of the results. More recent literature is suggestive of a significant shift from the purely 7 8 d e s c r i p t i v e approach to model-based, s t a t i s t i c a l l y - o r i e n t e d methods of d e s c r i b i n g biomass. Th i s approach not only leads n a t u r a l l y to the need to address quest ions r e l a t i n g to cho ice of proper model f o r use i n a g iven s i t u a t i o n , but more impor tan t l y perhaps, a t taches p a r t i c u l a r importance to choosing es t imates tha t are s t a t i s t i c a l l y reasonab le. The f o l l o w i n g rev iew of the l i t e r a t u r e on developments that l ed to the a d d i t i v i t y p rob - lem w i l l be b r i e f and, h o p e f u l l y , i n f o rmat i ve r a the r than exhaus t i ve . For a more complete rev iew, see Chiyenda (1974) or Kurucz (1969). More recent comprehensive reviews are g iven by Smith (1979) and Smith and W i l l i ams (1980). The f o r e s t r y l i t e r a t u r e c r e d i t s Tu f t s (1919) w i t h the f i r s t repor ted work on t ree component biomass. In that work, Tu f t s c o r r e l a t e d trunk c i r - cumference of f r u i t t r ee s w i t h the weight of t h e i r tops (or crowns). F o l - lowing that work, many workers i n f o r e s t r y engaged i n biomass s tud ie s of one form or another. A c co rd i n g l y , cons ide rab le work has been repor ted i n the genera l area of t o t a l biomass p roduc t i on of va r i ous t r ee spec ies (Honer, 1971; Ke l l o gg and Keys, 1968; Young and Chase, 1965) and of f o r e s t e co - systems (Ovington, 1962). Some of t h i s work was c a r r i e d out as pa r t of ongoing inventory programmes ( e . g . , Honer, 1971) wh i l e others were conducted i n re search connected w i t h f o r e s t f i r e - h a z a r d abatement e f f o r t s ( e . g . , K i i l , 1967, 1968; Loomis et a l . , 1966; Storey e t a l . , 1969). In a biomass study of 13 North American t ree spec ie s , Storey et a l . (1955) found that dry crown weight , branchwood weight , and f o l i a g e weight were s i g n i f i c a n t l y r e l a t e d to stem diameter at the base of l i v e crown f o r a l l the spec ie s . On the other hand, Ovington (1956) i n v e s t i g a t e d and compared the forms, we ight s , and p r o d u c t i v i t y of t ree spec ies grown i n c l o se stands. Th is study was mot ivated by s i l v i c u l t u r a l and e c o l o g i c a l c o n s i d e r a t i o n s . In a study s i m i l a r to that of 1955, Storey and Pong 9 (1957) i n v e s t i g a t e d and compared crown c h a r a c t e r i s t i c s of a number of hardwood species. Fahnestock (1960) used data c o l l e c t e d from nine coniferous tree species i n the Northern Rocky Mountain area to f i t r e g r e s s i o n equations to p r e d i c t crown weight and proceeded to construct crown weight t a b l e s f o r the species. Among the species i n v e s t i g a t e d were D o u g l a s - f i r (Pseudotsuga m e n z i e s i i [Mirb.j Franco), western hemlock (Tsuga heterophyla [Raf.] Sarg.) and western red-cedar (Thuja p l i c a t a Donn). Tadaki et a l . (1961) i n v e s t i g a t e d the p r o d u c t i v i t y of a young stand of b i r c h (Betula p l a t y p h y l l a ) i n southern Hokkaido, Japan, and e s t a b l i s h e d l i n e a r r e l a t i o n s h i p s on l o g a r i t h m i c axes between ba s a l area and stem biomass, ba s a l area and branch biomass, and between ba s a l area and f o l i a g e biomass. They al s o reported that estimated f r e s h and dry f o l i a g e weights d i d not vary w i t h stand density but that branch biomass decreased w i t h stand d e n s i t y . Brown (1963) i n v e s t i g a t e d the r e l a t i o n s between crown weight and diameter i n some Lake States red pine (Pinus r e s i n o s a A i t . ) p l a n t a t i o n s and a l s o studied the i n f l u e n c e of s i t e q u a l i t y and stand d e n s i t y on the weight of i n d i v i d u a l t r e e crowns. Keen (1963) analysed average green weights and centres of g r a v i t y of samples of black spruce (Picea mariana [ M i l l . ] B.S.P.), white spruce (Picea glauca [Moench.] Voss.), and balsam f i r (Abies balsamae [L.] M i l l . ) and i n v e s t i g a t e d t h e i r v a r i a t i o n w i t h species, season, and l o c a t i o n . He a l s o derived a t a b u l a t i o n of weights and centres of g r a v i t y of the t r e e s . Young et a l . (1964) used r e g r e s s i o n equations to c o n s t r u c t f r e s h and dry f i b r e weight t a b l e s f o r i n d i v i d u a l tree components, groups of components, and complete trees f o r seven tre e species. Brown (1965) 10 i n v e s t i g a t e d the e f f e c t of s i t e and stand den s i t y on the crown s i z e of i n d i v i d u a l red p ine and j a ck p ine (Pinus banks iana Lamb.) t rees and s tud ied ways of e s t ima t i n g crown f u e l we ight s . The study i n d i c a t e d that e s t i - mated amounts of f o l i a g e and branchwood per u n i t area v a r i e d w i t h the age and growing c ond i t i o n s of the s tand. Loomis e t a l . (1966) used a n a l y s i s of covar iance to t e s t the e f f e c t of stand den s i t y on dry f o l i a g e and branchwood weights i n s h o r t l e a f p ine (Pinus ech ina ta M i l l . ) and found that reg res s ions of dry f o l i a g e and branchwood weights f o r d i f f e r e n t stand d e n s i t i e s were not s i g n i f i c a n t l y d i f f e r e n t . Dyer (1967) prepared p r e l i m i n a r y f r e s h and dry weight t ab l e s f o r no r thern whi te cedar (Thuja o c c i d e n t a l i s L.) and de r i ved l i n e a r r eg re s s i on equat ions f o r p r e d i c t i n g f r e s h and dry wood weights of va r i ou s t r ee components as percentages of t o t a l t r ee f r e s h and dry we ights . K i i l (1967) used r eg re s s i on a n a l y s i s to con s t ruc t f u e l weight t ab l e s f o r white spruce and lodgepole p ine (Pinus c on t o r t a Dougl.) i n wes t - c e n t r a l A l b e r t a and found that a combinat ion of diameter at b r ea s t - he i gh t and e i t h e r crown w id th or crown l eng th gave the most p r e c i s e e s t ima t i n g equat ion f o r f u e l weight. In a f o l l ow -up study, K i i l (1968) s tud ied the f u e l complex of 70 -yea r -o ld lodgepole p ine i n the same area w i t h a v iew to f a c i l i t a t i n g measurement and p r e d i c t i o n of weight and s i z e d i s t r i b u t i o n of f u e l components. Kurucz (1969) obta ined p r e d i c t i v e r eg re s s i on equat ions f o r t o t a l and component biomass of D o u g l a s - f i r , western hemlock, and western r e d - cedar grown on the U n i v e r s i t y of B r i t i s h Columbia Research Fores t near Haney, B r i t i s h Columbia. In a study that was probably mot ivated as much by Ku ruc z ' s (1969) study as by o the r s , Kozak (1970) cons ider s the problem of a d d i t i v i t y of component biomass r eg re s s i on equat ions f o r purposes of p r e d i c t i o n . The r e a l essence o f .Kozak ' s (1970) work does not l i e i n the uniqueness of the problem he poses bu t , r a t h e r , i n the s t a t i s t i c a l problems that i t r a i s e s and the p o t e n t i a l p r a c t i c a l impact that a s o l u t i o n to these problems might have. Other s tud ie s conducted f o l l o w i n g ko zak ' s (1970), wh i l e e s s e n t i a l l y underscor ing the importance of the biomass e s t ima t i on problem i n f o r e s t r y and r e l a t e d d i s c i p l i n e s , d i d not address the a d d i t i v i t y aspect of the problem d i r e c t l y . I n s tead, many i n v e s t i g a t o r s cont inued to look f o r the best set of v a r i a b l e s g i v i n g the most pars imonious p r e d i c t i v e equat ion f o r t o t a l and component biomass ( e . g . , Crow, 1971; Honer, 1971; Johnstone, 1971; Muraro, 1971; Z a v i t k o v s k i , 1971; Sando and Wick, 1972). Biomass s tud ie s and methods of e f f e c t i v e l y p r e d i c t i n g i n d i v i d u a l t r ee biomass cont inued to i n t e r e s t app l i ed q u a n t i t a t i v e b i o l o g i s t s i n the m id - and l a t e - s e v e n t i e s and w e l l i n t o the e i g h t i e s . Th i s i n t e r e s t i n biomass i s a s c r i b a b l e to the a p p l i c a b i l i t y of i n d i v i d u a l t ree biomass i n f o rmat i on i n address ing a wide range of e c o l o g i c a l and f o r e s t management problems. These i nc lude l a r g e - s c a l e biomass i n v e n t o r i e s (Young, 1978; Ker and Van R a a l t e , 1980), n u t r i e n t - c y c l i n g problems (Marks and Bormann, 1972; Kimmins, 1977; Kimmins and K ruml i k , 1976; Kimmins e t a l . , 1979), as w e l l as the determinat ion of net p r o d u c t i v i t y of f o r e s t ecosystems (wh i t taker et a l . , 1974). Many s t u d i e s , such as Jacobs and Cun ia (1980), J oke l a et a l . (1981), Keyes and G r i e r (1981), Schmitt and G r i g a l (1981), Yandle and Wiant (1981), Z a v i t k o v s k i et a l (1981), Chaturved i and Singh (1982), Freedman et a l . (1982), and Singh (1982), have used r eg re s s i on methods to address the biomass p r e d i c t i o n problem. I t i s worth ment ioning that biomass s tud ie s of v a r i ou s d e s c r i p t i o n s are be ing conducted to date . Some of these are e s s e n t i a l l y computer- 12 based s imu la t i on s of v a r i ou s aspects of the biomass problem. An example of t h i s i s the FORCYTE study be ing undertaken by Kimmins and h i s a s s oc i a te s at the U n i v e r s i t y of B r i t i s h Columbia (see Kimmins and S c o u l l a r , 1979; Kimmins e t__a l . , 1980). As desc r ibed by i t s au thor s , "FORCYTE i s an i n t e r - a c t i v e s imu l a t i o n model designed to examine, on a s i t e - s p e c i f i c b a s i s , the long- term e f f e c t s on n u t r i e n t budgets and p r o d u c t i v i t y of va r i ou s i n t en s i v e f o r e s t management and ha r ve s t i n g p r a c t i c e s . " Other s tud ie s are conducted as pa r t of ongoing n a t i o n a l programmes aimed at i d e n t i f y i n g u s e f u l model- l i n g procedures f o r p r e d i c t i n g o r , o the rw i se , d e s c r i b i n g biomass. An example of t h i s i s the study, aga in at the U n i v e r s i t y of B r i t i s h Columbia, by Smith (1979) and Smith and W i l l i ams (1980) o r i g i n a l l y commissioned by the Canadian Fo r e s t r y Se rv i ce to propose the development of a comprehensive f o r e s t biomass growth model. That p roposa l has s ince been approved and work i s c u r r e n t l y under way to develop such a model. S u r p r i s i n g l y , most of the s tud ie s c i t e d e a r l i e r do not cons ide r the a d d i t i v i t y problem except f o r pass ing re fe rence to Kozak ' s a d d i t i v i t y r e s u l t i n a few ins tances ( e . g . , Ker and Van R a a l t e , 1980; S ingh, 1982). One might surmize that t h i s apparent l ack of i n t e r e s t i n the a d d i t i v i t y problem might be l a r g e l y due to the f a c t that a d d i t i v i t y has, s ince i t s i n t r o d u c t i o n i n t o the f o r e s t r y l i t e r a t u r e by Kozak (1970), been r e s t r i c t e d to s i t u a t i o n s i n which each component equat ion conta ins the same independent v a r i a b l e s . Th i s prec ludes the use of a d d i t i v i t y i n the more common and important s i t u a t i o n s where on ly s t a t i s t i c a l l y important independent v a r i - ab les are used i n any component equat ion . An ex tens ion of the a d d i t i v i t y problem to such s i t u a t i o n s a long w i t h i t s corresponding s t a t i s t i c a l theory would obv iou s l y be of i n t e r e s t . Th i s i s what i s intended to be done i n suc- ceeding chapters of the d i s cou r se . One hopes that s tud ie s such as have been 12a c i t e d above w i l l , i n t ime , b e n e f i t from or a t l e a s t f i n d u s e f u l comple- mentary methodology i n the theory to be presented i n t h i s d i s s e r t a t i o n . CHAPTER IV 4.0 ADDITIVITY IN THE CASE r. = p J We consider f i r s t the models y. = XB. + e. (i =1, 2, .... k) and 3 3 3 = £. , y. = XB_ + e_ where i t is understood that each of these k + 1 I j=l J 2 T T models involves the same matrix X. This is the case considered by Kozak (1970). Consider the estimation of 3^ assuming £j % (<!>> o r e. (d), Vo2.) where d> is an n x 1 null vector, I is an identity matrix 3 3 of dimension n and V is a known symmetric positive definite matrix of dimension n. Note that we have not for now specified the form of the distribution function of e. as this is not necessary to obtain estimates 3 of desirable properties. We restrict attention in this chapter to the situation where X is of f u l l rank. Under the assumption that ^ (<}>, ̂ °j)» ordinary least squares (OLS) f i t t i n g of the k component models yields Gauss-Markoff estimators Bj = ( X ^ ) " 1 xV. (j = 1, 2, k) . (4.0.1) The result given by (4.0.1) is completely basic and warrants no further comment except to note that the resulting Bj are best linear unbiased estimators (BLUE's) in the sense of the Gauss-Markoff theorem (see Gray- b i l l , 1976, p. 219; Kempthorne, 1975, p. 32; Searle, 1971, p. 88). When £j ̂ (<|>, Vo?), generalized least squares (GLS) f i t t i n g applied to each component model leads to Gauss-Markoff estimators 14 B\ = ( X t V ~ 1 X ) " 1 X t V ~ 1 y j ( j = 1, 2, k ) . . (4.0.2) Note that i n (4.0.2) the ex i s t ence of V ^ i s guaranteed by the p o s i t i v e d e f i n i t e n e s s of V. I t i s worth p o i n t i n g out that the OLS e s t imator of 3^ i s i n genera l d i f f e r e n t from the GLS e s t ima to r , except i n the s p e c i a l case where there e x i s t s a (p + 1) x (p + 1) nons ingu lar mat r i x F such that VX = XF. Th is i s a very s p e c i a l r e s u l t and i s due to Zyskind (1962). See G r a y b i l l (1976), Kempthorne (1975), and Sear le (1971) f o r re fe rences to t h i s r e s u l t . Now cons ider the t o t a l model. I n t roduc ing the expres s i on f o r y j g iven e a r l i e r i n t o the expres s i on f o r the t o t a l model, one gets k y T = Z (XB. + e.) = XB T + e T (4.0.3) j = l J or k k y = X I B• + I e. = XB T + e T (4.0.4) j = l J j = i J k k from where i t i s c l e a r that £_, = £. -, B- and £_, = ! . n e. . Hence, the T j = l J T j = l j k - l e a s t squares e s t imato r of B j i s g iven by B^ = 8., where 8- i s g iven by (4.0.1) or (4.0.2) accord ing as o rd i na r y l e a s f squares or gene ra l i z ed l e a s t squares f i t t i n g i s used to ob t a i n BLUE ' s . Thus the t o t a l equat ion i s complete ly determined by c o e f f i c i e n t s of the component equat ions , as Kozak (1970) po in ted out . Having shown ( f o r = p) that the t o t a l equat ion i s complete ly determined by the parameters of the component equat ions , i t might be of i n t e r e s t to i n v e s t i g a t e whether s t a t i s t i c s de r i ved from the component analyses can be u t i l i z e d to make i n fe rences p e r t a i n i n g to the t o t a l equa- t i o n . I t w i l l be shown, i n the seque l , that t h i s i s the case i n genera l . The r e s u l t s presented below w i l l be u s e f u l f o r t e s t i n g hypotheses 15 concern ing the t o t a l equat ion and f o r c o n s t r u c t i n g conf idence i n t e r v a l s . In order to s i m p l i f y the d e r i v a t i o n s , we s h a l l assume that the random v a r i a b l e s y. and y are independent f o r each j * I ( j , A = 1, 2, . . . , k ) . J * As we po in t out l a t e r , t h i s assumption may be q u i t e u n r e a l i s t i c f o r the phenomena be ing model led and t h i s may cons ide rab l y a f f e c t the u t i l i t y of the theory to be developed on the ba s i s of t h i s assumption. 4.1 In ferences f o r T o t a l Model when E j ^ N(<j>, I a l ) We now address the problem of i n fe rence f o r the t o t a l model when the £j f o l l o w a m u l t i v a r i a t e normal d i s t r i b u t i o n w i t h e xpec ta t i on vec to r <j> and covar iance mat r i x l a ? , where $ and I have been de f i ned e a r l i e r . Note that we have e x p l i c i t l y s p e c i f i e d the form of the d i s t r i b u t i o n of e. here s ince such s p e c i f i c a t i o n i s necessary f o r i n f e r e n c e . J Let us beg in by supposing that i t i s de s i r ed to d i s cove r how w e l l the independent v a r i a b l e s i n the t o t a l equat ion e x p l a i n the observed v a r i a t i o n i n the components of y^,. To answer t h i s que s t i on , one needs to determine the amount of v a r i a t i o n i n the components of y^, that can be a t t r i b u t e d j o i n t l y to these independent v a r i a b l e s . Th is i s the usua l sum of squares due to r e g r e s s i o n . Denote the uncor rected sum of squares due to f i t t i n g the u n r e s t r i c t e d t o t a l equat ion by SS (3 T) and that due to f i t t i n g a v e r s i o n of t h i s model r e s t r i c t e d so that a l l components of ĝ , other than the i n t e r c e p t component are set equal to zero by SS^Cg^) . I t i s easy to show that s s R ( e T ) = = ( Z j = 1 I. ') x ' t t ^ y.) (4.1.1) and S S R ( B ( ) T ) = g ( I y ) - ny 2 (4.1.2) i = l where y . T i s the i ^ component of y T and y T i s the mean of the components 16 of y,p. I t f o l l o w s that the de s i r ed r eg re s s i on sum of squares f o r f i t t i n g the t o t a l equat ion i s g iven by k t k SS R ( (3 * T |B 0 T ) = ( Z 0 ) X t ( Z y ) - n y T 2 . (4.1.3) j = l j = l As u s u a l , the co r r ec ted t o t a l sum of squares f o r the t o t a l equat ion i s _ n _ SS T = y£ y T - n y T 2 = Z y | T - n y T 2 (4.1.4) i = l and the e r r o r sum of squares i s g iven by SS E = y£ y T - S S R ( 3 T ) (4.1.5) or as the r e s u l t of s ub t r a c t i n g (4.1.3) from (4 .1 .4 ) , that i s , SS E = SS T - S S R ( B * T | B 0 T ) . I t can be shown e a s i l y that S S R ( 3 * T | B Q T ) i s a s s oc i a ted w i t h p degrees of freedom wh i l e SS_, i s a s s o c i a t ed w i t h n - p - 1 degrees of freedom. Furthermore, s ince SSrj, i s a s soc i a ted w i t h n - 1 degrees of freedom and SS T i s the sum of S S R ( B * T j and SS E , i t f o l l o w s from Cochran ' s theorem (see Hogg and C r a i g , 1970, p.393; Kempthorne, 1975, p. 57; Montgomery, 1976, p. 37) that s s r ( 3 * x | B 0 T ) and SSg are independent. By our d i s t r i b u t i o n a l assumption on ( j = 1, 2, k) we have that S S R ( g * T | B Q T ) / a 2 and SSg/of a re , r e s p e c t i v e l y , noncen t ra l ch i - square w i t h p degrees of freedom and non- t t ? c e n t r a l i t y parameter g*^ X Xg*^,/2a^, and c e n t r a l ch i - square w i t h n - p - 1 degrees of freedom. We have a l ready noted that they are i nde - pendent, whence i t f o l l o w s that a t e s t of s i g n i f i c a n c e f o r the t o t a l equat ion can be obta ined u s ing data and est imates r e l a t i n g to the component models w i thout a c t u a l l y f i t t i n g the t o t a l equat ion . An R 2 a s soc i a ted w i t h the t o t a l equat ion i s s i m i l a r l y obta ined. For examining the hypothes i s that some component of i s equal to ze ro , one i s o f t en i n t e r e s t e d i n c o n s t r u c t i n g conf idence i n t e r v a l s about the component or performing a direct t-test (see Montgomery, 1976, p. 325). In either case, one requires an estimate of the covariance matrix of 3 T > Denote the true covariance matrix of g\ by ^ and that of 8 T by Then we have that $j = ( X ^ ) - 1 ^ (4.1.6) and by the fact that B T = E ^ = 1 j-L and that the (j = 1, 2, .. ., k) are independent, we have J T = ( X ^ ) " 1 Z j = 1 o2.. (4.1.7) The results given i n (4.1.6) and (4.1.7) are completely basic and we s h a l l not venture to prove them here. The estimator of $ w i l l be given by | T = ( X t X ) " 1 . Z j = 1 5^ (4.1.8) „ 2 , th where O j i s the mean square error associated with f i t t i n g the j com- ponent equation. Hence confidence l i m i t s on the relevant component' of 3 T , say g , w i l l be given by hr 1 './a.n-p-l I t e j - l 5 P V ! (*-1-9> where t ,„ . i s the (1 - a/2) 100-th percentile of the central t -a/2,n-p-l r d i s t r i b u t i o n with n - p - 1 degrees of freedom and C i s the (£ + l ) s t diagonal element of the matrix (XfcX) ^. Note that £ takes integer values in the range 0 to p inclusive. The corresponding test based on the t - d i s t r i b u t i o n i s performed by computing and rejecting the n u l l hypothesis that 8 0 T = 0 i f | t 0 | > t ,„ _ _-. • Note also that the test in (4.1.10) can be derived as a special case of a more general approach i n the context of general linear hypothesis theory, as w i l l be shown i n a l a t e r part of the discourse. The inferences based on (4.1.9) and (4.1.10) are v a l i d i f (XfcX) ^ 18 is a diagonal matrix. In general, however, (XfcX) ^ is not diagonal and so results obtained from (4.1.9) and (4.1.10) can be misleading (see Montgomery, 1976, p. 326). This is so be cause both (4.1.9) and (4.1.10) and are based on the assumption that the elements of 3 ^ , such as 8 ^ 3j T for i * j , are independent. When (XtX) ^ is not diagonal, this assumption does not hold in general. Therefore, a test for 8 „ T = 0 Xi 1 versus 3 ^ * 0 must be constructed using the 'extra sum of squares' prin- ciple used in deriving (4.1.3) or using the general linear hypothesis theory alluded to above. The extra sum of squares principle is described in Draper and Smith (1981, p. 97 cf.) and in Montgomery (1976, pp. 326- 328) . Finally, for constructing a confidence interval about a true value, y ^ , corresponding to the x-coordinate, X ^ = (1, x - ^ r p » • • • » > where X _ is a row vector, one needs the covariance matrix $~ of y . Since y^ = X8,r,, we know from theory (e.g., Morrison, 1976, pp. 83-84) that i> = X(± T) Xfc = X(X tX)" 1 X t E k . a2.. (4.1.11) T Y T rT j=l j Hence the estimator of is given by yT L = X(X tX)" 1 Xfc . o2. (4.1.12) and, in particular, a- , the estimated standard error of y corres- y£T 1 1 ponding to the x-coordinate X ^ is 8 , n - l y A ) " 1 4^.1 ¥ • < 4 - l - 1 3 ) Therefore, the desired confidence interval for y is given by y 4 T ± [ x^Cx 'x ) " 1 x[T z)ml o]l* t a l 2 ^ x • (4.1.14) Results given above show that tests concerning specific components of 3 r r , and corresponding confidence intervals can be constructed using information obtained from analyses relating to the component equations. Thus there is no need to f i t the total equation in order to make infer- ences about i t . These observations pertain to the situation where e. <\i N(c|), la2.). We show below that the same holds when e. ̂ N(<f>, Va2) . 4.2 Inferences for Total Model when e. ̂ N(<j>, Va2.) J J Consider now the situation where e. ̂ N(6, Va2.) with V as defined J J earlier. It was stated in (4.0.2) that the BLUE for gj under this dis- tributional assumption is (XtV ̂ X) ̂ Xt V ^ . One must add here that this estimator is also a maximum likelihood estimator (MLE). It is desirable to motivate the derivation of (XtV ''"X) ̂ Xt V ^y^ , mainly to clear the way for it s use in making inferences. Accordingly, consider the model specification y. = Xg. + e . with e. ̂ K(<j>, Va2.) . Since V J J J J J is positive definite, there exists an n x n nonsingular matrix P such that V = PfcP. (4.2.1) Hence we have that (P1") lV P"1 = I. (4.2.2) Suppose one pre-multiplies the model y^ = Xg.. + by (Pfc) ̂ ; then one has (P t)~ 1y. = (P*) lX&. + (P*) 1 e. (4.2.3) which may be given equivalently by y*. = X*g. + e*.. (4.2.4) J J J Note that E y*. = X*g . = (P t)~ 1X B., since E e*. = $, and J J J J E(e*. e*5) = Io 2 so that we now have that z*. ̂ N(d>, Io 2). Therefore J J J J J a l l the theory developed in section 4.1 applies to (4.2.4). Applying OLS to (4.2.4) one obtains 20 &. = ( X * C X * ) " 1 X* f c y*. J J = [ x t ( p t p ) " 1 x f 1 x t ( p t p ) " 1 y j = ( X t V - 1 X ) " 1 X f c V _ 1 y , (A.2.5) which i s the gene ra l i z ed l e a s t squares e s t imato r g iven e a r l i e r . Note that t h i s type of f i t t i n g (GLS) which produces (A.2.5) i s a l s o commonly r e f e r r e d to as weighted l e a s t squares f i t t i n g and, i n more genera l p r e - s en t a t i on s , as minimum V-norm f i t t i n g (see Kempthorne, 1975). Re su l t s u s e f u l i n making i n fe rences when ^ N(<j>, Va^) are b a s i c a l l y s i m i l a r to those de r i ved above f o r E J ̂ N(<j>, l a p w i t h a few important d i s t i n c t i o n s as a r e s u l t of our t r an s fo rmat i on of y^ above. In t e s t i n g f o r s i g n i f i c a n c e of the t o t a l equat i on , f o r i n s t ance , the r eg re s s i on sum of squares i s g iven by k t- t- k S S_ (B*|e n T ) = ( Z |p X * ( 1 y V " n y V (4.2.6) ° T j = l J j = l 3 1 where X* and y * . are as de f ined above and y * i s the mean of the components of v*m = Z. ., y * . . Furthermore, the co r r e c t ed t o t a l sum of squares T J=l J f o r the t o t a l equat ion i s SS T = y * ^ y * T - n y * T 2 (4.2.7) and the e r r o r sum of squares i s SS E = y * T t y * T - S S R ( B T ) , (4.2.8) where SS^g^) i s g iven by the f i r s t term on the r i gh t -hand s i de of (4 .2 .6 ) . In these sums of squares, g\ i s as de f ined i n (4 .2 .5 ) . Aga in , SSr, (3* T I B__) la% and SS^/a 2 w i l l be d i s t r i b u t e d as noted e a r l i e r except K 1 1 (Jl 1 l i 1 that the n o n c e n t r a l i t y parameter a s soc i a ted w i t h the d i s t r i b u t i o n of the former i s now g r j , * t x t V ̂ XB*rr ,/2a 2 . As be fo re , i t i s c l e a r that a t e s t of s i g n i f i c a n c e f o r the t o t a l r e g r e s s i on equat ion and the a s soc i a ted R 2 are ob ta i nab le from data and a n c i l l a r y q u a n t i t i e s de r i ved from f i t t i n g 21 the component equat ions wi thout recourse to a c t u a l l y f i t t i n g the t o t a l equat ion . In l i g h t of (4.2.5) the covar iance mat r i x of Sj i s now given by = (X tV _ 1X) a 2 (4.2.9) and that of by $ T = (XfcV X X) 1 Z k = 1 o 2 (4.2.10) w i t h ^ and ^ obta ined r e s p e c t i v e l y by s imply r e p l a c i n g a? by a? i n the express ions ' f o r ^ and In t h i s con tex t , a 100(1 - a )% c o n f i - dence i n t e r v a l f o r 3 ^ i s g iven by hT ± t a / 2 , n - p - l [ C V Z j = l 5 j ] i ( 4 ' 2 - n ) wh i l e a corresponding t - t e s t i s obta ined by c a l c u l a t i n g = & * T " C \ £ E j = l ( 4 - 2 ' 1 2 ) and r e j e c t i n g the n u l l hypothes i s that 3 _ = 0 i f It* 0I > t ,„ £T a /2 , n - p - l In (4.2.11) and (4 .2 .12) , C* r e f e r s to the (£ + l ) s t d iagona l element of ( X f c V '''X) , w i t h £ s p e c i f i e d as be fo re . The l i m i t a t i o n s of the r e s u l t s g iven by (4.2.11) and (4.2.12) when ( X t V ''"X) ^ i s not d iagona l are equa l l y v a l i d here so that one must r e s o r t to us ing the e x t r a sum of squares p r i n c i p l e or genera l l i n e a r hypothes i s theory to ob ta i n v a l i d t e s t s . To con s t ruc t a 100(1 - a )% conf idence i n t e r v a l about a t rue va lue of y T , say y^, corresponding to the x - c o o r d i n a t e , X £ T = (1, x 1 T Xp,p) , f o r X ^ T a row v e c t o r , one can show e a s i l y that the covar iance mat r i x of y^, = P*" yj* i s es t imated by L = X ( X t v " 1 X ) ~ 1 X s E k . a 2 . (4.2.13) r y T J=l J There fo re , a conf idence i n t e r v a l f o r y i s g iven by *£T * [ X l I ( x V l « _ 1 4 * j = l 5 j ^ ' a / Z . n - p - l * ( 4 ' 2 ' U ) Once aga in , r e s u l t s obta ined i n t h i s s e c t i o n show that t e s t s con - ce rn ing s p e c i f i c components of B.J-. and a s soc i a ted conf idence i n t e r v a l s 22 are c o n s t r u c t i b l e from i n fo rmat i on r e l a t i n g to analyses of the component equat ions . Th is r e s u l t ho lds when ^ N((|>, V a 2 ) . We a l ready showed that i t ho lds when e. ^ N(d>, l a 2 ) . In both cases , our d e r i v a t i o n s are J J based on the assumption that y . and y . are independent f o r each j * I ( j , il = 1, 2, . . . , k) . Th is e s s e n t i a l l y completes our c o n s i d e r a t i o n of the problem of e s t ima t i on and i n fe rence f o r the t o t a l model when r_. = p. The problem of a d d i t i v i t y as de f i ned here when r^ = p i s mathematic- a l l y n i c e and, i n a sense, t r i v i a l . The problem, however, has obvious p r a c t i c a l l i m i t a t i o n s as a r e s u l t of r e q u i r i n g that r j = p s i n c e , i n p r a c t i c e , one would l i k e to r e t a i n i n each component equat ion on ly those of the p independent v a r i a b l e s that are s t a t i s t i c a l l y important . I t i s , t h e r e f o r e , of i n t e r e s t to cons ider the consequences f o r e s t i m a t i o n and i n fe rence f o r the t o t a l model when r. ^ p w i t h the p o s s i b i l i t y that J r. < p f o r a l l j . Th i s i s the s i t u a t i o n not cons idered by Kozak (1970). We cons ider t h i s case i n the next chapte r . CHAPTER V 5.0 ADDITIVITY WHEN r. £ p WITH r. < p FOR SOME j J J We now relax the requirement that each component model contain a l l the independent variables in the total equation. Specifically, i f the total equation contains p independent variables (each assumed important), we shall allow the component equations to contain only statis- t i c a l l y important independent variables among the p variables. Thus , the number of independent variables in component equation j , may be less than p and s t r i c t l y so for at least one j . This admits the possibility that r_. < p for each j provided that in that case each inde- pendent variable in the total equation is contained in at least one com- ponent equation. This w i l l be consistent with additivity as defined here. With X = (i>|Xi | . . . |X ) as defined earlier, we now consider models of the form yj = Xgj + e (j = 1, 2, k) (5.0.1) and Y X = E j = 1 Yj = XB T + e T (5.0.2) where the matrix X is common to the k + 1 models. However, the latter models differ from those specified earlier in the following respects. In (5.0.1), is a (p + 1) x 1 vector defined so that g^ has an inter- cept component with r^ £ p of the remaining p components nonzero and the other p - r. equal to zero. The relative positions of the zero J 23 24 and nonzero components in the last p cells of g.. w i l l be such as correspond to the presence or absence of particular X^ ( i = 1, 2, ..., p) in the j t n component equation. g T w i l l be as defined before with p + 1 nonzero components. Our definition of g.. implies that the effective X matrix, say Xj, corresponding to component equation j is necessarily different for each j except when g. and g have nonzero components in identical positions for j * £ ( j , I = 1, 2, k). Assume that e.. and e^, are distributed as specified in chapter IV and also that y. and y (hence J * e. and e ) are independent for j * £. J £ 5.1 Estimation when e. a* N(<j) l a 2 ) J 1_ We f i r s t consider the problem of estimation for the total equation when e. ̂ N (d>, la2.). F i r s t , we state an intuitive result for the e s t i -J J mation problem and then demonstrate i t s validity. Note that model (5.0.1) is equivalent to model (2.2.1) with a condition adjoined, namely, y. = Xg. + e. .3. b = <(> (j = 1, 2, k) (5.1.1) J J J where b is a vector of zeros corresponding to the vector of components q j of gj in (5.0.1) which are set equal to zero in the j*"* 1 component equation. It is clear that estimation relating to (5.1.1) can be achieved via con- strained minimization. Denote the resulting solution by g*^ (j = 1, 2, k) and the corresponding conditioned f i t by Yj = Xg*.. (j = 1, 2, k). (5.1.2) Furthermore, let ŷ , be as defined in chapter IV; we shall occasionally refer to ŷ , as the unconditioned total model. Also, let ŷ ,̂ , be termed the 'conditioned' total equation in a sense to be defined momentarily k - . . . . . and let y * T r = E. , Xg*.. Then the conditioned total predictive equation IL. j-1 j is given by 25 y TC = y T ~' ^ y T ~ y *TC^ (5.1.3) where the f a c t o r - 9*JQ i s a c o r r e c t i o n or c o n d i t i o n i n g f a c t o r which c o r r e c t s the uncond i t ioned f i t t e d t o t a l equat ion (obta ined i n chapter IV) f o r parameters that are set equal to zero i n some of the component equat ions . Note that (5.1.3) imp l i e s that y T C = Y * T C = X ^ ^ = 1 3*j a n d that ^TC - Z j = l * * j " g T " C*T ~ Z j = l ( 5 - 1 ' 4 ) In (5 .1 .4 ) , B ^ and 0^ are the e s t imato r of the parameter vec to r of the t o t a l cond i t i oned equat ion and that of the uncond i t ioned t o t a l equa t i on , r e s p e c t i v e l y . Equat ion (5.1.4) imp l i e s that the parameter vec to r of the cond i t i oned t o t a l equat ion i s es t imated by adding the est imates of the parameter vec to r s corresponding to the cond i t i oned component equa- t i o n s . Our immediate task i s to demonstrate that t h i s r e s u l t i s mathe- m a t i c a l l y v a l i d . To do so, we s t a te our problem as one of m in im i za t i on subject to c o n s t r a i n t s . Let us so lve the e s t ima t i on problem a s soc i a ted w i t h f i t t i n g y T = XB T + E t (5.1.5) subject to B T = Z j = ] _ &*.. (5.1.6) Note that 8*j i n (5.1.6) r e f e r s to the parameter vec to r corresponding to the cond i t i oned component equat ion j . Th i s problem i s so lved by m in im i z i ng the Lagrangian f u n c t i o n S ( g T , 6) = ( y T - X g T ) t ( y T - XB T ) + 2 9 ^ - Z * = 1 & * . ) , (5.1.7) where 2 6 t i s a vec to r of Lagrange m u l t i p l i e r s . Now, d i f f e r e n t i a t i n g (5.1.7) w i t h respect to the elements of B̂ , and 6, r e s p e c t i v e l y , one ob- t a i n s 9 S = 2X t XB T + 26 - 2X ty r j, (5.1.8) 98j 26 f f = ~ e T " z j = i **y (5-1-9) Equat ing both (5.1.8) and (5.1.9) to zero leads to X C X B T + 6 = X ^ (5.1.10) 3 T = Z j = 1 B*.. (5.1.11) Our s o l u t i o n vec to r g T must s a t i s f y both (5.1.10) and (5.1.11) . From (5.1.10) one has xcx g T = x ty T - e =>g T = (x^)-1 x̂ x - (x'x)"1 0 = 3 T - (,A)~h. (5.1.12) Now, ( 5 . 1 . 1 2 ) = * § = X t X(g_ - g_) and s ince 8 T = E k g * . by (5 .1 .11) , T T T J=l J one has that 6 = X t X (g - E k = g * . ) . (5.1.13) Hence, p u t t i n g (5.1.13) i n t o (5.1.12) one gets § T = g T - ( X ^ ' V x d j - E k = 1 g*.) = g T - ( g T - E k = 1 %*.), (5.1.14) a r e s u l t g iven e a r l i e r i n equat ion (5 .1 .4 ) . This e s t a b l i s h e s the v a l i d i of that r e s u l t . S ince g^, the parameter vec to r of the t o t a l cond i t i oned equa t i on , i s determined by g*^ ( j = 1, 2, k ) , i t i s important to d i s cus s the e s t i m a t i o n of S*. here. I t should be emphasized that g*. i s a J J (p + 1) x 1 vec to r hav ing p - r^ of i t s components equal to ze ro . We are e x p l i c i t l y assuming that the i n t e r c e p t component of g*^ i s nonzero. In e s t ima t i n g 6*j> i t i s important to recogn ize that one does so c o n d i - t i o n a l l y on some s p e c i f i e d components be ing assumed equal to ze ro . In the f o l l o w i n g , we cas t the problem w i t h i n the framework of genera l l i n e a r hypothes i s theory . Consider testing the general hypothesis H : = m, where 3. . is a (p + 1 ) x 1 parameter vector of the model ( 2 . 2 . 1 ) , K C is any matrix of s rows and p + 1 columns and m is a vector, of order s, of specified constants. We shall require that K*" be of f u l l row rank, that is r(K t) = s, where r(«) denotes the rank of the argument. One is inter- ested here in estimating 3 j under the null hypothesis H : = m. Designate the parameter vector under the null hypothesis by 3 * j . Using constrained least squares (see Searle, 1 9 7 1 , pp. 1 1 3 - 1 1 4 ) , the desired estimator is given by 3 * j = fL - ( X ^ ) " 1 K[K t(X tX)~ 1 K]" 1 (K'JL - m), ( 5 . 1 . 1 5 ) where 3 j is the unconstrained estimator of 3 j (with a l l independent vari- ables included) . When the hypothesis is of the form H : b^ = <j> for b a subset of 3 . of order q., we have Kfc = [I 0], m = s = q.. (5.1.16) Now partition 3^, g\ and (XfcX) 1 as follows where p. + q. = p + 1. Then the estimator of 3 * . is J J J f * 3 * . = \ • (5.1.17) J Vb " T T \ p . p .q . q .q . If X is partitioned as X= (X JX ), then the estimator in (5.1.17) is equivalently given by V P j P j P j P j ^ j q j P j P j y 28 Observe that i f the columns of X are orthogonal to those of X , then (XfcX) is block-diagonal so that Equation (5.1.19) expresses the expected fact that when the columns of X and those of X are orthogonal, then f i t t i n g only the last p. com- ponents of X w i l l yield the same b as f i t t i n g a l l components of X con- P j ditionally on the f i r s t qj having zero coefficients. The consequence for estimation of having a l l columns of X mutually orthogonal should be obvious from this. In general, the estimator of 3 for any linear model depends upon variables not included in the model, including those that are not known. When a subset of a set of predictor variables is s t a t i s t i c a l l y unimportant, i t is common practice to f i t an equation which simply ignores the unimportant subset. Unless the latter subset is. orthogonal to the important one in the original set, the resulting f i t w i l l not be conditioned in the sense defined above. Hence, the corresponding estimator of the parameter vector is different from that of a correspond- ing conditioned f i t , again unless orthogonality holds. .The estdmator is also of smaller order and is given by the non-null part of B* j in (5.1.19). It follows from (5.1.18) and (5.1.19) that one can correct the latter estimator and f i l l i t out appropriately to obtain the corres- ponding conditioned estimator. In view of this, the remainder of this chapter w i l l be based on conditioned component equations in the sense just defined. (5.1.19) 29 5.2 Inference when e. ^ N(<J>,Ia2) J J When t e s t i n g the s i g n i f i c a n c e of the t o t a l cond i t i oned equat i on , the r eg re s s i on sum of squares i s g iven by S S R ( 8 * T C | B 0 T C ) = ( Z j = 1 8 * ^ ) X t ( E k = 1 y . ) - n y T 2 . (5.2.1) Other sums of squares r e l a t i n g to t h i s p a r t i c u l a r t e s t i n g problem are obta ined i n an obvious way. We omit f u r t h e r d e t a i l s wh ich, aga in , are complete ly obv ious. We con s i de r , i n s t ead , the problem of t e s t i n g s pec i f hypotheses r e l a t i n g to the t o t a l cond i t i oned equat ion . I t has been demonstrated above that the e s t imator of the parameter vec to r g,^ f o r the t o t a l cond i t i oned equat ion i s determined by summing corresponding components of cond i t i oned component equat ions . A n a l y t i c - a l l y , there are two p o s s i b l e ways i n which a p a r t i c u l a r component of 8 might tu rn out to be zero under a d d i t i v i t y . F i r s t , a component of 8^^, may be zero as a r e s u l t of the c a n c e l l a t i o n law when adding negat ive and p o s i t i v e elements. While t h i s i s a r i t h m e t i c a l l y p l a u s i b l e , i t i s ' not reasonable g iven that we have assumed a p r i o r i that each independent v a r i a b l e i s s t a t i s t i c a l l y important. Secondly, i f a p a r t i c u l a r i n d e - pendent v a r i a b l e has an est imated c o e f f i c i e n t of zero i n each i n d i v i d u a l component equat i on , then the corresponding component of B ^ w i l l be zero . Aga in , t h i s i s u n l i k e l y s ince i t i s con t ra ry to the hypothes i s that each independent v a r i a b l e i s important. The fo rego ing suggests that an hypothes i s which s ta te s that some component (or vec to r of a subset of components of ^^Q) i s zero i s not a reasonable hypothes i s . In s tead, an hypothes i s that some component (or vec to r of a subset of components of &JQ) i s equal to c , where c i s a known s c a l a r (vector w i t h a l l i t s components) d i f f e r e n t from zero , i s a reasonable hypothes i s . The s p e c i f i c a t i o n of c may be based on 30 past exper ience or r e l a t e d ana ly se s . There are two ways i n which c may be s p e c i f i e d . The s p e c i f i c a t i o n of c may be d i r e c t as desc r ibed i n the preced ing paragraph. On the other hand, c may be s p e c i f i e d i n d i r e c t l y by s imply s p e c i f y i n g c . ( j = 1, 2, . . . , k) i n a s e r i e s of subhypotheses r e l a t i n g to each com- ponent cond i t i oned equat ion . We emphasize that the va lue of c i s not s p e c i f i c a l l y determined i n the l a t t e r case, but i t i s determined under a d d i t i v i t y of the cond i t i oned component equat ions when they are f i t t e d under the subhypotheses s p e c i f i e d by c^. We s h a l l r e f e r to the t e s t of hypothes i s concern ing c when c i s s p e c i f i e d d i r e c t l y as a d i r e c t t e s t of the hypothes i s c . The corresponding t e s t when c i s s p e c i f i e d i n - d i r e c t l y through the c^ ( j = 1, 2, . . . , k) w i l l be r e f e r r e d to as an i n d i r e c t t e s t . We use t h i s terminology on ly f o r i t s mnemonic appea l . Let the cond i t i oned t o t a l model s p e c i f i e d by g ^ be des ignated k as the f u l l model. R e c a l l tha t g r = g* . and that the sum of square r e g re s s i on due to f i t t i n g t h i s model i s g iven by S S R ( g * ^ | g ^ ^ J as g iven i n (5 .2 .1 ) . Now cons ider f i t t i n g a v e r s i o n of the f u l l model r e s t r i c t e d so that some s p e c i f i e d component(s) of g ^ i s ( a r e ) g iven by the s c a l a r (vector ) c . Le t the l a t t e r model be indexed by the parameter vec to r c ^ c • • g r . Note that g i s obta ined by f i t t i n g the model indexed by g „ X L I L x L sub ject to the f u r t h e r r e s t r i c t i o n that some s p e c i f i e d component(s) of g ^ i s ( a r e ) equal to c. Designate the sum of squares r eg re s s i on due • C C i c to f i t t i n g the model indexed by g^- by SS D (g* g ^ , ) . Th is sum of square I L K O I L i s g iven by U C I ̂ C \ _ / o C \ t „f-.-„k SSR(g*TC|g T̂C) = (g^r X t ( Z j = 1 y j) - n y T 2 . (5.2.2) c . S ince the model indexed by 3T(_, i s a r e s t r i c t i o n on the model indexed by gTC, i t f o l l o w s that SSR(g*^c | g^) ^ S S R ^ * T C I B 0TC ) * H e n C e t h e 31 sum of squares HO = S S R ( B * T C | B 0 T C ) - SS R(3* T C|3^ T C) (5.2.3) can be i n t e r p r e t e d as the e x t r a sum of squares f o r t e s t i n g the hypothes i s that a s p e c i f i e d component (s) of $ ^ i s (are) equal to c. Thus, us ing ijj(c) and the e r r o r sum of squares f o r the f u l l model leads to a d i r e c t t e s t "for c. Suppose, now, that c i s s p e c i f i e d i n d i r e c t l y by s p e c i f y i n g c^ ( j = 1, 2, . . . , k ) . As i n an e a r l i e r pa r t of the seque l , l e t 3*̂ be the parameter vec to r corresponding to cond i t i oned component model j . Suppose we f i t the model indexed by 3*j under the subhypothesis that ~ c some component of 3*j 1 S equal to c^. Th i s leads to e s t imato r s 3*j ( j = 1, 2, . . . , k ) , where the s u p e r s c r i p t c on 3*j denotes the f u r t h e r r e s t r i c t i o n c^. Again i f we denote the t o t a l cond i t i oned model w i t h the f u r t h e r r e s t r i c t i o n c imposed i n d i r e c t l y through the c^ as be ing c indexed by 3̂ ,̂, i t f o l l ows by our r e s u l t on a d d i t i v i t y that c k. c 3__ = 6*. • The r e g re s s i on sum of squares due to f i t t i n g the l a t t e r model i s g iven by SS R(3* T C|3^ T C) = ( ^ ( B ' . V ) X C ( Z k = 1 y . ) - n y / . (5.2.4) Once aga in , one has that S S R ( 3 * X C I 3 Q T C ) i S S R ( 3 * T C | 3 0 T C ) , so that again the sum of squares * * ( c ) = S S R ( B* T C|3 0 T C) " S S R(3* T C|3^ T C) (5.2.5) can be i n t e r p r e t e d as the e x t r a sum of squares f o r t e s t i n g the hypothes i s that a s p e c i f i e d component (s) of 3 r i s (are) equal to c v i a the c . Thus, us ing ^ * (c ) and the sum of squares e r r o r f o r the f u l l model leads to an i n d i r e c t t e s t f o r c. I t i s worth emphasizing that the procedure f o r t e s t i n g a hypothes i s concern ing c d i r e c t l y or i n d i r e c t l y i s a p p l i c a b l e whether c i s a s c a l a r 32 or a vector. Tests of hypotheses concerning individual components of 3 ^ ,Q using the usual t - s t a t i s t i c and/or univariate confidence intervals about such components can be obtained in a manner similar to that described in chapter IV provided an estimate of the covariance matrix of is available. These tests and confidence intervals are especially likely to be mislead- ing here, however, since the conditioning makes the components of 3 ^ cor- related unless the columns of X are orthogonal. Therefore, tests on individual components of B̂ -̂, are best performed as outlined in the preced- ing paragraph. On the other hand, a confidence interval can be constructed about y^x> say, corresponding to a given x-coordinate in an obvious way. 5.3 Estimation and Inference when e. ̂ N(<j>, Vo2.) J i _ Results presented in sections 5.1 and 5.2 relate to the distribu- tional assumption ^ N((j), la?). These results carry over to the case £j ^ N(<j>, Vo?), for V as defined in chapter IV with the obvious modifica- tion that wherever X, y^ and y^ occur, in the various expressions, they are replaced by (P1") ''"X, (P*") ̂y^ and (P*") ^y^, in that order, where V^F = V with P as defined in that chapter. Thus, most of the results w i l l involve V ^ as demonstrated before. In the following chapter, i t is demonstrated that the problem of additivity when T j = p can be treated as a special case of additivity when r^ i p. Such a demonstration provides a basis for constructing a unified theory relating to the additivity problem. CHAPTER VI 6.0 A GENERALIZATION OF THE ADDITIVITY PROBLEM Within the framework of the conditioning p r i n c i p l e described i n the preceding chapter, the f i t t i n g of the component and corresponding t o t a l equations when r^ = p can be considered as a problem of f i t t i n g subject to ' n u l l ' or ' t r i v i a l ' conditions. By n u l l or t r i v i a l condi- tions we mean here that no further conditions are imposed on the g. (j = 1, 2, k) beyond the basic a d d i t i v i t y requirement that Bj = BT- The point to observe here i s that there i s no require- ment that any component(s) of g.. be equal to zero. In terms of r e s u l t (5.1.3) i n the preceding chapter, t h i s implies that y* TQ = y T so that (5.1.3) reduces to y T C = y T- (6.0.1) Note that ŷ ,, ŷ ,̂ , and y* T£ were a l l defined i n the previous chapter. Thus the co r r e c t i o n factor or the conditioning factor i s i d e n t i c a l l y zero when r . = p. J In terms of a r e s u l t given i n (5.1.14), the above implies that g T = I k g*. (6.0.2) T J=l J so that (5.1.14) then reduces to B T = 8 r (6.0.3) Equations (6.0.1) through (6.0.3) suggest that the problem of a d d i t i v i t y when r j E p can be treated as a sp e c i a l case of a d d i t i v i t y when r.. ^ p. 33 34 In t h i s connection, the theory of estimation and inference described i n chapter V reduces to that presented i n chapter IV when r^ = .p. This gene r a l i z a t i o n i s s i g n i f i c a n t , at least t h e o r e t i c a l l y , since i t makes i t possible to v i s u a l i z e the a d d i t i v i t y problem as defined here as one very general problem which can be studied under one u n i f i e d theory of estimation and inference. CHAPTER VII 7.0 OTHER ASPECTS OF THE ADDITIVITY PROBLEM The development of the theory has, thus f a r , been based upon the assumption that y. and y (and hence e. and e ) are independent f o r each j ^ £ ( j , i = 1, 2, ..., k). As indicated e a r l i e r i n the t h e s i s , how- ever, there are examples of applications where t h i s assumption i s simply not tenable. The implications of t h i s , e s p e c i a l l y for inference, are well worth considering and w i l l be examined i n t h i s chapter. There w i l l be occasion also to consider other general complements of the add- i t i v i t y problem such as that of the matrix X not of f u l l column rank and of V not nec e s s a r i l y p o s i t i v e d e f i n i t e . These l a t t e r generalizations are useful when considering c e r t a i n classes of the general l i n e a r model. In p a r t i c u l a r , they permit the extension of the theory of a d d i t i v i t y as developed here to c l a s s i f i c a t o r y models which are generally associated with designed experiments and are ro u t i n e l y analysed using analysis of variance procedures. Furthermore, i t i s noteworthy that i t i s generally assumed i n most applications of regression analysis that the X matrix i s f i x e d (that i s , that the independent v a r i a b l e s are eit h e r known or are measured without e r r o r ) . Yet i t i s quite conceivable that the inde- pendent va r i a b l e s may themselves be random, j u s t as the dependent v a r i a b l e y, or they may be fixed but measured with erro r . It i s reasonable to consider b r i e f l y the implications f o r analysis of these p o s s i b i l i t i e s , at least f o r completeness. 35 36 7.1 The Case y., y (j * £) Dependent J As a preamble, suppose that X i , X2, . .. , X^ are multivariate normal random vectors, each of dimension m, with mean vectors ( i = 1, 2, k) and corresponding covariance matrices ( i = 1, 2, ..., k ) . k Now define U = E. , X. and suppose one i s interested i n the d i s t r i b u t i o n 1=1 1 of U. I f , in addition, i t i s assumed that the vectors X^ ( i = 1, 2, k) are independent, then'by a well-known theorem i n multivariate analysis (see Muirhead, 1982, p. 14), i t follows that U i s m-variate k normally d i s t r i b u t e d with mean vector u = E. -, u. and covariance matrix J i=l 1 i T T = £ k , % .. On the other hand, i f the vectors X. ( i = l , 2, ...,k) TU 1=1 T i 1 are dependent, we know only that EU = E k . y. (7.1.1) i=l 1 and t. = E k . t. + E E i . . , (7.1.2) TU 1=1 T i . . T i l where t . i s the variance-covariance matrix of X. and t . . i s the covariance T i 1 T i j matrix of X^ and X. ( i * j ) . Hence i t follows that k k U ^ (E. , u., E. , I. + E E I . . ) . Note that we have only s p e c i f i e d the 1=1 1 1=1 T i . . T i j parameters of the d i s t r i b u t i o n of U and not i t s form. Indeed, as far as i s known, without making any assumptions concerning the form of depen- dency of the summands in the d e f i n i t i o n of U, the exact d i s t r i b u t i o n of U under dependence i s la r g e l y an outstanding problem i n mathematical s t a t i s t i c s . However, as suggested above, given some knowledge of the form of dependency among the k vectors X^, i t i s possible to obtain a d i s t r i b u t i o n for U (Olkin, 1983, personal communication). Furthermore, one might surmize that the d i s t r i b u t i o n of U under dependence of the component vectors might be derivable as a multivariate generalization of the univariate analogue considered by Springer (1979, pp. 72-75). Even so, however, the explicit representation of such a distribution is likely to be nontrivial. The representation of $ given in (7.1.2) derives from a simple generalization of the univariate case to the multivariate case. For the analogous univariate result, see Mood, Graybill, and Boes (1974, p. 179). Note that independence of the vectors ( i = 1, 2, ..., k) implies that H 0, the null matrix, in (7.1.2). Now let U = y_ and X. = y. (j = 1, 2, ..., k); then for y., y J- J J J *• dependent for each j * I, i t follows that k k k J, x y„ = Z. - y. ^ (XE. .. 3., E. . t. + H I..), For simplicity in what T j=l J 2 J=l J J=l T J T i J follows, we shall mostly use t T to designate £ k i . + Z Z i , .-. Even given that the y^ (j = 1, 2, ..., k) are individually multivariate normal under dependence i t is not known what the distribution of ŷ , is exactly. What is known is that ŷ , is either multivariate normal or is not multi- variate normal (see Kale, 1970). Examples are found in the literature of linear combinations of normal random variables which are themselves (the linear combinations, that is) not normal (e.g., Rosenberg, 1965; Behboodian, 1972) and of marginally normal random variables whose joint distributions are not normal (e.g., Ruymgaart, 1973). These results of course generalize to vector random variables. The overall implica- tion of this is that lack of knowledge of the exact distribution of ŷ , and, in particular, i t s probable non-normality renders the construction of a small-sample theory of inference considerably more d i f f i c u l t . Given lack of knowledge of the exact distribution of ŷ ,, a small- sample theory of inference for the total regression model is constriictibl on the basis of normality of ŷ , i f one can demonstrate that the vectors 38 V j (j = 1, 2, ..., k) are j o i n t l y multivariate normal. This i s the case because i t i s well-known that i f yx, yz , ..., y, are j o i n t l y m u l t i - K. variate normal, then every l i n e a r function of these y ^ ' s i s multivariate normal. (Note that they.'s are vectors here.) This r e s u l t follows from a cha r a c t e r i z a t i o n of the b i v a r i a t e normal d i s t r i b u t i o n which gener- a l i z e s to other j o i n t multivariate normal d i s t r i b u t i o n s (see Rao, 1965, pp. 437-438). Therefore, i n our instance, under dependence of the y^'s, normal theory can be used to construct inferences concerning y^ or or both i f i t can be shown that y i , y2» •••> y, are j o i n t l y multivariate K. normal. This suggests the need for methods of assessing multivariate normality based upon r e a l i z a t i o n s of the vectors y i , y2> •••> y^- Graphical methods of assessing multivariate normality have been proposed in t h e \ l i t e r a t u r e (e.g., Healy, 1968; Cox, 1968; Andrews, Gnanadesikan, and Warner, 1973). Other authors have proposed a n a l y t i c a l s i g n i f i c a n c e tests for t e s t i n g for multivariate normality (e.g., Malkovich and A f i f i , 1973; Hawkins, 1981). More recently, however, Koziol (1982) introduced a test for assessing multivariate normality which i s f a i r l y easy to use and has some nice properties. If a test for j o i n t m u l tivariate normality such as Koziol's (1982) leads one to entertain j o i n t multivariate normal- i t y , then one proceeds to make inferences concerning ŷ , or based upon the usual normality assumptions. I f , on the other hand, j o i n t m u l t i v a r i - ate normality i s rejected, then one can eit h e r appeal to asymptotic r e s u l t s to construct approximate tests or resort to nonparametric approaches. We s h a l l discuss the l a t t e r approach only b r i e f l y i n th i s t h e s i s . But f i r s t , l e t us tackle the problem of estimation. 39 7.1.1 Estimation for Total Model under Dependence As observed e a r l i e r , estimation should not, in general, be hampered by lack of knowledge of the d i s t r i b u t i o n of ŷ , and, in p a r t i c u l a r , by i t s non-normality. Consider estimation for the t o t a l model when e. a, N(d>, la2.) and when e. ̂ N(<b, V a 2 ) . It i s demonstrated i n t h i s J J J J section that when the y^'s are dependent, the concept of a d d i t i v i t y , as defined here, does not hold. This follows from the following reasoning. When the e. are dependent, one has that k e m ^ (<f>, £• •, $• + £ E t • •) in general. In the case where T T i=l ' i . . T i i i * j J ^ N((j>, l a 2 ) (j = 1, 2, k) , the f i r s t term i n the variance-covar i - ance matrix of ê , reduces to l a 2 , , where a 2 = £j=]_ ° j • Furthermore, i t i s easy to see that t . . = Ip..a.a. so that E E I.. = I E E p..a.a.. + i j i j 1 J ^ + U i 5 e j i J i J Therefore, under dependence of the with ^ N(4>, Io 2) , ^ (<j>,Ia2) where a 2 = £ k a 2 + E E p..a.a.. Thus the variance-T Y 1=1 i . . i i l l i * j J J covariance matrix of Erj, i s diagonal. On the other hand, when e. ^ N(d), V a 2 ) , the f i r s t term i n the variance-covariance matrix of e_ J J T becomes Va2,, with a2, as defined above, so that e ^ (cf>, Va2, + E E $^.) where i . . i s not diagonal. Recall that under the assumption of independence of the y^'s and V p o s i t i v e d e f i n i t e , the transformation matrix P, such that PfcP.= V, ••was the same for each component model and the t o t a l model. As a consequence, a d d i t i v i t y followed n a t u r a l l y since § T = (X t v" 1X)" 1X t v" 1y T = ( X t v " 1 X ) " 1 x V 1 E k = 1 y.. = (X tv" 1X)" 1X tv" 1[yi + ... + yfc] = £ k (7.1.1.1) J=l J 40 Note that the above r e s u l t holds a l s o when V = I. Now suppose that under dependence of the ( j = 1, 2, ..., k ) , the variance-covariance matrix of may be w r i t t e n i n the form V^o 2, f o r some matrix Vj. F i r s t , note that when ^ N(<t>, Vo-^) w i t h V p o s i t i v e d e f i n i t e , there i s no guaran- tee that i s p o s i t i v e d e f i n i t e , though we know that i t i s at l e a s t p o s i t i v e s e m i - d e f i n i t e . Secondly, even i f were p o s i t i v e d e f i n i t e , i t i s obvious that the matrix P^ such that = P^P^ i s not n e c e s s a r i l y equal to the matrix P which transforms each of the component equations. This holds when ^ N(<|>, i a ^ ) because Io" T 4 Io2. Consequently, when the y^ are dependent, 3^ cannot be determined from the a d d i t i v i t y property. This r e s u l t must obviously hold f o r more general V and Vj. Assuming p o s i t i v e d e f i n i t e n e s s of V and V^, n o n a d d i t i v i t y i s demonstrated as f o l - lows. Note that § T = ( X t V ^ 1 X ) " 1 X t V T " 1 y T = (X f cV T lX) 1 X t V T 1 Z k = i y = E k 6 * j = l j + E k § = ( x t v" 1X)" 1X t V _ 1 2 k y.. (7.1.1.2) J=l J J=l J Indeed, e q u a l i t y holds only i f V\j, = V which i m p l i e s that E „ = 0 f o r a l l i 4 . j and, t h e r e f o r e , independence of the Y j ' s - The above r e s u l t s suggest that when the Y j ' s a r e dependent, the parameters of the t o t a l equation must be determined by a c t u a l l y f i t t i n g the corresponding t o t a l equation r a t h e r than from a d d i t i v i t y . An exception to t h i s would be i n those cases where dependence i s so weak that the f i r s t term i n the covariance matrix dominates the second term (namely E I t..) i n the sense that the e n t r i e s of each are clo s e to zero. But t h i s simply i m p l i e s that independence l a r g e l y o b t a i n s . 41 S ince i n f e r ence f o r the t o t a l model i s e s s e n t i a l l y l i n k e d to e s t i m a - t i o n , i t f o l l o w s that i n fe rences concern ing parameters of the t o t a l model when the y a re dependent can on l y be made a f t e r f i t t i n g the r e l e van t t o t a l models d i r e c t l y . There fo re , under dependence of the y , one must a c t u a l l y f i t the t o t a l model i n order to es t imate i t s parameters and make i n f e rence s about them. Let us r e t u r n to the problem of i n f e rence under dependence. S i n ce , as observed e a r l i e r , e ,̂ may or may not be m u l t i v a r i a t e normal , one has s e v e r a l op t i on s . F i r s t , one can t e s t f o r j o i n t m u l t i v a r i a t e no rma l i t y (JMVN) of y 1 } y 2 , y, along the l i n e s of K o z i o l (1982). I f the t e s t shows that j o i n t m u l t i v a r i a t e no rma l i t y i s t enab l e , then one uses normal theory to make i n fe rence s concern ing the t o t a l equat ion as desc r ibed below. I f j o i n t m u l t i v a r i a t e no rma l i t y i s not t enab l e , one may examine y T = Yj d i r e c t l y f o r n o r m a l i t y , s i n ce l a c k of j o i n t m u l t i v a r i a t e no rma l i t y does not r u l e out the p o s s i b i l i t y t ha t y^ i s normal . I f both j o i n t m u l t i v a r i a t e no rma l i t y of the y . ' s ( j = 1, 2, k) and no rma l i t y of y^, by d i r e c t examinat ion of E j - j Yj a r e n o t t enab l e , then one may use non- parametr ic procedures. I f c e r t a i n c ond i t i o n s a re met, one may use asymp- t o t i c r e s u l t s t o a r r i v e at approximate i n f e rence s (see A r no l d , 1981, s e c - t i o n s 10.1, 10.3) when y T i s not normal . Th is l a t t e r i s not cons idered f u r t h e r here . However the other approaches a re cons idered b r i e f l y below. 7.1.2 In ference f o r T o t a l Model when y j , y 2 , y^ are JMVN When a procedure f o r a s se s s ing j o i n t m u l t i v a r i a t e no rma l i t y such as K o z i o l ' s (1982) leads to the conc l u s i on tha t the assumption of j o i n t m u l t i - v a r i a t e no rma l i t y i s reasonable f o r y^, y 2 , y^> one t r e a t s y m = E. , y. as m u l t i v a r i a t e normal w i t h mean Xg,,, and v a r i ance - cova r i ance T J= l J T mat r i x r = E k , + E E The no rma l i t y of y when y i , y 2 , y, i J 1 3 X J I K are j o i n t l y m u l t i v a r i a t e normal i s a standard r e s u l t i n mathematical s t a t i s t i c s as suggested i n section 7.1. Recall that i f e. ̂ N(4>, l a 2 ) (j = 1, 2, .... k) , then with the e. 3 3 3 k ? j o i n t l y m u l t i v a r i a t e normal, £ ^ N(4>, I(£. - erf + E E P..o".a,)). J ' ' T ' j = l J ^ i j l j " With these conditions, f i t t i n g y^ = X6 T + d i r e c t l y by ordinary least squares y i e l d s BLUE's for 3 „ and a^ = E. a f + E E p . . a . a . i n the sense 4 T j = l j ± ^ i ] i ] of the Gauss-Markoff theorem. These estimators, denoted by 3 ^ and cr = MSE, r e s p e c t i v e l y , are also maximum l i k e l i h o o d estimators. I t i s note- worthy that the components of o are not estimable d i r e c t l y from the t o t a l model. However, the usual analysis of variance t e s t s f o r the t o t a l model based upon the above estimates are v a l i d and confidence i n t e r v a l s may be constructed on i n d i v i d u a l components of 3 ^ , and on y^ e s s e n t i a l l y as explained i n chapter IV. This approach i s also a p p l i c a b l e i f a d i r e c t examinat ion of ŷ , = ^ j = 1 Yj suggests that i t i s normally d i s t r i b u t e d . When e. ̂ N(<j>, Va?) and the £. are j o i n t l y m u l t i v a r i a t e normal, then k E _ i/ N(<|>, VE. - ar + E E $..). I f i t i s possible to write J = 1 3 1*3 1J VE. - a? + E E i . . as V^a 2 for some p o s i t i v e d e f i n i t e matrix V m, then i t 1=1 J ± ? £ j i j T T T' i s c l e a r by r e s u l t s given i n chapter IV that there e x i s t s a matrix P^ such that V^ = P ^ P r p . Therefore, generalized l e a s t squares applied to the t o t a l equation y i e l d s BLUE's which are also maximum l i k e l i h o o d estimators. Tests of hypotheses concerning 3,^, or i t s components can be achieved as outlined i n section 4.2 I f , on the other hand, the variance-covariance matrix of £rri i s simply p o s i t i v e semi-definite rather than p o s i t i v e d e f i n i t e , then an approach such as i s used by Zyskind (1967) and Zyskind and Martin (1969) may be employed f o r estimation and inference. 43 7.1.3 I n fe rence f o r T o t a l Model when y^, y 2 , . . . . y^. a re not JMVN I t has been noted above that when y^, y 2 , y, a re not j o i n t l y K m u l t i v a r i a t e normal , i t i s s t i l l p o s s i b l e tha t y_ = E. .. y . i s m u l t i - T j = l v a r i a t e normal. I t has a l s o been i n d i c a t e d tha t when y^ i s m u l t i v a r i a t e normal, then e s t ima t i on and i n fe rence s r e l a t i n g to the t o t a l equat ion f i t t e d d i r e c t l y can be c a r r i e d out as o u t l i n e d i n the preceding s e c t i o n . When y i , y 2 , y a re not j o i n t l y m u l t i v a r i a t e normal and K. y T = Yj I s n o t normal, one of the opt ions l e f t f o r i n fe rence s f o r the t o t a l model i s v i a use of nonparametr ic procedures. I t i s not the o b j e c - t i v e here to pursue the subject i n d e t a i l bu t , f o r purposes of completeness, to i n d i c a t e what procedures a re a v a i l a b l e and p o s s i b l e r e f e r ence s . Randies and Wolfe (1979) g i ve a nonparametr ic approach to t e s t i n g the s lope i n s imple l i n e a r r e g r e s s i o n . C l e a r l y , t h i s i s of l i m i t e d use i n our con tex t . However, both the e s t ima t i on problem and i n f e r e n c e procedures f o r more genera l r e g r e s s i o n problems are cons idered i n chapter 9 of Ho l lander and Wolfe (1973). Other methods of d e a l i n g w i t h non-normal i t y of y^, i n e s t i m a t i o n and i n f e r ence f o r the t o t a l equat ion i s to use any of a number of s o - c a l l e d robust r e g r e s s i o n techn iques . One such techn ique i s known as robust r i d g e r e g re s s i on proposed by Hogg (1979). For f u r t h e r re fe rences to some of these robust r e g r e s s i o n techn iques see Montgomery and Peck (1982, s e c t i o n 9 .3) . F i n a l l y , be fore l e a v i n g the subject of nonparametr ic approaches and how they might be a p p l i e d to f i t t i n g a t o t a l model under non -no rma l i t y , i t i s worth mentioning two f a i r l y nove l non - parametr ic methods which a re a p p l i c a b l e to r e g re s s i on s i t u a t i o n s . These are the j a c k k n i f e and the boo t s t r ap . For a r e fe rence to use of the j a c k - k n i f e i n r e g re s s i on see M i l l e r (1974) and f o r a p p l i c a t i o n of boot s t rap techniques i n r e g r e s s i o n see E f ron (1979). 44 7.2 The Case of X Random or Measured w i t h E r r o r I t i s g e n e r a l l y assumed i n r e g r e s s i o n a p p l i c a t i o n s t ha t the i n d e - pendent v a r i a b l e s are e i t h e r f i x e d and known or t ha t they a re measured without e r r o r . Indeed, i n most of the development i n t h i s t h e s i s , t h i s has been assumed i m p l i c i t l y . In such s i t u a t i o n s , on l y the dependent v a r i a b l e y i s assumed random. In many b i o l o g i c a l a p p l i c a t i o n s , however, i t i s o f t en the case tha t both y and the independent v a r i a b l e s a re random. A l t e r n a t i v e l y , i t may w e l l be that the independent v a r i a b l e s a re i n f a c t f i x e d but a re measured w i t h e r r o r . In the f o l l o w i n g , these two p o s s i - b i l i t i e s a re cons idered i n the l i g h t of t h e i r i m p l i c a t i o n f o r e s t i m a t i o n and i n f e r e n c e f o r t he t o t a l model. A t t e n t i o n i s r e s t r i c t e d i n both cases to the s i t u a t i o n of independent v j ' s ' Without l o s i n g s i g h t of the a d d i t i v i t y problem i t w i l l s u f f i c e here to examine the consequence f o r e s t i m a t i o n and i n f e r ence on a component equat ion which, f o r s i m p l i c i t y , w i l l be r e f e r r e d to without the j s ub s c r i p t as y = X3 + e. 7.2.1 The Case of X Random Sampson (1974) d i s t i n g u i s h e s between two r e l a t e d r e g re s s i on schemes. One scheme i s that i n which the independent v a r i a b l e s a re constant o r f i x e d , as i s o f t e n assumed. He r e f e r s t o t h i s s imply as r e g re s s i on a n a l y s i s . The other i s t ha t i n which the independent v a r i a b l e s a re random v a r i a b l e s (or r e a l i z a t i o n s of random v a r i a b l e s ) . Th i s l a t t e r r e g r e s s i o n scheme i s r e f e r r e d to as m u l t i v a r i a t e a n a l y s i s of r e g r e s s i o n . We concern ou r se l ve s i n the present s e c t i o n w i t h the l a t t e r scheme. The o b j e c t i v e here i s not to p rov ide a d e t a i l e d a n a l y s i s of the s i t u a t i o n but to h i g h l i g h t the e f f e c t t ha t randomness of X may have on e s t ima t i on and i n f e r ence f o r component model j and, hence, f o r the t o t a l equat ion . The f o l l o w i n g i s l a r g e l y due to Sampson (1974). 45 The m u l t i v a r i a t e a n a l y s i s of regression scheme assumes that the vector y and the vectors c o n s i s t i n g of the columns of the matrix X form a mu l t i v a r i a t e random v a r i a b l e (or a r e a l i z a t i o n of a m u l t i v a r i a t e random v a r i a b l e ) . In the present case, i t w i l l be assumed that the j o i n t d i s t r i - bution i s m u l t i v a r i a t e normal. Denote the continuous random v a r i a b l e cor- responding to the independent v a r i a b l e by X (where X i s p-dimensional) and l e t X* be a r e a l i z a t i o n of X. The random v a r i a b l e corresponding to the dependent v a r i a b l e i s denoted by y and i t s r e a l i z a t i o n by y*. With a t t sample of s i z e n ((y^, x^) , i = 1, 2, n), l e t ẑ , = (y^, x./) and the corresponding r e a l i z a t i o n s {(y^, x * ) , i = 1, 2, n) and (z* , i = 1, 2, n). In the mu l t i v a r i a t e analysis of regression, i t i s assumed that f o r 1 _< i _< n, z^ are independently and i d e n t i c a l l y d i s t r i b u t e d according to N(<|>, $) . In the mu l t i v a r i a t e analysis of regression model, the parameters equivalent to 8 and a i n the regression a n a l y s i s model are $22 $21 and $ n - $ i 2 $22 $21, where squared error l o s s . Thus, when one speaks of regression c o e f f i c i e n t s i n the m u l t i v a r i a t e analysis of regression s i t u a t i o n , one speaks of (7.2.1.1) and $21 i s p x 1. As stated by Sampson (1974), the j u s t i f i c a t i o n f o r the appropriateness of $ 2 2 $ 2 i as a parameter vector i s that f o r 1 <̂ i _< n, E(y^ - x^ y ) 2 i s minimized f o r y = $22 $21» so that x^ $22 $21 i s the best l i n e a r predictor of y. i n the sense of minimizing 46 probability distribution law of the argument. The relationship between regression analysis and (7.2.1.2) should be f a i r l y obvious. Without going into further technical details, we state results concerning estima- tion and inference in multivariate analysis of regression and how they relate to corresponding results in regression analysis with fixed or nonrandom X. An important result concerning estimation in multivariate analysis of regression i s that although the maximum likelihood (ML) estimators are necessarily d i f f erent from those in regression analysis (mainly because they are defined on different sample spaces), the corresponding ML e s t i - mates under the two models are exactly the same. Thus estimation under the two model formulations is the same. However, Sampson (1974) shows that for testing hypotheses in the two situations, the power functions are different. This i s a significant result in that i t stresses the importance of using a correct model in order to obtain tests with the correct power. This result is of considerable relevance in the present and other biological applications where X may in fact be random rather than fixed as i s often assumed in regression situations. The implication for additivity is that testing i s obviously affected by randomness of X but not estimation. 7.2.2 The Case when X is Measured with Error In regression situations where the independent variables may reason- ably be considered fixed, i t is conceivable that an error may be intro- duced when measuring X at i t s fixed value. It i s noteworthy that this problem i s not necessarily the same as that of random X unless further assumptions are made about both X and y. The main objective here is to demonstrate the effect upon estimation and inference when X is measured w i t h e r r o r . For s i m p l i c i t y , r e s t r i c t a t t e n t i o n to the s imple l i n e a r r e g r e s s i o n model y = B 0 + B iX + E , (7.2.2.1) where i t i s assumed t ha t e ^ N ( 0 , a 2 ) and cov (e . , e.) = 0 f o r ±4 j . Now i f X i s measured w i t h e r r o r , one does not observe X d i r e c t l y but r a the r observes X ' = X + 6 (7.2.2.2) where X i s the t r u e v a l ue of X and 6 i s a measurement e r r o r . Suppose tha t 6 ^ N (0 ,o 2 ), X ^ N(u , a2.) and tha t e, 6, and X a re independent. Then Y 0 A A and X ' f o l l o w a b i v a r i a t e normal d i s t r i b u t i o n (Snedecor and Cochran, 1973) and the r e g re s s i on of Y on X ' i s l i n e a r w i t h r e g r e s s i o n c o e f f i c i e n t Bj = B i/(1 + X ) , (7.2.2.3) where X = o 2 / a 2 . Thus i t i s the case tha t when X i s measured w i t h e r r o r , £ X our l e a s t squares es t imate of the r e g re s s i on of Y on X i s b ia sed i n tha t i t underest imates the t r u e r e g r e s s i o n c o e f f i c i e n t from f i t t i n g Y on X. When X i s not normal , t he above r e s u l t ho lds i n l a r g e samples and holds a p p r o x i - mate ly i n sma l l samples i f X i s sma l l (see Snedecor and Cochran, 1973). I n fe rences concern ing y or the r e g re s s i on c o e f f i c i e n t a re v a l i d i f X i s measured w i t h e r r o r p rov ided that £, 5, and the t r u e X are approx imate ly normal . However, p r e d i c t i o n s of y a re l e s s p r e c i s e because of the i nc rea se i n r e s i d u a l s as a r e s u l t of e r r o r s i n X. The r e s u l t s g i ven above have some re levance i n the a d d i t i v i t y problem. More i m p o r t a n t l y , they po i n t to the need f o r a proper r e g re s s i on approach i f proper es t imates and i n fe rences are to be made. For other aspects of t h i s problem see Wald (1940), Berkson (1950), and Madansky (1959). 48 7.3 Other General Complements In more general applications of the general l i n e a r model, i t i s not uncommon that the matrix X i s not of f u l l column rank. Suppose, f i r s t , that ^ N(<f>, l°j) • Observe that XfcX i s singular and, therefore, a unique solution does not e x i s t f o r the least squares problem. An optimal solution i s obtainable, however, by using the well-known concept of a generalized inverse. Let us begin by considering a p a r t i c u l a r generalized inverse, one commonly ref e r r e d to as the Moore-Penrose inverse (Moore,1920; Penrose, 1955) but also often c a l l e d the pseudo-inverse or, simply, p-inverse. Attention i s r e s t r i c t e d to r e a l X throughout; but f i r s t some d e f i n i t i o n s are in order, given here as theorems. Theorem 1. Suppose X i s a r e a l n x (p+.l) matrix with rank (X) = r < p + 1. Then the (p + 1) * (p + 1) matrix XtX has exactly 2 2 2 r p o s i t i v e eigenvalues Xi 5 A2 5 ••• = ^ r > 0 plus the zero eigenvalue with m u l t i p l i c i t y p + 1 - r . The next theorem i s based on a well-known theorem in matrix algebra c a l l e d the Singular-Value Decomposition theorem. Theorem 2. With X s a t i s f y i n g theorem 1, one can always f i n d an n x n orthogonal matrix U and a (p + 1) x (p + 1) orthogonal matrix G such that A = uScG and X = UAG*" with A the n x (p + 1) matrix 'D 0' A =( where D i s an r x r diagonal matrix with i 1 " * 1 diagonal element d.. = X. > 0 for 1 S i i r . The expression of X i n the form 11 1 X = UAG*" i s termed the singular-value decomposition of X. One must remark that U and G i n the above theorem are not neces- s a r i l y unique. The r e a l importance of theorem 2 la that a.decomposition 49 of the matrix X e x i s t s , a r e s u l t which leads to d e f i n i t i o n of the Moore- Penrose inverse as follows. Theorem 3 . I f , from the n x (p + 1) matrix A of theorem 2 , one defines A + a s the (p + 1) x n matrix then the Moore-Penrose inverse (pseudo-inverse) of the matrix X i s given by X + = G A V 1 , where G and U are as s p e c i f i e d i n theorem 2 . With X + defined as above, an optimal least squares solution for a model of the form y = X 8 + £ i s given by B = X +y. ( 7 . 2 . 1 ) Pertaining to the a d d i t i v i t y problem of the discourse, r e s u l t ( 7 . 2 . 1 ) implies that £. = X +y. ( 7 . 2 . 2 ) J J for component equation j (j = 1 , 2 , ..., k) with g T = £ k -j B. ( 7 . 2 . 3 ) T J=l J for the t o t a l model. Note that inference theory r e l a t i n g to the t o t a l model as discussed elsewhere i n the thesis now incorporates X + in an obvious way. Note, for instance, that when ^ N(t(>, I°j) » i t i s the case that the covariance matrix of 8^ i s given by $ S = X + X + t Z k . a2., ( 7 . 2 . 4 ) T 3 T j = l j a r e s u l t which can be derived e a s i l y from ( 7 . 2 . 2 ) and ( 7 . 2 . 3 ) . 50 When y. ^ N(<f>, Vo.), using our usual transformation, g. i s given by J (7.2.5) so that (7.2.6) In (7.2.5) and (7.2.6), X T i s the Moore-Penrose inverse of * t -1 * t - 1 X = (P ) X and y^ = (P ) y . Again, the incorporation of these r e s u l t s i n inference theory i s a straightforward exercise and i s omitted here. However, some remarks are in order with respect to X (or X ). F i r s t , i t i s noteworthy that although U and G are not necess a r i l y unique in the decomposition of X given by theorem 2, the Moore-Penrose inverse X (or X ) i s unique. Therefore, d i f f e r e n t U and G w i l l lead to the same X + and, hence, the same optimal solution §. It was mentioned e a r l i e r that when X i s not of f u l l column rank, there i s no unique solution to the least squares problem of f i t t i n g y = X£ + e or i t s corresponding transform. While t h i s i s so, i t i s remarkable that the solution (7.2.1) or i t s transformed version i s optimal i n the sense that i t i s the only solution giving least 2-norm; that i s , i t i s the best so l u t i o n to the least squares problem. When X i s square and nonsingular, then X + = X \ the unique inverse of X. F i n a l l y , i t i s important to mention that the r e a l p r a c t i c a l usefulness of X + hinges upon the ease with which i t can be determined in any one problem. It turns out that X + i s r e l a - t i v e l y easy to compute when X has a few columns. However, the task of computing X + becomes increasingly more d i f f i c u l t with an increasing number of columns i n X. Since many p r a c t i c a l problems tend to involve an X matrix with a f a i r l y large number of columns ( e s p e c i a l l y i n c l a s s i - f i c a t o r y models), the use of X + often presents a computational b a r r i e r . Largely because of t h i s , a more general (weaker) generalized inverse 51 i s used in many singular s i t u a t i o n s . The following i s only an introduction to t h i s type of inverse. For more complete treatments, see Searle (1971) and Rao and Mitra (1966). The Moore-Penrose inverse described above s a t i s f i e s the following conditions: (i) xx+x = X ( i i ) x+xx+ • = x+ ( i i i ) (x+x)fc = x+x (iv) (xxV = xx+ (7.2.7) If one defines a matrix X s a t i s f y i n g only condition (i) in (7.2.7), *t" *t" that i s s a t i s f y i n g XX X = X, then X i s termed a generalized inverse of X (see Searle, 1971). Unlike X , the Moore-Penrose inverse, X i s not unique. However, X^ i s considerably easier to compute than X +. Furthermore, any X^ has the property that i t generates a l l possible solutions r e l a t i n g to any given estimation problem and these solutions are invariant under a f f i n e transformations. The l a t t e r property i s of value with regard to estimation and inference for l i n e a r functions of the parameters in a given problem. It should be pointed out that X^ enters into inference theory i n much the same way that the Moore-Penrose inverse does. Further d e t a i l s r e l a t i n g to the use of x"*" are omitted here as they can be found elsewhere (e.g., Searle, 1971; Rao and Mitra, 1971). Most of the r e s u l t s presented so f a r are based upon the assumption that whenever e. ^ N(<f>, V a 2 ) , then V i s p o s i t i v e d e f i n i t e and known. J . J While t h i s i s commonly true and lends i t s e l f to f a i r l y straightforward mathematical manipulations, there are instances in which V i s not neces- s a r i l y p o s i t i v e d e f i n i t e . In addition, the elements of V may be unknown. The general approach i s indicated here for the case where V i s nonnegative d e f i n i t e and known. The case of V unknown i s considerably more d i f f i c u l t . As was the case when X was not of f u l l column rank, the concept of the generalized inverse i s employed when dealing with models where V i s not p o s i t i v e d e f i n i t e . In the general case, a sol u t i o n for model j (j = 1, 2, k) would be given by 3 . = (X tV tX) +X tV +y. (7.2.8) J J where V i s any generalized inverse of V. Corresponding r e s u l t s f o r a d d i t i v i t y and the associated inference problem generally correspond t . to those presented e a r l i e r with the obvious modification that V i s used in place of V ^. Other d e t a i l s are given i n Searle (1971, section 5.8) while another f a i r l y i n s t r u c t i v e approach i s given by Zyskind (1967) and Zyskind and Martin (1969). CHAPTER VIII 8.0 SOME EXTENSIONS OF THE THEORY It i s well-known that data sets generated under experimental condi- tions according to a predetermined design model can be analysed using the general regression approach. Indeed, although the conventional anal- y s i s of variance approach i s used in analysing most such data sets, the general regression approach often represents the most e f f i c i e n t and, at times, the only exact method of a n a l y s i s , e s p e c i a l l y for unbalanced s i t u a - t i o n s . In view of t h i s l i n k between regression analysis and conventional analysis of variance, i t i s reasonable to ask whether the problem of addit- i v i t y , as defined here, cannot be envisaged within the context of c l a s s i - f i c a t o r y models. It i s shown, i n t h i s chapter, that an extension of the a d d i t i v i t y problem to c l a s s i f i c a t o r y models i s not only t h e o r e t i c a l l y p l a u s i b l e but also makes sense in some p r a c t i c a l s i t u a t i o n s . Secondly, in view of the growing i n t e r e s t in the use of nonlinear models i n many branches of applied biology, i t i s of i n t e r e s t to investigate the extent to which the concept of a d d i t i v i t y , as understood here, can be expected to hold i n nonlinear s i t u a t i o n s . In a f o r e s t r y context, such an i n v e s t i - gation has an important bearing upon the determination of t o t a l volume or weight biomass of i n d i v i d u a l trees from corresponding component biomass using any of the well-known nonlinear models, such as the Chapman-Richards function. The main thrust of the development in t h i s chapter w i l l , 53 54 then, be directed towards e s t a b l i s h i n g the extent to which the theory of a d d i t i v i t y , as developed i n chapters IV, V, and VI, applies to c l a s s i - f i c a t o r y and nonlinear models. 8.1 Extension to C l a s s i f i c a t o r y Models A t h e o r e t i c a l j u s t i f i c a t i o n for the extension of the theory of addi- t i v i t y to c l a s s i f i c a t o r y models i s based upon the fact that any c l a s s i f i - catory model can be equivalently expressed i n l i n e a r regression form. A s p e c i a l feature to note about such a model i s that the incidence matrix, otherwise known as the design matrix, i s , i n general, not of f u l l column rank. Therefore, no unique solution e x i s t s for the estimation problem using least squares. Hence, one eit h e r uses the unique Moore-Penrose inverse to obtain an optimal solution or uses a generalized inverse to a r r i v e at a so l u t i o n . As indicated elsewhere i n the discourse, the d e c i - sion to use the Moore-Penrose inverse or a generalized inverse w i l l depend upon considerations of computational e f f i c i e n c y . Furthermore, the incor- poration of r e s u l t s from the estimation problem into inference involves the simple s u b s t i t u t i o n of expressions involving the appropriate general- ized inverse into s t a t i s t i c s derived i n e a r l i e r chapters. To indicate the p r a c t i c a l i t y of the a d d i t i v i t y problem i n the context of a c l a s s i f i - catory model, we describe below how such a problem might a r i s e i n p r a c t i c e . We draw our example from the f i e l d of a g r i c u l t u r e . For s i m p l i c i t y , consider a c o n t r o l l e d f i e l d crop experiment involv- ing a treatments, each r e p l i c a t e d n times. It i s a simple matter to recognize the design here as a completely randomized design. We s h a l l suppose that the leaf component of the biomass of the crop under i n v e s t i - gation i s used f o r human consumption as a vegetable. Further, suppose that the f l o r a l component of the crop i s used as a d i f f e r e n t type of 55 vegetable food. Next, suppose that the seed component i s used as another type of food while the remaining unusable above-ground part of the plant i s burned as a f u e l . If one i s interested i n the e f f e c t of treatment upon the accumulation of biomass (on a green weight basis) i n component j (j = 1, 2, 3, 4) and the corresponding e f f e c t on t o t a l biomass accumula- t i o n ( r e s u l t i n g from adding the four components), then one has the follow- ing problem. Designate the observed biomass of the j*"* 1 component corres- ponding to the r r e p l i c a t e by y.. ( i = 1, 2, ..., a; j = 1, 2, 3, 4; x j r r = 1, 2, ..., n). Then one would be interested i n models of the form y. . = y. + T.. + e . . (8.1.1) i j r j i j i j r and ? i T r = Z ] = l ? i j r " *T + T i T + E i T r > ( 8 ' 1 ' 2 ) with appropriate assumptions on the er r o r s . Note that by using an approp- r i a t e incidence matrix X, one may write (8.1.1) and (8.1.2), r e s p e c t i v e l y , in matrix form as and y.. = X6j + £j (j = 1, 2, 3, 4) (8.1.3) y T = z j = 1 y.. = X0 T + e r (8.1.4) We remark that 6. = ( u . , i , ,, r~ •, T . ) t while 6 T = ( y T , x 1 T , x 2 T , T ^ ) 1 " . C l e a r l y , the estimation and inference theory presented elsewhere i n t h i s thesis can be applied to (8.1.3) and (8.1.4) subject only to the proviso that a generalized inverse or the Moore-Penrose inverse i s used i n place of (X*"X) ^ or (XfcV ''"X) ̂ . The regression formulation (8.1.3) and (8.1.4) of the analysis of variance models (8.1.1) and (8.1.2) makes i t e s p e c i a l l y easy to estimate 9̂ sub- j e c t to the condition that c e r t a i n of i t s components are equal to zero. Furthermore, there i s no reason why the a d d i t i v i t y concept cannot be 56 applied to more complicated c l a s s i f i c a t o r y models such as the randomized complete block design, L a t i n square design, and other designs since, i n each case, one can write the corresponding models i n l i n e a r regression form. This demonstrates that the extension of the notion of a d d i t i v i t y of component regression equations to c l a s s i f i c a t o r y models i s not only t h e o r e t i c a l l y p l a u s i b l e but also appears to make sense i n p r a c t i c e . 8.2 Extension to Nonlinear Models A d e f i n i t i o n was given i n chapter II for a regression model in general and for a l i n e a r regression model i n p a r t i c u l a r . From those d e f i n i t i o n s , i t follows that any regression model r * e R* s a t i s f y i n g R* i L i s a non- l i n e a r regression model. Recall that L was defined i n chapter II as the set of a l l l i n e a r regression models. Conventionally, nonlinear r e - gression models are divided into two groups, namely the c l a s s of nonlinear regression models that can be made l i n e a r by applying an appropriate trans- formation to the nonlinear model and the clas s of nonlinear regression models for which there e x i s t s no known l i n e a r i z i n g transformation. The two types of nonlinear regression models are generally referred to i n the l i t e r a t u r e as i n t r i n s i c a l l y l i n e a r and i n t r i n s i c a l l y nonlinear, re s p e c t i v e l y (see Draper and Smith, 1981). The main objective in t h i s section i s to investigate whether the notion of a d d i t i v i t y does make sense for these two types of nonlinear models. We consider models of the form y.. = f(X, B..) + E j (j = 1, 2, k) (8.2.1) and y T = £ k = 1 Yj = f(X, B T) + e T. (8.2.2) Attention i s directed here toward discovering the extent to which model (8.2.2) i s a r i t h m e t i c a l l y determined by the models s p e c i f i e d i n (8.2.1). 57 For purposes of s i m p l i c i t y * we r e s t r i c t d e t a i l e d analysis to two types of nonlinear models, namely B, -X y.. = $ e i J e (j = 1, 2, k) (8.2.3) and B, -X y.. = 3 Q : j e i J + z (j = 1, 2, k ) . (8.2.4) Note that models s p e c i f i e d by (8.2.3) are i n t r i n s i c a l l y l i n e a r , so that for purposes of estimation, one may transform them to l i n e a r form using a logarithmic transformation. This leads to Zn y. = £.n(BQj) + g^X + UTI e^ (j = 1, 2, . .., k) (8.2.5) or simply y j * = B 0 j * + B l j X + ej* ( j = l ' 2 ' '•' k ) ' (8- 2- 6) On the other hand, models s p e c i f i e d by (8.2.4) cannot be so transformed t e c h n i c a l l y although the f i r s t member of the expression on the r i g h t of t h i s model i s l i n e a r i z a b l e . Indeed, (8.2.4) s p e c i f i e s the more simple forms of an i n t r i n s i c a l l y nonlinear model. Now consider f i t t i n g the l i n e a r i z e d form of (8.2.3) and suppose one i s interested i n t h i s ' l i n e a r i z e d form for p r e d i c t i o n purposes. I f , in addition, one i s interested i n the p r e d i c t i v e equation for the sum of the transformed form of the components, then the parameters of. the l a t t e r model are determined by a d d i t i v i t y from the component models. This i s the case since we have y T* - y.* - Z%x (6oj* + B l jx + Ej*) " B0j* + hi* + 'i* = B0T* + B1TX + eT*. (8.2.7) Thus, as i n the ordinary model, a d d i t i v i t y holds here as long as the variable of i n t e r e s t i s the transformed version of the dependent variable y.. 58 Kozak (1970) made t h i s point i n h i s paper and, therefore, i t i s not new. Suppose, on the other hand, that one i s r e a l l y interested i n the o r i g i n a l nonlinear form as a p r e d i c t i v e equation. In th i s case, the l i n e a r i z a - t i o n i s only an intermediate step aimed l a r g e l y at si m p l i f y i n g the process of estimation. Note that i n t h i s case some of the parameters estimated from the l i n e a r i z e d equation would need to be further transformed before being inserted i n a nonlinear p r e d i c t i v e equation. Now, i f the nonlinear f i t t e d analogue of (8.2.7) were desired, note that using a d d i t i v i t y one would need to back-transform the expression y T * = E k = 1 £n~y\ = Z k = 1 A n " ^ + ( E j = 1 5 l j)X (8.2.8) to obtain k * k ~ ( Z i = l g l i ) X y = n . y. = ( I I . 6 )e J J y T J L j = l y j U 1 j = l P 0 j ; e 3 X = I I k = 1 [ § o j e l j ]. (8.2.9) The r e s u l t i n (8.2.9) warrants some comment. Perhaps the most important of such comments i s the following. If one i s interested i n p r e d i c t i n g t o t a l biomass, say, as a sum of the components y^ using a model of the form (8.2.3), then one must not do so by invoking a d d i t i v i t y of the trans- formed version of (8.2.3) and then re-transform (that i s , back-transform). If one does so, then one gets an equation which predicts the product of the components rather than t h e i r sum. In a n u t s h e l l , one gets the wrong pr e d i c t i v e equation. Herein l i e s the r e a l v i r t u e of a proper analysis of a modelling s i t u a t i o n . Thus f or models of the form (8.2.3), a t o t a l p r e d i c t i v e equation of the same form cannot be determined from a d d i t i v i t y of the parameters of the transformed component equations. It may be determined at least a r i t h m e t i c a l l y , however, as a simple sum of the corresponding f i t t e d nonlinear component equations, though the p r e d i c t i v e 59 merits of such an equation may be debatable. With respect to models of the form (8.2.4), i t has been indicated that such models do not admit transformation to l i n e a r form. Therefore, the parameters would be estimated using any of a number of known i t e r a t i v e search techniques. As with models of the form (8.2.3), however, a pre- d i c t i v e t o t a l equation cannot be obtained here by appealing to the addit- i v i t y property since the parameters of the t o t a l p r e d i c t i v e equation cannot be determined by adding corresponding parameters in the component equations. However, i f simple p r e d i c t i o n was the objective, then a p r e d i c t i v e t o t a l equation may be obtained by simply adding up the p r e d i c t i v e component equations. Once again, the p r e d i c t i v e usefulness of such a model i s l a r g e l y an open question. The foregoing discussion indicates that the a d d i t i v i t y property, which holds almost u n i v e r s a l l y f or l i n e a r models, does not carry over, in general, to the class of nonlinear models. This precludes, for instance, the use of the notion of a d d i t i v i t y i n inventory and/or biomass studies i f nonlinear models are used for p r e d i c t i o n . CHAPTER IX 9.0 COMPUTATIONAL CONSIDERATIONS We have attempted to present, in preceding chapters, a theory of estimation and inference for the a d d i t i v i t y problem and to indicate general- i z a t i o n s and extensions to other types of models. The e s s e n t i a l objective of the discourse has been to present the a d d i t i v i t y problem as perceived here within the general framework of l i n e a r model theory. One hopes that t h i s objective has been achieved to a large extent. However, our derivation of expressions for estimators and associated s t a t i s t i c s , p a r t i - u l a r l y i n chapter V, leaves one important question l a r g e l y unanswered. This question i s : Does the conditioning p r i n c i p l e introduced to handle the a d d i t i v i t y problem in general c a l l f o r new computing subroutines or algorithms i n order to obtain estimates and other s t a t i s t i c s ? We show, i n t h i s chapter, that no such subroutines or algorithms are required. A l l estimates and associated s t a t i s t i c s can be computed using e x i s t i n g system-based software such as i s provided by the various s t a t i s t i c a l packages. Examples of such packages are MIDAS (The Michigan Interactive Data Analysis System, The University of Michigan, Ann Arbor, Michigan), BMDP (Biomedical computer programmes P-series, University of C a l i f o r n i a Press, Los Angeles, C a l i f o r n i a ) , SAS ( S t a t i s t i c a l Analysis System, SAS I n s t i t u t e , Raleigh, North C a r o l i n a ) , and SPSS ( S t a t i s t i c a l Package for the Social Sciences, McGraw-Hill Inc., New York). At computing 60 61 i n s t a l l a t i o n s where SAS i s not a v a i l a b l e , BMDP i s perhaps the most commend- a b l e package to use main ly because i t has opt ions f o r generat ing d i a g n o s t i c p l o t s and other i n f o rmat i on v a l u a b l e i n model cho i ce and v a l i d a t i o n . To mot i va te the d e r i v a t i o n of the main r e s u l t of t h i s chapte r , cons ide r the e s t ima t i on problem a s s o c i a t ed w i t h f i t t i n g component equat ion j ( j = 1, 2, k) as presented i n chapter V. More s p e c i f i c a l l y , r e c a l l t ha t i n f i t t i n g equat ion j , where equat ion j con ta i n s on ly s t a t i s - t i c a l l y important independent v a r i a b l e s , the e s t imato r f o r B..* i s g i ven by M - L i . l ' o.o.i) ib - T T b p., p,q, q 4q., q. 3 3 3 3 3 3 where b and b a re g iven by the p a r t i t i o n q • P • 3 3 A q \ B. = 3 b (9.0.2) of B obta ined from f i t t i n g the f u l l model (w i th a l l independent v a r i a b l e s ) w h i l e T and T a re obta ined from a corresponding p a r t i t i o n i n g P j q 3 q3 q3 of ( X t X ) ~ 1 i n the form /T T \ t -1 ^ q J P J | (X CX) 1 = I I . (9.0.3) \T T / V p . q . p.p./ 3 3 3 3 With a cor respond ing p a r t i t i o n i n g of the X ma t r i x i n the form X = (X |x ), the r e s u l t i n (9.0.1) i s g iven e q u i v a l e n t l y by (see q j P J equat ion (5.1.18) i n chapter V) § . * = ( , ) • (9.0.4) ( T _ T T ~ T )X y. p.p. p.q. q.q. q-p. p. J * 62 Consider now f i t t i n g y. on the set X on l y ( i . e . , so that the 3 P j c o e f f i c i e n t s of the components of a re set i d e n t i c a l l y equal to z e r o ) . I t i s shown below tha t the e s t imato r 6* , say, from the l a t t e r f i t i s p r e c i s e l y equal t o the n o n - n u l l pa r t of * i n ( 9 . 0 . 4 ) . Hence B \ * i s complete ly s p e c i f i e d by s imply f i t t i n g y. on the set X . The conse- 2 P j quence of t h i s r e s u l t i s t ha t no s p e c i a l a l go r i t hm i s necessary to ob t a i n 0j* beyond those a l r eady a v a i l a b l e on standard s t a t i s t i c a l packages. We demonstrate t h i s r e s u l t f o r m a l l y by s t a t i n g and p rov ing the f o l l o w i n g theorem. Theorem: Let x = ( X i , X2, . . . , ^} be a set of p r e d i c t o r v a r i a b l e s and cons ider f i t t i n g the l i n e a r model (w i t h i n t e r c e p t ) y. = Xg + e. J j J s ub i ec t to a subset b of 8. of order q. being equal to the zero v e c t o r . q j J 2 Here X = ( X Q | X X | . . . |Xm_^) i s an n x m (n > m) ma t r i x of f u l l column rank and we assume, as u s u a l , that ê . <v N(<J>, l f f ? ) • Denote the s o l u t i o n from f i t t i n g t h i s con s t r a i ned model by §..*• Now p a r t i t i o n X so tha t X = ( X I x ) f o r q + p . =m, 0 < q . < m i n accordance w i t h the c o n - q j p i i 2 2 s t r a i n i n g of 8.. so tha t (xtx)"1 Then B . * i s g i ven by - ( * 8. * = I J V T -T T - . - ; p i p j p i q i q j q i q j p i p j J _ 1 T ) X t y. J 63 Furthermore, i f b * i s the s o l u t i o n from f i t t i n g y. on the set X P- J P- J 3 ( tha t i s , i gno r i ng X ), then q j b * = (T -T T - 1 T ) X C y. . p. p.p. p.q. q.q. q.p. p. i J 3 3 3 3 3 3 H3 3 3 We prove the above theorem by u t i l i z i n g a wel l-known theorem i n l i n e a r a l geb ra concern ing the i n ve r se of a p a r t i t i o n e d m a t r i x . Th i s theorem i s s t a ted here as a lemma, w i thout p roo f , and we r e f e r the reader to G r a y b i l l (1976, p. 19) or any standard t e x t i n l i n e a r a l geb ra f o r a p roo f . Lemma; Let W be an n x n nons ingu la r ma t r i x t ha t i s p a r t i t i o n e d as f o l l o w s : /wn W 1 2 w = I \ W 2 i w22> where W„ has s i z e n^ x n_. f o r i , j = 1, 2 (n i + n 2 = n, 0 < n j < n) . I f |Wn| 4 0 and |w22| 4 0, then W ̂ i s g i ven by W -1 [ W I J - W I 2 W 2 2 "H?21 ] -Wu ^Wl2 [W22-W2iWi i "H/i2] ^ -W22 "Hj2j [Wii-Wj2W22 ^W2ll ^ [ W 2 2 - W 2 I W J I ^Wi2l ^ Proof of Theorem: The expres s ion f o r i§.* g i ven i n the theorem f o l l o w s 3 from Sea r l e (1971, pp. 113-114), as demonstrated e a r l i e r i n the t h e s i s . I t remains to show tha t the second pa r t of the theorem ho ld s . Wi th (X t X) p a r t i t i o n e d as i n the theorem and us ing the above lemma, i t f o l l o w s tha t ( X t X ) " 1 : 64 where T 1 —. q.q. 3 3 [xt x - xt x (xfc x )~1xt x ] 1 q. q. q. p. p. p. p. q. J 3 3 j J J J 3 Vj _ -(xfc x )_1xt x [xfc x -y?- x (xt x ) q. q. q- p. p. p. p. q- q- q. J j 3 3 3 3 3 3 2 3 x l"1 q3 Pj = -(xt x )~1xt x [xfc x -x11 x (x11 X ) p . p . p . q . q - q . q. p. •p. p -Vx r 1 P3 q3 [xfc x -xfc x (xfc x )"1xt x ]_1 . p. p. p. q. q. q. q. p. J J J J 3 3 3 3 Note that the existence of (X*" X ) and (Xt X ) ^ follows from q- q. P. P. the f a c t that X i s of f u l l column rank. Note a l s o that the s o l u t i o n b * obtained from f i t t i n g y. on the set X must also be given by P. J P. *J J 3 b * = ( X 1 X )" 1X t y. . P. P- P. P- 3 F J F3 3 3 Therefore, to complete the proof of the theorem, we need only show that (T -T T _ 1 T ) = (X t X To t h i s end, note that p.p. p.q. q.q. q.p. p- p- 3 3 3 3 3 3 H3 3 3 3 T - T T " 1T = [X C X -x' X (X t X )" 1X t X ] _ 1 p.p. p.q. q.q. q-p- p. p- p- q- q- q- q- P- 3 3 3 3 3 3 3 3 3 3 3 H3 3 3 3 3 - {(x t X ) V X p. p. p- q. J 3 3 3 [xt x -x* x (xt x )"1xt x ] -1 q. q. q. p. p. p. p- q. 3 3 3 3 3 3 3 3 [X t X -x' X (x1 x )~1xt x ](xfc X )"1xt X q j q j q J P J P J P J P J q J q3 q3 q J P J [xfc x -xc x (xfc x )"1xt x p. p. p. q. q- q- q- p- 3 3 3 3 3 3 3 3 65 = [x f c X -X* x (x f c X ) " 1 x t X l " 1 p. p. p- q. q- q. q. p. 3 3 3 3 3 3 3 3 - {(x f c x ) " 1 x t X (x f c x ) " 1 x t X P j P J P J q j q j q j q3 P J [x* x -xz x (x f c x ) " 1 x t x p. p. p. q. q. q. q. p. J J J 3 3 3 3 3 = [ i - ( x t X )" 1X t X (Xt X ) " 1 x t X ] P J P J P J qj qj qj ^ P j [x t x - x c x ( x t x ) " 1 x t x ] _ 1 p. p. p. q. q. q. q. p. J 3 3 3 3 3 3 3 = ( x c x ) " 1 [ x t x -x f c x (x c x ) " 1 x t X ] p. p. p. p. p. q. q. q. q. q. 3 3 3 3 3 H J J J 3 3 [Xfc X -X 1 X (x c X )~ 1 x t X ] _ 1 p. p. p. q. q. q. q. p. 3 * J 3 3 3 3 3 3 P - P . J J which i s what we set out to show. Q.E.D. Thus, we have established by the above theorem that the estimation problem associated with the generalized a d d i t i v i t y problem as developed i n chapter V does not require that new computing algorithms be developed to obtain estimates and rela t e d s t a t i s t i c s . One simply f i t s a model containing what are construed to be s t a t i s t i c a l l y important independent v a r i a b l e s . The parameter estimate corresponding to such a f i t can then be augmented to the corresponding estimate for a f u l l model constrained so that the unimportant independent v a r i a b l e s have c o e f f i c i e n t s of zero. The v i r t u e of the estimator of the parameter vector f o r component equation j (j = 1, 2 k) given i n chapter V i s that i t i s of appropriate s i z e f o r a d d i t i v i t y . However, whether g\* i s obtained d i r e c t l y as i n chapter V or i n d i r e c t l y by f i t t i n g a subset X and then augmenting the P j r e s u l t i n g estimator, i t s components may need to be permuted before invoking a d d i t i v i t y to obtain the estimator for the corresponding t o t a l equation. Such permuting ensures that appropriate components of B\ (j = 1, 2, k) are added to obtain (3,^. F i n a l l y , the theorem proved above i s based upon the d i s t r i b u t i o n a l assumption E^ ̂ N(<J>, I^j) • C l e a r l y , obvious modifications i n the theorem would make i t hold for the case E. ̂ N(<)>, Vo?) for V p o s i t i v e J J d e f i n i t e . Other generalizations are also p o s s i b l e . CHAPTER X 10.0 SOME ILLUSTRATIVE EXAMPLES In t h i s chapter, some examples are given that i l l u s t r a t e the a p p l i c a - t i o n of the theory presented i n the discourse. F i r s t , however, a some- what d e t a i l e d analysis i s given aimed at assessing the tenacity of the k assumption of m u l t i v a r i a t e normality of ŷ , = Z^_^ y^ f o r each of three data sets. As was stated i n chapter VII, the usual inferences f o r the k t o t a l model depend c r i t i c a l l y upon the assumption that y^, = y^ i s mu l t i v a r i a t e normal. Since there i s no p r i o r knowledge that the assump- t i o n holds, i t i s necessary to assess for mu l t i v a r i a t e normality i n order to more appropriately q u a l i f y any inference statements i n the examples. 10.1 Assessing M u l t i v a r i a t e Normality of y Koziol's (1982) method f or assessing j o i n t m u l t i v a r i a t e normality of the components y j , y 2 , y^ was used on three data sets. The f i r s t of these data sets i s that used by Kozak (1970) to i l l u s t r a t e the a d d i t i v i t y r e s u l t presented i n h i s paper. The second data set i s B r i t i s h Columbia co a s t a l western hemlock data used by Kurucz (1969). The t h i r d data set i s western hemlock data from various parts of B r i t i s h Columbia and was obtained from the ENFOR project (Williams, 1983, personal communication). Note that ENFOR i s an acronym for ENergy from FORests. Koziol's (1982) method f or assessing m u l t i v a r i a t e normality i s based upon a Cramer-von Mises type s t a t i s t i c , which i s computed as follows: 68 1. Given X T , X? , •••> X are random k-dimensional vectors, c a l c u l a t e n X = (Xi, X2» •••> X^) 1" and S. Here S i s the sample variance- covariance matrix of the n vectors and i s k x k. 2. Calculate the sample Mahalanobis squared distances Y\, Y2, Y n defined by Y. = (X. - X) tS~ 1(X. - X). 1 1 1 3. Put Z. = F. . (Y.), i = 1, 2, n and order the Z. i n ascending 1 (k) 1 1 order so.that Z / l N S Z.„. S . . . S Z. N . (Y.) here denotes (1) (2) (n) (k) 1 the area under the chi-square density function with k degrees of freedom between the l i m i t s of zero and Y_̂ ( i . e . , F ( k ) ( V = P r [ Y = Y i ] ) ' C alculate J using n 2 . f n s 1 J = £ [Z - ( i - h)lnV + (12n) i = l K ' Note that with three components, k i s equal to three i n our case. The three data sets are reproduced i n Appendix I (a,b,c). The f i r s t two of the data sets i n the appendix are reported i n imperial u n i t s , while the t h i r d data set i s given i n metric units. However a l l analyses reported i n t h i s d i s s e r t a t i o n were c a r r i e d out i n metric u n i t s . Before presenting d e t a i l s of the test for j o i n t m u l t ivariate nor- mality of the Yj' s> i t i s worth pointing out some te c h n i c a l considerations which s i m p l i f y considerably the computation of the K o z i o l s t a t i s t i c J ^ . In p a r t i c u l a r , t h i s s i m p l i f i e s the computational formula for the sample Mahalanobis squared distances Y_̂ ( i = 1, 2, n). Observe that the j o i n t d i s t r i b u t i o n of the Y j ' s l n t h i s case i s conditional upon the independent variables (the X's). As a r e s u l t of t h i s and based upon the notion that the regression of y on the X i s important ( s i g n i f i - cant), the sample Mahalanobis squared distances are given by 68a Y. = (y. - y . ) t S ~ 1 ( y . - y.) or by -t - I -Y. = e. S E . x i x where y. - y. = £. = ( y ^ - y . ^ y ± 2 - y . ^ . . . , y ± k - y ^ ) ' and S = (n - 1) ^ EE*" (note that E = ( E I , £ 2 » E 3 ) t ) . The d i s t i n c t i o n between Y. and y. must be borne in mind here. The r e s i d u a l vectors used i n l x computing Y^ above are those obtained from f i t t i n g component equations using only s t a t i s t i c a l l y s i g n i f i c a n t independent v a r i a b l e s for each .data set. These equations are those used to obtain conditioned t o t a l p redic- t i v e equations i n examples 1, 2, and 3 that follow. It should be empha- sized that a test f o r multivariate normality of E J , £ 2 , ..., E ^ i s equivalent here to a test for multivariate normality of y i , y 2 » •••> y^- Thus i f E I , £ 2 , ...» e. are j o i n t l y multivariate normal, one can speak K. of the multivariate normality of ylt yz, y^ and hence of E ^ and y T- In computing the s t a t i s t i c for each data set, an APL programme was used to calculate the Y^ as s p e c i f i e d above using an IBM 5100 Portable Computer. This computer i s located i n the Mathematics Annex at the University of B r i t i s h Columbia. APL i s an extremely e f f i c i e n t language when one i s dealing with matrix computations. The computation of the chi-square p r o b a b i l i t i e s i n step 3 was achieved by c a l l i n g the IMSL (International Mathematical and S t a t i s t i c a l L i b r a r i e s ) subroutine MDCH which computes cumulative c h i - square p r o b a b i l i t i e s . A short f o r t r a n programme was written to c a l l MDCH (see Appendix Id, PROGRAMME 1). Note that although DF i n programme 1 i s s p e c i f i e d as 2.0, DF = 3.0 for the f i r s t part of t h i s assessment problem. F i n a l l y , the ordered chi-square p r o b a b i l i t i e s from step 3 were used i n another f o r t r a n programme to c a l c u l a t e (see PROGRAMME 2 i n Appendix Id). The r e s u l t s of Koziol's (1982) test on the three data sets are summarized i n Table 1 below. Table 1. Results of Koziol's (1982) te s t f o r m u l t i v a r i a t e normality on three data sets Data Set Sample DF Computed K o z i o l Size S t a t i s t i c (J ) p-value n Kozak (1970) 10 3.0 0.05799 > 0.15 Kurucz (1969) 18 3.0 0.86078 < 0.01 ENFOR 48 3.0 4.27682 « 0 . 0 1 The p-values i n Table 1 are obtained by comparison with Koziol's Table 1 (K o z i o l , 1982). I t i s to be emphasized that due to small sample siz e s associated with Kozak's (1970) and Kurucz's (1969) data, our p- values may be somewhat o f f . However, on the basis of these r e s u l t s , the assumption of j o i n t m u l t i v a r i a t e normality w i l l be entertained f o r Kozak's data but not f o r the other two data sets. Note that t h i s conclusion i s quite reasonable for the ENFOR data because of the moderate (n = 48) sample s i z e . In view of the above r e s u l t s (ignoring the small sample sizes i n the f i r s t two data sets) i t i s reasonable to expect that y^ i s m u l t i v a r i a t e normal for Kozak's data since i t i s reasonable that y i , y2, y3 are j o i n t l y 70 mult i v a r i a t e normal i n t h i s data set. On the other hand, the above r e s u l t s suggest only that f o r the two other data sets y T may or may not be normal, since i t i s possible f o r y^ to be m u l t i v a r i a t e normal even when Yl> Yl> Y 3 a r e n o t j o i n t l y m u l t i v a r i a t e normal. For both the Kurucz (1969) and ENFOR data, i t was considered of some in t e r e s t to check f o r j o i n t b i v a r i a t e normality of the y 's. Accordingly, Koziol's test f o r m u l t i v a r i a t e normality was applied to pair-wise yj' s» thus three t e s t s were performed on each data set. The r e s u l t s are sum- marized i n Table 2 below. Table 2. Results of Ko z i o l 's test for b i v a r i a t e normality DF Sample Size Koziol's Computed S t a t i s t i c (J ) p-value n Kurucz's (1969) data ( e i , ?2) 2.0 18 0.8484 <0.01 2.0 18 0.5023 <0.01 (£2, £3) 2.0 18 0.6873 <0.01 ENFOR data (©1, £2) 2.0 48 5 .4432 « 0 . 0 1 ( e i , £3) 2.0 48 3.7356 « 0 . 0 1 (£2. £3) 2.0 48 3 .1189 «0.0l The r e s u l t s i n Table 2 indi c a t e that the assumption of j o i n t b i v a r i - ate normality i s rejected e s s e n t i a l l y i n every case. This r e s u l t i s not unexpected since, having rejected t r i v a r i a t e normality, one expects that b i v a r i a t e normality should f a i l to obtain i n at le a s t one of the three cases. It i s also probably adequate to check b i v a r i a t e normality and rej e c t t r i v a r i a t e normality the f i r s t time b i v a r i a t e normality f a i l s to 71 h o l d . I t has been shown so f a r that j o i n t m u l t i v a r i a t e no rma l i t y of Yl> Y2» Y3 does not appear to ho ld f o r the Kurucz (1969) and ENFOR data w h i l e m u l t i v a r i a t e n o r m a l i t y w i l l be en te r t a i ned f o r the Kozak (1970) da t a . I t should be emphasized aga in t h a t , i n g ene r a l , one should be more cau t i ou s i n accept ing m u l t i v a r i a t e no rma l i t y f o r the Kozak data because of the very sma l l sample s i z e . However, f o r purposes of the examples to f o l l o w , m u l t i v a r i a t e no rma l i t y w i l l be e n t e r t a i n e d . Once a ga i n , i t i s reasonab le then to assume y m u l t i v a r i a t e normal f o r the Kozak da t a . However, one i s unable to dec ide whether or not y^ i s m u l t i - v a r i a t e normal f o r the Kurucz and ENFOR da ta . A d i r e c t examinat ion of the behaviour of y^ = v j i-s necessary to make a judgement concern ing i t s n o r m a l i t y or non -no rma l i t y . One way i n which i n f o rmat i on can be obta ined concern ing the m u l t i - v a r i a t e n o r m a l i t y o r l a c k of i t f o r y^ = ^^aj_ Yj * s t o f i t t n e component models and i n v e s t i g a t e the behaviour of the e m p i r i c a l d i s t r i b u t i o n of = e j • Th i s can be ach ieved , i n p a r t , by p l o t t i n g a h i s togram of or a normal p r o b a b i l i t y p l o t of e^. Un f o r t una te l y , these p r o c e - dures r e q u i r e l a r g e enough sample s i z e s i n order f o r the p l o t s to be reasonably i n t e r p r e t a b l e . L a r ge l y because of t h i s , i t was p o s s i b l e to examine such p l o t s i n t h i s study on ly f o r the ENFOR data because of i t s moderate sample s i z e (n = 48) . The Kurucz data were obv i ou s l y too sma l l t o be examined by t h i s procedure. Three component models were f i t t e d u s ing the ENFOR da ta . Bo le biomass was regressed on D 2H and DCL, where D denotes d iameter at b r e a s t - he i gh t , H denotes he i gh t , and CL crown l e n g t h . Bark biomass was regressed on D 2H and HCL and crown biomass was regressed on DCL and HCL. The 72 resi d u a l s from these f i t t e d component equations were added up and a h i s t o - gram and normal p r o b a b i l i t y p l o t constructed using the BMDP P:5D subroutine. If the histogram of e T looks s u f f i c i e n t l y bell-shaped, i t i s reasonable to conclude that y T i s normal. S i m i l a r l y , normality of y T would be suggested by a s u f f i c i e n t l y l i n e a r normal p r o b a b i l i t y p l o t . The plots are given i n Appendix II(a,b) and both suggest that e T and hence y T i s normal for the ENFOR data. Based upon r e s u l t s of t h i s section, we can proceed as though y^, were multivariate normal for the Kozak and ENFOR data but are unable to say whether y^ i s multivariate normal or not for the Kurucz data. 10.2 Example 1 In t h i s section, the data given by Kozak (1970) are used to apply a d d i t i v i t y theory as presented i n chapters IV and V of t h i s t h e s i s . As indicated e a r l i e r , though the data are reproduced i n the appendix i n imperial u n i t s , a l l c a l c u l a t i o n s here are i n metric u n i t s . It i s further assumed throughout that the biomass components are independent. Admit- tedly, t h i s may be a tenuous assumption; however, we use i t l a r g e l y for purposes of demonstrating the a p p l i c a t i o n of the theory. For the e f f e c t of dependence on estimation and inference see the discussion i n chapter VII. Let us assume further that ^ N(cj), l°j)» F i r s t , consider f i t t i n g the component equations using both diameter and the square of diameter as independent v a r i a b l e s . R e c a l l that t h i s i s the case considered by Kozak (1970). Then the f i t t e d component equa- tions are given by 73 y i = 131.39 - 19.037X + 0.95195X 2, R 2 = 0.9923 y 2 = -1.12 + 0.205X + 0.02980X 2, R 2 = 0.9965 y 3 = -13.08 + 1.136X + 0.08361X 2 , R 2 = 0.9605. The corresponding t o t a l f i t t e d equat ion i s y T = 117.1.9 - 17.696X + 1.06540X 2, R 2 = 0.9948. One can check e a s i l y t ha t the c o e f f i c i e n t s of the t o t a l equat ion are obta ined by adding corresponding c o e f f i c i e n t s of the f i t t e d component equat ions , as Kozak (1970) demonstrated. The r e g r e s s i o n sum of squares f o r the t o t a l equat ion i s 289060, to f i v e - d i g i t accuracy, and g i ven that X t y r f = (2300.6, 56183, 1448300) t , one can check e a s i l y t ha t t h i s i s the r e s u l t one ob ta in s us ing a d d i t i v i t y and equat ion (4.2.6) of the t h e s i s . Next, Kozak (1970) r epo r t s t ha t when on l y s t a t i s t i c a l l y s i g n i f i c a n t ( important) independent v a r i a b l e s are used i n f i t t i n g the component equa- t i o n s , the f i r s t equat ion ( y i ) i n vo l ve s both X and X 2 , the second ( y 2 ) on l y X 2 , and the l a s t (y3) o n l y X 2 a l s o . The me t r i c analogues of Kozak ' s s p e c i f i c a t i o n of these f i t t e d equat ions a re y i = 131.39 - 19.037X + 0.95195X 2, R 2 = 0.9923 y 2 = 0.822 + 0.03471X 2, R 2 = 0.9962 y 3 = -2.342 + 0.11079X 2, R 2 = 0.9588. In accordance w i t h the ex tens ion of the concept o f a d d i t i v i t y as developed i n chapter V, t he re i s a t o t a l equat ion determined by a d d i t i v i t y of the c o e f f i c i e n t s of the preceding equat ions . In f a c t , t h i s equat ion i s g iven by y T C = 129.87 - 19.037X + 1.09745X 2. The sum of squares r e g re s s i on f o r t h i s cond i t i oned t o t a l equat ion i s g i ven by B T C X t y T - n y 2 = 818660 - 531860.4564 = 286799.5436. 74 S ince the t o t a l c o r r e c t ed sum of squares i s 290580, i t f o l l o w s that R 2 corresponding to the t o t a l cond i t i oned equat ion i s g i ven by R 2 = 0.9870. I t i s worth remarking tha t i n terms of R , t h i s model f i t s the data almost as w e l l as the u n r e s t r i c t e d t o t a l model, w i t h an R of 0.9948. Other a spect s of t h i s problem, i n c l u d i n g computat iona l d e t a i l s , a re prov ided i n a more d e t a i l e d example i n Appendix I I I . In connect ion w i t h t h i s problem and r e l a t e d problems concern ing a d d i t i v i t y , the ques t ion n a t u r a l l y a r i s e s whether the v a r i a b l e s i n the cond i t i oned t o t a l equat ion remain s t a t i s t i c a l l y s i g n i f i c a n t a f t e r being i nco rpo ra ted i n t o the t o t a l cond i t i oned equat ion . The answer appears to be tha t they would be s t a t i s t i c a l l y s i g n i f i c a n t i f the v a r i a b l e s i n the cond i t i oned component equat ions a re not ve ry h i g h l y c o r r e l a t e d . However, t h i s may not be the case i f the v a r i a b l e s a re h i g h l y c o r r e l a t e d . I t should be po in ted out tha t t h i s has not been checked thoroughly and, thu s , should be viewed here as l a r g e l y a c o n j e c t u r e . For the Kozak d a t a , how- ever , the c o n t r i b u t i o n to the t o t a l cond i t i oned equat ion of each v a r i a b l e was checked by computing the i n c rea se i n r e s i d u a l sum of squares when a p a r t i c u l a r v a r i a b l e i s omitted from the t o t a l cond i t i oned equat ion . The f o l l o w i n g p a r t i a l F -va lues were c a l c u l a t e d : F x = 96.27, F x2 = 15.64 The degrees of freedom f o r these p a r t i a l F va lues a re 1 and 7, r e s p e c t i v e l y . I t i s c l e a r from these r e s u l t s that both diameter at b reas t he ight and i t s square are s t a t i s t i c a l l y important i n the t o t a l cond i t i oned equat ion . 75 10.3 Example 2 In t h i s example, use i s made of the western hemlock data from coa s t a l B r i t i s h Columbia to go through the basic computational r e s u l t s as i n the previous example. These data were used by Kurucz (1969) and are given i n Appendix 1(b). F i r s t i t should be noted that, as discussed i n section 10.1, i t has not been pos s i b l e to determine whether y^ i s m u l t i v a r i a t e normal f o r these data or not. Therefore, i n f e r e n t i a l r e s u l t s given i n t h i s section r e l a t i n g to these data must not be viewed as s t r i c t l y v a l i d . The essence of t h i s example i s mainly to demonstrate use of the concept of a d d i t i v i t y computationally. One would need to check that y T i s reasonbly m u l t i - v a r i a t e normal for inference statements to carry f u l l weight. The c a l - culations here are c a r r i e d out i n metric units and the assumption i s made that the components y^ (j = 1, 2, 3) are independent. An all-combinations ( a l l subsets) procedure provided by the BMDP package (P:9R) was used to f i n d the best v a r i a b l e subsets for p r e d i c t i n g component biomass. Three components were recognized for purposes of t h i s a n a l y s i s , namely bole ( y i ) , bark ( y 2 ) , and crown (branches + f o l i a g e = y 3) . The crown component was obtained by simply adding branch and f i n e branch components f o r i n d i v i d u a l trees. Using R 2 as a s e l e c t i o n c r i - - t e r i o n , the best equations were found to be y i = -75.708 + 0.01330X2, R 2 = 0.9907 y 2 = -25.782 + 0.00203X2, R 2 = 0.9309 y 3 = -24.765 + 0.095895X!, R 2 = 0.8122 where Xj = ( h e i g h t ) 2 and X 2 = (height)(diameter) 2. A corresponding un r e s t r i c t e d t o t a l f i t t e d equation, using X\ and X 2, i s given by y T = 33.542 + 0.26819Xx + 0.01993X2, R 2 = 0.9655. 76 The t o t a l corrected sum of squares corresponding to the l a t t e r f i t i s 182559547. As i n the preceding example, a conditioned t o t a l f i t t e d equation i s obtained by a d d i t i v i t y as y T_ = -126.255 + 0.95895X! + 0.01533X2. Since x'y = (53205, 113670000, 15112000000)t, i t follows that the sum of squares regression associated with the l a t t e r conditioned equation i s § T C X t y T - ny 2 = 332291089.2 - 18(2955.8)2 = 175029523.7. Therefore, the R 2 associated with t h i s t o t a l conditioned equation i s 0.9588. Again, i f the assumption of normality of y^ held, one would conclude from t h i s that the conditioned t o t a l equation performs well when compared with the un r e s t r i c t e d t o t a l equation. I t may be noted that because of the great v a r i a b i l i t y i n the s i z e of the trees i n t h i s data set, one needs to be c a r e f u l about the p r e d i c t i v e goodness of these models. Indeed, as mentioned elsewhere i n t h i s t h e s i s , we are much less concerned here with using the best equations i n a p a r t i c u l a r sense than with demonstrating c e r t a i n aspects of a d d i t i v i t y . 10.4 Example 3 The next example i s based upon the ENFOR data for western hemlock (see Appendix 1(c)). We r e s t r i c t d e t a i l s to the l e v e l of previous examples. Using an all-combinations procedure as i n the previous example, the following equations were found to be the best for p r e d i c t i n g component biomass: yi = 6.49538 + 0.01541D2H - 0.12258DCL, R 2 = 0.9836 y 2 = 0.93179 + 0.00247D2H - 0.03112HCL, R 2 = 0.9561 y 3 = -4.82066 + 0.31477DCL - 0.23344HCL, R 2 = 0.8424. The unrestricted total biomass equation i s given by y T = 2.05138 - 0.000074D2H + 0.09319DCL - 0.0667HCL, R2 = 0.8674. The corresponding total conditioned equation is given by y„n = 2.60651 + 0.01788D2H + 0.19219DCL - 0.26456HCL. The sum of squares regression due to f i t t i n g the latter equation is given by f£ x'y - ny 2 = 2876.506. Hence the R2 corresponding to this model J. L» J. J. is 0.8215 which compares favourably with that of the unrestricted total equation. Note that even when crown variables are used for predicting crown biomass, the R2 is s t i l l in 0.80-0.90 range for that component. In Appendix III, the computational details relating to the additivity problem are given using Kozak's (1970) data again. The objective there is to show how the various statistics are computed, especially the var i - ances of the parameters in the total conditioned equation. CHAPTER XI CONCLUSIONS AND REMARKS In the discourse we have generalized the a d d i t i v i t y problem as o r i g i n a l l y posed i n the context of f o r e s t r y by Kozak (1970). It has been shown that the s t a t i s t i c a l theory of estimation and inference f o r the generalized a d d i t i v i t y problem as defined here i s c o n s t r u c t i b l e within the general framework of general l i n e a r model theory. I t i s important to recog- nize that both estimation and inference theory i s , i n general, dependent upon d i s t r i b u t i o n a l assumptions f o r the £_. (j = 1, .. ., k) and upon whether the e_. are dependent or not. When the are dependent, i t has been shown that a d d i t i v i t y as defined here, does not hold. Furthermore, inference theory r e l a t i n g to the t o t a l model i s complicated by the fac t that although the components may follow normal d i s t r i b u t i o n s , i t does not follow automatically that t h e i r sum i s also normal. This suggests a need to inv e s t i g a t e , or otherwise, j u s t i f y the normality of y^, before inference can be drawn about i t when dependence obtains among the components. In p a r t i c u l a r , i t would be u s e f u l i f future studies i n t h i s area could address the problem r e l a t i n g to the d i s t r i b u t i o n of ŷ , d i r e c t l y uSing large enough data sets along the l i n e s indicated i n section 10.1. Large data sets that might become a v a i l a b l e through projects such as the ENFOR project might make such studies possible and worthwhile. Other d i r e c t i o n s of further i n v e s t i g a t i o n might be the determination of the form of the dependence among components. This might si m p l i f y the problem of determining the 78 79 d i s t r i b u t i o n a l behaviour of y^,. The problem of a d d i t i v i t y has also been seen to lead to i n t e r e s t i n g but, as yet, unsolved problems i n m u l t i v a r i a t e d i s t r i b u t i o n theory. This i s obviously a f r u i t f u l l i n e of further research for those who are theor- e t i c a l l y i n c l i n e d . One of the i n t e r e s t i n g r e s u l t s obtained here i s that the a d d i t i v i t y problem i s n a t u r a l l y extendible to the cl a s s of l i n e a r models known as c l a s s i f i c a t o r y models generally encountered i n designed experiments. This extension must not be construed to be a c c i d e n t a l since any c l a s s i f i c a t o r y model can, i n general, be expressed i n regression form. The a d d i t i v i t y problem does not, however, extend to the c l a s s of i n t r i n s i c a l l y nonlinear models. Hence the usefulness of theory r e l a t i n g to the a d d i t i v i t y prob- lem i n mensurational studies involving nonlinear functions would, at best, be minimal. However, the theory should f i n d wide a p p l i c a b i l i t y among ecol o g i s t s and quantitative s c i e n t i s t s interested i n the assessment of biomass. The a d d i t i v i t y problem does not require the construction of new com- puting subroutines as c l e a r l y demonstrated i n chapter IX. This should make i t e s p e c i a l l y easy to use the theory of a d d i t i v i t y as developed here. F i n a l l y , the examples given i n the preceding chapter show that the concept of a d d i t i v i t y i s quite p r a c t i c a l and r e a l i s t i c and s t a t i s t i c a l l y appealing. 80 REFERENCES Andrews, D. F., Gnanadesikan, R., and Warner, J . L. 1973. Methods of a s ses s ing m u l t i v a r i a t e n o r m a l i t y . In M u l t i v a r i a t e A n a l y s i s 3, K r i s h n a i a h , P. R. ( e d . ) , pp. 95-116, Academic P re s s , New York. A r no l d , S. F. 1981. The theory of l i n e a r models and m u l t i v a r i a t e a n a l y s i s . John Wi ley and Sons, I n c . , New York. Behboodian, J . 1972. A s imple example of some p r o p e r t i e s of normal random v a r i a b l e s . Amer. Math. 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On best l i n e a r e s t ima t i on and a genera l Gauss-Markoff theorem i n l i n e a r models w i t h a r b i t r a r y non- nega t i ve covar iance s t r u c t u r e . SIAM J . A p p l . Math. 17: 1190-1202. APPENDIX 1(a) Kozak's (1970) Biomass Data DBH Bole Bark Branches Total (Inches) (lbs.) (lbs.) (lbs.) YT=Y1+Y2+Y3 X Yi Y 2 Y 3 (lbs.) 7.2 254 29 73 356 11.0 749 60 192 1001 9.8 519 49 151 719 7.5 217 31 115 363 12.2 1025 76 222 1323 6.7 242 26 76 344 5.9 136 18 37 191 5.5 127 18 39 184 3.5 62 6 11 79 8.7 375 39 98 512 Conversion f a c t o r s : 1 i n = 2.54 cm, 1 l b . = 0 .4535924 kg 87 APPENDIX 1(b) Western Hemlock Biomass Data (Kurucz, 1969) Fine Branch Height DBH Branch + Foliage Bole Bark Total (Feet) (Inches) (lbs.) (lbs.) (lbs.) (lbs.) (lbs.) 20.0 3.5 25.23 36.26 15.56 3.57 80.62 14.0 1.6 3.43 7.27 3.32 0.82 14.84 34.0 6.1 55.62 71.95 60.18 16.21 203.96 53.0 11.8 314.77 309.58 318.24 52.21 994.80 41.0 8.8 130.36 168.71 169.58 35.45 504.10 55.0 10.5 231.26 260.52 278.94 44.52 815.24 55.0 11.7 289.72 275.71 346.04 56.06 967.53 93.0 21.9 1210.37 859.97 2095.34 220.94 4395.23 73.0 15.9 701.20 430.48 916.22 120.91 2168.81 92.0 17.0 529.87 423.46 1329.83 188.24 . 2471.40 117.0 24.5 1201.44 468.42 3806.64 487.50 5964.00 167.0 30.5 3188.59 993.78 7971.30 1221.49 13375.16 123.0 23.7 2424.59 831.81 3332.09 369.61 6958.10 131.0 26.0 1658.45 522.55 4323.78 357.79 6862.57 176.0 34.1 3876.44 2013.48 11187.62 1447.37 18524.91 175.0 36.4 4027.03 1535.28 13874.81 2337.72 21774.84 148.0 31.1 2870.00 1204.30 8920.92 1196.20 14191.62 151.0 29.0 5926.09 2149.10 7537.04 1415.22 17027.45 Conversion f a c t o r s : 1 in = 2.54 cm, 1 foot = 0.3048 m, 1 l b . = .4535924 kg APPENDIX 1(c) DBH (cm) 19.20 28.00 22.70 23.90 28.90 23.90 14.60 26.80 11 .20 3.10 5.70 14.00 12.10 7.00 16.40 4.50 9.50 10.50 17.60 18.20 15.40 14.50 32.70 31 .30 16.10 1 1 .20 42.40 17.40 16.40 29.90 1 1 .30 21 .50 12.20 14.20 7.80 9.40 I 1 .00 19.80 13.60 II .20 9.80 9.70 12.90 17.70 1 1.40 14.00 18.70 11.10 Height (m) 22.40 25.20 26.70 28.30 28.20 22.40 12.80 26.60 9.80 3.50 4.90 11 .20 10.20 6.10 13.40 3.90 7.10 8.80 10.70 12.40 8.70 8.40 20.60 21 .30 13.40 10.00 22.40 1 1 .60 12.60 17.00 8.30 13.50 6.70 9.70 5.90 7.30 7.60 9.70 6.90 6.10 6.40 8.10 9.20 10.70 9.70 9.70 9.80 6.90 ENFOR Biomass Data Crown Crown Length Width (m) (m) 14.50 4.30 20.20 6.80 16.70 4.20 15.40 4.10 11.40 5.60 12.10 4.00 U.20 3.30 23.60 4.80 7.80 2.60 3.10 1.30 4.50 1.20 9.70 4.40 9.30 3.10 5.60 2.40 11.90 5.20 3.80 1.40 7.00 2.30 8.80 2.70 10.60 14.00 10.80 4.40 7.00 3.50 7.20 4.00 16.30 3.80 15.30 6.70 10.80 4.80 8.30 3.70 19.70 6.50 10.60 2.80 11.30 3.80 16.20 5.20 6.20 2.40 11.60 3.60 5.50 3.10 9.10 4.30 5.90 2.60 7.30 2.00 6.60 2.60 7.60 2.60 5.20 1.20 4.40 2.30 5.50 2.10 8.10 2.70 9.20 4.40 10.70 4.70 8.80 4.30 9.70 2.50 9.80 3.50 6.30 2.30 88 T o t a l Bole Bark Crown Biomass (kg) (kg) (kg) (kg) 94.43 13.96 9.29 117.69 253.21 27.19 57.67 338.08 184.65 21.67 7.21 213.52 224.57 27.48 11.15 263.21 324.82 34.56 45.79 405.17 155.99 20.84 15.00 191.83 28.85 4.58 18.80 52.22 234.19 32.09 28.43 294.72 19.77 2.97 5.84 28.58 0.64 0.12 0.71 1.48 2.79 0.37 2.56 5.73 24.04 3.85 15.75 43.64 17.34 2.82 9.73 29.89 5.43 0.74 2.09 8.26 43.07 6.66 24.29 74.02 1 .30 0.32 2.16 3.79 7.89 1 .41 4.23 13.53 13.80 2.08 6.77 22.65 36.52 5.51 17.54 59.58 46.94 7.57 24. 17 78.68 25.01 4.09 10.86 39.96 19.94 3.20 12.19 35.32 232.63 43.90 89.67 366.20 326.34 50.43 86.41 463.18 39.00 7.68 26.77 73.45 16.15 2.52 1 1 .29 29.95 523.92 96.28 129.66 749.86 46.42 6.46 28.42 81 .30 41.12 6.07 8.65 55.84 131.27 18.81 142.89 292.96 13.23 3.65 4.81 21 .70 82.87 13.57 16.55 1 12.99 14.35 3.30 5.14 22.79 22.44 3.31 16.59 42.34 3.93 0.53 2.04 6.50 6.07 1.00 5.04 12.11 10.94 1.16 4.42 16.51 38.36 4.72 13.77 56.85 15.57 2.18 8.54 26.29 9.91 1.50 5.22 16.62 7.17 1.57 4.56 13.30 8.40 0.82 1 .98 11.19 18.47 1.93 6.25 26.65 31.71 3.53 14.63 49.87 14.91 1 .60 6.88 23.39 21 .87 3.64 6.61 32.12 35.83 4.62 19.28 59.73 9.01 1.40 5.28 15.69 89 APPENDIX 1(d) PROGRAMME 1: Fortran programme c a l l s IMSL subroutine MDCH to compute chi-square p r o b a b i l i t i e s INTEGER IER REAL. XC18) READ(5< 10) <X< J)< J= l » 18) 10 FORMAT OX, 1SFS. 5) DO 15 I - l i 18 DF=2. O CALL MDCH(X(I> • DF« P. IER) WRITE<6,20) X(I),P 20 FORMAT( ' X < I )= ', F8. 5. 5X. 'P= ', F10. 5) 15 CONTINUE STOP END PROGRAMME 2: Fortran programme computes and p r i n t s K o z i o l s t a t i s t i c s REAL X( 18 >, JN READ(5i 10) (X(J)» <J~li IB) 10 FORMAT (18F8. 5) SUM=0. O DO 20 1=1/ 18 SUM=SUM+(XU>-< (1-0. 5)/18. 0) )**2 20 CONTINUE JN=SUM+ < 1.0/216.0) WRITE(6, 30) JN 30 FORMAT( ' JN=',F10. 6) STOP END INTERVAL NAME 5 10 15 +. + + +. *-42.OOO +X *-38.500 + •-35.OOO + *-31.500 + •-28.OOO + *-24.500 + *-21.000 +XX •-17.500 +X *-14.000 +X •-10.500 +X •-7.OOOO +XX •-3.5000 +XXXXXX •0.00000 +XXXXXXXXXXXXX •3.50000 +XXXXXXXX •7.OOOOO +XXX •10.5000 +XXXXX •14.0000 +XXX •17.50CO +X •21.OOOO + •24.5000 + •28.0000 + •31.5000 + •35.0000 + •38.5000 + •42.OOOO + •45.5000 + •49.OOOO • •52.5000 + •56.OOOO + •59.5000 + •63.OOOO + •66.5000 + •70.OOOO + •73.5000 + •77.OOOO + •80.5000 + •84.OOOO +X •87.5000 + + - 20 -- + - 25 -- + - 30 --+- 35 -- + - 40 --+- 45 -- + - 50 --+- 55 60 --+ FREQUENCY PERCENTAGE INT. CUM. INT. CUM. 1 0 0 0 0 O 2 1 1 1 2 6 13 8 3 5 3 1 O 0 0 O 0 0 0 O O O O 0 0 0 0 0 0 0 1 0 3 4 5 6 8 14 27 35 38 43 46 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 47 48 48 2.1 0.0 0.0 0.0 0.0 0.0 .2 . 1 . 1 . 1 .2 .5 4 2 2 2 4 12 27. 1 16.7 6.3 10.4 6.3 2.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 O.O 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.1 0.0 2.1 2.1 2.1 2.1 2.1 2.1 6.3 8.3 10.4 12.5 16.7 29.2 56.3 72.9 79.2 89.6 95.8 97.9 97 .9 97 97. 97. 97. 97. 97.9 97 .9 97 .9 97 .9 97.9 97 .9 97. 97 97. 97. 97. 97.9 100.0 100.0 .9 .9 .9 .9 .9 .9 .9 .9 .9 .9 a H> CO rt O CW H g M J. CO l-h O H CD M a o p 55 W C J H X - + - 5 • - + - 10 • - + - 15 20 --+- 25 -- + - 30 --+- 35 -- + - 40 --+- 45 50 -- + - 55 60 KD O APPENDIX 1 1(b) Normal p r o b a b i l i t y p l o t of re s i d u a l s (e_ = e.) f o r the ENFOR data 2 . 4 1 . 8 1 . 2 . 6 0 0 . 0 - . 6 0 1 . 2 •1.8 - 2 . 4 - 3 7 . 5 - 5 0 . 0 - 2 5 . 0 RESIDUAL 92 APPENDIX H T A SOMEWHAT DETAILED COMPUTATIONAL EXAMPLE In t h i s part of the t h e s i s , i t i s intended to use Kozak's (1970) data to show some of the computational d e t a i l s r e l a t i n g to the generalized a d d i t i v i t y problem. These data are chosen p a r t l y because of t h e i r small s i z e , making the computational exercise f a i r l y straightforward yet making possible a demonstration of the computations involved. The computations for larger data sets (with more independent variables) are performed as t y p i c a l l y shown here. P a r t i c u l a r attention i s given i n the following to aspects of the computational d e t a i l s not given i n section 10.2. In the following, Xi = diameter, X2 = (diameter) 2, and the matrix X re f e r s to the 10x3 matrix X = (X 0|Xi|x 2) where X 0 i s a column vector of l ' s . Also l e t X* = (X0IX2). The following matrices w i l l be of use i n t h i s discussion. (8.2031 -0.82845 0.019172 \ 0.82845 0.087614 -0.002093 0.019172 -0.002093 0.0000514/ , / 0.36948 -0.0006217 \ /2300.6\ (X* tX*) 1 = , X y = 56060 V^O.0006217 0.00000143/ \ 1443700 / The un r e s t r i c t e d t o t a l f i t t e d equation i s given by y T = 117.19 - 17.696Xi + 1.06540X2. The estimated variance-covariance matrix of 8 ^ for t h i s model i s ( X t X ) ~ 1 r j 2 = (X tX)~ 1(217.64) , where 217.64 i s the MSE associated with f i t t i n g y^. Thus the standard errors for the parameters of t h i s model can be obtained from the estimated variance-covariance matrix. In t h i s case they are 42.25, 4.37, 0.1058 for § Q T , § 1 T» and res p e c t i v e l y . The component equations containing only important independent v a r i - ables are given i n section 10.2. The corresponding t o t a l conditioned equation as determined by a d d i t i v i t y i s y T C = 129.87 - 19.037X! + 1.09745X2. The covariance matrix of 3^, for the above equation, i s obtained, under the assumption of independence of the y^'s, as the sum of the covariance matrices of the estimated parameters of the component equations. In the present case, the estimated covariance matrix i s given by '8.94206 -0.82845 0.01793 \ i? . a2, J * 3 j> = |-0.082845 0.087614 -0.002093 BTC \0.01793 -0.002093 0.0000543; where i s the mean square associated with f i t t i n g component model j . In the present case, a 2 = 194.04, a 2 = 0.55375, and a 2 = 42.198. There- fore E? , a 2 = 236.792. Hence the estimated standard errors for the J=l J parameter estimates i n y ^ are, i n order, 46.02, 4.55, 0.1134, which compare favourably with those given above for the u n r e s t r i c t e d t o t a l model. It should be pointed out here that before adding the covariance matrices, they are f i l l e d up with zeros to bring them to the f u l l s i z e corresponding to a l l variables i n the conditioned t o t a l equation and the elements are permuted to correspond to the same parameters p r i o r to a d d ition. This part has not been exhibited i n the above derivations. F i n a l l y , the predicted values generated by the u n r e s t r i c t e d t o t a l equation y^ and i t s residuals are compared with those obtained using the t o t a l conditioned equation y T r,. The r e s u l t s are given i n Table 3(a,b) and i n Figure 1. For these data, at l e a s t , the t o t a l conditioned equation performs r e l a t i v e l y w e l l . Of course t h i s may p a r t l y be a s c r i b - able to the very small sample and narrow sample range. Table. 3 Comparison of predicted values and r e s i d u a l s of unrestricted t o t a l equation with those of t o t a l conditioned equation a. Unrestricted Total Equation OBSERVED TOTAL BIOMASS PREDICTED TOTAL BIOMASS RESIDUAL 161 .48 166 .53 -5 .05 454 .05 454 .27 -0 .23 326 . 13 336 .65 -10. .51 164 .65 166 .53 -1 . 87 600, . 10 591 .69 8 .41 156 .04 124 .40 31 , .63 86 .64 91 .08 -4. .44 83 .46 77 .71 5. .75 35 .83 43 .88 -8. .05 232 .24 246 .21 -13. .97 b. Total Conditioned Equation OBSERVED TOTAL BIOMASS 161.48 454.05 326. 13 164.65 600.10 156.04 86.64 83.46 35.83 232.24 PREDICTED TOTAL 165.48 454.69 335.99 165.48 593.78 123.73 91 .05 78. 10 47.36 245. 10 BIOMASS RESIDUAL -4.00 -0.65 -9.86 -0.83 6.32 32.30 -4.41 5.36 -11.53 -12.86 95 Figure 1. Scatter of Residuals from Total Unrestricted Equation and from Total Conditioned Equation a 4- C7 . ¥ Key + = (y T» e T) X = ( y T C , e T C ) a CC O o . CO LU a: q a a a f—% . 1 X + X , 1 1 1 I 1 0.0 10.0 20.0 30.0 40.0 , 50.0 60.0 PREDICTED TOTAL BJOMflSS ( X 1 0 3 )
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