A General Model for Multivariate Analysis (International Series in Decision Processes)

A General Model for Multivariate Analysis

INTERNATIONAL SERIES IN DECISION PROCESSES INGRAM OLKIN, Consulting Editor Statistical Theory of Reliability and Life Testing: Probability Models, R. E. Barlow and F. Proschan Probability Theory and Elements of Measure Theory, H. Bauer Time Series, R. Brillinger Decision Analysis for Business, R. Brown Probability and Statistics for Decision Making, Ya-lun Chow A Guide to Probability Theory and Application, C. Derman, L. J. Gieser, and I.Oikin Introduction to Statistics and Probability, E. Dudewicz A General Model for Multivariate Analysis, J.D. Finn Statistics: Probability, Inference, and Decision, 2d ed., W. L. Hays and R. L. Winkler Statistics: Probability, Inference, and Decision, Volumes I and II and Combined ed., W. L. Hays and R. L. Winkler Introduction to Statistics, R. A. Hultquist Introductory Statistics with FORTRAN, A. Kirch Reliability Handbook, B. A. Kozlov and I. A. Ushakov (edited byJ. T. Rosenblatt and L. H. Koopmans) An Introduction to Probability, Decision, and Inference, I. H. LaValle Elements of Probability and Statistics, S. A. Lippman Modern Mathematical Methods for Economics and Business, R. E. Miller Applied Multivariate Analysis, S. J. Press Fundamental Research Statistics for the Behavioral Sciences, J. T. Roscoe Applied Probability, W. A. Thompson, Jr. Quantitative Methods and Operations Research for Business, R. E. Trueman Elementary Statistical Methods, 3d ed., H. M. Walker and J. Lev An Introduction to Bayesian Inference and Decision, R. L. Winkler FORTHCOMING TITLES

A Basic Course in Statistics with Sociological Applications, 3d ed., T. R. Anderson and M. Zelditch, Jr. Fundamentals of Decision Analysis, I. H. LaValle

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A General ~ode I tor Multivarifte Analysis I

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JEREMY D. State University pf New York at Buffalo I

HOLT, RINEHART' AND WINSTON, INC. New York Chicago S~n Francisco Atlanta Dallas Montreal ToroJto London Sydney

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Library of Congress Cataloging in Publication Data Finn, Jeremy D. A general model for multivariate analysis. (International series in decision processes) Bibliography: p. 410 1. Multivariate analysis. 2. Analysis of variance. I. Title. OA278.F56 519.5'3 74-8629 ISBN 0-03-083239-X Copyright© 1974 by Holt, Rinehart and Winston, Inc. All rights reserved Printed in the United States of America

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Preface Scientists faced with the task of analyzing and understanding human behavior are in constant need ~=Jf models by which they may test hypotheses involving a greater quantity and complexity of behavioral variables. In response to this need, multivariate mod~ls for the application of such techniques as analysis of variance, regression analysis, and analysis of covariance are gaining in accessibility and usage. Multivariate analysis may be' conceptualized in two ways. First, it is a means of analyzing behavioral phenomena. It is based upon the realization that hardly any form of human behavior wor thy of study has only a single facet; that behind any measurable trait are compor]lents that covary only partially; that a "better" scientific description of any behavior is derived through some degree of finer analysis. Further, no observable pehavior results from a single antecedent. The "principle of multiple causes" is one we confront in all except the smallest analytic units (for example, th~ "one-gene, one-enzyme theory" of genetic . . action). The second conceptualizatibn of multivariate analysis is fitting a set of algebraic models to situations with multiple random variables, usually criterion or outcome variables, which are measures of the same sample(s) of subjects. Behavioral data are often of thi~ form. Intelligence is measured in terms of at least a quantitative and a verbal 'ability. Following Guilford (1959), creativity is measured by the administration of at least six separate scales. Often two achievement scores, one for speed and one for power, are assigned to a test respondent. The class of experimental designs known as "repeated measures" designs denotes a multivariate situation. In this case each subject is measured on a given scale at two or more points in time or under differing experimental conditions. The evaluation of the attainment of course objectives in either experimental or traditional instru~tional settings is likely to require multivariate analysis procedures. With multivariate models, the simultaneous consideration of the attainment of several co9nitive levels or of both cognitive and noncognitive instructional outcomes is facilitated. In each instance, the analysis of a single summary measure-for example, a total or average score-will result in the loss of the information conveyed by the individual scales. Statistical analysis of each of a series of measures separately will result in redundancy, which in turn will threaten the validity of the interpretations drawn from the data. Use of the appropriate multivariate model will allow the researcher to retain the multiple scores and to treat them simultaneously, giving appropriate consideration to the correlations among them. 1

vii

viii

Preface

Multivariate techniques comprise two related methodologies, each with its own objectives. The first of these is concerned with the discovery of an underlying structure of response data that have been collected, or of the behaviors they represent. Factor analysis is employed in the attempt to locate and isolate sets of measures with the properties that the tests or scales within the set have relatively high intercorrelations with each other, and that scales of one set have small or zero intercorrelations with those of another set. If sets of variables that contribute very little to discrimination among subjects can be identified, they are often ignored or eliminated. In this sense, factor analysis or a simpler technique, component analysis, is used as a data-reduction device. Recent contributions by Bock and Bargmann (1966) and by Joreskog (1969) allow the researcher to hypothesize, from psychological theory or from prior analysis, a given latent structure underlying a set of measures, and to apply statistical criteria to test the fit of the observed outcomes to the hypothesis. A second set of multivariate procedures, which constitutes the primary focus of this book, includes multivariate extensions of such commonly used estimation and hypothesis-testing procedures as analysis of variance, analysis of covariance, and regression analysis. Through these methods questions may be answered about the contribution of structured and identifiable independent variables to the explanation of between-individual or between-group variation in one or more criterion measures. Examples of such questions in multivariate form might include: "Does intelligence predict these four achievement measures?'' "Are there significant differences between control and experimental groups on speed as well as accuracy of learning, or on four body dimensions?" "Does the mean growth curve of the group administered an experimental drug differ from that of a placebo group?" The dependent or criterion variables generally have nonzero intercorrelations. The model implied by each question is of a form familiar to most behavioral scientists. A primary purpose of this book is to describe the multiple-criterion form of these models, and to provide the computational tools which facilitate data analysis under that form. A general model for multivariate analysis describes the analysis of quantitative data through application of a "general linear model." Linear estimation and tests of hypotheses are discussed, which are univariate and multivariate forms of the following techniques: The summary of raw and transformed multivariable data Multiple correlation and regression Canonical correlation Principal components Analysis of variance, with equal or unequal subclass frequencies Analysis of covariance Discriminant analysis Step-down analysis To provide sequence with other statistical materials, the univariate multiple regression model is introduced first, and is discussed in greatest mathematical detail. Multivariate regression and univariate and multivariate analysis of

Preface

ix

variance are presented as extensions of that basic model. The analysis-ofvariance presentation is less detailed and contains more exemplary material. The remaining techniques are viewed as by-products of the formulation of the regression and variance analysis models. Particular emphasis is given to topics that have been inadequately described in current journals and texts in the social sciences. For example, lengthy discussion is devoted to reparameterization in the analysis of variance and to the estimation of parameters in linear models. Five sample problems are introduced in the first chapter and are described throughout the text as the appropriate analysis techniques are encountered. These are relatively large problems. Hand computation in multivariate analysis is not feasible for any but the most trivial examples. The sample problems were selected instead to exemplify a variety of real design and analysis problems. The analyses for the five samples were originally performed on the MULTIVARIANCE program (Finn, 1972d). The computer input-output listings are provided as an appendix to the text (separately numbered C.1 through C.166), along with a brief version of the MULTIVARIANCE user's manual. The statistical results are transcribed to the earlier chapters as they are discussed. This book is addressed to users and potential users of multivariate statistical techniques. Readers should have familiarity with univariate statistical theory, to a degree provided by, say, a good one-year course in applied statistical methods. Topics that are especially requisite are estimation and significance testing in fixed-effects analysis of variance; the design and estimation of planned contrasts in analysis-of-variance models; simple univariate regression models and analysis; and the basic concepts of covariance and correlation. The book relies entirely on the statement and formulation of linear models, and some facility with these skills is essential. Knowledge of the algebra of matrices is desirable but not necessary. Those aspects of linear algebra employed in the book are discussed briefly in Chapter 2. This book may be read in several ways. As a text in multivariate analysis, the organization provides sequence for detailed study of the general linear model and its applications. Supplementary material on matrix algebra plus computer routines for class exercises are recommended. As a reference, the examples may be studied by themselves as illustrations of (a) the data for which multivariate models are appropriate and (b) the presentation and interpretation of the outcomes. Toward this end, study of the computer runs and the respective problem discussions is likely to be especially useful. This book has been in preparation a long time. In that time there have been many people who have helped in one way or another. I wish to thank them all. In particular, I owe a great deal to Professor Darell Bock of The University of Chicago. Without his teachings this book, and more, would not have been possible. I wish to thank Professor Ingram Olkin, who has been continually supportive. His reviews and comments have had a major impact on the form of the book. Two students, Kathleen VanEvery and Nancy Breland, have provided useful reviews and suggestions for improvements. Also there are those who have lent their data as examples and are acknowledged in the first chapter, those who have helped in the development of the MUL TIVARIANCE program, and many who have made individual suggestions

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Preface

which are incorporated in the book. Thank you. Computer time and assistance in running the examples were provided by the Computing Center of the State University of New York at Buffalo. In the preparation of the manuscript, Jeanette Ninas Johnson and the staff at Holt, Rinehart and Winston spent many difficult hours with the material. The manuscript was typed by Jacqueline Rance and Diana Webster. I am glad it is they who have their jobs. I am especially grateful to my wife, Joyce, who sat up many evenings reading and re-reading galley proofs with me. Although her knowledge of statistics increased only a little, her knowledge of Greek has grown immensely. Stockholm, Sweden June 1974

Jeremy D. Finn

Contents Page vii

Preface

Section I Chapter 1 1.1 1.2 1.3 1.4

2.3 2.4 2.5 2.6 2.7

Multivariate Analysis

The Algebra of Matrices

Chapter 3

3.2 3.3

2 5 9

10

19

Notation Simple matrix operations Transposition Addition, Subtraction Multi plication Scalar functions of matrices Rank Determinant Trace Matrix factoring and inversion Triangular factorization Inversion Orthonormalization Matrix derivatives Characteristic roots and vectors Exercises Matrices Problems

Section II 3.1

2

Perspective The multivariate general linear model Application Approach to the general multivariate model Five sample problems Sample Problem 1-Creativity and achievement Sample Problem 2-Word memory experiment Sample Problem 3- Dental calculus reduction Sample Problem 4- Essay grading study Sample Problem 5- Programmed instruction effects

Chapter 2 2.1 2.2

Introduction

20 23 30 36 47 48 50

Method

Summary of Multivariate Data

Vector expectations Standardization The multivariate normal distribution Samples of multivariate data One sample More than one sample Note on within-group variances and correlations Linear combinations of variables

54 54 61 66

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xii

Contents

3.4

Sample problems Sample Problem 1- Creativity and achievement Sample Problem 3- Dental calcu Ius reduction

Chapter 4 4.1 4.2

4.3 4.4

4.5 4.6

Chapter 5 5.1

5.2

5.3 5.4

6.4 6.5 6.6

7.2 7.3 7.4

92 96

108 110

123 127

Multiple Regression Analysis: Tests of Significance

134 135

Correlation

Simple correlation Partial correlation Multiple correlation Multiple criteria Canonical correlation Sample Problem 1- Creativity and achievement Condensing the variates: Principal components Sample Problem 1 -Creativity and achievement Sample Problem 3-Dental calculus reduction

Chapter 7 7.1

92

Separating the sources of variation Model and error Subsets of predictor variables Order of predictors Test criteria Hypotheses Likelihood ratio criterion Hotelling's P Univariate statistics Step-down analysis Multiple hypotheses Reestimation Sample Problem 1 -Creativity and achievement

Chapter 6 6.1 6.2 6.3

Multiple Regression Analysis: Estimation

Univariate multiple regression model Estimation of parameters: Univariate model Conditions for the estimability of (3 Properties of~ Estimating dispersions Some simple cases Prediction Summary Multivariate multiple regression model Estimation of parameters: Multivariate model Properties of B Estimating dispersions Some simple cases Prediction Summary Computational forms Sample Problem 1- Creativity and achievement

85

Analysis of Variance: Models

Constructing the model Univariate case Multivariate case Least-squares estimation tor analysis~ot-variance models Rep"arameterization Conditions tor the selection of contrasts Some simple cases The selection of contrasts One-way designs Bases tor one-way designs Higher-order designs Interpretation of contrast weights

145

160 165

173 175 181 182 187 193 198

205 205 215 219 228

Contents

Chapter 8 8.1 8.2 8.3 8.4

Chapter 9 9.1 9.2

9.3

Analysis of Variance: Estimation

Point estimation Properties of 8 Conditions for the estimation of 0 Estimating dispersions Predicted means and residuals A simple case Sample problems Sample Problem 2-Word memory experiment Sample Problem 3-Dental calculus reduction Sample Problem 4-Essay grading study

Analysis of variance: Tests of Significance

Separating the sources of variation Some simple cases Test criteria Hypotheses Likelihood ratio criterion Hotelling's P Univariate F tests Step-down analysis Multiple hypotheses Notes on estimation and significance testing Sample problems Sample Problem 2-Word memory experiment Sample Problem 3-Dental calculus reduction Sample Problem 4- Essay grading study Sample Problem 5-Programmed instruction effects

Chapter 10 Analysis of Variance: Additional Topics 10.1 10.2

Discriminant analysis Sample Problem 3-Dental calculus reduction Analysis of covariance The Models Estimating 0 and B Estimating dispersions Prediction Tests of hypotheses Sample Problem 2-Word memory experiment

Appendix A.

Answers to matrix algebra exercises (Section 2.7)

xiii

251 252 260 267 273

296 297 308

328

357 357 368

394

Appendix B. Program user's guide

397

Appendix C. Input-output listings

409

Computer printout C.1-C.166

References

410

Index

417

A General Model for Multivariate Analysis

Section

I

Introduction

CHAPTER

I

Multivariate Analysis 1.1

PERSPECTIVE

This book describes the application of one general statistical model to behavioral data. The model frequently has high appeal. It is general; it is simple; there are available computer programs; and almost any behavioral data can be analyzed according to one form or another of the model. This same appeal function also requires that we be cautious. For we must ask whether this model is the correct one for our particular research assumptions and hypotheses. It is not always clear whether mathematical models that contain estimable parameters, and that reflect the behavioral models we assume, actually exist. Frequently they do not, and models must be constructed for specific cases. Often enough, these endeavors culminate in formulations that have a more general applicability. Statistical journals publicize large numbers of such cases. In the discipline of psychology, no instance is more outstanding than Thurstone's attempts to discover processes basic to the then-held concept of general intelligence. The by-product of these endeavors was the development and dissemination of a widely used technique, multiple factor analysis. Multivariate linear models are not always applicable to the specific problems at hand. And indeed the subset of multivariate procedures presented in this book represents only a small portion of those conceivable. Yet the procedures discussed in this book under the rubric "multivariate analysis techniques" share a resemblance to models of behavior in their very representation of behavior as having multiple antecedents and experimental outcomes as having multiple facets. Students of human behavior, with frames of reference from the extremes of atomism to those of holism, find themselves with multiple observations of each subject. For the atomist, this may involve tracing the development of a specific trait over time, or of its variants with specific imposed or natural stimulation. Bloom (1964) has summarized more than a thousand longitudinal studies of the development of physical characteristics, cognitive achievement, interests and attitudes, and personality measures through the childhood years. Each characteristic is represented by responses to the same or parallel tests, at different ages, by the same individuals. The data are of a naturally multivariate form, and various multivariate growth models are useful for describing the trends over time. Similarly, traits from a variety of disciplines are studied over time or under vary2

Multivariate Analysis

3

ing experimental conditions: the effectiveness of drugs with given diseases, over time or after repeated administrations; the change in value of certain preferred stocks overtime or with modifications in the company's and competitors' products; the gradual consumption of the nation's natural resources; changes in national birth rates; and so on. For the holist there are problems of a different multivariate nature. Here the outcome of an experiment or comparative study, at a single point in time, is completely represented only by multiple measurement scales. This may occur when the construct of interest is composed of well-defined but conceptually smaller units or when there is some lack of certainty about the definition of the construct, and a subsequent need to measure it in several ways. Generally the multiple measures are moderately to highly intercorrelated, as aspects of the same behavioral phenomena. Examples of such cases abound. All useful theories of personality attribute behavior to a multiplicity of underlying components. Murray (1938) has postulated a series of idiosyncratic "needs" as the driving forces in observed human behavior. For example, with respect to the seeking, giving, or withholding of affection, the individual will respond in a manner determined largely by his needs for affiliation, rejection, ni.Jrturance, and succorance. Thus the individual's capacity to exhibit a given degree of affection is reflected in tour measurements on these partial constructs. They may in turn be analyzed simultaneously for between-individual or between-group variation. Academic achievement is best described in terms of behaviors of progressively greater complexity. Both The Conditions of Learning (Gagne, 1966) and The Taxonomy of Educational Objectives (Bloom, 1956; Krathwohl, Bloom, and Masia, 1964) define categories of intellectual achievement. They are ordered according to the ability of an individual to achieve a given level of content mastery, only after having mastered the behaviors of lower or simpler levels. The Taxonomy pertaining to cognitive achievements lists six general levels of content mastery: knowledge, comprehension, application, analysis, synthesis, and evaluation. Each level is defined further in terms of subcategories. For example, comprehension includes the abilities to translate materials from one form of communication to another; to interpret, explain, or summarize a communication; and to extrapolate trends or sequences beyond given data to determine implications and consequences. Although an individual may achieve at one level in the hierarchy only after having mastered prior levels, the progression is tar from absolute. As a result, a person's achievement with respect to any curriculum is adequately described only by providing estimates of achievement at every level. For analysis, the resulting data are both multivariate and naturally ordered by complexity. There are additional situations in which multivariate analysis is particularly useful. A tester may be interested in the simultaneous reliability of a series of items or tests, which may not be independent or equally intercorrelated. Variable-reducing analyses, such as component or discriminant analysis, are of practical value tor placing individuals into homogeneous groupings, from multiple behavioral measures. Multivariate procedures may be applied to sets of measures that have been identified through cluster or factor analysis to have

4

Introduction

common components. This may be accomplished without the necessity of forming arbitrary linear composites ofthe measures, such as the summation of scores on scales having high intercorrelations with a particular "factor." Finally, since most computer programs for multivariate methods also provide results for each of the criterion measures separately, their use facilitates the simultaneous performance of a number of univariate analyses. This feature is of value, for example, in the comparison of results from raw and transformed data (see Pruzek and Kleinke, 1967). In every case, it is critical that the variables of any set share a common conceptual meaning, in order for the multivariate results to be valid. It is an easy matter to abuse, say, an extensive computer program to perform analyses on sets of variables which bear no "real-life" counterpart as a group. Likewise, an extensive program, such as MULTIVARIANCE, may be used to produce tests of significance that are quantitatively correct but do not conform to assumed probability statements. This may be due either to the quantity of nonindependent results, or to their exploratory nature. When research yields multiple response measures, the employment of rigorous scientific methodology resting on strong design formulations is more important than ever. One of the most concise treatments of the design and conduct of quantitative evaluation of behavioral data is provided by Federer in the introductory chapter of Experimental Design (Macmillan, 1955). Federer's brief but important chapter is recommended reading for anyone concerned with problems in the behavioral sciences. The evaluation process may be conceptualized as having six aspects: 1. Discovery of a behavioral problem. 2. Searching for existing solutions to the problem. 3. Selection of an approach to the study of the unknowns and the statement of expectations. 4. Formulation of the technical methodology to be employed in the evaluation. 5. Execution of the technical formulations. 6. Interpretation of the research outcomes. Quantitative analysis forms only a small portion of evaluation methodology, and we might be chagrined at the disproportionate quantity of reference material we have for this one aspect. However, quantitative thinking modes can form a basis for formulation of all phases of the evaluation process. Unresolved research problems are subjected to evaluation through the principles of scientific method. The hypothetico-deductive approach to empirical investigation has been well described by Ellis (1952). The primary assumption is the existence of a research hypothesis, or expected solution to the problem, prior to the collection of quantitative data. Although unexpected findings have often been generated for verification by hypothesis-seeking approaches to data analysis, the validation of such findings through replication is essential. Testing hypotheses drawn from earlier studies and from behavioral theory has the advantage of providing two sets of confirmatory data, one logical and one empirical. When the two agree, the conclusions form a firm base and are likely to replicate. If one has a large set of data, dividing it into two parts-one for

Multivariate Analysis

5

generating hypotheses and the other for confirmation-can maintain this confirmatory power. Researchers often disavow any prior knowledge from which to draw. hypotheses. Yet in informal discussion, the same individuals may admit that they really believe the new approach to be superior to the old or that their results will be essentially the same as another investigator's, in a different situation. These are hypotheses. That is, they are informed best guesses as to the experimental outcomes. Frequently a problem yields competing hypotheses, each of which would suggest a different outcome. These too should be stated and tested, as competing explanations. Nothing is lost, for no amount of exploration or estimation is precluded by testing prior beliefs.

1.2 THE MULTIVARIATE GENERAL LINEAR MODEL* A model of an object or event is any attempt at representing that object or event other than the original occurrence or representation. The general linear model is a very specific sort of model, involving the algebraic representation of relationships among observable human characteristics. In most studies the symbolic representation of such relationships is the second model applied to the observed behavior. The first is the modeling of behavioral constructs through algebraic or quantitative representation. This occurs in the process of measurement. In contrast, we will restrict ourselves here to the analysis of the already quantified responses. If the measurements are objective, reliable, and valid, we will be willing to assume that such quantitative indicators correspond in important ways to the constructs of interest and will yield insight into their behavior. Let us denote as y1 the quantified response of subject ito a single outcome or criterion measure y. Subject i may have been assigned to, or selected from, a population of observations identified by sharing common attributes on one or more exactly observable traits. In addition to the y1 then, subject i is identified by having values on a set of antecedent or independent variables X;, hypothesized to be related in some way toy. The X; may be categorical variables defining the population, or measured variables having ordinal, interval, or ratio scales, or both. x;1 is the value on variable X; for subject i. The process of fitting a linear model to data is one of determining a set of coefficients, a;, that multiply the X;1 in order to reproduce y1 as closely as possible for a set of observations. The model may be written as

Yi =

~; a;Xn+ei

= a1xli+a2x21+ · · ·+a3x;i+et

(1.2.1)

e1 represents the extent to which y1 cannot be reproduced by the weighted function for the particular subject. If y is a random variable and x 31 represents a fixed *For a more extensive discussion of linear statistical models, Chapter 5 of Introduction to Linear Statistical Models, Volume 1 (Graybill, 1961) is highly recommended.

6

Introduction

value of X;, then E will also be a random variable. It represents both the extent to which the model is incompletely specified and the measurement error iny. To state that "the model fits the data" implies that theE; are small, and that y; can be known from knowledge of the xii. Researchers in the behavioral sciences are perhaps more accustomed to asking, "Is there a significant difference between means?" or "Is a significant amount of variation in the dependent variable attributable to the predictors?" than "Does the model fit the data?" Yet when the components of the linear model are clearly specified and understood, it will be seen that the two sets of questions are in fact the same. Equation 1.2.1 is a model in two senses. First,_ the equation specifies the components into which the observation is partitioned-that is, some function of the particular X;i and all else. The original event y; is represented as the sum of two components, one being itself a function of j= 1, 2, ... , J additional events chosen in advance by the researcher. Second, the relationship between the additional events X; and the rest of the model is linear; that is, all weights aj are to unit power only. In still another sense, however, Eq. 1.2.1 is not truly a model. The sum of the right-hand components of 1.2.1 is exactly y;, and not some other representation of it. Thus, for consistency, we will refer to the portion of 1.2.1 exclusive of Ei as the linear model. Indeed, the most important modeling in behavioral research is the representation of the outcome y;, by other selected and weighted measures, X;. E; is commonly relegated the function of depicting unknown factors, hypothesized to be of a random and/or trivial nature in influencing y;, at least when compared to the purposefully selected antecedent measures. The variables X; may be of several types. When the X; are entirely categorical measurements and have values 0 and 1, the linear model is usually referred to as the analysis-of-variance model. The question of fit to sample data-that is, of the relative contributions of I.;aiX;; and E; in yielding information about Yi- is most commonly phrased as "Are there significant differences among means of the J populations represented by the samples?" When X; are scores on J measured variables, the model is that of regression. The question asked most often is one of the percentage of variation in y attributable to one or more of the X;. Finally, the analysis-of-covariance model is the form of Eq. 1.2.1 when some of the X; are categorical and others are measured. The variables X; may themselves be nonunit powers or cross products without destroying the linearity of the model. The variables X; in unpowered form comprise the additive portion of the model. Any x; that are non unit powers of a measure, or the cross products of two or more other X;, comprise the nonadditive or interactive portion of the model. Thus the general linear model as represented by 1.2.1 will suffice for a variety of polynomial analyses, as well as all linear analyses of variance, including interaction terms. MPd.lti_j2"_Lls mYHi'!!}!iate when y is a V€lQt~>r variable having more than a single outcome measure. A separcife'set ofweig hts a~ is necessary each out~- -"• ..,......~_'"""'"',..,.,..,~-~----,