Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1980 Randomization analysis of covariance and change- over designs Winston A. Richards Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Statistics and Probability Commons Recommended Citation Richards, Winston A., "Randomization analysis of covariance and change-over designs " (1980). Retrospective Theses and Dissertations. 7121. https://lib.dr.iastate.edu/rtd/7121 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This was produced from a copy of a document sent to us for microfilming. 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Ml 48106 18 BEDFORD ROW. LONDON WCIR 4EJ. ENGLAND 8106046 RICHARDS, WINSTON A. THE RANDOMIZATION ANALYSIS OF COVARIANŒ AND CHANGE­ OVER DESIGNS Iowa State UniveTsity PHJD. 1980 University Microfilms Internstionsi 300N'.ZeebRoad.AjmAItor.MI48106 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this document have been identified here with a check mark . 1. Glossy photographs 2. Colored illustrations 3. Photographs with dark background '4. Illustrations are poor copy 5. orint shows through as there is text on both sides of page 6. Indistinct, broken or small print on several pages 7. Tightly bound copy with print lost in spine 8. Computer printout pages with indistinct print ^ 9. Page(s) lacking when material received, and not available from school or author 10. Page(s) seem to be missing in numbering only as text follows 11. Poor carbon copy 12. Not original copy, several pages with blurred type 13. Appendix pages are poor copy 14. Original copy with light type 15. Curling and wrinkled pages 16. Other Universiiv MiaoFiims Intemanonal -nrv -rr —r sr\ AMK* 'voanc lOi AA iCl .4 inn Randomization analysis of covariance and change-over designs by Winston A. Richards A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Major: Statistics Approved; Signature was redacted for privacy. In Charge Mayor Work Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. For the Graduée College Iowa State University Ames, Iowa 1980 ii TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. REVIEW OF LITERATURE ON RANDOMIZATION THEORY 19 2.1 Basic Concepts 19 2.1.1 General Ideas for Randomization Analysis 20 2.1.2 Additivity 21 2.1.3 Unbiasedness in Designs 22 2.1.4 Randomization Tests 23 2.2 Formal Theory 23 2.3 Conditions for Convergence 33 2.3.1 Conditions W of Wald and Wolfowitz 33 2.3.2 Madow's Conditions W'" 34 2.3.3 Noether Conditions 37 2.3.4 Cramer's Conditions 46 3. THE ANALYSIS OF COVARIANCE 48 3.1 GMN Linear Regression 48 3.1.1 Maximum Likelihood Estimators 48 3.1.2 The Analysis of Covariance from the Linear Regression Viewpoint 49 3.1.3 Classificatory Models: The Univariate Analysis of Covariance in the Comparative Experiment with Gauss-Markoff Assumptions 54 3.2 Randomization Theory 57 3.2.1 A Randomization Permutation Representation 57 3.2.2 Bias (AOV) in the Analysis of Covariance 61 3.2.3 Unbiasedness of Treatment Contrasts 66 3.2.4 Expectations of Various Statistics Under Randomization 70 3.2.5 Weighted Randomization 73 3.3 Limiting Expectations under Randomization for Large Finite Populations 75 3.4 The Analysis of Multiple Covariance 85 3.5 Normal Law Approximations to the Limiting Distribu­ tions under Randomization Theory 94 iii Page 3.6 The Analysis of Covariance in the Randomized Block Design 102 3.7 Other Related Work 109 3.8 Summary and Conclusions 110 A. DESIGNS FOR TREATMENTS APPLIED IN SEQUENCE 114 4.1 Latin Square Designs 116 4.1.1 Designs for Treatments Applied in Sequence in the Literature 118 4.1.2 Randomization Theory of the Latin Square Design 122 4.2 The General Balanced Change-Over Design [cod] and Extra Period Design 128 4.2.1 Randomization Analysis of the General Balanced COD 130 4.2.2 Change-Over Designs in the Presence of First Order Residual Effects 138 4.2.3 Least Squares Analysis 139 4.2.3.1 The Reduced Normal Equations for T and r 142 4.2.3.2 Time Trends and Serial Correlations 147 4.2.4 Expectations Under Randomization Only 151 4.2.5 Balanced Extra Period Change-Over Designs 159 4.3 Summary 166 4.3.1 Expectation of Mean Squares 166 4.3.2 Conclusions 171 5. APPENDIX A ORDER IN PROBABILITY 174 6. APPENDIX B NORMS AND SPECTRAL RADII OF MATRICES 182 7. APPENDIX C DESIGN RANDOM VARIABLES 186 8. APPENDIX D CALCULATIONS FOR CODS 192 8.1 Calculations for Extra Period cods 196 9. APPENDIX E A MONTE CARLO SIMULATION OF THE ANALYSIS OF COVARIANCE 198 iv Page 10. BIBLIOGRAPHY 208 11. ACKNOWLEDGEMENTS 212 1 1. INTRODUCTION This thesis is concerned with the analysis of comparative experi­ ments. The general idea of these is that one has a collection of experimental units and one has a number of treatments that one wishes to compare. A conqjarative experiment consists of assigning each of the treatments to a subset of the collection of experimental units. The outcome of this process in the simplest case is that one will have observations, say {y.,}, where the subscript j indexes the treatments and the subscript k indexes experimental units within treatments. The task of the experimenter and of the statistician consists of two parts: (a) to decide on an assignment of treatments to experimental units and (b) to decide on modes of interpreting the resultant data. We assume that the experimental units will have some intrinsic responses under some defined basic conditions. We also assume that the experimenter has some control over the treatments themselves of their application to the experimental units and we wish to determine what differential effects, if any, the treatment exert or appear to exert on the variation of responses of the experimental units. We hope, with some degree of optimism, that the relationship between the responses and the treatment effects can be described in a simple functional algebraic equation, the terms in vrtiich represent appropriate variables which are defined over some (limited) specified ranges. By examining the algebraic function we hope to leam more about the under­ lying true relationships and appreciate the differential affects, if 2 any, on the response variable that are produced by changes in the treatment quantities. In our developments we shall assume the algebraic functional relationship is linear in unknown parameters which represent the effects of interest. These unknown parameters are estimated under certain other assumptions and the fitted equation is obtained from the data by the method of least squares. The theory and method of least squares is well-known and so will not be discussed her per se. Associated with the least squares procedure is the analysis of variance in which we have a Pythagorean partition of the metric Zy^^ = y'y into "orthogonal" components corresponding to the treatment effects and the error or intrinsic variability of the experimental units. The data is then "interpreted" by means of various statistical tests and inferences are made about the treatment effects. In the situation where the distribution of the underlying popu­ lation of interest is assumed, e.g., normal distribution say, the validity of the procedures used, or the probabilistic functions based on the resulting sampling distribution will depend, perhaps critically, on the assumptions. Violations of the assumptions may have very serious consequence on the correctness of justification of the inference or decision involved.