Analysis of Messy Data VOLUME III: ANALYSIS OF COVARIANCE George A Milliken Dallas E Johnson CHAPMAN & HALL/CRC A CRC Pr ess Compan y Boca Raton London Ne w York Washington, D.C C0317fm frame Page Monday, July 16, 2001 7:52 AM Library of Congress Cataloging-in-Publication Data Milliken, George A., 1943– Analysis of messy data / George A Milliken, Dallas E Johnson v : ill ; 24 cm Includes bibliographies and indexes Contents: v Designed experiments v Nonreplicated experiments Vol has imprint: New York : Van Nostrand Reinhold ISBN 0-534-02713-X (v 1) : $44.00 ISBN 0-442-24408-8 (v 2) Analysis of variance Experimental design Sampling (Statistics) I Johnson, Dallas E., 1938– II Title QA279 M48 1984 519.5′352 dc19 84-000839 This book contains information obtained from authentic and highly regarded sources Reprinted material is quoted with permission, and sources are indicated A wide variety of references are listed Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use Apart from any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored or transmitted, in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage or retrieval system, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licenses issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of the license issued by the appropriate Reproduction Rights Organization outside the UK The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for creating new works, or for resale Specific permission must be obtained in writing from CRC Press LLC for such copying Direct all inquiries to CRC Press LLC, 2000 N.W Corporate Blvd., Boca Raton, Florida 33431 Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation, without intent to infringe Visit the CRC Press Web site at www.crcpress.com © 2002 by Chapman & Hall/CRC No claim to original U.S Government works International Standard Book Number 1-584-88083-X Library of Congress Card Number 84-000839 Printed in the United States of America Printed on acid-free paper C0317fm frame Page Monday, June 25, 2001 1:04 PM Table of Contents Chapter Introduction to the Analysis of Covariance 1.1 Introduction 1.2 The Covariate Adjustment Process 1.3 A General AOC Model and the Basic Philosophy References Chapter One-Way Analysis of Covariance — One Covariate in a Completely Randomized Design Structure 2.1 2.2 2.3 2.4 The Model Estimation Strategy for Determining the Form of the Model Comparing the Treatments or Regression Lines 2.4.1 Equal Slopes Model 2.4.2 Unequal Slopes Model-Covariate by Treatment Interaction 2.5 Confidence Bands about the Difference of Two Treatments 2.6 Summary of Strategies 2.7 Analysis of Covariance Computations via the SAS® System 2.7.1 Using PROC GLM and PROC MIXED 2.7.2 Using JMP® 2.8 Conclusions References Exercise Chapter 3.1 3.2 3.3 3.4 3.5 3.6 Examples: One-Way Analysis of Covariance — One Covariate in a Completely Randomized Design Structure Introduction Chocolate Candy — Equal Slopes 3.2.1 Analysis Using PROC GLM 3.2.2 Analysis Using PROC MIXED 3.2.3 Analysis Using JMP® Exercise Programs and Initial Resting Heart Rate — Unequal Slopes Effect of Diet on Cholesterol Level: An Exception to the Basic Analysis of Covariance Strategy Change from Base Line Analysis Using Effect of Diet on Cholesterol Level Data Shoe Tread Design Data for Exception to the Basic Strategy © 2002 by CRC Press LLC C0317fm frame Page Monday, June 25, 2001 1:04 PM 3.7 Equal Slopes within Groups of Treatments and Unequal Slopes between Groups 3.8 Unequal Slopes and Equal Intercepts — Part 3.9 Unequal Slopes and Equal Intercepts — Part References Exercises Chapter Multiple Covariates in a One-Way Treatment Structure in a Completely Randomized Design Structure 4.1 4.2 4.3 4.4 4.5 Introduction The Model Estimation Example: Driving A Golf Ball with Different Shafts Example: Effect of Herbicides on the Yield of Soybeans — Three Covariates 4.6 Example: Models That Are Quadratic Functions of the Covariate 4.7 Example: Comparing Response Surface Models Reference Exercises Chapter Two-Way Treatment Structure and Analysis of Covariance in a Completely Randomized Design Structure 5.1 5.2 5.3 Introduction The Model Using the SAS® System 5.3.1 Using PROC GLM and PROC MIXED 5.3.2 Using JMP® 5.4 Example: Average Daily Gains and Birth Weight — Common Slope 5.5 Example: Energy from Wood of Different Types of Trees — Some Unequal Slopes 5.6 Missing Treatment Combinations 5.7 Example: Two-Way Treatment Structure with Missing Cells 5.8 Extensions Reference Exercises Chapter 6.1 6.2 6.3 6.4 6.5 6.6 Beta-Hat Models Introduction The Beta-Hat Model and Analysis Testing Equality of Parameters Complex Treatment Structures Example: One-Way Treatment Structure Example: Two-Way Treatment Structure © 2002 by CRC Press LLC C0317fm frame Page Monday, June 25, 2001 1:04 PM 6.7 Summary Exercises Chapter Variable Selection in the Analysis of Covariance Model 7.1 Introduction 7.2 Procedure for Equal Slopes 7.3 Example: One-Way Treatment Structure with Equal Slopes Model 7.4 Some Theory 7.5 When Slopes are Possibly Unequal References Exercises Chapter Comparing Models for Several Treatments 8.1 Introduction 8.2 Testing Equality of Models for a One-Way Treatment Structure 8.3 Comparing Models for a Two-Way Treatment Structure 8.4 Example: One-Way Treatment Structure with One Covariate 8.5 Example: One-Way Treatment Structure with Three Covariates 8.6 Example: Two-Way Treatment Structure with One Covariate 8.7 Discussion References Exercises Chapter Two Treatments in a Randomized Complete Block Design Structure 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 Introduction Complete Block Designs Within Block Analysis Between Block Analysis Combining Within Block and Between Block Information Determining the Form of the Model Common Slope Model Comparing the Treatments 9.8.1 Equal Slopes Models 9.8.2 Unequal Slopes Model 9.9 Confidence Intervals about Differences of Two Regression Lines 9.9.1 Within Block Analysis 9.9.2 Combined Within Block and Between Block Analysis 9.10 Computations for Model 9.1 Using the SAS® System 9.11 Example: Effect of Drugs on Heart Rate 9.12 Summary References Exercises © 2002 by CRC Press LLC C0317fm frame Page Monday, June 25, 2001 1:04 PM Chapter 10 More Than Two Treatments in a Blocked Design Structure 10.1 10.2 10.3 Introduction RCB Design Structure — Within and Between Block Information Incomplete Block Design Structure — Within and Between Block Information 10.4 Combining Between Block and Within Block Information 10.5 Example: Five Treatments in RCB Design Structure 10.6 Example: Balanced Incomplete Block Design Structure with Four Treatments 10.7 Example: Balanced Incomplete Block Design Structure with Four Treatments Using JMP® 10.8 Summary References Exercises Chapter 11 Covariate Measured on the Block in RCB and Incomplete Block Design Structures 11.1 11.2 11.3 11.4 11.5 11.6 Introduction The Within Block Model The Between Block Model Combining Within Block and Between Block Information Common Slope Model Adjusted Means and Comparing Treatments 11.6.1 Common Slope Model 11.6.2 Non-Parallel Lines Model 11.7 Example: Two Treatments 11.8 Example: Four Treatments in RCB 11.9 Example: Four Treatments in BIB 11.10 Summary References Exercises Chapter 12 Random Effects Models with Covariates 12.1 12.2 12.3 12.4 Introduction The Model Estimation of the Variance Components Changing Location of the Covariate Changes the Estimates of the Variance Components 12.5 Example: Balanced One-Way Treatment Structure 12.6 Example: Unbalanced One-Way Treatment Structure 12.7 Example: Two-Way Treatment Structure 12.8 Summary References Exercises © 2002 by CRC Press LLC C0317fm frame Page Monday, June 25, 2001 1:04 PM Chapter 13 Mixed Models 13.1 13.2 13.3 13.4 Introduction The Matrix Form of the Mixed Model Fixed Effects Treatment Structure Estimation of Fixed Effects and Some Small Sample Size Approximations 13.5 Fixed Treatments and Locations Random 13.6 Example: Two-Way Mixed Effects Treatment Structure in a CRD 13.7 Example: Treatments are Fixed and Locations are Random with a RCB at Each Location References Exercises Chapter 14 Analysis of Covariance Models with Heterogeneous Errors 14.1 14.2 14.3 Introduction The Unequal Variance Model Tests for Homogeneity of Variances 14.3.1 Levene’s Test for Equal Variances 14.3.2 Hartley’s F-Max Test for Equal Variances 14.3.3 Bartlett’s Test for Equal Variances 14.3.4 Likelihood Ratio Test for Equal Variances 14.4 Estimating the Parameters of the Regression Model 14.4.1 Least Squares Estimation 14.4.2 Maximum Likelihood Methods 14.5 Determining the Form of the Model 14.6 Comparing the Models 14.6.1 Comparing the Nonparallel Lines Models 14.6.2 Comparing the Parallel Lines Models 14.7 Computational Issues 14.8 Example: One-Way Treatment Structure with Unequal Variances 14.9 Example: Two-Way Treatment Structure with Unequal Variances 14.10 Example: Treatments in Multi-location Trial 14.11 Summary References Exercises Chapter 15 Analysis of Covariance for Split-Plot and Strip-Plot Design Structures 15.1 15.2 15.3 15.4 Introduction Some Concepts Covariate Measured on the Whole Plot or Large Size of Experimental Unit Covariate is Measured on the Small Size of Experimental Unit © 2002 by CRC Press LLC C0317fm frame Page 10 Monday, June 25, 2001 1:04 PM 15.5 Covariate is Measured on the Large Size of Experimental Unit and a Covariate is Measured on the Small Size of Experimental Unit 15.6 General Representation of the Covariate Part of the Model 15.6.1 Covariate Measured on Large Size of Experimental Unit 15.6.2 Covariate Measured on the Small Size of Experimental Units 15.6.3 Summary of General Representation 15.7 Example: Flour Milling Experiment — Covariate Measured on the Whole Plot 15.8 Example: Cookie Baking 15.9 Example: Teaching Methods with One Covariate Measured on the Large Size Experimental Unit and One Covariate Measured on the Small Size Experimental Unit 15.10 Example: Comfort Study in a Strip-Plot Design with Three Sizes of Experimental Units and Three Covariates 15.11 Conclusions References Exercises Chapter 16 Analysis of Covariance for Repeated Measures Designs 16.1 16.2 16.3 16.4 Introduction The Covariance Part of the Model — Selecting R Covariance Structure of the Data Specifying the Random and Repeated Statements for PROC MIXED of the SAS® System 16.5 Selecting an Adequate Covariance Structure 16.6 Example: Systolic Blood Pressure Study with Covariate Measured on the Large Size Experimental Unit 16.7 Example: Oxide Layer Development Experiment with Three Sizes of Experimental Units Where the Repeated Measure is at the Middle Size of Experimental Unit and the Covariate is Measured on the Small Size Experimental Unit 16.8 Conclusions References Exercises Chapter 17 Analysis of Covariance for Nonreplicated Experiments 17.1 17.2 17.3 17.4 17.5 17.6 17.7 17.8 Introduction Experiments with A Single Covariate Experiments with Multiple Covariates Selecting Non-null and Null Partitions Estimating the Parameters Example: Milling Flour Using Three Factors Each at Two Levels Example: Baking Bread Using Four Factors Each at Two Levels Example: Hamburger Patties with Four Factors Each at Two Levels © 2002 by CRC Press LLC C0317fm frame Page 11 Monday, June 25, 2001 1:04 PM 17.9 Example: Strength of Composite Material Coupons with Two Covariates 17.10 Example: Effectiveness of Paint on Bricks with Unequal Slopes 17.11 Summary References Exercises Chapter 18 Special Applications of Analysis of Covariance 18.1 18.2 18.3 18.4 Introduction Blocking and Analysis of Covariance Treatments Have Different Ranges of the Covariate Nonparametric Analysis of Covariance 18.4.1 Heart Rate Data from Exercise Programs 18.4.2 Average Daily Gain Data from a Two-Way Treatment Structure 18.5 Crossover Design with Covariates 18.6 Nonlinear Analysis of Covariance 18.7 Effect of Outliers References Exercises © 2002 by CRC Press LLC C0317fm frame Page 13 Monday, June 25, 2001 1:04 PM Preface Analysis of covariance is a statistical procedure that enables one to incorporate information about concomitant variables into the analysis of a response variable Sometimes this is done in an attempt to reduce experimental error Other times it is done to better understand the phenomenon being studied The approach used in this book is that the analysis of covariance model is described as a method of comparing a series of regression models — one for each of the levels of a factor or combinations of levels of factors being studied Since covariance models are regression models, analysts can use all of the methods of regression analysis to deal with problems such as lack of fit, outliers, etc The strategies described in this book will enable the reader to appropriately formulate and analyze various kinds of covariance models When covariates are measured and incorporated into the analysis of a response variable, the main objective of analysis of covariance is to compare treatments or treatment combinations at common values of the covariates This is particularly true when the experimental units assigned to each of the treatment combinations may have differing values of the covariates Comparing treatments is dependent on the form of the covariance model and thus care must be taken so that mistakes are not made when drawing conclusions The goal of this book is to present the structure and philosophy for using the analysis of covariance by including descriptions of methodologies, illustrating the methodologies by analyzing numerous data sets, and occasionally furnishing some theory when required Our aim is to provide data analysts with tools for analyzing data with covariates and to enable them to appropriately interpret the results Some of the methods and techniques described in this book are not available in other books, but two issues of Biometrics (1957, Volume 13, Number 3, and 1982, Volume 38, Number 3) were dedicated to the topic of analysis of covariance The topics presented are among those that we, as consulting statisticians, have found to be most helpful in analyzing data when covariates are available for possible inclusion in the analysis Readers of this book will learn how to: • Formulate appropriate analysis of covariance models • Simplify analysis of covariance models • Compare levels of a factor or of levels of combinations of factors when the model involves covariates • Construct and analyze a model with two or more factors in the treatment structure • Analyze two-way treatment structures with missing cells • Compare models using the beta-hat model • Perform variable selection within the analysis of covariance model © 2002 by CRC Press LLC Special Applications of Analysis of Covariance 49 TABLE 18.44 PROC GLM Code to Fit the Model with the Data for Order One of Person Three Also Removed and the Resulting Box-Plot of the Residuals data wide; set wide; if person=3 proc glm data=wide; class person model resp=prod order order*prod output out=glmres r=glmr; proc sort data=glmres; by order; proc univariate data=glmres plot 0.5 –0.5 –1 order and order=1 then delete; order prod; mild*prod person; normal; var glmr;by order; | + | | | | | | + | | | | | | | | | | + -+ + -+ + -+ | | | | | | * -* + -+ + * + * * + * | + | * + * | | | | | | | + -+ | | | + -+ | | | | + -+ | + -+ | + | | | | | | | | | | + -+ -+ -+ + -1 adjust=simulate option was used to control the experiment wise error rate There are significance differences (p = 0.05) between products and 2, and 4, and and This example demonstrates the drastic effects that outliers can have on the analysis Additionally, outliers can occur for any size of experimental unit or random effect in the model This example illustrated that residual outliers can have a disabling effect on the convergence for complex covariance structures as well as on the estimates of the covariance structure parameters When outliers occur for other random effects, the estimates of the corresponding variance components are greatly effected as well as possibly having an impact on some of the tests for fixed effects The data analyst needs to make sure the data set has been screened for possible outliers and when such observations or levels of random effects are identified, then the researcher needs to go back to the records to determine if there is an identifiable cause If no cause can be identified, then there is no basis to remove the data from the analysis © 2002 by CRC Press LLC 50 Analysis of Messy Data, Volume III: Analysis of Covariance TABLE 18.45 PROC MIXED Code to Fit the Unequal Slopes Model with the UN(1)Covariance Structure to the Data with Two Outliers Removed and the Results to be Used as Starting Values for the ARH(1) Covariance Structure proc mixed ic scoring=20 data=wide; class person order prod; model resp=prod order order*prod mild*prod/ddfm=kr; random person; repeated order/subject=person type=un(1); CovParm person UN(1,1) UN(2,1) UN(2,2) UN(3,1) UN(3,2) UN(3,3) UN(4,1) UN(4,2) UN(4,3) UN(4,4) Subject person person person person person person person person person person Estimate 1.680400 0.088202 0.000000 0.200085 0.000000 0.000000 0.764544 0.000000 0.000000 0.000000 0.014563 Effect prod order order*prod mild*prod NumDF 3 DenDF 29.1 17.0 28.3 26.1 © 2002 by CRC Press LLC FValue 1.95 1.35 1.97 2.61 ProbF 0.1433 0.2904 0.0811 0.0586 Special Applications of Analysis of Covariance 51 TABLE 18.46 PROC MIXED Code to Fit the Unequal Slopes Model with the ARH(1) Covariance Structure Using Starting Values from the UN(1) Covariance Structure to the Data with Two Outliers Removed and the Results proc mixed ic data=wide; class person order prod; **starting values from un(1); model resp=prod order order*prod mild*prod/ddfm=kr outp=predicted; random person/solution; repeated order/subject=person type=arh(1); parms 1.6804 0.0882 2001 7645 01456 1/lowerb = 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4; lsmeans prod/diff; ods output solutionr=randomsol; CovParm person Var(1) Var(2) Var(3) Var(4) ARH(1) Subject person person person person person Estimate 1.663316 0.041829 0.463647 1.631961 0.180904 0.843573 Effect prod order order*prod mild*prod NumDF 3 DenDF 25.4 13.8 20.8 29.2 © 2002 by CRC Press LLC FValue 0.79 0.57 1.73 2.33 ProbF 0.5127 0.6468 0.1451 0.0790 52 Analysis of Messy Data, Volume III: Analysis of Covariance TABLE 18.47 The EBLUPs of the Person Effects from the ARH(1) Model Fit to the Data Set with Outliers Removed Person 10 11 12 13 14 15 16 17 18 19 20 Estimate –1.578 0.617 0.139 –0.181 –0.962 3.926 –0.239 –1.244 –0.329 0.117 0.263 0.942 –2.344 0.385 0.097 0.844 –0.166 –0.619 –0.201 0.532 StdErrPred 0.510 0.436 0.519 0.550 0.519 0.459 0.515 0.583 0.461 0.501 0.620 0.534 0.538 0.707 0.477 0.452 0.482 0.554 0.604 0.435 df 21 14 29 19 17 15 16 20 11 16 20 20 17 22 18 14 18 19 20 12 tValue –3.10 1.42 0.27 –0.33 –1.85 8.55 –0.46 –2.13 –0.71 0.23 0.42 1.76 –4.35 0.54 0.20 1.87 –0.34 –1.12 –0.33 1.22 Probt 0.0055 0.1791 0.7905 0.7454 0.0813 0.0000 0.6484 0.0456 0.4909 0.8176 0.6762 0.0933 0.0004 0.5915 0.8404 0.0831 0.7353 0.2782 0.7434 0.2460 TABLE 18.48 Box-Plot of the t-Values Corresponding to the EBLUPs in Table 18.47 proc univariate normal plot data=randomsol; var tvalue Variable: tValue (t Value) Stem –0 –2 –4 Leaf # Boxplot * 223452489 9175333 11 + + + * -* | + + + + Multiply Stem.Leaf by 10**–1 © 2002 by CRC Press LLC Special Applications of Analysis of Covariance 53 TABLE 18.49 PROC MIXED Code to Fit the Unequal Slopes Model with UN(1) Covariance Structure to the Data Set with Person Six Removed to Obtain Starting Values for the ARH(1) Covariance Structure data wide; set wide; if person=6 then delete; proc mixed ic scoring=20 data=wide; class person order prod; model resp=prod order order*prod mild*prod/ddfm=kr; random person; repeated order/subject=person type=un(1); CovParm person UN(1,1) UN(2,1) UN(2,2) UN(3,1) UN(3,2) UN(3,3) UN(4,1) UN(4,2) UN(4,3) UN(4,4) Subject person person person person person person person person person person Estimate 0.686824 0.094153 0.000000 0.166396 0.000000 0.000000 0.754291 0.000000 0.000000 0.000000 0.029599 Effect prod order order*prod mild*prod NumDF 3 DenDF 26.5 15.1 25.4 24.2 © 2002 by CRC Press LLC FValue 1.84 0.96 1.72 6.62 ProbF 0.1647 0.4366 0.1360 0.0010 54 Analysis of Messy Data, Volume III: Analysis of Covariance TABLE 18.50 PROC MIXED Code to Fit the Unequal Slopes Model with ARH(1) Covariance Structure Using Starting Values from the UN(1) Structure in Table 18.49 proc mixed ic data=wide; class person order prod; **starting values from un(1); model resp=prod order order*prod mild*prod/ddfm=kr outp=predicted; random person/solution; repeated order/subject=person type=arh(1); parms 6868 09414 1664 7543 0296 1/lowerb = 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4; ods output solutionr=randomsol; CovParm person Var(1) Var(2) Var(3) Var(4) ARH(1) Subject person person person person person Estimate 0.689719 0.096455 0.223257 0.840527 0.031573 0.375293 Effect prod order order*prod mild*prod NumDF 3 DenDF 26.3 12.4 23.0 25.2 © 2002 by CRC Press LLC FValue 1.18 0.81 1.36 6.02 ProbF 0.3360 0.5135 0.2609 0.0015 Special Applications of Analysis of Covariance 55 TABLE 18.51 EBLUPs from the Analysis without Person Six Person 10 11 12 13 14 15 16 17 18 19 20 Estimate –1.167 0.305 0.414 0.432 –0.311 0.026 –0.888 –0.513 0.076 0.496 1.539 –1.831 –0.149 0.243 1.170 0.123 –0.461 –0.131 0.627 StdErrPred 0.354 0.329 0.332 0.383 0.366 0.392 0.415 0.326 0.377 0.421 0.386 0.386 0.477 0.340 0.351 0.371 0.415 0.409 0.347 df 15 12 7.6 19 18 24 24 13 22 21 19 20 23 14 16 19 24 21 18 tValue –3.30 0.93 1.25 1.13 –0.85 0.07 –2.14 –1.57 0.20 1.18 3.99 –4.74 –0.31 0.71 3.34 0.33 –1.11 –0.32 1.81 Probt 0.0047 0.3719 0.2494 0.2733 0.4058 0.9481 0.0424 0.1402 0.8414 0.2519 0.0008 0.0001 0.7577 0.4865 0.0041 0.7440 0.2780 0.7528 0.0874 TABLE 18.52 Box-Plot of the t-Values Corresponding to the EBLUPs for the Analysis without Person Six proc univariate normal plot data=randomsol; var tvalue; Variable: Resid Stem –0 –2 –4 –6 –8 Leaf 22355 121 9328 40 10 # 1 2 + + + + Multiply Stem.Leaf by 10**–1 © 2002 by CRC Press LLC Boxplot | | + -+ * + * | | + -+ | | | 56 Analysis of Messy Data, Volume III: Analysis of Covariance TABLE 18.53 PROC MIXED Code to Fit the Unequal Slopes Model with Test for Equal Slopes proc mixed ic data=wide; class person order prod; **starting values from un(1); model resp=prod order order*prod mild mild*prod/ddfm=kr; random person; repeated order/subject=person type=arh(1); parms 6868 09414 1664 7543 0296 1/lowerb = 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4; CovParm person Var(1) Var(2) Var(3) Var(4) ARH(1) Subject person person person person person Estimate 0.689719 0.096455 0.223257 0.840527 0.031573 0.375293 Effect prod order order*prod mild mild*prod NumDF 3 DenDF 26.3 12.4 23.0 16.2 20.7 © 2002 by CRC Press LLC FValue 1.18 0.81 1.36 22.92 0.12 ProbF 0.3360 0.5135 0.2609 0.0002 0.9499 Special Applications of Analysis of Covariance 57 TABLE 18.54 PROC MIXED Code to Fit the Equal Slopes Model with ARH(1) Covariance Structure to the Data Set with Outliers Removed proc mixed ic data=wide; class person order prod; **starting values from un(1); model resp=prod order order*prod mild /ddfm=kr outp=predicted; random person/solution; repeated order/subject=person type=arh(1); parms 6868 09414 1664 7543 0296 1/lowerb = 1e-4, 1e-4, 1e-4, 1e-4, 1e-4, 1e-4; CovParm person Var(1) Var(2) Var(3) Var(4) ARH(1) Subject person person person person person Estimate 0.722087 0.000100 0.247060 1.113557 0.105590 0.676322 Effect prod order order*prod mild NumDF 3 DenDF 27.1 18.9 28.1 16.8 FValue 8.96 0.70 2.14 24.30 ProbF 0.0003 0.5661 0.0602 0.0001 TABLE 18.55 Adjusted Means and Pairwise Comparisons of the Means Using the Simulate Adjustment for Multiple Comparisons lsmeans prod/diff adjust=simulate; Effect prod prod prod prod prod Estimate 9.79 10.40 10.01 10.53 StdErr 0.24 0.24 0.24 0.23 df 32.9 31.6 32.5 29.5 tValue 40.14 43.69 40.93 45.27 Probt 0.0000 0.0000 0.0000 0.0000 prod 1 2 _prod 4 Estimate –0.61 –0.21 –0.74 0.40 –0.13 –0.53 StdErr 0.17 0.17 0.15 0.17 0.14 0.17 df 27.2 22.6 28.4 28.3 28.4 31.1 tValue –3.61 –1.25 –4.96 2.35 –0.92 –3.13 Probt 0.0012 0.2231 0.0000 0.0261 0.3672 0.0037 © 2002 by CRC Press LLC Adjp 0.0056 0.5866 0.0003 0.1057 0.7887 0.0184 58 Analysis of Messy Data, Volume III: Analysis of Covariance REFERENCES Conover, W J and Iman, R L (1982) Analysis of covariance using the rank transformation, Biometrics 38(3):715–724 R Littell, G A Milliken, W Stroup, and R Wolfinger (1996) SAS System for Mixed Models SAS Institute Inc., Cary, NC Milliken, G A and Johnson, D E (1992) Analysis of Messy Data, Volume I: Design Experiments, Chapman & Hall, London SAS Institute Inc (1999) SAS/STAT ® User’s Guide, Version 8, Cary, NC Urquhart, N S (1982) Adjustment in covariance when one factor affects the covariate, Biometrics 38(3):651–660 EXERCISES EXERCISE 18.1: A nutritionist designed a study to evaluate the ability of three diets (A, B, and C) to affect weight loss for both males and females At the start of the study, the males and females were randomly assigned to the diets and each subject’s initial weight and percent body fat were measured The subjects were given the diet and were to return every month to measure their weight They were also given a food diary in which they recorded all of the items they ate during the third week of the month and the number of grams of fat and the milligrams of salt on a daily basis were determined Each subject was weighed after 1, 2, 3, and months The data in the table, salti, dfati, and wti, are the measurements after the ith month Use the initial weight (iwt), percent body fat (bfat), diet salt (salti), and diet fat (dfati) as possible covariates and carry out an thorough analysis of this data set © 2002 by CRC Press LLC diet A A A A A A A A A B B B B B B B B B B B sex F F F F M M M M M F F F F F F M M M M M sub 4 5 Iwt 175 185 172 174 200 223 193 194 236 172 157 158 161 178 169 230 230 203 204 209 bfat 27.1 32.4 30.5 27.2 27.4 32.4 25.2 23.7 35.7 27.8 26.8 26.8 30.2 31.1 29.1 30.8 34.1 29.9 27.7 30.6 salt1 1932 2088 2246 2897 2098 2257 2078 2059 2795 2620 2269 2703 2809 2099 2398 2690 2310 2482 1953 2698 dfat1 111 70 89 79 110 104 84 85 81 58 107 103 109 76 73 86 65 122 75 115 wt1 179 181 174 178 206 224 192 198 239 168 159 156 164 175 166 231 229 201 202 206 salt2 1929 2011 2081 2887 2081 2322 2153 1851 2997 2628 2173 2811 2882 2127 2491 2714 2328 2550 1953 2707 dfat2 105 77 96 86 112 110 86 83 85 65 108 102 116 80 70 92 58 122 80 110 wt2 177 180 171 176 204 222 186 190 233 167 161 159 164 173 164 229 225 203 197 209 salt3 1904 2035 2173 2846 2010 2180 2325 1795 2976 2746 2323 2753 2963 2230 2606 2612 2330 2660 1898 2693 dfat3 102 72 103 83 115 105 89 77 92 62 102 95 111 79 74 86 59 129 85 103 wt3 176 175 171 173 209 225 185 188 228 169 167 163 172 171 165 232 224 207 190 213 salt4 1828 2149 2261 2956 2043 2182 2326 1980 2941 2774 2211 2833 2868 2248 2733 2585 2217 2518 1880 2659 dfat4 98 77 103 79 119 103 88 72 87 65 96 99 109 84 70 82 58 125 81 106 wt4 174 173 172 172 210 221 184 185 226 162 162 156 166 160 157 226 210 197 172 197 Special Applications of Analysis of Covariance Data for Exercise 18.1 59 © 2002 by CRC Press LLC 60 diet C C C C C C C C C C C C C sex F F F F F F F F M M M M M sub © 2002 by CRC Press LLC Iwt 165 168 159 171 174 152 173 173 216 235 224 184 234 bfat 25.5 29.5 24.4 30.7 30.1 23.0 28.2 29.2 31.5 31.7 30.2 24.8 33.6 salt1 2576 2487 2247 2395 2487 2740 2327 2304 2411 2178 1876 2164 2394 dfat1 116 96 81 73 70 128 108 117 117 111 111 60 122 wt1 168 170 160 169 173 152 174 172 214 233 220 189 234 salt2 2662 2489 2210 2418 2543 2694 2215 2324 2349 2174 1558 2210 2444 dfat2 113 101 77 69 64 130 110 114 115 117 111 53 116 wt2 166 168 160 167 172 152 173 168 211 233 217 187 228 salt3 2608 2539 2276 2351 2583 2658 2297 2217 2306 2137 1681 2167 2301 dfat3 107 96 82 67 61 138 109 115 117 122 116 55 119 wt3 164 161 157 161 167 147 166 163 205 224 209 176 216 salt4 2609 2592 2369 2330 2649 2574 2179 2243 2257 2096 1522 2149 2378 dfat4 109 94 84 61 68 142 103 111 117 129 123 61 118 wt4 165 165 154 162 167 151 164 162 201 226 208 174 216 Analysis of Messy Data, Volume III: Analysis of Covariance Data for Exercise 18.1 Special Applications of Analysis of Covariance 61 EXERCISE 18.2: Use the rank transformation and provide an analysis of the data in Exercise 3.1 EXERCISE 18.3: Use the rank transformation and provide an analysis of the data in Exercise 4.1 EXERCISE 18.4: Use the rank transformation and provide an analysis of the data in Section 5.5 EXERCISE 18.5: Use the rank transformation and provide an analysis of the data in Section 5.7 EXERCISE 18.6: The data in the table below are times to dissolve a piece of chocolate candy A class of 48 students was available for the study Each student was given a butterscotch chip and the time to dissolve the chip by mouth was recorded The students were put into blocks of size six based on their time to dissolve the butterscotch chip (bst) Then the six students in each block were randomly assigned to one of the chocolate candy types and the time to dissolve the candy piece (time) by mouth was determined Provide a complete analysis of this data set Analyze the data without using the bst as a covariate Reanalyze the data ignoring the blocking factor but use bst as a covariate Compare the three analyses Data for exercise 18.6 block Choc Chip Red M&M® Small M&M® bst 15 16 21 23 26 28 31 35 bst 16 18 21 23 27 30 31 36 bst 16 21 22 24 27 30 33 36 time 15 16 26 20 31 31 43 45 time 24 19 31 19 33 34 38 42 time 14 16 21 16 28 31 28 33 Button bst 16 21 22 25 27 31 34 36 time 32 35 35 28 37 35 41 48 Blue M&M® Snow Cap bst 16 21 23 25 28 31 35 36 bst 16 21 23 25 28 31 35 39 time 24 28 30 30 33 31 45 42 time 14 24 25 27 33 37 40 42 EXERCISE 18.7: The data for this example involved finding ten groups of three persons each and then randomly assigning one of the three shapes of hard candy to each person within a group Each person was given a butterscotch chip to dissolve by mouth and the time was recorded Then each person recorded the time to dissolve the assigned piece of hard candy The groups form blocks as they were observed on different days, bst is the time to dissolve the butterscotch piece, and time is the time to dissolve the hard candy piece Carry out a thorough analysis of this data set © 2002 by CRC Press LLC 62 Analysis of Messy Data, Volume III: Analysis of Covariance Data for Exercise 18.7 Round Block 10 bst 25 23 31 32 38 23 22 20 19 32 time 99 92 123 115 188 96 94 88 99 113 Square bst 34 19 25 17 21 27 30 32 23 21 time 184 92 102 88 93 110 125 148 98 96 Flat bst 30 36 35 21 25 37 33 20 25 22 time 105 127 125 91 92 133 118 93 95 91 EXERCISE 18.8: Use the beta-hat model approach described in Chapter to test each of the following hypotheses for the data in Table 18.33 and compare the results obtained by PROC NLMIXED H o1: α1 = α = α = α vs Ha1: ( not H o1:) H o : β1 = β2 = β3 = β vs Ha : ( not H o :) H o 3: γ = γ = γ = γ vs Ha 3: ( not H o 3:) H o : α1 + β e − γ insect = α + β e − γ insect = α + β e − γ insect = α + β e − γ insect vs Ha : ( not H :) EXERCISE 18.9: Use the model comparison approach described in Chapter to test each of the hypotheses in Exercise 18.8 Also provide a test of the equal model hypothesis © 2002 by CRC Press LLC Special Applications of Analysis of Covariance 63 EXERCISE 18.10: The data in in the table below are from an experiment similar to that in Section 16.5 Determine an appropriate covariance matrix and then provide a detailed analysis of the data set Data for Exercise 18.10 Where IBP Is Initial Blood Pressure and bp1, …, bp6 are the Blood Pressure Readings for the Six Time Periods Drug No No No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes No No No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Exercise No No No No No No No No No No No No No No No No Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes © 2002 by CRC Press LLC Person 8 8 IBP 133 137 148 136 140 139 154 152 150 147 142 144 140 137 143 137 150 151 151 142 149 132 134 151 131 148 151 150 144 151 152 135 bp1 146 136 148 139 141 140 185 146 145 147 133 139 132 133 144 122 143 144 155 149 142 133 136 150 95 138 146 137 129 140 149 123 bp2 144 135 148 137 143 136 181 147 134 139 121 128 121 122 133 111 142 137 149 101 140 127 129 145 85 130 134 126 123 138 141 115 bp3 142 134 148 138 145 135 183 146 136 139 122 128 121 122 131 111 139 134 147 140 136 123 125 141 78 124 129 119 118 134 138 109 bp4 141 134 143 139 147 135 182 145 134 132 120 122 115 118 124 104 137 131 140 140 132 122 122 137 76 120 130 119 117 131 132 107 bp5 142 134 143 139 147 132 180 146 132 134 165 119 117 119 123 101 134 128 136 137 126 117 123 131 76 122 134 120 118 126 132 111 bp6 143 137 144 142 147 135 181 143 134 135 116 116 116 121 124 101 128 121 131 129 126 116 115 130 79 123 134 121 117 129 131 109 ... 2001 1:46 PM Analysis of Messy Data, Volume III: Analysis of Covariance TABLE 1.3 Analysis of Covariance to Provide the Estimates of the Slope and Intercepts to be Used in Adjusting the Data Source... to use software to carry out the analysis of covariance © 2002 by CRC Press LLC C0317c01 frame Page 10 Sunday, June 24, 2001 1:46 PM 10 Analysis of Messy Data, Volume III: Analysis of Covariance. .. frame Page Sunday, June 24, 2001 1:46 PM Analysis of Messy Data, Volume III: Analysis of Covariance TABLE 1.5 Analysis of the Adjusted Yields (Too Many Degrees of Freedom for Error) Source Model Error