Design of Experiments

Transcription

1 Design of Experiments Statistical Principles of Research Design and Analysis Second Edition Robert O. Kuehl The University of Arizona Duxbury Thomson Learning* Pacific Grove Albany Belmont Bonn Boston Cincinnati Detroit Johannesburg London Madrid Melbourne Mexico City New York Paris Singapore Tokyo Toronto Washington

2 Contents Research Design Principles The Legacy of Sir Ronald A. Fisher Planning for Research Experiments, Treatments, and Experimental Units Research Hypotheses Generate Treatment Designs Local Control of Experimental Errors Replication for Valid Experiments How Many Replications? Randomization for Valid Inferences Relative Efficiency of Experiment Designs From Principles to Practice: A Case Study 26 Getting Started with Completely Randomized Designs Assembling the Research Design How to Randomize Preparation of Data Files for the Analysis A Statistical Model for the Experiment Estimation of the Model Parameters with Least Squares Sums of Squares to Identify Important Sources of Variation A Treatment Effects Model Degrees of Freedom Summaries in the Analysis of Variance Table Tests of Hypotheses About Linear Models Significance Testing and Tests of Hypotheses Standard Errors and Confidence Intervals for Treatment Means Unequal Replication of the Treatments How Many Replications for the F Test? 63 Vll

3 Vlii CONTENTS 2A.1 Appendix: Expected Values 70 2A.2 Appendix: Expected Mean Squares 71 Treatment Comparisons Treatment Comparisons Answer Research Questions Planning Comparisons Among Treatments Response Curves for Quantitative Treatment Factors Multiple Comparisons Affect Error Rates Simultaneous Statistical Inference Multiple Comparisons with the Best Treatment Comparison of All Treatments with a Control Pairwise Comparison of All Treatments Summary Comments on Multiple Comparisons A Appendix: Linear Functions of Random Variables 121 Diagnosing Agreement Between the Data and the Model Valid Analysis Depends on Valid Assumptions Effects of Departures from Assumptions Residuals Are the Basis of Diagnostic Tools Looking for Outliers with the Residuals Variance-Stabilizing Transformations for Data with ICnown Distributions Power Transformations to Stabilize Variances Generalizing the Linear Model Model Evaluation with Residual-Fitted Spread Plots 141 4A Appendix: Data for Example Experiments to Study Variances Random Effects Models for Variances A Statistical Model for Variance Components Point Estimates of Variance Components Interval Estimates for Variance Components Courses of Action with Negative Variance Estimates Intraclass Correlation Measures Similarity in a Group Unequal Numbers of Observations in the Groups How Many Observations to Study Variances? Random Subsamples to Procure Data for the Experiment Using Variance Estimates to Allocate Sampling Efforts Unequal Numbers of Replications and Subsamples 164 5A Appendix: Coefficient Calculations for Expected Mean Squares in Table

4 6 Factorial Treatment Designs Efficient Experiments with Factorial Treatment Designs Three Types of Treatment Factor Effects The Statistical Model for Two Treatment Factors The Analysis for Two Factors Using Response Curves for Quantitative Treatment Factors Three Treatment Factors Estimation of Error Variance with One Replication How Many Replications to Test Factor Effects? Unequal Replication of Treatments 208 6A Appendix: Least Squares for Factorial Treatment Designs Factorial Treatment Designs: Random and Mixed Models Random Effects for Factorial Treatment Designs Mixed Models Nested Factor Designs: A Variation on the Theme Nested and Crossed Factors Designs How Many Replications? Expected Mean Square Rules 255 CONTENTS ix 8 Complete Block Designs Blocking to Increase Precision Randomized Complete Block Designs Use One Blocking Criterion Latin Square Designs Use Two Blocking Criteria Factorial Experiments in Complete Block Designs Missing Data in Blocked Designs Experiments Performed Several Times 292 8A. Appendix: Selected Latin Squares Incomplete Block Designs: An Introduction Incomplete Blocks of Treatments to Reduce Block Size Balanced Incomplete Block (BIB) Designs How to Randomize Incomplete Block Designs Analysis of BIB Designs Row-Column Designs for Two Blocking Criteria Reduce Experiment Size with Partially Balanced (PBIB) Designs Efficiency of Incomplete Block Designs 325 9A.1 Appendix: Selected.Balanced Incomplete Block Designs 330 9A.2 Appendix: Selected Incomplete Latin Square Designs 332 9A.3 Appendix: Least Squares Estimates for BIB Designs 336

5 X CONTENTS 10 Incomplete Block Designs: Resolvable and Cyclic Designs _ Resolvable Designs to Help Manage the Experiment Resolvable Row-Column Designs for Two Blocking Criteria Cyclic Designs Simplify Design Construction Choosing Incomplete Block Designs A.1 Appendix: Plans for Cyclic Designs A.2 Appendix: Generating Arrays for a Designs Incomplete Block Designs: Factorial Treatment Designs Taking Greater Advantage of Factorial Treatment Designs " Factorials to Evaluate Many Factors Incomplete Block Designs for 2 n Factorials A General Method to Create Incomplete Blocks Incomplete Block Designs for 3" Factorials Concluding Remarks A Appendix: Incomplete Block Design Plans for 2 n Factorials Fractional Factorial Designs Reduce Experiment Size with Fractional Treatment Designs The Half Fraction of the 2" Factorial Design Resolution Related to Aliases Analysis of Half Replicate 2 n-1 Designs The Quarter Fractions of 2 n Factorials Construction of 2 n ~ p Designs with Resolution III and IV Genichi Taguchi and Quality Improvement Concluding Remarks A Appendix: Fractional Factorial Design Plans Response Surface Designs Describe Responses with Equations and Graphs Identify Important Factors with 2 n Factorials Designs to Estimate Second-Order Response Surfaces Quadratic Response Surface Estimation Response Surface Exploration Designs for Mixtures of Ingredients Analysis of Mixture Experiments A.I Appendix: Least Squares Estimation of Regression Models A.2 Appendix: Location of Coordinates for the Stationary Point A.3 Appendix: Canonical Form of the Quadratic Equation 467

6 CONTENTS xi 14 Split-Plot Designs Plots of Different Size in the Same Experiment Two Experimental Errors for Two Plot Sizes The Analysis for Split-Plot Designs Standard Errors for Treatment Factor Means Features of the Split-Plot Design Relative Efficiency of Subplot and Whole-Plot Comparisons The Split-Split-Plot Design for Three Treatment Factors The Split-Block Design Additional Information About Split-Plot Designs Repeated Measures Designs Studies of Time Trends Relationships Among Repeated Measurements A Test for the Huynh-Feldt Assumption A Univariate Analysis of Variance for Repeated Measures Analysis When Univariate Analysis Assumptions Do Not Hold Other Experiments with Repeated Measures Properties Other Models for Correlation Among Repeated Measures A.I Appendix: The Mauchly Test for Sphericity A.2 Appendix: Degrees of Freedom Adjustments for Repeated Measures Analysis of Variance Crossover Designs Administer All Treatments to Each Experimental Unit Analysis of Crossover Designs Balanced Designs for Crossover Studies Crossover Designs for Two Treatments A.I Appendix: Coding Data Files for Crossover Studies A.2 Appendix: Treatment Sum of Squares for Balanced Designs Analysis of Covariance Local Control with a Measured Covariate Analysis of Covariance for Completely Randomized Designs The Analysis of Covariance for Blocked Experiment Designs Practical Consequences of Covariance Analysis 570 References 576 Appendix Tables 587 Answers to Selected Exercises 633 Index 661