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Topics in Experimental Design Ronald Christensen Professor of Statistics Department of Mathematics and Statistics University of New Mexico Copyright c 2016 Topics in Experimental Design Springer Preface This (non)book assumes that the reader is familiar with the basic ideas of experi- mental design and with linear models. I think most of the applied material should be accessible to people with MS courses in regression and ANOVA but I had no hesi- tation in using linear model theory, if I needed it, to discuss more technical aspects. Over the last several years I’ve been working on revisions to my books Analysis of Variance, Design, and Regression (ANREG); Plane Answers to Complex Ques- tions: The Theory of Linear Models (PA); and Advanced Linear Modeling (ALM). In each of these books there was material that no longer seemed sufficiently relevant to the themes of the book for me to retain. Most of that material related to Exper- imental Design. Additionally, due to Kempthorne’s (1952) book, many years ago I figured out p f designs and wrote a chapter explaining them for the very first edition of ALM, but it was never included in any of my books. I like all of this material and think it is worthwhile, so I have accumulated it here. (I’m not actually all that wild about the recovery of interblock information in BIBs.) A few years ago my friend Chris Nachtsheim came to Albuquerque and gave a wonderful talk on Definitive Screening Designs. Chapter 5 was inspired by that talk along with my longstanding desire to figure out what was going on with Placett- Burman designs. I’m very slowly working on R code for this material. See http://www. stat.unm.edu/˜fletcher/R-TD.pdf. Also, a retypeset version of the first edition of ANREG (ANREG-I) is available at http://www.stat.unm.edu/ ˜fletcher/anreg.pdf. R computer code for the second edition of ANREG is available at http://www.stat.unm.edu/˜fletcher/Rcode.pdf. As I have mentioned elsewhere, the large numbers of references to myself are as much due to sloth as ego. Ronald Christensen Albuquerque, New Mexico November 12, 2016 March 30, 2019 vii Contents Preface ............................................................ vii Table of Contents ................................................... ix 1 Fundamentals ................................................. 1 1.1 Notation..................................................3 2 2 f Factorial Systems ............................................ 5 2.0.1 Effect contrasts in 2 f factorials.........................6 2.1 Confounding..............................................9 2.1.1 Split plot analysis.................................... 17 2.1.2 Partial confounding.................................. 18 2.2 Fractional replication....................................... 22 2.2.1 Fractional replication with confounding................. 27 2.3 Analysis of unreplicated experiments.......................... 28 2.3.1 Balanced ANOVA computing techniques................ 36 2.4 More on graphical analysis................................... 37 2.4.1 Multiple maximum tests.............................. 38 2.4.2 Simulation envelopes................................. 40 2.4.3 Outliers............................................ 41 2.5 Lenth’s method............................................ 42 2.6 Augmenting designs for factors at two levels.................... 42 2.7 Exercises................................................. 43 3 p f Factorial Treatment Structures ............................... 45 3.1 3 f Factorials............................................... 46 3.2 Column Space Considerations................................ 52 3.3 Confounding and Fractional Replication....................... 54 3.3.1 Confounding........................................ 55 3.3.2 Fractional Replication................................ 57 3.4 Taguchi’s Orthogonal Arrays................................. 61 ix Contents ix 3.4.1 Listed Designs...................................... 63 3.4.2 Inner and Outer Arrays............................... 64 3.4.3 Signal-to-Noise Ratios................................ 65 3.4.4 Taguchi Analysis.................................... 66 3.5 Analysis of a Partially Confounded 33 ......................... 69 3.5.1 The Expanded ANOVA Table.......................... 72 3.5.2 Interaction Contrasts................................. 74 3.6 5 f Factorials............................................... 78 4 Mixed Factor Levels ............................................ 83 4.1 Fractionated Partial Confounding............................. 83 4.2 More Mixtures of Prime Powers.............................. 86 4.3 Taguchi’s L9 with an Outer Array............................. 87 4.4 Factor Levels that are Powers of Primes........................ 89 5 Screening Designs .............................................. 93 5.1 Designs at Two Levels...................................... 93 5.2 Theory for Designs at Two Levels............................. 97 5.2.1 Blocking........................................... 100 5.2.2 Hare’s Full model.................................... 103 5.2.3 Construction of Hadamard Matrices..................... 104 5.3 Designs for Three Levels.................................... 106 5.3.1 Definitive Screening Designs.......................... 106 5.3.2 Some Linear Model Theory........................... 108 5.3.3 Weighing out of conference............................ 113 5.4 Notes..................................................... 114 6 Response Surface Maximization .................................115 6.1 Approximating Response Functions........................... 117 6.2 First-Order Models and Steepest Ascent........................ 119 6.3 Fitting Quadratic Models.................................... 130 6.4 Interpreting Quadratic Response Functions..................... 136 6.4.1 Noninvertible B ...................................... 143 7 Recovery of Interblock Information in BIB Designs . 147 7.1 Estimation................................................ 148 7.2 Model Testing............................................. 152 7.3 Contrasts................................................. 154 7.4 Alternative Inferential Procedures............................. 155 7.5 Estimation of Variance Components........................... 156 7.5.1 Recovery of interblock information..................... 158 References .........................................................161 Index .............................................................165 Chapter 1 Fundamentals Three fundamental ideas in experimental design are replication, blocking, and ran- domization. Replication is required so that we have a measure of the variability of the observations. Blocking is used to reduce experimental variability. Blocking has inspired a number of standard designs including randomized complete blocks, Latin squares, balanced incomplete blocks, Youden squares, balanced lattice designs, and partially balanced incomplete blocks. Randomly assigning treatments to experimen- tal units gives one a philosophical basis for inferring that the treatments cause the results observed in the experiment, cf. Peirce and Jastrow (1885) and Fisher (1935). An extremely useful concept in experimental design is the use of treatments with factorial structure. For example, if you are interested in the effect of two factors, say the effect of alcohol and the effect of sleeping pills, rather than performing separate experiments on each, you can incorporate both factors into the treatments of a single experiment. Briefly, suppose we are interested in two levels of alcohol, say, a0 – no alcohol and a1 – a standard dose of alcohol, and we are also interested in two levels of sleeping pills, say, s0 – no sleeping pills and s1 – a standard dose of sleeping pills. A factorial treatment structure involves forming 4 treatments, a0s0 – no alcohol, no sleeping pills, a0s1 – no alcohol, sleeping pills, a1s0 – alcohol, no sleeping pills, a1s1 – alcohol and sleeping pills. A factorial treatment structure obtains when the treatments in an experiment consist of all possible combinations of the levels of some factors. The point of using factorial treatment structures is that they allow one to look very efficiently at the main effects of the factors but also that they allow examination of interactions between the factors. If there is no interaction, i.e., if the effect of alcohol does not depend on whether or not the subject has taken sleeping pills, you can learn as much from running one experiment using factorial treatment structure on, say, 20 people as you can from two separate experiments, one for alcohol and one for sleeping pills, each involving 20 people. This is a 50% savings in the number of observations needed. Second, if interaction exists, i.e., if the effect of alcohol depends on the amount of sleeping pills a person has taken, you can study that interaction in an experiment with factorial treatment structure but you cannot study 1 2 1 Fundamentals interaction in an experiment that was solely devoted to looking at the effects of alcohol. An experiment such as this involving two factors each at two levels is referred to as a 2×2 = 22 factorial treatment structure. Note that 4 is the number of treatments we end up with. If we had 3 levels of sleeping pills, it would be a 2 × 3 factorial, thus giving 6 treatments. If we had three levels of both alcohol and sleeping pills it would be a 3 × 3 = 32. If we had three factors, say, alcohol, sleeping pills, and benzedrine each at 2 levels, we would have a 2×2×2 = 23 structure.
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