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1 STA 536 – Experiments with More Than One Factor 3.6 Latin Square Design - Two Blocking Variables Wear Experiment : Testing the abrasion resistance of rubber-covered fabric, y = loss in weight over a period of time. One treatment factor : Material type A, B, C, D. Two blocking factors : (1) four positions on the tester, (2) four applications (four different times for setting up the tester) Latin square design of order k : Each of the k Latin letters (i.e., treatments) appears once in each row and once in each column. It is an extension of RBD to accommodate two blocking factors. Randomization applied to assignments to rows, columns, treatments. (Collection of Latin Square Tables given in Appendix 3A of WH).

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2 STA 536 – Experiments with More Than One Factor Table 14: Latin Square Design (columns correspond to positions, rows correspond to applications and Latin letters correspond to materials), Wear Experiment Table 15: Weight Loss Data, Wear Experiment

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3 STA 536 – Experiments with More Than One Factor A Latin square design can study two blocking factors (rows and columns) and one experimental factor (Latin letters) with all factors having k levels.

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4 STA 536 – Experiments with More Than One Factor Constrains and Estimates

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5 STA 536 – Experiments with More Than One Factor ANOVA for Latin Square Design

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6 STA 536 – Experiments with More Than One Factor The ANOVA decomposition is:

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7 STA 536 – Experiments with More Than One Factor Wear Experiment

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8 STA 536 – Experiments with More Than One Factor F Test and Multiple Comparisons

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9 STA 536 – Experiments with More Than One Factor

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10 STA 536 – Experiments with More Than One Factor Therefore blocking can make a difference in decision making if treatment effects are smaller.

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11 STA 536 – Experiments with More Than One Factor Exist of Latin Square Exists for any k Why? How to run randomization? Randomly permuting rows permuting columns assigning treatments to letters

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12 STA 536 – Experiments with More Than One Factor 3.7 Graeco-Latin Square Design Two Latin squares are said to be orthogonal if the two squares when superimposed have the property that each pair of letters appears once. For example, by superimposing two 3 × 3 squares, each of the nine combinations of A, B and C appears once. To distinguish the variables represented by the letters in the two squares, it is customary to replace the Latin letters A, B and C in the second square by the Greek letters α, β, and γ. The superimposed squareas shown is called a Graeco-Latin square.

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13 STA 536 – Experiments with More Than One Factor Graeco-Latin Square Design The Graeco-Latin square design can be used for studying 4 variables, e.g., treatment comparisons with Latin letters representing treatments, and rows, columns and Greek letters representing three blocking variables. The linear model for a Graeco-Latin square design is

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14 STA 536 – Experiments with More Than One Factor ANOVA

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15 STA 536 – Experiments with More Than One Factor Hyper-Graeco-Latin squares Hyper-Graeco-Latin squares can be obtained as extensions of Graeco-Latin squares by superimposing three or more mutually orthogonal Latin squares. A collection of Latin squares, Graeco-Latin squares and hyper-Graeco-Latin squares is given in Appendix 3A. Graeco-Latin squares may not exist for some k, such as k=6 Similarly, hyper-Graeco-Latin squares may not exist for some k, such as k=6 For a prime or a power of a prime k, there exist k-1 orthogonal Latin squares

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16 STA 536 – Experiments with More Than One Factor 3.8 Balanced Incomplete Block Design The catalyst experiment To study the effect of 4 catalysts on the reaction time of a chemical process. The experimental procedure consists of selecting a batch raw material, loading the pilot plant, applying each catalyst in a separate run and observing the reaction time. Batches of raw material is a nuisance factor (blocking variable). Ideally, should apply all types of catalyst to each batch of raw material (complete RBD). However, each batch is only large enough to permit 3 catalysts to be run.

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17 STA 536 – Experiments with More Than One Factor A balanced incomplete block design (BIBD) is used. – incomplete: not all catalysts are run for each batch. – balanced: each pair of catalysts appear together twice. The order in which the catalysts are run in each block is randomized.

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18 STA 536 – Experiments with More Than One Factor BIBD - Notation number of treatments t number of blocks b number of replicates for each treatments r block size k (incomplete: k < t) number of replicates for each pair of treatments λ, balanced: each pair of treatments is compared in λ blocks. t=4, b=4, r=3, k=3, λ=2 for this example Two basic relations among the parameters:

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19 STA 536 – Experiments with More Than One Factor The linear model for a BIBD is (Model I): Because not all the treatments are compared within a block, data are not observed for all (i, j) pairs. This complicates the analysis but can still be handled by GLM. Model II be: Then testing H0 is equivalent to testing Model I against Model II.

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20 STA 536 – Experiments with More Than One Factor SAS Code data catalyst; input catalyst $ x1-x4; batch=1; y=x1; output; batch=2; y=x2; output; batch=3; y=x3; output; batch=4; y=x4; output; drop x1-x4; cards; 1 73 74. 71 2. 75 67 72 3 73 75 68. 4 75. 72 75 ; proc GLM; class catalyst batch; model y=batch catalyst/solution; run; proc GLM; class catalyst batch; model y=catalyst batch/solution; run;

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21 STA 536 – Experiments with More Than One Factor Source DF Type I SS Mean Square F Value Pr > F batch 3 55.00000000 18.33333333 28.21 0.0015 catalyst 3 22.75000000 7.58333333 11.67 0.0107 Error 5 3.25000000 0.65000000 The ANOVA table shows that after adjusting the batch (block) effects, there are differences between the catalysts with a p-value of 0.0107(= Prob(F 3,5 >11.667)). How to test the blocking effects? We should compare Model I with:

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22 STA 536 – Experiments with More Than One Factor Source DF Type I SS Mean Square F Value Pr > F catalyst 3 11.66666667 3.88888889 5.98 0.0415 batch 3 66.08333333 22.02777778 33.89 0.0010 Error 5 3.25000000 0.65000000 After adjusting the catalyst effects, there are differences between the batches (block) with a p-value of 0.001(= Prob(F 3,5 > 33.89)). If blocking is ignored in the analysis, the treatment factor, catalyst, would not be as significant. proc GLM; /*wrong analysis*/ class catalyst batch; model y=catalyst/solution; run;

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23 STA 536 – Experiments with More Than One Factor Sum of Source DF Squares Mean Square F Value Pr > F catalyst 3 11.66666667 3.88888889 0.45 0.7251 Error 8 69.33333333 8.66666667 Thus, blocking is effective here because it removes the significant batch-to-batch variation. If the hypothesis of no treatment differences is rejected, then the multiple comparison procedures can be used to identify which treatments are different.

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24 STA 536 – Experiments with More Than One Factor lsmeans catalyst / PDIFF ADJUST=TUKEY CL TDIFF; Least Squares Means for Effect catalyst t for H0: LSMean(i)=LSMean(j) / Pr > |t| Dependent Variable: y i/j 1 2 3 4 1 -0.35806 -0.89514 -5.19183 0.9825 0.8085 0.0130 2 0.358057 -0.53709 -4.83378 0.9825 0.9462 0.0175 3 0.895144 0.537086 -4.29669 0.8085 0.9462 0.0281 4 5.191833 4.833775 4.296689 0.0130 0.0175 0.0281

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25 STA 536 – Experiments with More Than One Factor 3.10 Analysis of Covariance - Incorporating Auxiliary Information The starch experiment To study the breaking strength (y) in grams of three types of starch film. The breaking strength is also known to depend on the thickness of the film (x) as measured in 10 4 inches. Film thickness varies from run to run and its values cannot be controlled or chosen prior to the experiment. Should treat film thickness as a covariate and use ANalysis of COVAriance (ANCOVA)

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26 STA 536 – Experiments with More Than One Factor x= thickness y= breaking strength

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27 STA 536 – Experiments with More Than One Factor The linear model is

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28 STA 536 – Experiments with More Than One Factor data Starch; input y1 x1 y2 x2 y3 x3; thickness=x1; y=y1; Starch='Canna '; output; thickness=x2; y=y2; Starch='Corn '; output; thickness=x3; y=y3; Starch='Potato'; output; keep thickness y Starch; cards; 791.7 7.7 731.0 8.0 983.3 13.0 610.0 6.3 710.0 7.3 958.8 13.3 710.0 8.6 604.7 7.2 747.8 10.7 940.7 11.8 508.8 6.1 866.0 12.2 990.0 12.4 393.0 6.4 810.8 11.6 916.2 12.0 416.0 6.4 950.0 9.7 835.0 11.4 400.0 6.9 1282.0 10.8 724.3 10.4 335.6 5.8 1233.8 10.1 611.1 9.2 306.4 5.3 1660.0 12.7 621.7 9.0 426.0 6.7 746.0 9.8 735.4 9.5 382.5 5.8 650.0 10.0 990.0 12.5 340.8 5.7 992.5 13.8 862.7 11.7 436.7 6.1 896.7 13.3.. 333.3 6.2 873.9 12.4.. 382.3 6.3 924.4 12.2.. 397.7 6.0 1050.0 14.1.. 619.1 6.8 973.3 13.7.. 857.3 7.9.... 592.5 7.2.. ; /*corrected analysis*/ proc GLM; class Starch; model y=thickness Starch; run; /*wrong analysis*/ proc GLM; class Starch; model y=Starch; run;

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29 STA 536 – Experiments with More Than One Factor There is no difference between the three starch types with a p-value of 0.3597 (=Prob(F 2,45 > 1.05)). The thickness is very significant.

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30 STA 536 – Experiments with More Than One Factor Wrong analysis incorrectly ignoring the covariate. The results incorrectly indicate a significant difference between the starch types.

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31 STA 536 – Experiments with More Than One Factor For the starch experiment, with the baseline constraint, the model takes the following form:

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32 STA 536 – Experiments with More Than One Factor 3.11 Transformation of the Response

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33 STA 536 – Experiments with More Than One Factor Power (Box-Cox) Transformation

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34 STA 536 – Experiments with More Than One Factor Variance Stabilizing Transformations

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35 STA 536 – Experiments with More Than One Factor Analysis of Drill Experiment

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36 STA 536 – Experiments with More Than One Factor 3.12 Practical Summary Read it!

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