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Latin Square Design Traditionally, latin squares have two blocks, 1 treatment, all of size n Yandell introduces latin squares as an incomplete factorial design instead – Though his example seems to have at least one block (batch) Latin squares have recently shown up as parsimonious factorial designs for simulation studies Latin Square Design Student project example – 4 drivers, 4 times, 4 routes – Y=elapsed time Latin Square structure can be natural (observer can only be in 1 place at 1 time) Observer, place and time are natural blocks for a Latin Square Latin Square Design Example – Region II Science Fair years ago (7 by 7 design) – Row factor—Chemical – Column factor—Day (Block?) – Treatment—Fly Group (Block?) – Response—Number of flies (out of 20) not avoiding the chemical Latin Square Design Chemical Control Piperine Black Pepper Lemon Juice Hesperidin Ascorbic Acid Citric Acid 1 A 19.8 B 13.0 C 13.0 D 7.8 E 13.6 F 15.0 G 14.5 2 G 16.8 A 5.3 B 11.0 C 6.0 D 16.0 E 12.2 F 14.7 3 F 16.7 G 14.0 A 12.3 B 5.3 C 10.7 D 11.7 E 11.0 Day 4 E 15.8 F 7.2 G 8.6 A 6.0 B 10.0 C 12.2 D 11.2 5 D 17.3 E 14.1 F 14.5 G 8.3 A 16.2 B 13.2 C 9.5 6 C 18.1 D 10.8 E 15.8 F 5.8 G 14.3 A 16.0 B 17.2 7 B 18.0 C 14.7 D 12.7 E 6.5 F 14.2 G 11.8 A 15.7 Power Analysis in Latin Squares For unreplicated squares, we increase power by increasing n (which may not be practical) The denominator df is (n-2)(n-1) H o : 0 Ho : L 0 n 2 i 2 nL2 2 c 2 i Power Analysis in Latin Squares For replicated squares, the denominator df depends on the method of replication; see Montgomery H o : 0 Ho : L 0 sn 2 i 2 2 snL 2 ci2 Graeco-Latin Square Design Suppose we have a Latin Square Design with a third blocking variable (indicated by font color): A B C D B C D A C D A B D A B C Graeco-Latin Square Design Suppose we have a Latin Square Design with a third blocking variable (indicated by font style): A B C D B C D A C D A B D A B C Graeco-Latin Square Design Is the third blocking variable orthogonal to the treatment and blocks? How do we account for the third blocking factor? We will use Greek letters to denote a third blocking variable Graeco-Latin Square Design A B C D B A D C C D A B D C B A Graeco-Latin Square Design A B C D B A D C C D A B D C B A Graeco-Latin Square Design 1 Row 2 3 4 Column 1 2 3 4 Aa Bb Cg Dd Bd Ag Db Ca Cb Da Ad Bg Dg Cd Ba Ab Graeco-Latin Square Design Orthogonal designs do not exist for n=6 Randomization – – – – – Standard square Rows Columns Latin letters Greek letters Graeco-Latin Square Design Total df is n2-1=(n-1)(n+1) Maximum number of blocks is n-1 – n-1 df for Treatment – n-1 df for each of n-1 blocks--(n-1)2 df – n-1 df for error Hypersquares (# of blocks > 3) are used for screening designs Conclusions We will explore some interesting extensions of Latin Squares in the text’s last chapter – Replicated Latin Squares – Crossover Designs – Residual Effects in Crossover designs But first we need to learn some more about blocking…