Transcript Latin Square Design
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) avoiding the chemical not
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)
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Power Analysis in Latin Squares For replicated squares, the denominator df depends on the method of replication; see Montgomery
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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 D A A B 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 C D A D A B 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 1 A a B d C b D g 2 B b A g D a C d Column 3 4 C D A B g b d a D d C a B g A b
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 n 2 -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…