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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)

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Latin Square Design

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  1. 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

  2. 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

  3. 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

  4. Latin Square Design

  5. 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)

  6. Power Analysis in Latin Squares • For replicated squares, the denominator df depends on the method of replication; see Montgomery

  7. Graeco-Latin Square Design • Suppose we have a Latin Square Design with a third blocking variable (indicated by font color): ABCD BCDA CDAB DABC

  8. Graeco-Latin Square Design • Suppose we have a Latin Square Design with a third blocking variable (indicated by font style): ABCD B CDA C D AB D A B C

  9. 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

  10. Graeco-Latin Square Design ABCD BADC CDAB DCBA

  11. Graeco-Latin Square Design A B CD BA D C C DAB DCB A

  12. Graeco-Latin Square Design Column 1 2 3 4 1 Aa Bb CgDd Row 2 Bd AgDbCa 3 Cb Da AdBg 4 Dg Cd BaAb

  13. Graeco-Latin Square Design • Orthogonal designs do not exist for n=6 • Randomization • Standard square • Rows • Columns • Latin letters • Greek letters

  14. 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

  15. 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…

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