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Completely Randomized Design

Completely Randomized Design. Cell means model:. Effects Model. GLM for Effects Model. CRD Contrasts. Balanced case (n i =n) -A linear combination L has the form: -A contrast is a linear combination with the additional constraint:. Cotton Fiber Example.

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Completely Randomized Design

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  1. Completely Randomized Design • Cell means model:

  2. Effects Model

  3. GLM for Effects Model

  4. CRD Contrasts Balanced case (ni=n) -A linear combination L has the form: -A contrast is a linear combination with the additional constraint:

  5. Cotton Fiber Example • Treatment--% cotton by weight (15%, 20%, 25%, 30%, 35%) • Response--Tensile strength Montgomery, D. (2005) Design and Analysis of Experiments, 6th Ed. Wiley, NY.

  6. Cotton Fiber Example

  7. Cotton Fiber Example

  8. Contrast Test Statistic Under Ho:L1=0,

  9. Unbalanced CRD Contrast

  10. Orthogonality • Contrasts are orthogonal if, for contrasts L1 and L2, we have

  11. Orthogonality • The usual a-1 ANOVA contrasts are not orthogonal (though columns are linearly independent) • Orthogonality implies coefficients will not change if terms are deleted from model

  12. Orthogonality • Sums of squares for orthogonal contrasts are additive, allowing treatment sums of squares to be partitioned • Mathematically attractive, though not all contrasts will be interesting to the researcher

  13. Cotton Fiber Example • Two sets of covariates (orthogonal and non-orthogonal) to test for linear and quadratic terms

  14. Cotton Fiber Example • For Orthogonal SS, L&Q=L+Q; Q=Q|L; L=L|Q • For Nonorthogonal SS, L&Q=L+Q|L=Q+L|Q

  15. Orthogonal polynomial contrasts • Require quantitative factors • Equal spacing of factor levels (d) • Equal ni • Usually, only the linear and quadratic contrasts are of interest

  16. Orthogonal polynomial contrasts • Cotton Fiber Example

  17. Orthogonal polynomial contrasts • Cotton Fiber Example

  18. Orthogonal polynomial contrasts

  19. Orthogonal polynomial contrasts

  20. Orthogonal polynomial contrasts Cotton Fiber Example Is a L+Q model better than an intercept model? Is a L+Q model not as good as a cell means model? (Lack of Fit test)

  21. Orthogonal polynomial contrasts • Yandell has an interesting approach to reconstructing these tests • Construct the first (linear) term • Include a quadratic term that is neither orthogonal, nor a contrast • Do not construct higher-order contrasts at all • Use a Type I analysis for testing

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