**The Cornfield-Tukey method for deriving expected mean squares.**

This unit presents a modified version of the Cornfield-Tukey method for deriving the symbolic values for the expected mean squares.

**Steps in deriving expected mean squares**

Step 1 - Write the linear model for the design. Step 2 - Construct a table with three parts. Step 3 - The row headings in part 1 contain each of the terms from the linear model leaving out μ. Step 4 - The column heading in part 2 contain the subscripts from the linear model along with the symbol for the levels. Step 5 - If a column heading appears as a row subscript in parentheses enter a 1 in part 2. Step 6 - If a column heading appears as a row subscript (not in parentheses) enter the appropriate sampling fraction (N', P', Q', etc.). Step 7 - If a column heading does not appear as a row subscript enter the letter for the number of levels Step 8 - In part 3 list a variance for each term in the linear model that contains all the row subscripts. Step 9 - Coefficients for variances are obtained by covering the column headed by subscripts that appear in the row but not including subscripts in parentheses. Step 10 - Resolve the sampling fractions and rewrite the expected mean squares P' = 0 if A is fixed and 1 if A is random Q' = 0 if B is fixed and 1 if B is random R' = 0 if C is fixed and 1 if C is random, etc

Step 1 - Y_{ijk}= μ + α_{j}+ β_{k}+ αβ_{jk}+ ε_{i(jk)}Part 1 Part 2 Part 3 i j k n p q ------------------------------------------------------------------- α_{j}n P' q σ^{2}_{ε}+ nQ'σ^{2}_{αβ}+ nqσ^{2}_{α}β_{k}n p Q' σ^{2}_{ε}+ nP'σ^{2}_{αβ}+ npσ^{2}_{β}αβ_{jk}n P' Q' σ^{2}_{ε}+ nσ^{2}_{αβ}ε_{i(jk)}N' 1 1 σ^{2}_{ε}Step 10 - Say that A is fixed and B is random: E(MSA) σ^{2}_{ε}+ nσ^{2}_{αβ}+ nqσ^{2}_{α}E(MSB) σ^{2}_{ε}+ npσ^{2}_{β}E(MSA*B) σ^{2}_{ε}+ nσ^{2}_{αβ}E(MSerror) σ^{2}_{ε}

**CRF-pqr Template**

Step 1 - Y_{ijkl}= μ + α_{j}+ β_{k}+ γ_{l}+ αβ_{jk}+ αγ_{jl}+ βγ_{kl}+ αβγ_{jkl}+ ε_{i(jk)}Part 1 Part 2 Part 3 i j k l n p q m ------------------------------------------------------------------- α_{j}β_{k}γ_{l}αβ_{jk}αγ_{jl}βγ_{kl}αβγ_{jkl}ε_{i(jkl)}Step 10 - Say that A is fixed and that B and C are random: E(MSA) E(MSB) E(MSC) E(MSA*B) E(MSA*C) E(MSB*C) E(MSA*B*C) E(MSerror)

**SPF-pr.q Template**

Step 1 - Y_{ijkl}= μ + α_{j}+ γ_{l}+ αγ_{jl}+ π_{i(jl)}+ β_{k}+ αβ_{jk}+ αβ_{jk}+ βγ_{kl}+ αβγ_{jkl}+ βπ_{ki(jl)}+ ε_{ijkl}Part 1 Part 2 Part 3 i j k l n p q m ------------------------------------------------------------------- α_{j}γ_{l}αγ_{jl}π_{i(jl)}β_{k}αβ_{jk}αβ_{jk}βγ_{kl}αβγ_{jkl}βπ_{ki(jl)}ε_{ijkl}Step 10 - Say that A, B & C are fixed and that subjects are random: E(MSA) E(MSC) E(MSA*C) E(MSblk(A*C)) E(MSB) E(MSA*B) E(MSB*C) E(MSA*B*C) E(MSB*blk(A*C)) E(MSerror)

Phil Ender, 7may06