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 - Yijk = μ + αj + βk + αβjk + εi(jk)

Part 1               Part 2           Part 3
                     i     j     k 
                     n     p     q 
-------------------------------------------------------------------
αj                   n     P'    q    σ2ε + nQ'σ2αβ + nqσ2α
βk                   n     p     Q'   σ2ε + nP'σ2αβ + npσ2β
αβjk                 n     P'    Q'   σ2ε + nσ2αβ
εi(jk)                N'    1     1    σ2ε

Step 10 - Say that A is fixed and B is random:

E(MSA)      σ2ε + nσ2αβ + nqσ2α
E(MSB)      σ2ε + npσ2β
E(MSA*B)    σ2ε + nσ2αβ
E(MSerror)  σ2ε

CRF-pqr Template

Step 1 - Yijkl = μ + αj + βk + γl + αβjk + αγjl + βγkl + αβγjkl + εi(jk)

Part 1               Part 2                Part 3
                     i     j     k     l
                     n     p     q     m
-------------------------------------------------------------------
αj                

βk 

γl

αβjk 

αγjl

βγkl

αβγjkl

εi(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 - Yijkl = μ + αj + γl + αγjl + πi(jl) + βk + αβjk + αβjk + βγkl + αβγjkl + βπki(jl) + εijkl


Part 1               Part 2                Part 3
                     i     j     k     l
                     n     p     q     m
-------------------------------------------------------------------

αj 

γl 

αγjl 

πi(jl) 

βk 

αβjk 

αβjk 

βγkl 

αβγjkl 

βπki(jl) 

εijkl

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)  

Linear Statistical Models Course