The rules in developing the EMS Table are explained using three random factors A & B fixed with C random and nested in B. This is a typical design of experiment known as split-plot/split-split-plot designs in agriculture, repeated measure in human subject studies, nested/hierarchical, sampling and sub-sampling designs. A model of three factors, A, B, and C (nested in B) is written as in Equation 3. As C is nested in B, the interaction BC and ABC do not exist.

Note. Where everything follows the usual notations such as “a” is the number of levels of the factor “A.” Alpha (α) is the effect due to the factor A and “i” indicates the ith level of factor A. k(j) is used to denote “k is nested in j.” C(B) is used to denote “C is nested in B.” 

Equation 8

While the variance for the random factor/effect is written as the sigma square with the associated subscript, the mean sum of square is used as the variance for the fixed effect factor. For example, the variance for A is written as

EMS Rule #1

Write all of the factors, factor types, levels, replications, factorial effects, their variances, and subscripts as in the table below.

EMS Rule #2

Insert the level/replication if the row subscript(s) does not contain the column subscript.

EMS Rule #3

Insert a “1” if the row subscript(s) contains the column subscript but dead (nested). [Replicates are always nested within the treatment combinations (=(ijk)l). B(A) notation is used when B is nested in A.]

EMS Rule #4

Unrestricted Model

(Fixed and random factors mixed interaction term is assumed random)

If the row subscript(s) contains the column subscript, insert a

1. “0” when both subscripts are fixed

2. “1” when at least one subscript is random

Restricted Model

(Fixed and random factors mixed interaction term is NOT assumed random)

If the row subscript(s) contains the column subscript, insert a

1. “0” when the column subscript is fixed

2. “1” when the column subscript is random

EMS Rule #5

1. Hide/mask/delete the column (s) for which the EMS is to be determined. For example, to find the Expected Mean Square for C, the C column is hidden/masked.

2. In the Expected Mean Square column, write the results by multiplying all the inserted values in a row and the associated variance of that row.  For example, the Expected Mean Square Component for C effect can be found as

EMS Rule #6

Add only the components which contain the C factor to find the EMS for C. Therefore, C, AC, and the Error rows are added to find the Expected Mean Square for C. 

Similarly, EMS for all other effects can be determined (Table 5). The correct divisor for the F-Statistics is the random error associated with the expected mean square for the respective effect (Table 5).

Table 5

EMS Table for Fixed Factor A and B with C Random and nested in B Mixed Unrestricted Model

Table 6

EMS Table for Fixed Factor A and B with C Random and nested in B Mixed Restricted Model