Expected Mean Square (EMS) is utilized to determine the correct analysis statistics, which is the F-Statistics. The rules in developing the EMS Table are explained using two fixed factors A and B with a random factor C. A model consisting of three factors, A, B, and C is written as in Equation 3.

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.  

Equation 6

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 EMS(BC), B and C columns are 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 BC effect can be found as

EMS Rule #6

To find the EMS for BC interaction term, add only the components which contain both B and C factors together. For example, neither B row nor C row contains both B and C together. Therefore, B and C rows should NOT be added to find the EMS for the BC interaction term. Both BC and the Error rows contain both B and C together. Therefore, BC and the Error rows are added to find the Expected Mean Square for BC.

Similarly, EMS for all other effects can be determined (Table 3). The correct divisor for the F-Statistics is the random error associated with the expected mean square for the respective effect (Table 3).

Approximate or Pseudo

F-Statistics/Tests

When the exact F-Statistics/F-test does not exist, pseudo F-test or approximate F-test can be performed by using any appropriate linear combinations of the expected mean squares, which produces all expected mean square components, but the interested effect. For example, the exact F-test does not exist for A, B or C in the random effect model described above. To perform an approximate F-test for C, the linear combination of “MS(BC)+MS(AC)-MS(ABC)” can be used, which results in all expected mean square components, but the C component. Therefore, the linear combination of “MS(BC)+MS(AC)-MS(ABC)” is the random error associated with the C effect, and can be used as a denominator to perform the F-test for C. The degree of freedom is calculated by Equation 1.

Equation 7

The EMS for the restricted model for the same mixed model is provided in Table 4.

Table 4

The EMS Table for Fixed Factor A & B with C Random Mixed Effect Restricted Model