# All Random Factors Design

Expected Mean Square (EMS) is utilized to determine the correct analysis statistics. In this example, the rules in developing the EMS Table are explained using three random factors, A, B, and 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 4

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

*For a random effect model (ANOVA Model II), there is no mixed interaction (fixed and random) term. Therefore, no restriction is necessary and by default, it is an unrestricted model. # 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, EMS 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. For example, A row does not contain C. Therefore, A should NOT be added to find the EMS for C. However, C, BC, AC, ABC and the Error rows contain C. Therefore, C, BC, AC, ABC and the Error rows are added to find the Expected Mean Square for C. Similarly, EMS for all other effects can be determined (Table 2). The correct divisor for the F-Statistics is the random error associated with the expected mean square for the respective effect (Table 2).

Table 2

The EMS Table for A, B, and C Random Effect Model. 