Suppose a model has two factors, A and B and the interaction between the 2 factors, A*B. A is a random factor and B is a fixed factor.
The following equation represents the response value at level i of factor A and level j of factor B:
is for the random factor A,
is for the fixed factor B,
is for the AB term
is for the error term
The equation for the conditional fitted value for follows:
where and represent the corresponding calculated coefficients for the fixed effect terms. Also, and are the calculated BLUP values for the random terms.
Mean of AB
The calculated mean of AB at level i of factor A and level j of factor B is .
Mean of A
The calculated mean of A at level i follows:
Mean of B
The calculated mean of B at level j follows:
General formula for means
In general, any mean can be represented by the following vector expression:
where is the calculated coefficient vector, is the calculated BLUP vector, and and are known vectors determined by the mean for a specific term. Then the mean formula has the same form as the conditional fit. Therefore, the formulas for standard error, the degrees of freedom, the confidence interval, the t-value, and the p-value for the mean are simlilarly derived as the corresponding formulas for the conditional fits. For more information on these formulas for the conditional fits, go to Methods and formulas for conditional fits and residuals in Fit Mixed Effects Model. By replacing the vector and the vector with the vector and the vector respectively in the conditional fit formulas, you get the corresponding formulas for the mean.