I am fitting mixed model with X fixed effect design matrix Z is random effect design matrix.
Consider the mixed model:
E[y|u] = X $\beta$ + Zu
with E[u] = 0,
y ~ (X$\beta$, ZDZ' + R), var(u) = D
V = var(y) = ZDZ' + R
We take y to be normally distributed:
y ~ N(X $\beta$, V),
estimating fixed effects for V known
According to McCulloch and Searle ( Generalized, Linear and Mixed models book, page 163)
To test the null hypothesis H0 : S'X $\beta$ = m, where S' is of full row rank (rs < rx), we can derive a chi-square statistic using
Under HQ, $\chi$ 2 has a central $\chi$ 2 distribution with rs = rank(S) degrees of freedom.
Under the null hypothesis, F has an F-distribution on rs and N — rx degrees of freedom.The null hypothesis is rejected at significance level a when F exceeds F N-rx, 1- $\alpha$ .
When I have single X variable, F test is strait forward as this include main effect only. In my case I have two X variables and want to test significance of both X variables and interactions. How can I achieve this ?