To test for the overall significance of carry-over effects, we can drop the carry-over covariates (\(x_1\) and \(x_2\) in our example) and re-run the ANOVA. Because the reduced model is a subset of the full model that includes the covariates, we can construct a likelihood ratio test.
\(\Delta G^2=(-2logL_{Reduced})-(-2logL_{Full})\)
with \(df_{Reduced}-df_{Full}\) degrees of freedom
The -2logL values are provided in the SAS Fit Statistics output for each model. For our example, the SAS output for the Full model with carry-over covariates is:
Fit Statistics | |||||
---|---|---|---|---|---|
-2 Res Log Likelihood | 148.1 | ||||
AIC (smaller is better) | 160.1 | ||||
AICC (smaller is better) | 165.0 | ||||
BIC (smaller is better) | 163.0 |
And for the reduced model without the carry-over covariates is:
Fit Statistics | |||||
---|---|---|---|---|---|
-2 Res Log Likelihood | 171.2 | ||||
AIC (smaller is better) | 183.2 | ||||
AICC (smaller is better) | 187.6 | ||||
BIC (smaller is better) | 186.1 |
So,
\(\Delta G^2 =171.2-148.1=23.1\)
and with
\(\chi^2_{0.05, 2}=5.991\)
we conclude that there are significant carry-over effects.