12.4: Testing the Significance of the Carry-Over Effect

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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. \[\Delta G^{2} = \left(-2 \log L_{Reduced}\right) - \left(-2 \log L_{Full}\right) \quad \text{with } df_{Reduced} - df_{Full} \text{ degrees of freedom}\]

The \(-2 \log L\) 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`	`122.5`
`AIC (smaller is better)`	`130.5`
`AICC (smaller is better)`	`132.6`
`BIC (smaller is better)`	`132.5`

And for the reduced model without the carry-over covariates is:

`Fit Statistics`
`-2 Res Log Likelihood`	`136.5`
`AIC (smaller is better)`	`144.5`
`AICC (smaller is better)`	`146.4`
`BIC (smaller is better)`	`146.4`

So, \[\Delta G^{2} = 136.5 - 122.5 = 14 \nonumber\] and with \[\chi_{.05, 2}^{2} = 5.991 \nonumber\] we conclude that there are significant carry-over effects.