7.4 Relative Model-Data Fit at Test Level (Cont’d)

Information criteria can be used to compare any two alternative models, as long as the same data are involved.

Nested models : Two models are nested if one model (the simpler or null model) is a special case of the complex mode. In case of nested models, the simple model can be obtained by constraining one or more parameters of the complex model. For instance, Linear Logistic Model is a special case of GDINA model, if we consider logit link and ignore interaction effects of attributes in GDINA model.

However, when two models are nested, we can use the likelihood-ratio (LR) test to perform a hypothesis test.

• \(H_0\): simpler model fits data as well as the complex model
• \(H_1\): the complex model fits data better

\[LR=-2\log\bigg[\frac{L_{\text{simpler}}(\mathbf{Y})}{L_{\text{complex}}(\mathbf{Y})}\bigg]=-2\bigg[\log L_{\text{simpler}}(\mathbf{Y})-\log L_{\text{complex}}(\mathbf{Y})\bigg] \] The \(LR\) statistic is \(\chi^2\) distributed with \(df\) being the number of parameters in the complex model minus the number of parameters in the simpler model.