Table 1 Influence analysis framework for MELS models
Influence measure | Influence sub-category | Formula |
|---|---|---|
Influence on model fit | ||
Difference in deviance | \(LR_{i^*} = 2\ln \left( \frac{\mathcal {L}_{(-i^*)}}{\mathcal {L}}\right)\) | |
Influence on point estimates of a group of parameters | ||
Cook’s distance | Fixed location effect estimates | \(C_{i^*}^{\beta } = \frac{1}{r_{\beta }}\left(\hat{\beta } - \hat{\beta }_{(-i^*)}\right)^T{\hat{\Sigma }}_{\hat{\beta }(-i^*)}^{-1}\left(\hat{\beta } - \hat{\beta }_{(-i^*)}\right)\) |
Fixed scale effect estimates | \(C_{i^*}^{\tau } = \frac{1}{r_{\tau }}\left(\hat{\tau } - \hat{\tau }_{(-i^*)}\right)^T\hat{\Sigma }_{\hat{\tau }(-i^*)}^{-1}\left(\hat{\tau } - \hat{\tau }_{(-i^*)}\right)\) | |
Variances and covariances of random effects | \(C_{i^*}^{\eta } = \frac{1}{r_{\eta }}\left(\hat{\eta } - \hat{\eta }_{(-i^*)}\right)^T\hat{\Sigma }_{\hat{\eta }(-i^*)}^{-1}\left(\hat{\eta } - \hat{\eta }_{(-i^*)}\right)\) | |
Influence on point estimate of a single parameter | ||
DFBETAS | \(\text {DFBETAS}_{i^*}^{\theta } = \frac{\hat{\theta } - \hat{\theta }_{(-i^*)}}{SE\left(\hat{\theta }_{(-i^*)}\right)}\) | |
Influence on variances and covariances of a group of parameters | ||
COVTRACE | Fixed location effect estimates | \({\text {COVTRACE}}_{i^*}^{\beta } = \left|{\textrm{Tr}}\left({\hat{\Sigma }}_{{\hat{\beta }}}^{-1}{\hat{\Sigma }}_{{\hat{\beta }}(-i^*)}\right) - r_{\beta }\right|\) |
Fixed scale effect estimates | \({\text {COVTRACE}}_{i^*}^{\tau } = \left|{\textrm{Tr}}\left(\hat{\Sigma }_{\hat{\tau }}^{-1}\hat{\Sigma }_{\hat{\tau }(-i^*)}\right) - r_{\tau }\right|\) | |
Variances and covariances of random effects | \({\text {COVTRACE}}_{i^*}^{\eta } = \left|{\textrm{Tr}}\left(\hat{\Sigma }_{\hat{\eta }}^{-1}\hat{\Sigma }_{\hat{\eta }(-i^*)}\right) - r_{\eta }\right|\) | |
COVRATIO | Fixed location effect estimates | \({\text {COVRATIO}}_{i^*}^{\beta } = \frac{\det \left(\hat{\Sigma }_{\hat{\beta }(-i^*)}\right)}{\det \left(\hat{\Sigma }_{\hat{\beta }}\right)}\) |
Fixed scale effect estimates | \({\text{COVRATIO}}_{i^*}^{\tau } = \frac{\det \left(\hat{\Sigma }_{\hat{\tau }(-i^*)}\right)}{\det \left(\hat{\Sigma }_{\hat{\tau }}\right)}\) | |
Variances and covariances of random effects | \({\text{COVRATIO}}_{i^*}^{\eta } = \frac{\det \left(\hat{\Sigma }_{\hat{\eta }(-i^*)}\right)}{\det \left(\hat{\Sigma }_{\hat{\eta }}\right)}\) | |