Background
Methods
Leave-one-out MELS model
Influence analysis
Influence on model fit
Influence on parameter estimates
Influence on the precision of parameter estimates
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 | \({\text{COVRATIO}}_{i^*}^{\eta } = \frac{\det \left(\hat{\Sigma }_{\hat{\eta }(-i^*)}\right)}{\det \left(\hat{\Sigma }_{\hat{\eta }}\right)}\) |
Results
Application to health behavior study example
Subject | Influence measure | Results |
---|---|---|
7 | Cook’s distance | - Largest \(C^{\tau }\) (0.069) |
- \(2^{nd}\) largest \(C^{\eta }\) (0.194) | ||
DFBETAS | - \(3^{rd}\) largest \(\text {DFBETAS}^{\beta _{SQ}}\) (0.055) | |
- Largest \(\text {DFBETAS}^{\tau _{SQ}}\) (0.163) | ||
- Largest \(\text {DFBETAS}^{\sigma ^2_{\omega }}\) (0.604) | ||
- Largest \(\text {DFBETAS}^{\sigma _{\nu \omega }}\) (0.325) | ||
COVRATIO | - Smallest \(\text {COVRATIO}^{\tau }\) (0.905) | |
- \(3^{rd}\) smallest \(\text {COVRATIO}^{\eta }\) (0.743) | ||
12 | Cook’s distance | - \(3^{rd}\) largest \(C^{\tau }\) (0.045) |
- \(4^{th}\) largest \(C^{\eta }\) (0.084) | ||
DFBETAS | - \(4^{th}\) largest \(\text {DFBETAS}^ {\beta _0}\) (0.096) | |
- \(2^{nd}\) largest \(\text {DFBETAS}^{\sigma ^2_{\omega }}\) (0.392) | ||
- \(2^{nd}\) smallest \(\text {DFBETAS}^{\sigma _{\nu \omega }}\) (-0.343) | ||
COVRATIO | - \(2^{nd}\) smallest \(\text {COVRATIO}^{\tau }\) (0.948) | |
- \(4^{th}\) smallest \(\text {COVRATIO}^{\eta }\) (0.862) | ||
49 | Cook’s distance | - Largest \(C^{\beta }\) (0.054) |
- \(4^{th}\) largest \(C^{\tau }\) (0.037) | ||
- Largest \(C^{\eta }\) (0.323) | ||
DFBETAS | - Largest \(\text {DFBETAS}^{\beta _0}\) (0.194) | |
- Smallest \(\text {DFBETAS}^{\tau _0}\) (-0.140) | ||
- Largest \(\text {DFBETAS}^{ \sigma ^2_{\nu }}\) (0.553) | ||
- \(4^{th}\) largest \(\text {DFBETAS}^{\sigma ^2_{\omega }}\) (0.325) | ||
- Smallest \(\text {DFBETAS}^{\sigma _{\nu \omega }}\) (-0.737) | ||
COVRATIO | - Smallest \(\text {COVRATIO}^{\beta }\) (0.925) | |
- \(4^{th}\) smallest \(\text {COVRATIO}^{\tau }\) (0.959) | ||
- Smallest \(\text {COVRATIO}^{\eta }\) (0.694) | ||
69 | Cook’s distance | - \(2^{nd}\) largest \(C^{\beta }\) (0.052) |
- \(3^{rd}\) largest \(C^{\eta }\) (0.102) | ||
DFBETAS | - Smallest \(\text {DFBETAS}^{\beta _0}\) (-0.281) | |
- \(2^{nd}\) largest \(\text {DFBETAS}^{\beta _{SQ}}\) (0.121) | ||
- \(2^{nd}\) largest \(\text {DFBETAS}^{\sigma ^2_{\nu }}\) (0.422) | ||
- \(3^{rd}\) largest \(\text {DFBETAS}^{\sigma _{\nu \omega }}\) (0.269) | ||
COVRATIO | - \(2^{nd}\) smallest \(\text {COVRATIO}^{\beta }\) (0.939) | |
- \(2^{nd}\) smallest \(\text {COVRATIO}^{\eta }\) (0.741) |
Simulation study
Simulation scenario 1
Simulation scenario 2
MELS models | MRMs | ||
---|---|---|---|
Single | Multiple (3) | Single | |
Scenario 1 | |||
\(C^{\tau }\) [large(%)] | 75.9 | 16.7 | - |
\(\text {COVRATIO}^{\tau }\)[large(%)/small(%)] | 0/100 | 0/99.6 | - |
\(\text {DFBETAS}^{\tau _0}\) [large(%)/small(%)] | 99.4/0 | 25.4/0.4 | - |
\(\text {DFBETAS}^{\tau _1}\) [large(%)/small(%)] | 5.4/34.6 | 1.4/15.7 | - |
\(\text {DFBETAS}^{\sigma ^2_{\omega }}\) [large(%)/small(%)] | 100/0 | 99.6/0 | - |
\(\text {C}^{\beta }\) [large(%)] | 3.5 | 5.8 | 0 |
\(\text {COVRATIO}^{\beta }\)[large(%)/small(%)] | 15.1/0.8 | 2.0/0.8 | 41.6/0 |
\(\text {DFBETAS}^{\beta _0}\) [large(%)/small(%)] | 0.8/1.7 | 0.6/0.6 | 0/0 |
\(\text {DFBETAS}^{\beta _1}\) [large(%)/small(%)] | 21.0/44.4 | 3.6/22.7 | 6.8/13.9 |
\(\text {DFBETAS}^{\sigma ^2_{\nu }}\) [large(%)/small(%)] | 0.6/0 | 1.0/0 | 0/20.1 |
Scenario 2 | |||
\(C^{\tau }\) [large(%)] | 99.8 | 62.3 | - |
\(\text {COVRATIO}^{\tau }\)[large(%)/small(%)] | 0/100 | 0/99.8 | - |
\(\text {DFBETAS}^{\tau _0}\) [large(%)/small(%)] | 99.8/0 | 69.9/0 | - |
\(\text {DFBETAS}^{\tau _1}\) [large(%)/small(%)] | 11.3/16.0 | 0.4/0.8 | - |
\(\text {DFBETAS}^{\sigma ^2_{\omega }}\) [large(%)/small(%)] | 100/0 | 100/0 | - |
\(\text {C}^{\beta }\) [large(%)] | 21.9 | 0 | 1.2 |
\(\text {COVRATIO}^{\beta }\) [large(%)/small(%)] | 0/2.6 | 0/0 | 2.0/1.2 |
\(\text {DFBETAS}^{\beta _0}\) [large(%)/small(%)] | 1.4/2.0 | 0/0 | 1.4/1.6 |
\(\text {DFBETAS}^{\beta _1}\) [large(%)/small(%)] | 38.9/44.9 | 0/0.8 | 1.4/0.6 |
\(\text {DFBETAS}^{\sigma ^2_{\nu }}\) [large(%)/small(%)] | 1.4/6.3 | 0/0 | 1.2/2.0 |