Conditional conceptual predictive statistic for mixed model selection

Model selection via conditional conceptual predictive statistic for mixed and stochastic restricted ridge estimators in linear mixed models

“…By following Kuran and Özkale, 17 we define transformed conditional Gauss discrepancies between the true model and the fitted candidate model, which are linear functions of the conditional Gauss discrepancies in ( 9) and (10) as…”

“…In LMMs, Wenren 8 and Wenren and Shang 10 proposed conditional $$$ {C}_p $$$ ($$$ {\mathrm{CC}}_p $$$) statistic. The true $$$ {\mathrm{CC}}_p $$$ ($$$ {\mathrm{TCC}}_p $$$) and $$$ {\mathrm{CC}}_p $$$ are the two versions of the conditional $$$ {C}_p $$$ which were, respectively, proved based on the conditional Gauss discrepancy corresponding to the two cases where the covariance matrix was known or unknown.…”

“…Both versions of the $$$ {\mathrm{CC}}_p $$$ statistic contain a goodness‐of‐fit function as a loss function plus a penalty term. Since the penalty term in the $$$ {\mathrm{CC}}_p $$$ is related to the effective number of parameters of a LMM proposed by Hodges and Sargent, 11 Wenren 8 and Wenren and Shang 10 computed the penalty term, say $$$ K $$$, in the $$$ {\mathrm{CC}}_p $$$ as the estimated effective degrees of freedom.…”

“…When multicollinearity problems that mentioned in the above paragraph are arose, one or more variables related to the fixed effects usually are deleted, but this could cause some not irrelevant consequences: the fitted candidate model could be misspecified and so, the underfitted or the overfitted candidate models result in large variances for the BLUE as well as a large variance for $$$ {\hat{y}}_i $$$. For this reason, Kuran and Özkale 17,18 enlarged Wenren, 8 Wenren and Shang 10 and Wenren et al's 9 studies for the ridge estimator and the ridge predictor under multicollinearity.…”

“…For this reason, Kuran and Özkale 17,18 enlarged Wenren, 8 Wenren and Shang 10 and Wenren et al's 9 studies for the ridge estimator and the ridge predictor under multicollinearity.…”