We have proposed an implementation of the RWLB addressing the estimation of uncertainty in prediction for the entire class of GLMM, either by estimating the MSEP or correcting the CV.
We have performed a simulation study that shows that this algorithm can be succesfully applied to both Gaussian and Non-Gaussian GLMM while displaying evidence of the good properties of our proposal when compared with competitors.
In the context of a LMM inspired by the Orthodonticdata, the RWLB-based methods that (i) are based onresamplingwhile accounting for the shrinkage of predictions (RWLB.V2) and (ii) rely on the simulation strategy (RWLB.V3), described in Section 3.3.3, display a relative Bias and Efficiency that is on par with the other Bootstrap-based methods and classic estimators based in second-order correct approximations e.g. Prasad & Rao (1990).
Moreover, these proposals seem well suited for the estimation of the uncertainty of Fitted Values.
In the simulated Mixed Logit Model inspired by the Toenail data, the method relying on the RWLB scheme that uses thesimulation strategy with EP obtained with Monte-Carlo approximations (RWLBE.v3.ep) produces Normal-based Confidence Intervals with Coverage Ratios that approximate the nominal levels especially when this level is “low” i.e. 90% or 95% and the number of observations per cluster increases. There is apparently a slight gain when using perturbed EP to create a correction for the effect of the uncertainty due to parameter estimation (GBC.v2.ep).
Further refinements of our appproach could include the use of RWLB replicates of the fixed effects to construct the samples at the basis of the RWLB methods, see Equations (27) and (28). Moreover, the theoretical connexions with the double bootstrap procedure of Hall
& Maiti (2006) would be worth studying. Finally, it could be of interest to provide LMM simulations with other settings for the Variance Components parameters.
Acknoweledgments
The computations were performed at University of Geneva on the Baobab cluster. DF acknoweledges the Travelling Support of the Societ´e Acad´emique de Gen`eve.
6 Appendix
6.1 Coverage Ratios for the Linear Predictor in GLMM study
0.90 0.95 0.99
Method v bui 7 14 21 7 14 21 7 14 21
RWLBE v1 cm 0.7203 0.8481 0.8898 0.7796 0.8979 0.9386 0.8537 0.9475 0.9795 ep 0.8889 0.9204 0.9147 0.9185 0.9483 0.9549 0.9557 0.9726 0.9853 v2 cm 0.7922 0.8673 0.8949 0.8402 0.9122 0.9421 0.8977 0.9548 0.9808 ep 0.7957 0.8682 0.8958 0.8521 0.9154 0.9429 0.9202 0.9618 0.9820 v3 cm 0.8160 0.8732 0.8982 0.8688 0.9191 0.9446 0.9308 0.9637 0.9827 ep 0.8923 0.9121 0.9099 0.9301 0.9468 0.9523 0.9666 0.9774 0.9856 REBE v1 cm 0.9201 0.9497 0.9862 0.9409 0.9652 0.9919 0.9650 0.9816 0.9969 ep 0.9847 0.9545 0.9873 0.9903 0.9692 0.9927 0.9954 0.9841 0.9973 v2 cm 0.9841 0.9558 0.9868 0.9898 0.9705 0.9924 0.9951 0.9857 0.9973 ep 0.9906 0.9573 0.9873 0.9941 0.9718 0.9928 0.9973 0.9865 0.9975 PBE cm 0.8869 0.9194 0.9139 0.9170 0.9478 0.9543 0.9544 0.9723 0.9851 ep 0.8903 0.9114 0.9093 0.9284 0.9463 0.9519 0.9658 0.9772 0.9855 Table 6: Average Coverage Ratios of Prediction Intervals for η with n= 300.
0.90 0.95 0.99
Method v bui 7 14 21 7 14 21 7 14 21
RWLBE v1 cm 0.7149 0.8454 0.8888 0.7750 0.8961 0.9377 0.8499 0.9458 0.9790 ep 0.8885 0.9198 0.9145 0.9182 0.9476 0.9545 0.9551 0.9721 0.9849 v2 cm 0.7855 0.8643 0.8939 0.8343 0.9098 0.9411 0.8930 0.9530 0.9803 ep 0.7797 0.8660 0.8948 0.8390 0.9140 0.9422 0.9110 0.9603 0.9816 v3 cm 0.7936 0.8707 0.8973 0.8509 0.9173 0.9439 0.9192 0.9620 0.9823 ep 0.8918 0.9119 0.9097 0.9291 0.9461 0.9520 0.9661 0.9767 0.9853 REBE v1 cm 0.8507 0.9314 0.9770 0.8837 0.9493 0.9846 0.9247 0.9704 0.9928 ep 0.9783 0.9347 0.9782 0.9856 0.9528 0.9856 0.9930 0.9734 0.9935 v2 cm 0.9766 0.9367 0.9775 0.9844 0.9542 0.9851 0.9922 0.9745 0.9933 ep 0.9886 0.9377 0.9781 0.9928 0.9552 0.9856 0.9967 0.9753 0.9936 PBE cm 0.8874 0.9195 0.9142 0.9174 0.9472 0.9542 0.9545 0.9721 0.9849 ep 0.8907 0.9116 0.9095 0.9284 0.9459 0.9518 0.9656 0.9765 0.9852 Table 7: Coverage Ratios of Prediction Intervals for η with n= 600.
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