2.4 Simulation Studies
2.6.1 Coverage Ratios for the Linear Predictor in GLMM study
0.90 0.95 0.99
Method v ubi 7 14 21 7 14 21 7 14 21
RWLB 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 2.6: Average Coverage Ratios of Prediction Intervals for η with n = 300.
0.90 0.95 0.99
Method v ubi 7 14 21 7 14 21 7 14 21
RWLB 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 2.7: Coverage Ratios of Prediction Intervals for η with n = 600.
List of Acronyms
PDF Probability Density Function PMF Probability Mass Function LM Linear Models
LMM Linear Mixed Models GLM Generalized Linear Models
GEE Generalized Estimating Equations GLMM Generalized Linear Mixed Models ML Maximum Likelihood
REML Residual orRestricted Maximum Likelihood LAML Laplace-approximated Maximum Likelihood
LALL Laplace-approximated log-Likelihood
LAWLL Laplace-approximated Weighted log-Likelihood RE Random Effect
BLUP Best Linear Unbiased Predictor REB Random Effect Bootstrap EP Empirical Predictions CM Conditional Modes
EBP Empirical Best Predictors CB Cluster Bootstrap
GCB Generalized Cluster Bootstrap PB Parametric Bootstrap
RWLB Random Weighted Laplace Bootstrap BCI Bootstrap-based Confidence Intervals CR Coverage Ratios
MSEP Mean Squared Error of the Prediction CV Conditional Variances
BP Best Predictor
MC Monte-Carlo
List of Tables
1.1 Parameters in the Simulation Study for LMM. . . 19
1.2 Coverage ratios : 1−α= 0.95 in the LMM Study. . . 23
1.3 Parameters in the second set of simulation studies. . . 25
1.4 Coverage ratios : 1−α= 0.95 in the Mixed Logit Study . . . 30
1.5 Coverage ratios for 1−α= 0.90 in the LMM Study.. . . 38
1.6 Coverage ratios : 1−α= 0.99 in the LMM Study. . . 38
1.7 Coverage ratios : 1−α= 0.90 in the Mixed Logit Study . . . 39
1.8 Coverage ratios : 1−α= 0.99 in the Mixed Logit Study . . . 40
2.1 Average relative bias of estimators of MSEP. . . 57
2.2 Average RRMSE estimators of MSEP. . . 58
2.3 Average coverage ratios for Prediction Intervals of level 1−α= 0.95. . . . 60
2.4 Average Coverage Ratios of Prediction Intervals forui with n = 300. . . 62
2.5 Average Coverage Ratios of Prediction Intervals forui with n = 600 . . . . 63
2.6 Average Coverage Ratios of Prediction Intervals forη with n = 300. . . 64
2.7 Coverage Ratios of Prediction Intervals forη with n = 600. . . 64
List of Figures
1.1 Original trajectories in the Orthodonticdataset . . . 18 1.2 Q-q plots of√
nβbk∗ −βbk vs. √
nβbk−βk fork ={0,1,2,3} and ni = 4 in the LMM study . . . 21 1.3 Q-q plots of √
n(σbk2∗−σbk2) vs. √
n(σbk2−σk2) for k ={1,2} and ni = 4 in the LMM study. . . 22 1.4 Q-q plots of √
nφb∗−φb vs. √
nφb−φin the LMM study. . . 22 1.5 Evolution of the average infection rate over the visits in the Toenail
dataset. . . 25 1.6 Q-Q plots of √
nβbk∗−βbk vs. √
nβbk−βk for k = {0,1,2,3}, n = 300 in the Mixed Logit Model. . . 27 1.7 Q-Q plots of √
nβbk∗−βbk vs. √
nβbk−βk for k = {0,1,2,3}, n = 600 in the Mixed Logit Model. . . 28 1.8 Q-Q plots of√
n(σb2∗−σb2) vs. √
n(σb2 −σ2) the Mixed Logit Model. . . . 29 1.9 Q-q plots of√
nβbk∗−βbkvs. √
nβbk−βkfork ={0,1,2,3}in the LMM study for ni = 12 . . . 36 1.10 Q-q plots of √
n(σb2∗k −σb2k) vs. √
n(σb2k−σk2) for k = {1,2} for ni = 12 in the LMM study . . . 37
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