Expectation-Maximization Algorithm for the Weibull Proportional Hazard Model under Current Status Data
Abstract
:1. Introduction
2. The Proposed Method
2.1. The Weibull PH Model
2.2. Data Augmentation
2.3. EM Algorithm
2.4. Asymptotic Variances and Covariance
3. Simulation Study
3.1. Simulation Study I: Compare the Proposed EM Method with the NR Method
- 1.
- For small sample sizes, such as , 60 and 100, both the Bias and ESE of the EM method are generally smaller than those of the NR method. For example, Table 2 presents the estimation results for . Among 54 cases, the EM method exhibits smaller biases in 35 cases, and a majority of the ESEs are also smaller than those of the NR method. Similar trends are observed in Table 1 and Table 3.
- 2.
- When the sample size is medium or large, such as and 400, the EM method still maintains an advantage in terms of the Bias, with over half of the 54 cases exhibiting smaller Bias. However, when considering ESEs, the EM method performs unsatisfactorily. For instance, when , the EM method produces smaller Bias and only smaller ESE.
- 3.
- With the increase in sample size, the Bias and ESE from both methods diminish towards zero, indicating that the performance of both methods improves with larger sample sizes, and the estimates can converge to the true values.
- 4.
- The estimated standard errors and sample standard deviations decrease and approach each other as the sample size increases, suggesting that the asymptotic covariance matrix obtained by Louis’ method performs well for finite sample sizes.
3.2. Simulation Study II: Compare the Weibull PH Model with the Semi-Parametric PH Model
- 1.
- The failure time follows a Weibull distribution, with
- 2.
- Referring to the simulation setting in McMahan et al. [9], the failure time follows a non-Weibull distribution, with
4. Real Data Analysis
4.1. Lung Tumor Data
4.2. The Data of IOL Calcification
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
PH | Proportional hazard model |
EM | Expectation-maximization algorithm |
NR | Newton–Raphson method |
MLE | Maximum likelihood estimator |
AIC | Akaike’s information criterion |
BIC | Bayesian information criterion |
GE | Germ-free environment |
CE | Conventional environment |
IOL | Intraocular lenses |
NPMLE | Non-parametric maximum likelihood estimation |
Appendix A
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EM | NR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
P = 0.3 | −0.5 | 0.0830 | 1.8440 | 2.2309 | 0.959 | 0.0836 | 1.9107 | 2.4515 | 0.968 | |
−0.5 | 0.0484 | 1.0089 | 1.0913 | 0.969 | 0.0328 | 1.2368 | 1.2633 | 0.970 | ||
−0.5 | 0.2092 | 1.9040 | 2.3507 | 0.961 | 0.2147 | 1.9659 | 2.5252 | 0.969 | ||
0 | 0.0206 | 0.9289 | 1.0585 | 0.962 | 0.0262 | 1.0142 | 1.1342 | 0.971 | ||
−0.5 | 0.1791 | 1.8637 | 2.2418 | 0.975 | 0.2132 | 1.9850 | 2.5868 | 0.981 | ||
0.5 | 0.1805 | 1.0694 | 1.1345 | 0.966 | 0.2178 | 1.5543 | 1.3574 | 0.970 | ||
0 | 0.0429 | 1.7708 | 2.1292 | 0.966 | 0.0392 | 1.8299 | 2.3199 | 0.973 | ||
−0.5 | 0.0800 | 0.9298 | 1.0756 | 0.968 | 0.0849 | 0.9939 | 1.1996 | 0.972 | ||
0 | 0.0028 | 1.8364 | 2.2490 | 0.961 | 0.0280 | 1.9182 | 2.5170 | 0.973 | ||
0 | 0.0007 | 0.8648 | 0.9720 | 0.963 | 0.0011 | 0.8916 | 1.0309 | 0.971 | ||
0 | 0.0102 | 1.8739 | 2.2762 | 0.970 | 0.0048 | 1.9260 | 2.4167 | 0.975 | ||
0.5 | 0.2084 | 0.9739 | 1.0988 | 0.966 | 0.2233 | 1.0044 | 1.1707 | 0.971 | ||
0.5 | 0.2492 | 1.8562 | 2.2518 | 0.966 | 0.2433 | 1.9033 | 2.3463 | 0.972 | ||
−0.5 | 0.1610 | 0.9248 | 1.0719 | 0.961 | 0.1584 | 0.9628 | 1.1288 | 0.970 | ||
0.5 | 0.1354 | 1.8649 | 2.3146 | 0.954 | 0.1446 | 1.9187 | 2.4616 | 0.964 | ||
0 | 0.0122 | 0.8776 | 0.9651 | 0.959 | 0.0095 | 0.9147 | 1.1997 | 0.962 | ||
0.5 | 0.2410 | 1.9257 | 2.3118 | 0.962 | 0.2821 | 2.0136 | 2.6029 | 0.969 | ||
0.5 | 0.2649 | 0.9864 | 1.0996 | 0.974 | 0.2984 | 1.0783 | 1.2327 | 0.981 | ||
P = 0.4 | −0.5 | 0.1729 | 1.7020 | 2.0879 | 0.963 | 0.2270 | 1.8282 | 2.5046 | 0.977 | |
−0.5 | 0.1813 | 0.8450 | 0.9767 | 0.956 | 0.2266 | 1.0621 | 1.3028 | 0.972 | ||
−0.5 | 0.2832 | 1.7186 | 2.0628 | 0.964 | 0.3393 | 1.8398 | 2.3169 | 0.973 | ||
0 | 0.0239 | 0.8061 | 0.9461 | 0.954 | 0.0420 | 0.8696 | 1.0821 | 0.959 | ||
−0.5 | 0.1104 | 1.7200 | 2.0367 | 0.960 | 0.1262 | 1.8137 | 2.2480 | 0.973 | ||
0.5 | 0.1507 | 0.8318 | 0.9381 | 0.970 | 0.1857 | 0.8719 | 1.0466 | 0.977 | ||
0 | 0.1023 | 1.6709 | 1.9692 | 0.957 | 0.1306 | 1.7874 | 2.2489 | 0.973 | ||
−0.5 | 0.1277 | 0.8259 | 0.9891 | 0.956 | 0.1544 | 0.9196 | 1.1636 | 0.970 | ||
0 | 0.0474 | 1.6652 | 2.0079 | 0.954 | 0.0963 | 1.8191 | 2.5094 | 0.971 | ||
0 | 0.0393 | 0.7970 | 0.9040 | 0.969 | 0.0522 | 0.8661 | 1.0923 | 0.976 | ||
0 | 0.0390 | 1.6845 | 1.9040 | 0.968 | 0.0403 | 1.8056 | 2.2031 | 0.978 | ||
0.5 | 0.1673 | 0.8339 | 0.9646 | 0.957 | 0.2160 | 0.8931 | 1.1492 | 0.966 | ||
0.5 | 0.1590 | 1.6926 | 1.9928 | 0.962 | 0.1831 | 1.7901 | 2.2137 | 0.979 | ||
−0.5 | 0.1838 | 0.8250 | 0.9279 | 0.963 | 0.1990 | 0.8734 | 1.0495 | 0.975 | ||
0.5 | 0.2011 | 1.6773 | 1.9646 | 0.966 | 0.2285 | 1.7751 | 2.1901 | 0.976 | ||
0 | 0.0177 | 0.7909 | 0.8990 | 0.971 | 0.0346 | 0.8421 | 1.0441 | 0.977 | ||
0.5 | 0.1129 | 1.7145 | 2.1417 | 0.960 | 0.1533 | 1.8410 | 2.5154 | 0.971 | ||
0.5 | 0.1499 | 0.8264 | 0.9115 | 0.972 | 0.2068 | 0.8921 | 1.1331 | 0.983 | ||
P = 0.5 | −0.5 | 0.1404 | 1.6381 | 1.9355 | 0.956 | 0.1861 | 1.9337 | 2.9388 | 0.983 | |
−0.5 | 0.1871 | 0.7933 | 0.8984 | 0.964 | 0.2680 | 0.9546 | 1.4432 | 0.980 | ||
−0.5 | 0.2128 | 1.6384 | 1.8534 | 0.962 | 0.2979 | 1.9294 | 2.6214 | 0.982 | ||
0 | 0.0253 | 0.7582 | 0.8298 | 0.959 | 0.0106 | 0.8936 | 1.2374 | 0.977 | ||
−0.5 | 0.0929 | 1.6298 | 1.7967 | 0.966 | 0.1344 | 1.8299 | 2.2020 | 0.979 | ||
0.5 | 0.0989 | 0.8058 | 0.8608 | 0.966 | 0.1651 | 0.9246 | 1.1044 | 0.980 | ||
0 | 0.0444 | 1.5998 | 1.8181 | 0.965 | 0.0435 | 1.8125 | 2.2458 | 0.983 | ||
−0.5 | 0.1861 | 0.7997 | 0.8878 | 0.969 | 0.2333 | 0.9375 | 1.2308 | 0.983 | ||
0 | 0.0304 | 1.6357 | 1.8762 | 0.956 | 0.0364 | 1.9037 | 2.5094 | 0.977 | ||
0 | 0.0535 | 0.7591 | 0.8195 | 0.953 | 0.0039 | 1.0509 | 1.1162 | 0.972 | ||
0 | 0.0308 | 1.6139 | 1.7789 | 0.956 | 0.0331 | 1.8368 | 2.3188 | 0.979 | ||
0.5 | 0.0903 | 0.8036 | 0.8984 | 0.970 | 0.1933 | 0.9571 | 1.2448 | 0.982 | ||
0.5 | 0.1448 | 1.6689 | 1.9246 | 0.964 | 0.2251 | 1.9309 | 2.7892 | 0.983 | ||
−0.5 | 0.1694 | 0.8016 | 0.8754 | 0.964 | 0.1987 | 0.9200 | 1.1835 | 0.977 | ||
0.5 | 0.1584 | 1.6408 | 1.8934 | 0.962 | 0.2116 | 1.8851 | 2.4036 | 0.979 | ||
0 | 0.0426 | 0.7553 | 0.8390 | 0.968 | 0.0198 | 0.8727 | 1.1649 | 0.978 | ||
0.5 | 0.1499 | 1.6196 | 1.8070 | 0.970 | 0.1770 | 1.8445 | 2.2799 | 0.984 | ||
0.5 | 0.0900 | 0.8012 | 0.8667 | 0.962 | 0.1762 | 0.9118 | 1.1601 | 0.977 |
EM | NR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
P = 0.3 | −0.5 | 0.1074 | 1.1225 | 1.2900 | 0.944 | 0.1125 | 1.1261 | 1.3128 | 0.944 | |
−0.5 | 0.0405 | 0.5654 | 0.5989 | 0.950 | 0.0446 | 0.5657 | 0.5983 | 0.952 | ||
−0.5 | 0.0499 | 1.1132 | 1.1935 | 0.953 | 0.0515 | 1.1145 | 1.2006 | 0.954 | ||
0 | 0.0155 | 0.5302 | 0.5485 | 0.966 | 0.0123 | 0.5304 | 0.5491 | 0.966 | ||
−0.5 | 0.0050 | 1.1140 | 1.2353 | 0.941 | 0.0056 | 1.1153 | 1.2443 | 0.941 | ||
0.5 | 0.0564 | 0.5686 | 0.6086 | 0.951 | 0.0548 | 0.5693 | 0.6139 | 0.951 | ||
0 | 0.0375 | 1.1005 | 1.1955 | 0.964 | 0.0375 | 1.1017 | 1.2020 | 0.963 | ||
−0.5 | 0.0888 | 0.5657 | 0.5999 | 0.948 | 0.0931 | 0.5662 | 0.6003 | 0.947 | ||
0 | 0.0507 | 1.0971 | 1.1601 | 0.957 | 0.0515 | 1.0978 | 1.1646 | 0.955 | ||
0 | 0.0171 | 0.5331 | 0.5844 | 0.948 | 0.0135 | 0.5331 | 0.5839 | 0.950 | ||
0 | 0.0093 | 1.1012 | 1.2218 | 0.956 | 0.0088 | 1.1025 | 1.2313 | 0.956 | ||
0.5 | 0.0819 | 0.5655 | 0.6033 | 0.951 | 0.0810 | 0.5666 | 0.6111 | 0.949 | ||
0.5 | 0.0867 | 1.1226 | 1.2370 | 0.951 | 0.0877 | 1.1249 | 1.2510 | 0.951 | ||
−0.5 | 0.0422 | 0.5666 | 0.5999 | 0.945 | 0.0460 | 0.5670 | 0.6010 | 0.946 | ||
0.5 | 0.0016 | 1.1134 | 1.2137 | 0.948 | 0.0006 | 1.1142 | 1.2196 | 0.948 | ||
0 | 0.0168 | 0.5332 | 0.5942 | 0.944 | 0.0132 | 0.5332 | 0.5934 | 0.946 | ||
0.5 | 0.0852 | 1.1258 | 1.2744 | 0.944 | 0.0889 | 1.1289 | 1.2905 | 0.943 | ||
0.5 | 0.0787 | 0.5677 | 0.6177 | 0.957 | 0.0770 | 0.5687 | 0.6211 | 0.953 | ||
P = 0.4 | −0.5 | 0.1149 | 1.0102 | 1.1372 | 0.951 | 0.1204 | 1.0180 | 1.1682 | 0.950 | |
−0.5 | 0.0569 | 0.5098 | 0.5621 | 0.949 | 0.0598 | 0.5132 | 0.5738 | 0.951 | ||
−0.5 | 0.0105 | 1.0175 | 1.1324 | 0.947 | 0.0131 | 1.0227 | 1.1519 | 0.947 | ||
0 | 0.0035 | 0.4781 | 0.5207 | 0.943 | 0.0034 | 0.4801 | 0.5284 | 0.944 | ||
−0.5 | 0.0945 | 1.0070 | 1.1029 | 0.944 | 0.0977 | 1.0110 | 1.1191 | 0.943 | ||
0.5 | 0.0455 | 0.5031 | 0.5380 | 0.959 | 0.0469 | 0.5048 | 0.5479 | 0.959 | ||
0 | 0.0298 | 1.0027 | 1.1316 | 0.947 | 0.0315 | 1.0089 | 1.1553 | 0.949 | ||
−0.5 | 0.0193 | 0.5074 | 0.5474 | 0.957 | 0.0220 | 0.5101 | 0.5605 | 0.958 | ||
0 | 0.0019 | 0.9944 | 1.1156 | 0.952 | 0.0022 | 1.0007 | 1.1410 | 0.951 | ||
0 | 0.0102 | 0.4780 | 0.5253 | 0.946 | 0.0111 | 0.4799 | 0.5321 | 0.944 | ||
0 | 0.0098 | 1.0039 | 1.1104 | 0.949 | 0.0098 | 1.0094 | 1.1322 | 0.951 | ||
0.5 | 0.0718 | 0.5100 | 0.5616 | 0.946 | 0.0730 | 0.5120 | 0.5716 | 0.942 | ||
0.5 | 0.0679 | 1.0202 | 1.2119 | 0.937 | 0.0702 | 1.0254 | 1.2306 | 0.936 | ||
−0.5 | 0.0792 | 0.5076 | 0.5621 | 0.948 | 0.0812 | 0.5098 | 0.5682 | 0.947 | ||
0.5 | 0.0610 | 1.0068 | 1.1049 | 0.944 | 0.0635 | 1.0113 | 1.1221 | 0.944 | ||
0 | 0.0165 | 0.4758 | 0.5096 | 0.944 | 0.0162 | 0.4769 | 0.5130 | 0.945 | ||
0.5 | 0.0678 | 1.0126 | 1.1113 | 0.948 | 0.0693 | 1.0184 | 1.1322 | 0.948 | ||
0.5 | 0.0865 | 0.5088 | 0.5526 | 0.953 | 0.0892 | 0.5116 | 0.5673 | 0.949 | ||
P = 0.5 | −0.5 | 0.0513 | 0.9776 | 1.0617 | 0.956 | 0.0602 | 0.9957 | 1.1085 | 0.957 | |
−0.5 | 0.0913 | 0.4900 | 0.5125 | 0.964 | 0.0871 | 0.4995 | 0.5419 | 0.963 | ||
−0.5 | 0.1144 | 0.9747 | 0.9752 | 0.964 | 0.1270 | 0.9908 | 1.0187 | 0.962 | ||
0 | 0.0224 | 0.4615 | 0.4980 | 0.953 | 0.0161 | 0.4703 | 0.5389 | 0.956 | ||
−0.5 | 0.0834 | 0.9733 | 1.0536 | 0.947 | 0.0979 | 0.9964 | 1.1363 | 0.949 | ||
0.5 | 0.0552 | 0.4915 | 0.5375 | 0.953 | 0.0822 | 0.5041 | 0.5997 | 0.949 | ||
0 | 0.0237 | 0.9685 | 1.0653 | 0.952 | 0.0242 | 0.9860 | 1.1128 | 0.951 | ||
−0.5 | 0.0763 | 0.4870 | 0.5234 | 0.958 | 0.0722 | 0.4965 | 0.5581 | 0.960 | ||
0 | 0.0472 | 0.9686 | 1.0561 | 0.949 | 0.0452 | 0.9913 | 1.1351 | 0.946 | ||
0 | 0.0096 | 0.4679 | 0.5137 | 0.937 | 0.0054 | 0.4775 | 0.5502 | 0.934 | ||
0 | 0.0181 | 0.9681 | 1.0731 | 0.954 | 0.0189 | 0.9899 | 1.1438 | 0.957 | ||
0.5 | 0.0525 | 0.4906 | 0.5291 | 0.950 | 0.0766 | 0.5018 | 0.5889 | 0.949 | ||
0.5 | 0.1028 | 0.9805 | 1.0581 | 0.959 | 0.1111 | 1.0013 | 1.1321 | 0.960 | ||
−0.5 | 0.0701 | 0.4887 | 0.5296 | 0.946 | 0.0696 | 0.5002 | 0.5764 | 0.949 | ||
0.5 | 0.0539 | 0.9740 | 1.0550 | 0.954 | 0.0616 | 0.9952 | 1.1130 | 0.954 | ||
0 | 0.0447 | 0.4655 | 0.5116 | 0.944 | 0.0325 | 0.4763 | 0.5572 | 0.945 | ||
0.5 | 0.0757 | 0.9826 | 1.0414 | 0.956 | 0.0832 | 1.0036 | 1.1060 | 0.957 | ||
0.5 | 0.0586 | 0.4884 | 0.5083 | 0.964 | 0.0830 | 0.4989 | 0.5545 | 0.961 |
EM | NR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
P = 0.3 | −0.5 | 0.0506 | 0.8134 | 0.8641 | 0.942 | 0.0522 | 0.8132 | 0.8661 | 0.941 | |
−0.5 | 0.0483 | 0.4208 | 0.4457 | 0.949 | 0.0523 | 0.4207 | 0.4453 | 0.950 | ||
−0.5 | 0.0867 | 0.8140 | 0.8513 | 0.954 | 0.0878 | 0.8137 | 0.8533 | 0.954 | ||
0 | 0.0061 | 0.3944 | 0.4099 | 0.945 | 0.0024 | 0.3942 | 0.4094 | 0.945 | ||
−0.5 | 0.0382 | 0.8190 | 0.8794 | 0.946 | 0.0392 | 0.8186 | 0.8815 | 0.946 | ||
0.5 | 0.0548 | 0.4191 | 0.4344 | 0.956 | 0.0512 | 0.4190 | 0.4347 | 0.956 | ||
0 | 0.0154 | 0.8024 | 0.8590 | 0.946 | 0.0152 | 0.8023 | 0.8612 | 0.946 | ||
−0.5 | 0.0147 | 0.4177 | 0.4349 | 0.947 | 0.0188 | 0.4175 | 0.4342 | 0.947 | ||
0 | 0.0028 | 0.8053 | 0.8456 | 0.953 | 0.0028 | 0.8049 | 0.8472 | 0.952 | ||
0 | 0.0193 | 0.3965 | 0.4129 | 0.953 | 0.0153 | 0.3963 | 0.4121 | 0.953 | ||
0 | 0.0258 | 0.8061 | 0.8418 | 0.946 | 0.0260 | 0.8056 | 0.8436 | 0.945 | ||
0.5 | 0.0350 | 0.4180 | 0.4458 | 0.941 | 0.0310 | 0.4179 | 0.4450 | 0.941 | ||
0.5 | 0.0352 | 0.8157 | 0.8365 | 0.953 | 0.0367 | 0.8156 | 0.8393 | 0.952 | ||
−0.5 | 0.0232 | 0.4188 | 0.4344 | 0.952 | 0.0271 | 0.4186 | 0.4339 | 0.953 | ||
0.5 | 0.0561 | 0.8169 | 0.8579 | 0.959 | 0.0572 | 0.8166 | 0.8599 | 0.958 | ||
0 | 0.0028 | 0.3950 | 0.4089 | 0.951 | 0.0010 | 0.3948 | 0.4083 | 0.951 | ||
0.5 | 0.0429 | 0.8208 | 0.8814 | 0.941 | 0.0439 | 0.8204 | 0.8838 | 0.941 | ||
0.5 | 0.0551 | 0.4190 | 0.4311 | 0.948 | 0.0515 | 0.4189 | 0.4310 | 0.948 | ||
P = 0.4 | −0.5 | 0.0033 | 0.7281 | 0.7429 | 0.959 | 0.0052 | 0.7290 | 0.7497 | 0.957 | |
−0.5 | 0.0168 | 0.3725 | 0.3818 | 0.953 | 0.0201 | 0.3730 | 0.3823 | 0.954 | ||
−0.5 | 0.0907 | 0.7382 | 0.7621 | 0.954 | 0.0929 | 0.7388 | 0.7678 | 0.952 | ||
0 | 0.0196 | 0.3545 | 0.3669 | 0.950 | 0.0161 | 0.3546 | 0.3674 | 0.951 | ||
−0.5 | 0.0772 | 0.7384 | 0.7902 | 0.942 | 0.0794 | 0.7387 | 0.7962 | 0.942 | ||
0.5 | 0.0524 | 0.3736 | 0.3938 | 0.943 | 0.0487 | 0.3738 | 0.3958 | 0.942 | ||
0 | 0.0115 | 0.7229 | 0.7605 | 0.948 | 0.0118 | 0.7239 | 0.7680 | 0.947 | ||
−0.5 | 0.0366 | 0.3725 | 0.3880 | 0.949 | 0.0396 | 0.3728 | 0.3891 | 0.950 | ||
0 | 0.0152 | 0.7260 | 0.7776 | 0.945 | 0.0157 | 0.7267 | 0.7848 | 0.942 | ||
0 | 0.0204 | 0.3533 | 0.3725 | 0.934 | 0.0243 | 0.3535 | 0.3733 | 0.935 | ||
0 | 0.0044 | 0.7292 | 0.7800 | 0.939 | 0.0049 | 0.7295 | 0.7862 | 0.938 | ||
0.5 | 0.0310 | 0.3726 | 0.3736 | 0.958 | 0.0270 | 0.3728 | 0.3758 | 0.956 | ||
0.5 | 0.0686 | 0.7347 | 0.8041 | 0.944 | 0.0711 | 0.7356 | 0.8109 | 0.941 | ||
−0.5 | 0.0185 | 0.3718 | 0.3893 | 0.956 | 0.0215 | 0.3721 | 0.3894 | 0.955 | ||
0.5 | 0.0215 | 0.7390 | 0.7908 | 0.949 | 0.0234 | 0.7397 | 0.7970 | 0.947 | ||
0 | 0.0057 | 0.3555 | 0.3735 | 0.956 | 0.0017 | 0.3557 | 0.3739 | 0.953 | ||
0.5 | 0.0266 | 0.7370 | 0.8072 | 0.931 | 0.0289 | 0.7378 | 0.8167 | 0.931 | ||
0.5 | 0.0383 | 0.3733 | 0.3984 | 0.940 | 0.0352 | 0.3736 | 0.4015 | 0.939 | ||
P = 0.5 | −0.5 | 0.0095 | 0.7045 | 0.7172 | 0.955 | 0.0065 | 0.7091 | 0.7345 | 0.955 | |
−0.5 | 0.0410 | 0.3555 | 0.3732 | 0.949 | 0.0352 | 0.3577 | 0.3807 | 0.946 | ||
−0.5 | 0.0552 | 0.7090 | 0.7134 | 0.963 | 0.0592 | 0.7138 | 0.7315 | 0.962 | ||
0 | 0.0003 | 0.3407 | 0.3488 | 0.957 | 0.0047 | 0.3424 | 0.3559 | 0.951 | ||
−0.5 | 0.0606 | 0.7058 | 0.7553 | 0.939 | 0.0662 | 0.7109 | 0.7783 | 0.936 | ||
0.5 | 0.0321 | 0.3564 | 0.3700 | 0.950 | 0.0410 | 0.3582 | 0.3851 | 0.942 | ||
0 | 0.0154 | 0.6972 | 0.7254 | 0.949 | 0.0153 | 0.7014 | 0.7414 | 0.947 | ||
−0.5 | 0.0563 | 0.3558 | 0.3697 | 0.953 | 0.0516 | 0.3578 | 0.3757 | 0.953 | ||
0 | 0.0004 | 0.6969 | 0.7125 | 0.967 | 0.0003 | 0.7014 | 0.7308 | 0.966 | ||
0 | 0.0008 | 0.3402 | 0.3669 | 0.931 | 0.0039 | 0.3417 | 0.3747 | 0.928 | ||
0 | 0.0220 | 0.7012 | 0.7403 | 0.950 | 0.0221 | 0.7059 | 0.7606 | 0.946 | ||
0.5 | 0.0008 | 0.3560 | 0.3761 | 0.946 | 0.0058 | 0.3575 | 0.3889 | 0.944 | ||
0.5 | 0.0198 | 0.7076 | 0.7766 | 0.944 | 0.0251 | 0.7128 | 0.7970 | 0.944 | ||
−0.5 | 0.0525 | 0.3571 | 0.3679 | 0.955 | 0.0478 | 0.3593 | 0.3753 | 0.954 | ||
0.5 | 0.0341 | 0.7077 | 0.7485 | 0.950 | 0.0380 | 0.7128 | 0.7702 | 0.948 | ||
0 | 0.0100 | 0.3404 | 0.3424 | 0.957 | 0.0054 | 0.3418 | 0.3494 | 0.955 | ||
0.5 | 0.0163 | 0.7048 | 0.7260 | 0.950 | 0.0208 | 0.7095 | 0.7481 | 0.948 | ||
0.5 | 0.0468 | 0.3558 | 0.3727 | 0.949 | 0.0555 | 0.3577 | 0.3903 | 0.943 |
EM | NR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
P = 0.3 | −0.5 | 0.0171 | 0.5542 | 0.5636 | 0.947 | 0.0159 | 0.5538 | 0.5644 | 0.947 | |
−0.5 | 0.0171 | 0.2881 | 0.2979 | 0.949 | 0.0210 | 0.2879 | 0.2977 | 0.950 | ||
−0.5 | 0.0377 | 0.5560 | 0.5725 | 0.943 | 0.0386 | 0.5556 | 0.5729 | 0.943 | ||
0 | 0.0105 | 0.2721 | 0.2862 | 0.951 | 0.0071 | 0.2720 | 0.2856 | 0.951 | ||
−0.5 | 0.0407 | 0.5549 | 0.5784 | 0.939 | 0.0414 | 0.5544 | 0.5788 | 0.940 | ||
0.5 | 0.0294 | 0.2875 | 0.2938 | 0.952 | 0.0258 | 0.2874 | 0.2932 | 0.951 | ||
0 | 0.0277 | 0.5474 | 0.5568 | 0.947 | 0.0278 | 0.5471 | 0.5575 | 0.947 | ||
−0.5 | 0.0141 | 0.2876 | 0.2920 | 0.957 | 0.0180 | 0.2875 | 0.2918 | 0.958 | ||
0 | 0.0262 | 0.5498 | 0.5774 | 0.950 | 0.0262 | 0.5494 | 0.5782 | 0.950 | ||
0 | 0.0069 | 0.2728 | 0.2800 | 0.943 | 0.0035 | 0.2726 | 0.2796 | 0.945 | ||
0 | 0.0097 | 0.5481 | 0.5652 | 0.942 | 0.0097 | 0.5476 | 0.5656 | 0.942 | ||
0.5 | 0.0350 | 0.2871 | 0.3066 | 0.941 | 0.0311 | 0.2869 | 0.3059 | 0.944 | ||
0.5 | 0.0216 | 0.5541 | 0.5484 | 0.957 | 0.0229 | 0.5537 | 0.5491 | 0.956 | ||
−0.5 | 0.0095 | 0.2866 | 0.2873 | 0.956 | 0.0134 | 0.2864 | 0.2871 | 0.956 | ||
0.5 | 0.0159 | 0.5543 | 0.5892 | 0.941 | 0.0150 | 0.5539 | 0.5898 | 0.941 | ||
0 | 0.0036 | 0.2712 | 0.2792 | 0.944 | 0.0002 | 0.2711 | 0.2788 | 0.944 | ||
0.5 | 0.0226 | 0.5556 | 0.5878 | 0.942 | 0.0234 | 0.5551 | 0.5882 | 0.942 | ||
0.5 | 0.0209 | 0.2878 | 0.3060 | 0.933 | 0.0171 | 0.2877 | 0.3053 | 0.935 | ||
P = 0.4 | −0.5 | 0.0231 | 0.4987 | 0.5155 | 0.955 | 0.0251 | 0.4988 | 0.5186 | 0.953 | |
−0.5 | 0.0257 | 0.2560 | 0.2611 | 0.951 | 0.0300 | 0.2560 | 0.2608 | 0.950 | ||
−0.5 | 0.0211 | 0.4958 | 0.4925 | 0.955 | 0.0226 | 0.4956 | 0.4950 | 0.954 | ||
0 | 0.0112 | 0.2430 | 0.2543 | 0.941 | 0.0065 | 0.2429 | 0.2541 | 0.939 | ||
−0.5 | 0.0265 | 0.4994 | 0.5045 | 0.950 | 0.0275 | 0.4990 | 0.5067 | 0.948 | ||
0.5 | 0.0227 | 0.2553 | 0.2637 | 0.949 | 0.0169 | 0.2553 | 0.2629 | 0.950 | ||
0 | 0.0153 | 0.4898 | 0.5004 | 0.957 | 0.0155 | 0.4898 | 0.5033 | 0.956 | ||
−0.5 | 0.0236 | 0.2565 | 0.2649 | 0.945 | 0.0278 | 0.2565 | 0.2644 | 0.945 | ||
0 | 0.0145 | 0.4904 | 0.4968 | 0.951 | 0.0147 | 0.4902 | 0.4993 | 0.950 | ||
0 | 0.0043 | 0.2422 | 0.2467 | 0.956 | 0.0090 | 0.2421 | 0.2462 | 0.955 | ||
0 | 0.0269 | 0.4928 | 0.5021 | 0.953 | 0.0271 | 0.4925 | 0.5042 | 0.953 | ||
0.5 | 0.0190 | 0.2561 | 0.2672 | 0.947 | 0.0133 | 0.2560 | 0.2667 | 0.948 | ||
0.5 | 0.0275 | 0.4981 | 0.5009 | 0.947 | 0.0297 | 0.4982 | 0.5042 | 0.946 | ||
−0.5 | 0.0149 | 0.2558 | 0.2654 | 0.948 | 0.0191 | 0.2558 | 0.2651 | 0.948 | ||
0.5 | 0.0428 | 0.4996 | 0.5153 | 0.947 | 0.0447 | 0.4994 | 0.5183 | 0.946 | ||
0 | 0.0113 | 0.2426 | 0.2510 | 0.937 | 0.0066 | 0.2425 | 0.2507 | 0.937 | ||
0.5 | 0.0086 | 0.4985 | 0.5189 | 0.945 | 0.0079 | 0.4981 | 0.5212 | 0.942 | ||
0.5 | 0.0442 | 0.2559 | 0.2641 | 0.947 | 0.0383 | 0.2558 | 0.2634 | 0.944 | ||
P = 0.5 | −0.5 | 0.0027 | 0.4753 | 0.4787 | 0.960 | 0.0005 | 0.4767 | 0.4868 | 0.958 | |
−0.5 | 0.0249 | 0.2427 | 0.2563 | 0.935 | 0.0213 | 0.2434 | 0.2577 | 0.933 | ||
−0.5 | 0.0508 | 0.4757 | 0.4795 | 0.958 | 0.0552 | 0.4770 | 0.4883 | 0.956 | ||
0 | 0.0021 | 0.2319 | 0.2333 | 0.955 | 0.0021 | 0.2323 | 0.2356 | 0.956 | ||
−0.5 | 0.0232 | 0.4747 | 0.4631 | 0.964 | 0.0263 | 0.4757 | 0.4718 | 0.962 | ||
0.5 | 0.0153 | 0.2424 | 0.2492 | 0.942 | 0.0149 | 0.2427 | 0.2542 | 0.939 | ||
0 | 0.0063 | 0.4699 | 0.4734 | 0.953 | 0.0063 | 0.4713 | 0.4812 | 0.949 | ||
−0.5 | 0.0256 | 0.2423 | 0.2434 | 0.954 | 0.0230 | 0.2429 | 0.2440 | 0.953 | ||
0 | 0.0195 | 0.4710 | 0.4612 | 0.964 | 0.0197 | 0.4722 | 0.4700 | 0.959 | ||
0 | 0.0098 | 0.2322 | 0.2453 | 0.936 | 0.0099 | 0.2327 | 0.2484 | 0.935 | ||
0 | 0.0181 | 0.4716 | 0.4720 | 0.954 | 0.0182 | 0.4726 | 0.4809 | 0.950 | ||
0.5 | 0.0179 | 0.2429 | 0.2558 | 0.947 | 0.0175 | 0.2432 | 0.2620 | 0.944 | ||
0.5 | 0.0171 | 0.4744 | 0.4878 | 0.950 | 0.0207 | 0.4758 | 0.4953 | 0.948 | ||
−0.5 | 0.0174 | 0.2422 | 0.2465 | 0.951 | 0.0146 | 0.2429 | 0.2475 | 0.949 | ||
0.5 | 0.0429 | 0.4766 | 0.4820 | 0.960 | 0.0470 | 0.4779 | 0.4911 | 0.956 | ||
0 | 0.0122 | 0.2318 | 0.2408 | 0.932 | 0.0123 | 0.2322 | 0.2433 | 0.930 | ||
0.5 | 0.0257 | 0.4752 | 0.4959 | 0.953 | 0.0283 | 0.4762 | 0.5055 | 0.947 | ||
0.5 | 0.0267 | 0.2427 | 0.2397 | 0.950 | 0.0261 | 0.2430 | 0.2446 | 0.946 |
EM | NR | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
P = 0.3 | −0.5 | 0.0120 | 0.3864 | 0.3899 | 0.950 | 0.0131 | 0.3861 | 0.3903 | 0.949 | |
−0.5 | 0.0027 | 0.2008 | 0.1964 | 0.954 | 0.0064 | 0.2006 | 0.1963 | 0.954 | ||
−0.5 | 0.0042 | 0.3881 | 0.4012 | 0.951 | 0.0049 | 0.3878 | 0.4015 | 0.951 | ||
0 | 0.0002 | 0.1899 | 0.1810 | 0.966 | 0.0033 | 0.1898 | 0.1808 | 0.965 | ||
−0.5 | 0.0223 | 0.3877 | 0.3857 | 0.954 | 0.0229 | 0.3873 | 0.3857 | 0.955 | ||
0.5 | 0.0195 | 0.2012 | 0.2049 | 0.937 | 0.0161 | 0.2012 | 0.2046 | 0.936 | ||
0 | 0.0078 | 0.3812 | 0.3774 | 0.949 | 0.0078 | 0.3809 | 0.3777 | 0.949 | ||
−0.5 | 0.0063 | 0.2006 | 0.2055 | 0.939 | 0.0100 | 0.2004 | 0.2054 | 0.938 | ||
0 | 0.0063 | 0.3800 | 0.3849 | 0.951 | 0.0063 | 0.3798 | 0.3851 | 0.951 | ||
0 | 0.0101 | 0.1897 | 0.1983 | 0.943 | 0.0069 | 0.1896 | 0.1980 | 0.942 | ||
0 | 0.0133 | 0.3809 | 0.3715 | 0.959 | 0.0133 | 0.3805 | 0.3717 | 0.960 | ||
0.5 | 0.0147 | 0.2003 | 0.2079 | 0.943 | 0.0112 | 0.2003 | 0.2074 | 0.946 | ||
0.5 | 0.0203 | 0.3860 | 0.3842 | 0.948 | 0.0213 | 0.3858 | 0.3847 | 0.948 | ||
−0.5 | 0.0007 | 0.2007 | 0.1969 | 0.956 | 0.0029 | 0.2005 | 0.1968 | 0.958 | ||
0.5 | 0.0010 | 0.3872 | 0.4031 | 0.936 | 0.0017 | 0.3869 | 0.4034 | 0.936 | ||
0 | 0.0004 | 0.1899 | 0.1929 | 0.944 | 0.0036 | 0.1898 | 0.1926 | 0.941 | ||
0.5 | 0.0171 | 0.3873 | 0.4030 | 0.943 | 0.0178 | 0.3869 | 0.4032 | 0.943 | ||
0.5 | 0.0204 | 0.2004 | 0.2038 | 0.942 | 0.0170 | 0.2004 | 0.2034 | 0.942 | ||
P = 0.4 | −0.5 | 0.0229 | 0.3475 | 0.3633 | 0.945 | 0.0250 | 0.3474 | 0.3651 | 0.945 | |
−0.5 | 0.0028 | 0.1785 | 0.1787 | 0.954 | 0.0075 | 0.1785 | 0.1784 | 0.954 | ||
−0.5 | 0.0050 | 0.3447 | 0.3499 | 0.949 | 0.0062 | 0.3445 | 0.3513 | 0.947 | ||
0 | 0.0025 | 0.1694 | 0.1748 | 0.948 | 0.0072 | 0.1693 | 0.1744 | 0.950 | ||
−0.5 | 0.0039 | 0.3450 | 0.3449 | 0.948 | 0.0044 | 0.3447 | 0.3457 | 0.947 | ||
0.5 | 0.0140 | 0.1782 | 0.1757 | 0.951 | 0.0078 | 0.1782 | 0.1746 | 0.955 | ||
0 | 0.0053 | 0.3405 | 0.3411 | 0.959 | 0.0052 | 0.3403 | 0.3428 | 0.957 | ||
−0.5 | 0.0079 | 0.1786 | 0.1825 | 0.954 | 0.0127 | 0.1786 | 0.1822 | 0.954 | ||
0 | 0.0032 | 0.3412 | 0.3425 | 0.951 | 0.0034 | 0.3410 | 0.3438 | 0.951 | ||
0 | 0.0013 | 0.1697 | 0.1725 | 0.938 | 0.0034 | 0.1697 | 0.1722 | 0.935 | ||
0 | 0.0070 | 0.3414 | 0.3452 | 0.959 | 0.0069 | 0.3410 | 0.3459 | 0.957 | ||
0.5 | 0.0174 | 0.1784 | 0.1795 | 0.956 | 0.0112 | 0.1784 | 0.1784 | 0.957 | ||
0.5 | 0.0105 | 0.3460 | 0.3415 | 0.960 | 0.0124 | 0.3459 | 0.3432 | 0.960 | ||
−0.5 | 0.0054 | 0.1787 | 0.1776 | 0.949 | 0.0007 | 0.1787 | 0.1773 | 0.952 | ||
0.5 | 0.0022 | 0.3458 | 0.3406 | 0.962 | 0.0009 | 0.3456 | 0.3419 | 0.962 | ||
0 | 0.0096 | 0.1697 | 0.1785 | 0.939 | 0.0048 | 0.1696 | 0.1782 | 0.938 | ||
0.5 | 0.0053 | 0.3461 | 0.3528 | 0.953 | 0.0059 | 0.3457 | 0.3536 | 0.952 | ||
0.5 | 0.0211 | 0.1787 | 0.1794 | 0.947 | 0.0149 | 0.1787 | 0.1782 | 0.949 | ||
P = 0.5 | −0.5 | 0.0043 | 0.3280 | 0.3215 | 0.958 | 0.0076 | 0.3287 | 0.3261 | 0.954 | |
−0.5 | 0.0122 | 0.1685 | 0.1762 | 0.945 | 0.0100 | 0.1689 | 0.1760 | 0.942 | ||
−0.5 | 0.0197 | 0.3286 | 0.3305 | 0.953 | 0.0233 | 0.3291 | 0.3361 | 0.950 | ||
0 | 0.0066 | 0.1612 | 0.1631 | 0.949 | 0.0041 | 0.1613 | 0.1637 | 0.948 | ||
−0.5 | 0.0103 | 0.3288 | 0.3458 | 0.943 | 0.0127 | 0.3291 | 0.3512 | 0.941 | ||
0.5 | 0.0183 | 0.1688 | 0.1694 | 0.952 | 0.0148 | 0.1689 | 0.1720 | 0.944 | ||
0 | 0.0009 | 0.3253 | 0.3371 | 0.956 | 0.0010 | 0.3259 | 0.3419 | 0.954 | ||
−0.5 | 0.0074 | 0.1687 | 0.1736 | 0.945 | 0.0051 | 0.1691 | 0.1735 | 0.947 | ||
0 | 0.0095 | 0.3255 | 0.3245 | 0.958 | 0.0096 | 0.3259 | 0.3298 | 0.955 | ||
0 | 0.0071 | 0.1613 | 0.1611 | 0.949 | 0.0046 | 0.1615 | 0.1619 | 0.949 | ||
0 | 0.0055 | 0.3259 | 0.3167 | 0.965 | 0.0056 | 0.3261 | 0.3218 | 0.960 | ||
0.5 | 0.0099 | 0.1688 | 0.1634 | 0.956 | 0.0059 | 0.1689 | 0.1652 | 0.956 | ||
0.5 | 0.0019 | 0.3298 | 0.3357 | 0.951 | 0.0051 | 0.3305 | 0.3404 | 0.945 | ||
−0.5 | 0.0093 | 0.1688 | 0.1718 | 0.946 | 0.0071 | 0.1692 | 0.1718 | 0.945 | ||
0.5 | 0.0004 | 0.3293 | 0.3208 | 0.951 | 0.0031 | 0.3298 | 0.3263 | 0.948 | ||
0 | 0.0069 | 0.1614 | 0.1709 | 0.935 | 0.0045 | 0.1615 | 0.1716 | 0.936 | ||
0.5 | 0.0133 | 0.3295 | 0.3433 | 0.938 | 0.0158 | 0.3298 | 0.3487 | 0.936 | ||
0.5 | 0.0147 | 0.1689 | 0.1738 | 0.950 | 0.0112 | 0.1691 | 0.1767 | 0.946 |
Weibull PH Model | Semi-Parametric PH Model | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
degree = 1 | −0.5 | 0.0022 | 0.5756 | 0.5697 | 0.962 | 0.0354 | 0.6093 | 0.6149 | 0.958 | |
−0.5 | 0.0137 | 0.2844 | 0.2796 | 0.948 | 0.0496 | 0.3090 | 0.2987 | 0.950 | ||
−0.5 | 0.0096 | 0.5559 | 0.5643 | 0.958 | 0.0481 | 0.5923 | 0.6142 | 0.952 | ||
0 | 0.0032 | 0.2707 | 0.2700 | 0.956 | 0.0075 | 0.2933 | 0.2830 | 0.964 | ||
−0.5 | 0.0435 | 0.5471 | 0.5661 | 0.952 | 0.0866 | 0.5852 | 0.6243 | 0.944 | ||
0.5 | 0.0155 | 0.2724 | 0.2656 | 0.954 | 0.0440 | 0.2937 | 0.2878 | 0.962 | ||
0 | 0.0203 | 0.5696 | 0.5649 | 0.960 | 0.0086 | 0.6031 | 0.6104 | 0.952 | ||
−0.5 | 0.0114 | 0.2837 | 0.2929 | 0.944 | 0.0428 | 0.3123 | 0.3122 | 0.940 | ||
0 | 0.0056 | 0.5511 | 0.5436 | 0.958 | 0.0004 | 0.5863 | 0.5916 | 0.948 | ||
0 | 0.0123 | 0.2700 | 0.2753 | 0.952 | 0.0049 | 0.2936 | 0.2961 | 0.960 | ||
0 | 0.0096 | 0.5436 | 0.5245 | 0.960 | 0.0084 | 0.5806 | 0.5714 | 0.962 | ||
0.5 | 0.0242 | 0.2728 | 0.2812 | 0.964 | 0.0573 | 0.2956 | 0.3097 | 0.958 | ||
0.5 | 0.0383 | 0.5731 | 0.6035 | 0.936 | 0.0086 | 0.6096 | 0.6400 | 0.944 | ||
−0.5 | 0.0121 | 0.2831 | 0.2844 | 0.966 | 0.0221 | 0.3074 | 0.3050 | 0.960 | ||
0.5 | 0.0006 | 0.5506 | 0.5546 | 0.950 | 0.0269 | 0.5877 | 0.6049 | 0.944 | ||
0 | 0.0085 | 0.2708 | 0.2638 | 0.958 | 0.0026 | 0.2925 | 0.2775 | 0.968 | ||
0.5 | 0.0218 | 0.5512 | 0.5378 | 0.954 | 0.0724 | 0.5904 | 0.5991 | 0.948 | ||
0.5 | 0.0219 | 0.2735 | 0.2760 | 0.956 | 0.0513 | 0.2966 | 0.3040 | 0.954 | ||
degree = 2 | −0.5 | 0.0484 | 0.5732 | 0.5917 | 0.952 | 0.0735 | 0.6211 | 0.6411 | 0.946 | |
−0.5 | 0.0386 | 0.2848 | 0.2827 | 0.954 | 0.0759 | 0.3399 | 0.3085 | 0.942 | ||
−0.5 | 0.0283 | 0.5526 | 0.5676 | 0.956 | 0.0695 | 0.5998 | 0.6258 | 0.946 | ||
0 | 0.0050 | 0.2715 | 0.2757 | 0.948 | 0.0126 | 0.3158 | 0.2964 | 0.944 | ||
−0.5 | 0.0248 | 0.5468 | 0.5494 | 0.970 | 0.0650 | 0.5994 | 0.6057 | 0.976 | ||
0.5 | 0.0225 | 0.2728 | 0.2620 | 0.974 | 0.0497 | 0.3200 | 0.2876 | 0.974 | ||
0 | 0.0127 | 0.5669 | 0.5739 | 0.960 | 0.0120 | 0.6178 | 0.6121 | 0.954 | ||
−0.5 | 0.0139 | 0.2813 | 0.2847 | 0.950 | 0.0245 | 0.3336 | 0.3087 | 0.944 | ||
0 | 0.0285 | 0.5458 | 0.5416 | 0.964 | 0.0313 | 0.5902 | 0.5855 | 0.966 | ||
0 | 0.0026 | 0.2702 | 0.2815 | 0.954 | 0.0098 | 0.3162 | 0.3003 | 0.956 | ||
0 | 0.0206 | 0.5450 | 0.5471 | 0.944 | 0.0176 | 0.5884 | 0.6037 | 0.946 | ||
0.5 | 0.0293 | 0.2729 | 0.2717 | 0.952 | 0.0592 | 0.3317 | 0.2906 | 0.962 | ||
0.5 | 0.0566 | 0.5789 | 0.5918 | 0.956 | 0.0960 | 0.6406 | 0.6362 | 0.956 | ||
−0.5 | 0.0035 | 0.2852 | 0.3038 | 0.942 | 0.0402 | 0.3485 | 0.3233 | 0.952 | ||
0.5 | 0.0121 | 0.5535 | 0.5423 | 0.964 | 0.0491 | 0.5996 | 0.5908 | 0.958 | ||
0 | 0.0016 | 0.2706 | 0.2913 | 0.936 | 0.0100 | 0.3174 | 0.3096 | 0.936 | ||
0.5 | 0.0691 | 0.5478 | 0.5645 | 0.956 | 0.1089 | 0.6086 | 0.6183 | 0.952 | ||
0.5 | 0.0305 | 0.2725 | 0.2785 | 0.952 | 0.0583 | 0.3370 | 0.3095 | 0.960 | ||
degree = 3 | −0.5 | 0.0005 | 0.5744 | 0.5983 | 0.942 | 0.0275 | 0.6254 | 0.6327 | 0.948 | |
−0.5 | 0.0281 | 0.2843 | 0.3043 | 0.938 | 0.0645 | 0.3296 | 0.3249 | 0.864 | ||
−0.5 | 0.0243 | 0.5497 | 0.5632 | 0.948 | 0.0650 | 0.5998 | 0.6210 | 0.940 | ||
0 | 0.0013 | 0.2714 | 0.2802 | 0.948 | 0.0298 | 0.3381 | 0.3099 | 0.914 | ||
−0.5 | 0.0356 | 0.5584 | 0.5848 | 0.942 | 0.0744 | 0.6160 | 0.6435 | 0.948 | ||
0.5 | 0.0107 | 0.2725 | 0.2775 | 0.934 | 0.0164 | 0.3191 | 0.2932 | 0.884 | ||
0 | 0.0341 | 0.5746 | 0.6104 | 0.932 | 0.0670 | 0.6332 | 0.6644 | 0.926 | ||
−0.5 | 0.0340 | 0.2854 | 0.3001 | 0.942 | 0.0692 | 0.3334 | 0.3262 | 0.876 | ||
0 | 0.0468 | 0.5475 | 0.5622 | 0.962 | 0.0893 | 0.6012 | 0.6137 | 0.956 | ||
0 | 0.0091 | 0.2713 | 0.2693 | 0.962 | 0.0189 | 0.3355 | 0.3009 | 0.918 | ||
0 | 0.0208 | 0.5576 | 0.5458 | 0.962 | 0.0596 | 0.6079 | 0.5873 | 0.960 | ||
0.5 | 0.0092 | 0.2725 | 0.2668 | 0.958 | 0.0198 | 0.3094 | 0.2809 | 0.914 | ||
0.5 | 0.0143 | 0.5727 | 0.5730 | 0.960 | 0.0178 | 0.6301 | 0.6142 | 0.964 | ||
−0.5 | 0.0027 | 0.2831 | 0.2879 | 0.946 | 0.0374 | 0.3252 | 0.3105 | 0.894 | ||
0.5 | 0.0165 | 0.5474 | 0.5586 | 0.956 | 0.0202 | 0.5923 | 0.6116 | 0.964 | ||
0 | 0.0021 | 0.2721 | 0.2896 | 0.936 | 0.0269 | 0.3156 | 0.3190 | 0.878 | ||
0.5 | 0.0234 | 0.5750 | 0.6228 | 0.942 | 0.0243 | 0.6327 | 0.6693 | 0.940 | ||
0.5 | 0.0360 | 0.2851 | 0.3033 | 0.944 | 0.0711 | 0.3465 | 0.3184 | 0.906 |
Weibull PH Model | Semi-Parametric PH Model | |||||||||
---|---|---|---|---|---|---|---|---|---|---|
Bias | ESE | SD | CP | Bias | ESE | SD | CP | |||
degree = 1 | −0.5 | 0.0451 | 0.5006 | 0.4966 | 0.960 | 0.0183 | 0.5508 | 0.5653 | 0.954 | |
−0.5 | 0.0323 | 0.2484 | 0.2578 | 0.928 | 0.0334 | 0.2794 | 0.2922 | 0.942 | ||
−0.5 | 0.0462 | 0.4860 | 0.4362 | 0.974 | 0.0348 | 0.5362 | 0.5162 | 0.962 | ||
0 | 0.0019 | 0.2397 | 0.2291 | 0.966 | 0.0027 | 0.2647 | 0.2625 | 0.952 | ||
−0.5 | 0.0462 | 0.4812 | 0.4395 | 0.966 | 0.0314 | 0.5310 | 0.5094 | 0.970 | ||
0.5 | 0.0427 | 0.2399 | 0.2146 | 0.972 | 0.0389 | 0.2679 | 0.2571 | 0.964 | ||
0 | 0.0207 | 0.4985 | 0.4739 | 0.954 | 0.0271 | 0.5483 | 0.5487 | 0.952 | ||
−0.5 | 0.0302 | 0.2473 | 0.2559 | 0.936 | 0.0434 | 0.2776 | 0.2967 | 0.942 | ||
0 | 0.0265 | 0.4878 | 0.5002 | 0.960 | 0.0273 | 0.5409 | 0.5906 | 0.946 | ||
0 | 0.0251 | 0.2403 | 0.2409 | 0.956 | 0.0299 | 0.2697 | 0.2824 | 0.952 | ||
0 | 0.0172 | 0.4800 | 0.4538 | 0.970 | 0.0283 | 0.5294 | 0.5377 | 0.952 | ||
0.5 | 0.0649 | 0.2404 | 0.2216 | 0.962 | 0.0133 | 0.2683 | 0.2609 | 0.954 | ||
0.5 | 0.0535 | 0.5007 | 0.5003 | 0.954 | 0.0159 | 0.5519 | 0.5806 | 0.948 | ||
−0.5 | 0.0363 | 0.2486 | 0.2494 | 0.946 | 0.0402 | 0.2811 | 0.2853 | 0.948 | ||
0.5 | 0.0468 | 0.4902 | 0.4690 | 0.966 | 0.0324 | 0.5436 | 0.5550 | 0.960 | ||
0 | 0.0017 | 0.2403 | 0.2385 | 0.956 | 0.0061 | 0.2655 | 0.2672 | 0.958 | ||
0.5 | 0.0675 | 0.4826 | 0.4342 | 0.970 | 0.0027 | 0.5301 | 0.4968 | 0.966 | ||
0.5 | 0.0514 | 0.2404 | 0.2176 | 0.968 | 0.0247 | 0.2680 | 0.2574 | 0.962 | ||
degree = 2 | −0.5 | 0.0086 | 0.5037 | 0.5100 | 0.956 | 0.0616 | 0.5580 | 0.5918 | 0.950 | |
−0.5 | 0.0397 | 0.2489 | 0.2369 | 0.962 | 0.0324 | 0.2847 | 0.2748 | 0.970 | ||
−0.5 | 0.0715 | 0.4850 | 0.4107 | 0.984 | 0.0077 | 0.5370 | 0.4818 | 0.974 | ||
0 | 0.0023 | 0.2395 | 0.2349 | 0.942 | 0.0018 | 0.2683 | 0.2708 | 0.942 | ||
−0.5 | 0.0872 | 0.4841 | 0.4895 | 0.950 | 0.0129 | 0.5369 | 0.5775 | 0.946 | ||
0.5 | 0.0628 | 0.2411 | 0.2208 | 0.958 | 0.0166 | 0.2722 | 0.2673 | 0.956 | ||
0 | 0.0030 | 0.4977 | 0.5126 | 0.946 | 0.0026 | 0.5517 | 0.5945 | 0.938 | ||
−0.5 | 0.0223 | 0.2486 | 0.2436 | 0.952 | 0.0500 | 0.2859 | 0.2845 | 0.952 | ||
0 | 0.0311 | 0.4885 | 0.4641 | 0.968 | 0.0385 | 0.5417 | 0.5457 | 0.954 | ||
0 | 0.0005 | 0.2408 | 0.2351 | 0.958 | 0.0010 | 0.2696 | 0.2697 | 0.956 | ||
0 | 0.0076 | 0.4813 | 0.4343 | 0.974 | 0.0154 | 0.5318 | 0.5157 | 0.962 | ||
0.5 | 0.0671 | 0.2411 | 0.2297 | 0.944 | 0.0122 | 0.2715 | 0.2733 | 0.944 | ||
0.5 | 0.0177 | 0.5027 | 0.4798 | 0.974 | 0.0696 | 0.5572 | 0.5592 | 0.964 | ||
−0.5 | 0.0485 | 0.2483 | 0.2382 | 0.966 | 0.0260 | 0.2848 | 0.2733 | 0.964 | ||
0.5 | 0.0369 | 0.4912 | 0.4734 | 0.960 | 0.0481 | 0.5459 | 0.5538 | 0.958 | ||
0 | 0.0156 | 0.2405 | 0.2404 | 0.960 | 0.0135 | 0.2689 | 0.2727 | 0.956 | ||
0.5 | 0.0353 | 0.4841 | 0.4587 | 0.950 | 0.0467 | 0.5390 | 0.5336 | 0.948 | ||
0.5 | 0.0527 | 0.2414 | 0.2294 | 0.950 | 0.0295 | 0.2742 | 0.2743 | 0.958 | ||
degree = 3 | −0.5 | 0.0045 | 0.5026 | 0.4690 | 0.974 | 0.0697 | 0.5629 | 0.5497 | 0.960 | |
−0.5 | 0.0227 | 0.2486 | 0.2355 | 0.964 | 0.0485 | 0.2920 | 0.2750 | 0.976 | ||
−0.5 | 0.0234 | 0.4889 | 0.4452 | 0.968 | 0.0674 | 0.5461 | 0.5250 | 0.954 | ||
0 | 0.0063 | 0.2401 | 0.2314 | 0.958 | 0.0123 | 0.2723 | 0.2673 | 0.956 | ||
−0.5 | 0.0416 | 0.4822 | 0.4445 | 0.976 | 0.0284 | 0.5341 | 0.5132 | 0.974 | ||
0.5 | 0.0690 | 0.2399 | 0.2276 | 0.948 | 0.0079 | 0.2736 | 0.2692 | 0.962 | ||
0 | 0.0061 | 0.4965 | 0.4888 | 0.966 | 0.0093 | 0.5557 | 0.5721 | 0.956 | ||
−0.5 | 0.0292 | 0.2479 | 0.2471 | 0.950 | 0.0438 | 0.2931 | 0.2825 | 0.958 | ||
0 | 0.0063 | 0.4861 | 0.4607 | 0.964 | 0.0090 | 0.5430 | 0.5525 | 0.956 | ||
0 | 0.0032 | 0.2400 | 0.2282 | 0.962 | 0.0059 | 0.2747 | 0.2662 | 0.958 | ||
0 | 0.0254 | 0.4795 | 0.4485 | 0.970 | 0.0324 | 0.5313 | 0.5234 | 0.968 | ||
0.5 | 0.0530 | 0.2405 | 0.2332 | 0.946 | 0.0269 | 0.2751 | 0.2757 | 0.968 | ||
0.5 | 0.0030 | 0.5040 | 0.4761 | 0.982 | 0.0702 | 0.5648 | 0.5647 | 0.974 | ||
−0.5 | 0.0297 | 0.2494 | 0.2517 | 0.956 | 0.0464 | 0.2917 | 0.2920 | 0.954 | ||
0.5 | 0.0579 | 0.4879 | 0.4614 | 0.972 | 0.0163 | 0.5414 | 0.5352 | 0.960 | ||
0 | 0.0186 | 0.2396 | 0.2390 | 0.962 | 0.0201 | 0.2713 | 0.2684 | 0.962 | ||
0.5 | 0.0741 | 0.4833 | 0.4801 | 0.956 | 0.0000 | 0.5361 | 0.5590 | 0.950 | ||
0.5 | 0.0494 | 0.2409 | 0.2098 | 0.962 | 0.0277 | 0.2778 | 0.2562 | 0.974 |
Weibull PH Model | Semi-Parametric PH Model | ||||
---|---|---|---|---|---|
AIC | BIC | AIC | BIC | ||
degree = 1 | 158.0671 | 171.2604 | 162.5652 | 188.9517 | |
159.3115 | 172.5048 | 164.0589 | 190.4455 | ||
153.2519 | 166.4451 | 158.3429 | 184.7294 | ||
158.5114 | 171.7047 | 163.0422 | 189.4287 | ||
159.2377 | 172.4310 | 164.0106 | 190.3972 | ||
153.1766 | 166.3699 | 158.3069 | 184.6934 | ||
159.3043 | 172.4976 | 164.0052 | 190.3918 | ||
158.5679 | 171.7612 | 163.3746 | 189.7611 | ||
153.1342 | 166.3275 | 158.1237 | 184.5102 | ||
degree = 2 | 157.5231 | 170.7164 | 163.7349 | 193.4198 | |
158.5837 | 171.7769 | 165.0200 | 194.7048 | ||
153.4395 | 166.6327 | 160.0308 | 189.7156 | ||
159.7065 | 172.8997 | 165.9552 | 195.6400 | ||
158.5474 | 171.7407 | 164.8709 | 194.5557 | ||
153.4417 | 166.6350 | 159.9141 | 189.5989 | ||
157.3260 | 170.5192 | 163.4532 | 193.1381 | ||
159.0321 | 172.2254 | 165.3613 | 195.0462 | ||
153.7928 | 166.9861 | 160.1865 | 189.8714 | ||
degree = 3 | 158.0656 | 171.2589 | 166.2319 | 199.2151 | |
155.7645 | 168.9577 | 164.1663 | 197.1495 | ||
157.2065 | 170.3998 | 165.3615 | 198.3447 | ||
157.1589 | 170.3521 | 165.2879 | 198.2711 | ||
154.8845 | 168.0777 | 163.3521 | 196.3353 | ||
157.3314 | 170.5247 | 165.5741 | 198.5573 | ||
158.9620 | 172.1552 | 167.2433 | 200.2265 | ||
153.8931 | 167.0864 | 162.4024 | 195.3856 | ||
157.6660 | 170.8592 | 165.8563 | 198.8395 |
Weibull PH Model | Semi-Parametric PH Model | ||||
---|---|---|---|---|---|
AIC | BIC | AIC | BIC | ||
degree = 1 | 178.4425 | 191.6358 | 180.6047 | 206.9913 | |
176.8683 | 190.0616 | 179.8233 | 206.2098 | ||
171.7429 | 184.9362 | 174.7805 | 201.1671 | ||
180.2393 | 193.4325 | 182.3858 | 208.7723 | ||
176.2032 | 189.3964 | 180.6287 | 210.3136 | ||
171.2697 | 184.4630 | 174.6974 | 201.0839 | ||
180.0552 | 193.2485 | 182.0399 | 208.4264 | ||
176.5421 | 189.7354 | 179.3505 | 205.7371 | ||
170.9358 | 184.1291 | 174.3874 | 200.7740 | ||
degree = 2 | 177.6850 | 190.8783 | 181.5645 | 211.2493 | |
177.4778 | 190.6711 | 181.9235 | 211.6084 | ||
170.4353 | 183.6285 | 175.1660 | 204.8509 | ||
178.6891 | 191.8824 | 182.5802 | 212.2651 | ||
175.4079 | 188.6011 | 179.8539 | 209.5387 | ||
169.7078 | 182.9011 | 174.7290 | 204.4138 | ||
179.2848 | 192.4781 | 182.8814 | 212.5662 | ||
175.9501 | 189.1434 | 180.4436 | 210.1285 | ||
169.5194 | 182.7126 | 174.4124 | 204.0973 | ||
degree = 3 | 179.3540 | 192.5473 | 184.6973 | 217.6805 | |
176.9312 | 190.1245 | 182.8466 | 215.8298 | ||
171.5355 | 184.7287 | 178.6081 | 211.5913 | ||
179.2683 | 192.4616 | 184.9653 | 217.9485 | ||
177.3079 | 190.5011 | 183.3511 | 216.3343 | ||
170.1874 | 183.3807 | 177.2666 | 210.2498 | ||
178.5908 | 191.7841 | 183.9859 | 216.9691 | ||
177.4439 | 190.6372 | 183.7068 | 216.6900 | ||
170.5001 | 183.6934 | 177.4353 | 210.4184 |
Variable | Description |
---|---|
Environment, GE = 1, CE = 0 | |
Presence of tumor at death time, yes = 1, no = 0 | |
The time of death |
AIC | BIC | |||
---|---|---|---|---|
Weibull PH model | 0.803 | 0.061 | 167 | 176 |
semi-parametric PH model (1) | 0.805 | 0.144 | 174 | 195 |
semi-parametric PH model (2) | 0.843 | 0.148 | 177 | 200 |
semi-parametric PH model (3) | 0.818 | 0.157 | 179 | 205 |
Variable | Description |
---|---|
Gender, male = 1, female = 0 | |
Calcified or not, calcified = 1, uncalcified = 0 | |
Examination time |
AIC | BIC | |||
---|---|---|---|---|
Weibull PH model | −0.269 | 0.097 | 287 | 299 |
semi-parametric PH model (1) | −0.251 | 0.130 | 295 | 323 |
semi-parametric PH model (2) | −0.254 | 0.108 | 297 | 328 |
semi-parametric PH model (3) | −0.256 | 0.210 | 299 | 334 |
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Chen, S.; Yang, F. Expectation-Maximization Algorithm for the Weibull Proportional Hazard Model under Current Status Data. Mathematics 2023, 11, 4826. https://doi.org/10.3390/math11234826
Chen S, Yang F. Expectation-Maximization Algorithm for the Weibull Proportional Hazard Model under Current Status Data. Mathematics. 2023; 11(23):4826. https://doi.org/10.3390/math11234826
Chicago/Turabian StyleChen, Sisi, and Fengkai Yang. 2023. "Expectation-Maximization Algorithm for the Weibull Proportional Hazard Model under Current Status Data" Mathematics 11, no. 23: 4826. https://doi.org/10.3390/math11234826
APA StyleChen, S., & Yang, F. (2023). Expectation-Maximization Algorithm for the Weibull Proportional Hazard Model under Current Status Data. Mathematics, 11(23), 4826. https://doi.org/10.3390/math11234826