In this simulation study, the finite sample properties of FIP W∗ (t, u) and the per-formance of inference procedure are investigated as terminal events arise. For the design of mixture rates of censoring and death, C is independently generated from an exponential distribution with parameter a and the terminal time D is designed to follow an exponential distribution with rate bI(X0 ≤ 1, Y0 ≤ 0.5) + b. The pa-rameters (a, b) are set to be (0.5, 0.01) and (0.6, 0.3) so that the mixture rates of 30% and 50% are achieved in the simulated data.
Tables 4.9-4.10 display the averages and standard deviations of 500 estimates, and the averages of 500 standard errors based on (3.1.2) at the considered points.
The averages of 500 estimates generally close to the true values of F∗(t, u)’s. The biases are apparently reduced when the sample size is large or the mixture rate is small. Moreover, the variation of estimator will decrease and the accuracy of estimated variances will be improved as the sample size increases or the mixture rate decreases. Table 4.11 exhibits the empirical coverage probabilities of 0.95 pointwise confidence intervals for F∗(t, u). The probabilities are generally around the nominal
level of 0.95. It is revealed in these tables that the closeness of empirical coverage probabilities to the nominal level relies on the sample size and the censoring rate.
Table 4.1: The averages and the standard deviations (SD) of 500 estimates FHL(t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 30%
n 200 400
t u F (t, u) Mean SD SE Mean SD SE
0.2231 0.25 0.158 0.156 0.0279 0.0269 0.156 0.0182 0.0190 0.5108 0.25 0.231 0.234 0.0320 0.0345 0.232 0.0227 0.0237 0.9163 0.25 0.247 0.249 0.0328 0.0395 0.248 0.0237 0.0278 1.6094 0.25 0.250 0.251 0.0331 0.0564 0.250 0.0238 0.0394 0.2231 0.50 0.196 0.194 0.0307 0.0295 0.193 0.0195 0.0209 0.5108 0.50 0.369 0.371 0.0374 0.0397 0.369 0.0262 0.0282 0.9163 0.50 0.469 0.469 0.0379 0.0512 0.470 0.0278 0.0362 1.6094 0.50 0.496 0.496 0.0385 0.0914 0.497 0.0281 0.0638 0.2231 0.75 0.200 0.197 0.0310 0.0297 0.197 0.0198 0.0211 0.5108 0.75 0.397 0.399 0.0374 0.0404 0.398 0.0261 0.0286 0.9163 0.75 0.581 0.579 0.0376 0.0477 0.580 0.0273 0.0338 1.6094 0.75 0.708 0.703 0.0400 0.0865 0.708 0.0280 0.0607 0.2231 1.0 0.200 0.198 0.0311 0.0298 0.197 0.0198 0.0211 0.5108 1.0 0.400 0.402 0.0376 0.0405 0.400 0.0263 0.0287 0.9163 1.0 0.600 0.598 0.0374 0.0461 0.598 0.0277 0.0327 1.6094 1.0 0.800 0.795 0.0357 0.0458 0.796 0.0265 0.0324
Table 4.2: The averages and the standard deviations (SD) of 500 estimates FIP W(t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 30%
n 200 400
t u F (t, u) Mean SD SE Mean SD SE
0.2231 0.25 0.158 0.158 0.0279 0.0265 0.158 0.0183 0.0187 0.5108 0.25 0.231 0.234 0.0321 0.0318 0.234 0.0227 0.0224 0.9163 0.25 0.247 0.250 0.0329 0.0334 0.250 0.0238 0.0236 1.6094 0.25 0.250 0.252 0.0332 0.0345 0.252 0.0239 0.0243 0.2231 0.50 0.196 0.196 0.0310 0.0289 0.196 0.0195 0.0204 0.5108 0.50 0.369 0.373 0.0375 0.0371 0.371 0.0263 0.0262 0.9163 0.50 0.469 0.471 0.0379 0.0409 0.471 0.0279 0.0289 1.6094 0.50 0.496 0.497 0.0386 0.0440 0.498 0.0281 0.0311 0.2231 0.75 0.200 0.200 0.0313 0.0292 0.200 0.0197 0.0206 0.5108 0.75 0.397 0.401 0.0374 0.0377 0.399 0.0262 0.0267 0.9163 0.75 0.581 0.583 0.0373 0.0416 0.584 0.0275 0.0294 1.6094 0.75 0.708 0.705 0.0400 0.0441 0.711 0.0279 0.0311 0.2231 1.0 0.200 0.200 0.0315 0.0292 0.200 0.0198 0.0206 0.5108 1.0 0.400 0.403 0.0377 0.0378 0.402 0.0264 0.0267 0.9163 1.0 0.600 0.602 0.0372 0.0415 0.602 0.0278 0.0294 1.6094 1.0 0.800 0.799 0.0358 0.0406 0.800 0.0264 0.0287
Table 4.3: The averages and standard deviations (SD) of 500 estimates FIM(t, u) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 30%
n 200 400
t u F (t, u) Mean SD Mean SD
0.2231 0.25 0.158 0.158 0.0279 0.158 0.0183 0.5108 0.25 0.231 0.234 0.0321 0.232 0.0227 0.9163 0.25 0.247 0.249 0.0328 0.248 0.0238 1.6094 0.25 0.250 0.251 0.0330 0.250 0.0238 0.2231 0.50 0.196 0.196 0.0310 0.196 0.0195 0.5108 0.50 0.369 0.372 0.0374 0.370 0.0263 0.9163 0.50 0.469 0.470 0.0378 0.471 0.0278 1.6094 0.50 0.496 0.495 0.0384 0.497 0.0280 0.2231 0.75 0.200 0.200 0.0313 0.200 0.0197 0.5108 0.75 0.397 0.400 0.0374 0.399 0.0262 0.9163 0.75 0.581 0.582 0.0373 0.583 0.0274 1.6094 0.75 0.708 0.702 0.0399 0.709 0.0279 0.2231 1.0 0.200 0.200 0.0314 0.200 0.0198 0.5108 1.0 0.400 0.403 0.0376 0.402 0.0264 0.9163 1.0 0.600 0.601 0.0372 0.602 0.0278 1.6094 1.0 0.800 0.795 0.0356 0.798 0.0264
Table 4.4: The averages and standard deviations (SD) of 500 estimates FHL(t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 50%
n 200 400
t u F (t, u) Mean SD SE Mean SD SE
0.2231 0.25 0.158 0.154 0.0263 0.0273 0.156 0.0192 0.0196 0.5108 0.25 0.231 0.230 0.0323 0.0349 0.230 0.0228 0.0248 0.9163 0.25 0.247 0.246 0.0344 0.0427 0.246 0.0244 0.0304 1.6094 0.25 0.250 0.248 0.0348 0.0696 0.249 0.0248 0.0488 0.2231 0.5 0.196 0.192 0.0298 0.0302 0.192 0.0210 0.0215 0.5108 0.5 0.369 0.367 0.0398 0.0417 0.366 0.0271 0.0296 0.9163 0.5 0.469 0.466 0.0431 0.0565 0.467 0.0306 0.0400 1.6094 0.5 0.496 0.494 0.0456 0.1181 0.494 0.0324 0.0821 0.2231 0.75 0.200 0.196 0.0298 0.0304 0.196 0.0212 0.0217 0.5108 0.75 0.397 0.395 0.0417 0.0423 0.394 0.0272 0.0300 0.9163 0.75 0.581 0.577 0.0457 0.0518 0.576 0.0307 0.0369 1.6094 0.75 0.708 0.703 0.0508 0.1123 0.703 0.0362 0.0790 0.2231 1.0 0.200 0.196 0.0299 0.0305 0.196 0.0213 0.0217 0.5108 1.0 0.400 0.398 0.0418 0.0423 0.396 0.0273 0.0300 0.9163 1.0 0.600 0.595 0.0458 0.0498 0.594 0.0310 0.0353 1.6094 1.0 0.800 0.794 0.0492 0.0537 0.793 0.0349 0.0382
Table 4.5: The averages and standard deviations (SD) of 500 estimates FIP W(t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 50%
n 200 400
t u F (t, u) Mean SD SE Mean SD SE
0.2231 0.25 0.158 0.157 0.0264 0.0274 0.158 0.0193 0.0193 0.5108 0.25 0.231 0.231 0.0325 0.0340 0.232 0.0229 0.0239 0.9163 0.25 0.247 0.247 0.0346 0.0371 0.248 0.0245 0.0260 1.6094 0.25 0.250 0.249 0.0350 0.0405 0.250 0.0249 0.0284 0.2231 0.5 0.196 0.195 0.0300 0.0300 0.195 0.0212 0.0211 0.5108 0.5 0.369 0.369 0.0400 0.0405 0.368 0.0273 0.0286 0.9163 0.5 0.469 0.469 0.0434 0.0481 0.470 0.0307 0.0340 1.6094 0.5 0.496 0.496 0.0458 0.0576 0.497 0.0325 0.0405 0.2231 0.75 0.200 0.199 0.0301 0.0302 0.199 0.0214 0.0213 0.5108 0.75 0.397 0.398 0.0419 0.0414 0.396 0.0273 0.0292 0.9163 0.75 0.581 0.581 0.0461 0.0503 0.581 0.0308 0.0355 1.6094 0.75 0.708 0.707 0.0512 0.0626 0.707 0.0365 0.0443 0.2231 1.0 0.200 0.200 0.0301 0.0302 0.199 0.0214 0.0213 0.5108 1.0 0.400 0.400 0.0419 0.0415 0.399 0.0274 0.0293 0.9163 1.0 0.600 0.600 0.0461 0.0503 0.600 0.0313 0.0356 1.6094 1.0 0.800 0.800 0.0494 0.0598 0.800 0.0349 0.0425
Table 4.6: The averages and standard deviations (SD) of 500 estimates FIM(t, u) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rate of 50%
n 200 400
t u F (t, u) Mean SD Mean SD
0.2231 0.25 0.158 0.157 0.0264 0.158 0.0193 0.5108 0.25 0.231 0.231 0.0324 0.231 0.0229 0.9163 0.25 0.247 0.245 0.0344 0.247 0.0244 1.6094 0.25 0.250 0.244 0.0345 0.247 0.0247 0.2231 0.5 0.196 0.195 0.0300 0.195 0.0211 0.5108 0.5 0.369 0.368 0.0399 0.368 0.0273 0.9163 0.5 0.469 0.467 0.0432 0.469 0.0306 1.6094 0.5 0.496 0.486 0.0450 0.492 0.0323 0.2231 0.75 0.200 0.199 0.0300 0.199 0.0213 0.5108 0.75 0.397 0.397 0.0418 0.396 0.0273 0.9163 0.75 0.581 0.578 0.0459 0.580 0.0308 1.6094 0.75 0.708 0.694 0.0507 0.701 0.0365 0.2231 1.0 0.200 0.199 0.0301 0.199 0.0214 0.5108 1.0 0.400 0.399 0.0419 0.398 0.0274 0.9163 1.0 0.600 0.597 0.0459 0.598 0.0312 1.6094 1.0 0.800 0.786 0.0490 0.793 0.0349
Table 4.7: The empirical coverage probabilities of FHL(t, u) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rates (c.r.) of 30% and 50%
c.r. 30% 50%
n 200 400 200 400
t u
0.2231 0.25 0.952 0.948 0.956 0.934 0.5108 0.25 0.958 0.978 0.964 0.948 0.9163 0.25 0.970 0.984 0.984 0.990 1.6094 0.25 0.990 1.000 0.998 1.000 0.2231 0.50 0.952 0.952 0.960 0.958 0.5108 0.50 0.976 0.966 0.972 0.966 0.9163 0.50 0.986 0.990 0.990 0.994 1.6094 0.50 1.000 1.000 1.000 1.000 0.2231 0.75 0.952 0.954 0.950 0.960 0.5108 0.75 0.968 0.972 0.954 0.958 0.9163 0.75 0.986 0.988 0.974 0.974 1.6094 0.75 1.000 1.000 0.998 1.000 0.2231 1.0 0.952 0.954 0.950 0.960 0.5108 1.0 0.968 0.970 0.958 0.952 0.9163 1.0 0.982 0.984 0.972 0.962 1.6094 1.0 0.994 0.976 0.970 0.968
Table 4.8: The empirical coverage probabilities of FIP W(t, u) at the selected points with the sample sizes (n) of 200 and 400, and the censoring rates (c.r.) of 30% and 50%
c.r. 30% 50%
n 200 400 200 400
t u
0.2231 0.25 0.946 0.952 0.954 0.946 0.5108 0.25 0.968 0.934 0.952 0.942 0.9163 0.25 0.976 0.934 0.966 0.950 1.6094 0.25 0.978 0.944 0.974 0.976 0.2231 0.50 0.962 0.948 0.950 0.964 0.5108 0.50 0.960 0.950 0.970 0.966 0.9163 0.50 0.960 0.942 0.962 0.964 1.6094 0.50 0.970 0.950 0.988 0.982 0.2231 0.75 0.954 0.948 0.944 0.956 0.5108 0.75 0.966 0.940 0.958 0.962 0.9163 0.75 0.956 0.954 0.956 0.966 1.6094 0.75 0.970 0.964 0.976 0.984 0.2231 1.0 0.954 0.948 0.946 0.958 0.5108 1.0 0.970 0.936 0.956 0.966 0.9163 1.0 0.968 0.956 0.968 0.962 1.6094 1.0 0.964 0.964 0.968 0.978
Table 4.9: The averages and standard deviations (SD) of 500 estimates FIP W∗ (t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the mixture rate of 30%
n 200 400
t u F∗(t, u) Mean SD SE Mean SD SE
0.2231 0.25 0.159 0.158 0.0282 0.0288 0.160 0.0196 0.0204 0.5108 0.25 0.233 0.232 0.0326 0.0351 0.235 0.0239 0.0248 0.9163 0.25 0.249 0.247 0.0337 0.0364 0.250 0.0246 0.0257 1.6094 0.25 0.252 0.250 0.0340 0.0366 0.252 0.0248 0.0259 0.2231 0.50 0.198 0.197 0.0316 0.0321 0.198 0.0219 0.0226 0.5108 0.50 0.372 0.370 0.0418 0.0439 0.373 0.0311 0.0309 0.9163 0.50 0.472 0.470 0.0474 0.0488 0.473 0.0336 0.0344 1.6094 0.50 0.498 0.497 0.0498 0.0499 0.499 0.0341 0.0351 0.2231 0.75 0.202 0.200 0.0318 0.0324 0.201 0.0220 0.0229 0.5108 0.75 0.400 0.398 0.0424 0.0453 0.401 0.0327 0.0320 0.9163 0.75 0.584 0.582 0.0503 0.0528 0.586 0.0367 0.0372 1.6094 0.75 0.711 0.708 0.0538 0.0545 0.713 0.0385 0.0384 0.2231 1.0 0.202 0.201 0.0318 0.0324 0.202 0.0219 0.0229 0.5108 1.0 0.403 0.400 0.0425 0.0455 0.404 0.0328 0.0321 0.9163 1.0 0.603 0.600 0.0504 0.0533 0.604 0.0373 0.0375 1.6094 1.0 0.803 0.799 0.0530 0.0539 0.805 0.0385 0.0380
Table 4.10: The averages and the standard deviations (SD) of 500 estimates FIP W∗ (t, u) and the averages of 500 standard errors (SE) at the selected points with the sample sizes (n) of 200 and 400, and the mixture rate of 50%
n 200 400
t u F∗(t, u) Mean SD SE Mean SD SE
0.2231 0.25 0.204 0.205 0.0348 0.0374 0.203 0.0256 0.0265 0.5108 0.25 0.286 0.287 0.0420 0.0442 0.285 0.0299 0.0314 0.9163 0.25 0.301 0.301 0.0439 0.0453 0.299 0.0308 0.0323 1.6094 0.25 0.303 0.303 0.0443 0.0454 0.301 0.0311 0.0324 0.2231 0.50 0.252 0.254 0.0385 0.0414 0.250 0.0285 0.0293 0.5108 0.50 0.444 0.448 0.0514 0.0534 0.441 0.0374 0.0382 0.9163 0.50 0.535 0.538 0.0567 0.0569 0.533 0.0404 0.0411 1.6094 0.50 0.559 0.561 0.0570 0.0575 0.558 0.0396 0.0416 0.2231 0.75 0.257 0.259 0.0387 0.0418 0.256 0.0289 0.0296 0.5108 0.75 0.478 0.481 0.0521 0.0549 0.476 0.0383 0.0394 0.9163 0.75 0.662 0.664 0.0592 0.0596 0.661 0.0417 0.0435 1.6094 0.75 0.782 0.783 0.0605 0.0575 0.779 0.0425 0.0428 0.2231 1.0 0.257 0.259 0.0388 0.0418 0.256 0.0289 0.0297 0.5108 1.0 0.482 0.485 0.0523 0.0550 0.479 0.0384 0.0395 0.9163 1.0 0.683 0.686 0.0592 0.0598 0.682 0.0417 0.0437 1.6094 1.0 0.870 0.873 0.0573 0.0526 0.869 0.0405 0.0404
Table 4.11: The empirical coverage probabilities of FIP W∗ (t, u) at the selected points with two sample sizes (n) of 200 and 400, and the mixture rates of censoring and death (m.r.) of 30% and 50%
m.r. 30% 50%
n 200 400 200 400
t u
0.2231 0.25 0.948 0.950 0.964 0.948 0.5108 0.25 0.960 0.956 0.966 0.960 0.9163 0.25 0.970 0.958 0.954 0.958 1.6094 0.25 0.962 0.954 0.956 0.958 0.2231 0.50 0.950 0.960 0.956 0.950 0.5108 0.50 0.962 0.944 0.944 0.962 0.9163 0.50 0.958 0.954 0.946 0.952 1.6094 0.50 0.958 0.964 0.936 0.962 0.2231 0.75 0.944 0.956 0.956 0.958 0.5108 0.75 0.966 0.938 0.958 0.958 0.9163 0.75 0.956 0.944 0.944 0.956 1.6094 0.75 0.948 0.942 0.904 0.960 0.2231 1.0 0.944 0.960 0.956 0.958 0.5108 1.0 0.962 0.936 0.954 0.954 0.9163 1.0 0.952 0.946 0.942 0.958 1.6094 1.0 0.944 0.940 0.864 0.944
Chapter 5
Application to Colorectal Cancer Data
The used colorectal cancer data arise from the SEER-Medicare database. A total of 71,519 patients with the SEER registries were systematically recruited since January 1, 1983. The repeated medicare reimbursements (dollars) and the corresponding times (months) were recorded between January 1, 1983 and August 31, 1993. Here, we apply our proposed methods to estimate the distribution of first pair of medicare reimbursement and claiming time of patients. The baseline covariates age and cancer stage are considered in our analysis. Moreover, the time to colorectal cancer-related death and last follow-up are included. The stage variable is the American Joint Committee on Cancer (AJC) clinical stage of disease, which ranges from 0 to 4 according to the severity of disease. The age variable is further categorizd into three layers (61-70, 71-80, and >80). More detailed explorations of data can be found in Bang (2005).
In this chapter, a random sample of size about 2000 is selected and analyzed. The
range of patients’ age in this sample is mainly from 65 to 103 years old. The features of sub-sample table 5.1 show the representative of whole data. The aim of our study is to estimate the joint distributions of claiming time and medicare reimbursement under different age layers and clinical stages of disease. Moreover, the mean medicare reimbursement and the probabilities of claiming time occurring before the death time are evaluated. Evidenced by the numerical studies, the low mixture rate of censoring and death (< 2%) in this sample will ensure the accuracy and precision of estimated distributions and related quantities. The results summarized in table 5.2 indicate that patients with older age or more severe disease stage tends to receive larger reimbursements from medicare. It is further detected that the greatest costs occur in the age layer of 71-80 and the disease stage 3. Those patients with older age or more severe disease stage are prone to claim reimbursements prior to death.
The reason might be that older or less healthy patients tend to be ailing and raise medical expenditure. In table 5.3, the probabilities of claiming time prior to death are generally high, especially in the groups of older age and more severe disease stage. Patients with disease stage 4 receive the greatest medicare reimbursements in the age layer of 61-70, while the greatest reimbursements for patients with age more than 70 occur in the disease stage 3.
The patterns of joint distributions in various age layers and disease stages are displayed in figures 5.1-5.2. The marginal distribution of claiming time and reim-bursement for patients with the first reimreim-bursement prior to death are also presented in figures 5.3-5.4. Figure 5.3 reveals that the claiming times of patients with age
Table 5.1: The characteristics of the colorectal cancer data and subsample
Full data Sub-sample
Male 51.5% 52.5%
Female 48.5% 47.5%
Age 61-70 23.0% 22.1%
71-80 47.4% 46.7%
> 80 29.6% 31.2%
Stage 0 6.7% 7.8%
1 22.6% 22.5%
2 31.0% 30.4%
3 22.5% 23.7%
4 17.1% 15.8%
more than eighty are shorter than those for younger patients. As for the reim-bursements, patients with age more than seventy receive more reimbursements than younger patients. In figure 5.4, no apparent difference between the estimated curves of claiming time is detected for various disease stages. In contrast, patients in dis-ease stage 0 reasonably incur less reimbursements than those in the more severe stages.
Table 5.2: The estimates of P (D > Xo) and E(Yo) under different age layers and
Table 5.3: The estimates of P (D > Xo) and E(Yo) under different age-disease stage groups
t
Figure 5.1: The joint distributions of claiming time and reimbursement for different age layers and disease stages
t
Figure 5.2: The joint distributions of claiming time and reimbursement for different age layers and disease stages
0 20 40 60 80 100
0.850.900.951.00
Distributions of Claiming Time
Time
Probability
10000 20000 30000 40000 50000
0.750.850.95
Distributions of Medicare Reimbursement
Dollars
Probability
Figure 5.3: The estimates of P (Xo ≤ t|D > Xo) and P (Yo ≤ u|D > Xo) for patients with age layers of 61-70(solid line), 71-80(dashed line) and > 80(dotted line)
0 20 40 60 80 100
0.800.850.900.951.00
Distributions of Claiming Time
Time
Probability
10000 20000 30000 40000 50000
0.700.800.901.00
Distributions of Medicare Reimbursement
Dollars
Probability
Figure 5.4: The estimates of P (Xo ≤ t|D > Xo) and P (Yo ≤ u|D > Xo) for patients with disease stages of 0(solid line), 1(dashed line), 2(dotted line), 3(dotted-dash line) and 4(long-3(dotted-dashed line)
Chapter 6 Discussion
In this thesis, we propose several estimators for the joint distribution function of claiming time and medicare reimbursement based on two types of cost data. The limiting Gaussian processes of the estimators are also developed with the uniformly consistent estimators of the asymptotic variance-covariances. Without the occur-rence of a terminal event, our numerical studies reveal that the IPW estimator surpasses the Huang-Louis and imputation estimators in computation cost. More-over, the IPW estimation has more accurate estimator of the variance-covariance than the Huang Louis estimation. The performance of inference procedures are shown to be satisfactory.
In our application, an appropriate regression model would be helpful to investi-gate the influences of ages and disease stages on the joint distribution of claiming time and medicare reimbursement. The nonparametric IPW estimation approach will be reasonably extended to the estimation of parameters in the considered re-gression model. To solve the problem of asymmetric information or moral hazard in
health economics, our further research will focus on seeking for the optimal compos-ite factors to minimize the medical cost conditioning on the claiming time of interest.
It is expected to help insurance companies to discriminate crafty policyholders.
As in the analyzed data, the claiming times and medicare reimbursements are intermittently occurring during the study period. Under the assumption that the recurrent pairs are independent and identically distributed conditioning on a la-tent variable, the estimation method of Huang and Wang (2005) can be applied to estimate the joint distribution of claiming time and medicare reimbursement. In biomedical contexts, this assumption seems to be out of reality. In our further study, we try to extend the developed methods to this issue with more suitable conditions being made.
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