Chapter 5 Simulations
5.2 Simulation Results
5.2.3 Results with Covariates
We have proposed to extend two inference approaches to a more complex situation that covariates affect the marginal distribution. Now we check the validity of the extension by simulations. Here we assume the Cox Proportional Hazard model to describe marginal heterogeneity. For the two-stage estimation approach, we only report the results that the marginal distributions are estimated non-parametrically. The parameter of α ranges from
2 .
1 to 19 and β1 =β2=0.8. We also evaluate the situation in presence of censoring with 30
% and 60 % censoring. To achieve the targeted censoring rates, we let C follow i )
5 . 3
exp(λ = and exp(λ =1.5), respectively (i=1,2). Two sample sizes with n=100 and
=500
n are evaluated. For each estimator, the average bias and standard deviation are reported based on 500 replications.
Table 2.1 summaries the result with covariates in absence of censoring. Our focus is on comparing the two methods after adjustment for the effects of β1 andβ2. The variation of the two approaches is close. However, the two-by-two table approach seems to produce less biased estimates. Table 2.2 and 2.3 are the results in presence of right censoring. The estimators of α have larger variation but still perform well. All of the estimators are consistent when the sample size increases. In the simulations not reported here, we find that the estimators of α become invalid if the marginal heterogeneity is ignored.
Chapter 6 Conclusion
In the thesis, we review three inference approaches for estimating the association parameter for copula models. The existing methods are originally developed for analyzing homogeneous data. Here we extend these methods to account for marginal heterogeneity explained by covariates.
The two-stage estimation procedure proposed by Shih and Louis (1995) is easy to implement but not applicable under more complicated data structures such as semi-competing risks data that involves dependent censoring. The proposed approach based on two-by-two tables is motivated by the Log-Rank statistics. In comparison, it is a simple procedure from both aspects of analytic derivations and computation. It also has nice performance in simulations. Since this approach only utilizes some moment conditions, it can be easily modified for different data structures. The estimator of Hsu and Prentice (1996) has poor performance in our simulations. If our numerical algorithm is correct, the poor performance may be caused by the plugged-in estimators of the nuisance functions.
The proposed method and the method by Hsu and Prentice (1996) are both moment-based procedures but their performances are very different. We have found that, for AC models, the odds ratio of the two-by-two table provides a better descriptive measure for the association. In contrast, the covariance function of martingale residuals proposed by Hsu and Prentice is much less natural. That is why it produces an estimating function that involves many nuisance parameters.
Table 1.1 Comparison of two approaches without external censoring.
) 10 (st.error 10
bias× −2 × -2
n=500 n=100 Two-Stage Two-Stage
α
parametric semi-parametric Two-by-Two Table
Parametric semi-parametric Two-by-Two Table 2
1. -0.213 (8.114) 0.669 (8.382) -0.026 (8.275) 0.968 (14.767) 3.8 (16.271) 0.383 (15.232) 1.5 -0.191 (10.047) 1.179 (10.531) 0.065 (10.536) 1.464 (19.040) 6.054 (20.966) 0.28 (20.749) 1.85714 -0.252 (12.514) 1.192 (13.326) 0.031 (13.401) 2.138 (23.801) 7.592 (26.739) -0.074 (27.063)
3
2. -0.393 (15.841) 0.844 (17.039) -0.042 (17.170) 2.793 (30.204) 7.993 (33.793) -0.028 (34.737)
3.0 -0.578 (20.526) 0.018 (22.080) -0.208 (22.244) 3.637 (39.343) 7.128 (43.565) -1.11 (45.182) 4.0 -0.882 (27.588) -1.561 (29.449) -0.306 (29.807) 4.775 (53.258) 4.507 (57.379) -2.252 (60.220)
6 .
5 -1.438 (39.439) -5.398 (41.709) -0.683 (42.362) 6.37 (76.751) -2.423 (81.810) -0.963 (85.493)
9.0 -2.532 (63.241) -16.53 (65.670) -1.721 (66.599) 9.213 (123.879) -29.453 (128.361) 0.159 (136.928) 19.0 -5.370 (134.733) -72.600 (138.037) -2.178 (142.870) 16.84 (264.620) -173.33 (271.930) 9.597 (303.381)
Table 1.2 Comparison of two approaches with censoring rate 0.3.
) 10 (st.error 10
bias× −2 × -2
n=500 n=100 Two-Stage Two-Stage
α
parametric semi-parametric Two-by-Two Table
Parametric semi-parametric Two-by-Two Table 2
1. 0.073 (9.088) 0.566 (9.182) 0.142 (9.068) 2.061 (16.984) 4.258 (18.502) 1.702 (17.562) 1.5 0.234 (11.437) 1.024 (11.641) 0.34 (11.635) 2.747 (22.384) 5.737 (23.998) 1.252 (23.918) 1.85714 0.19 (14.198) 0.965 (14.721) 0.334 (14.909) 3.487 (27.112) 6.941 (29.197) 1.423 (30.365)
3
2. 0.024 (17.916) 0.395 (18.709) 0.244 (19.105) 4.139 (34.322) 7.02 (36.550) 1.254 (39.061) 3.0 -0.314 (22.888) -0.743 (24.042) 0.255 (24.793) 4.853 (44.807) 5.003 (47.158) 0.051 (51.647) 4.0 -1.224 (30.484) -3.377 (31.640) 0.078 (33.127) 3.853 (60.379) -2.213 (61.451) 0.297 (69.924)
6 .
5 -2.872 (43.457) -9.567 (44.718) 0.114 (47.279) 0.406 (86.207) -21.197 (83.557) 5.469 (99.999) 9.0 -9.223 (68.992) -31.33 (69.585) 0.634 (74.918) -21.92 (141.180) -88.543 (128.043) 16.75 (165.275) 19.0 -70.26 (158.490) -189.043 (152.155) 3.405 (161.384) -212.22 (348.587) -481.44 (260.105) 33.59 (369.187)
Table 1.3 Comparison of two approaches with censoring rate 0.6.
) 10 (st.error 10
bias× −2 × -2
n=500 n=100 Two-Stage Two-Stage
α
parametric semi-parametric Two-by-Two Table
parametric semi-parametric
Two-by-Two Table
1. 2 0.44 (10.714) 0.584 (10.759) 0.53 (10.736) 4.018 (21.149) 5.067 (22.596) 0.846 (21.655) 1.5 0.323 (13.202) 0.703 (13.357) 0.442 (13.401) 4.091 (27.924) 6.032 (29.777) 0.639 (29.280) 1.85714 0.317 (16.350) 0.685 (16.582) 0.409 (16.822) 5.072 (34.088) 7.577 (36.085) 1.563 (36.934) 2. 3 0.199 (20.272) 0.409 (20.862) 0.416 (21.371) 6.378 (42.449) 8.277 (43.924) 0.798 (46.024) 3.0 0.073 (26.114) -0.205 (27.040) 0.577 (28.097) 7.54 (54.756) 7.428 (55.933) 0.861 (60.081) 4.0 -0.299 (34.732) -1.645 (36.026) 0.928 (37.965) 7.942 (72.487) 2.874 (73.717) 2.214 (83.575)
6 .
5 -1.692 (49.035) -5.835 (50.423) 1.059 (53.934) 6.253 (104.281) -10.142 (103.610) 7.152 (119.437) 9.0 -7.103 (78.580) -19.496 (79.399) 2.336 (86.260) -6.881 (166.990) -53.666 (159.488) 25.38 (196.765) 19.0 -54.14 (175.675) -124.01 (168.489) 6.457 (184.836) -165.24 (373.784) -377.02 (315.480) 32.51 (453.817)
Table 2.1 Comparison of two approaches under marginal heterogeneity without external censoring.
) 10 (st.error 10
bias× −2 × -2
n=500 n=100 Two-Stage Two-by-Two Two-Stage Two-by-Two
α β1 = β2
αˆ 1
βˆ βˆ2
αˆ 1
βˆ βˆ2 2
1. -0.501 (6.391) 0.059 (6.288) -0.385 (9.188) -0.188 (9.640) -0.694 (14.900) -0.396 (14.132) 0.022 (21.557) -0.758 (21.390) 1.5 -1.983 (7.925) -1.230 (7.934) -0.257 (9.150) -0.345 (9.759) -3.612 (17.381) -2.002 (17.066) -0.526 (20.297) 1.460 (21.300) 1.85714 -3.055 (9.437) -1.924 (9.747) 0.729 (9.110) 0.194 (9.501) -7.404 (21.619) -5.029 (21.922) 1.015 (21.017) 0.766 (22.258)
3
2. -4.229 (11.624) -2.016 (11.763) -0.265 (8.924) 0.400 (9.970) -11.885 (25.475) -6.569 (26.423) 1.496 (21.028) 1.476 (21.540) 3.0 -5.615 (15.486) -2.360 (15.760) 0.626 (9.976) 0.494 (9.766) -15.534 (34.260) -6.149 (36.794) 1.836 (21.810) 2.830 (22.377) 4.0 -7.863 (20.590) -2.752 (21.118) 1.845 (9.404) 1.580 (9.646) -25.774 (47.262) -10.616 (50.432) 1.717 (22.160) 2.244 (23.109)
6 .
5 -14.769 (26.894) -5.651 (27.678) -0.111 (9.259) 0.027 (9.180) -42.934 (59.398) -14.262 (66.131) -0.876 (21.050) -0.829 (21.554) 9.0 -31.589 (42.410) -11.676 (43.023) 0.402 (9.888) 0.364 (9.788) -98.841 (99.761) -34.205 (105.528) 0.519 (21.985) 0.929 (21.968) 19.0
0.8
-114.648 (89.175) -41.539 (93.940) 0.494 (9.542) 0.473 (9.435) -341.087 (219.164) -127.069 (245.475) 0.420 (23.016) 0.205 (22.920)
Table 2.2 Comparison of two approaches under marginal heterogeneity with censoring rate 0.3.
) 10 (st.error 10
bias× −2 × -2
n=500 n=100 Two-Stage Two-by-Two Two-Stage Two-by-Two
α β1 = β2 αˆ 1
βˆ βˆ2
αˆ 1
βˆ βˆ2 2
1. -0.267 (7.589) -0.250 (8.413) -0.345 (10.011) 0.080 (10.170) 0.638 (16.577) 1.978 (18.085) -0.614 (22.271) 0.593 (23.457) 1.5 -1.858 (9.418) -1.567 (10.379) 0.083 (9.424) -0.124 (10.435) -2.140 (21.365) -1.309 (23.301) -0.466 (22.783) 1.363 (25.114) 1.85714 -1.963 (10.775) -1.007 (11.722) 0.337 (9.852) 1.079 (10.411) -4.958 (23.924) -0.703 (28.945) 0.933 (22.150) 0.658 (23.185)
3
2. -3.272 (14.306) -0.576 (16.869) 0.232 (10.123) 0.704 (10.714) -10.582 (31.164) -3.821 (36.515) -0.872 (23.256) -0.840 (24.870) 3.0 -6.165 (17.556) -1.408 (20.790) 0.327 (10.057) 0.704 (10.849) -21.709 (37.096) -10.014 (45.931) 1.367 (24.254) 1.187 (23.103) 4.0 -7.834 (23.841) 0.285 (27.205) -0.431 (11.057) -0.273 (10.434) -24.520 (52.442) 3.311 (69.246) 1.304 (24.693) 1.672 (24.308)
6 .
5 -17.583 (31.227) -1.378 (36.732) 0.574 (10.675) 0.352 (10.524) -57.033 (69.257) -2.362 (98.199) 1.592 (23.854) 1.033 (23.943) 9.0 -109.124 (55.087) -58.938 (66.582) 0.586 (10.134) 0.349 (10.517) -259.518 (111.386) -135.602 (152.304) 2.719 (23.804) 0.992 (24.344) 19.0
0.8
-292.511 (127.019) -97.013 (136.070) 0.397 (11.115) 0.562 (11.293) -668.705 (210.578) -183.206 (335.635) 2.082 (24.015) 2.155 (24.233)
Table 2.3 Comparison of two approaches under marginal heterogeneity with censoring rate 0.6.
) 10 (st.error 10
bias× −2 × -2
n=500 n=100 Two-Stage Two-by-Two Two-Stage Two-by-Two
α β1 = β2 αˆ 1
βˆ βˆ2
αˆ 1
βˆ βˆ2 2
1. -0.022 (9.175) 0.647 (11.317) 0.480 (11.516) 0.484 (12.338) 0.535 (19.467) 5.426 (27.392) 0.403 (27.579) 1.314 (28.132) 1.5 -0.700 (11.438) 0.997 (14.396) -0.311 (11.200) 0.597 (12.072) 0.269 (27.556) 2.762 (34.610) -0.788 (25.720) 0.609 (27.642) 1.85714 -1.103 (13.657) 0.187 (16.934) -0.092 (11.824) 0.619 (11.884) -2.658 (29.413) 1.710 (39.691) 1.143 (26.587) 4.003 (29.461)
3
2. -2.726 (18.460) 0.039 (23.483) -0.025 (11.456) 0.230 (12.124) -4.620 (40.159) 5.159 (57.367) -0.679 (27.484) -0.800 (28.109) 3.0 -5.260 (21.654) -1.091 (27.524) -0.243 (11.857) 0.166 (11.525) -14.801 (51.198) -1.219 (74.896) 1.004 (28.668) 2.123 (27.966) 4.0 -13.133 (28.383) -5.477 (37.248) 0.005 (12.170) 0.090 (11.847) -34.456 (62.953) 2.777 (97.260) 1.595 (27.887) -0.243 (26.747)
6 .
5 -22.755 (42.830) -6.343 (56.938) -0.314 (11.934) 1.115 (13.131) -70.193 (94.419) -13.234 (133.771) 0.490 (27.494) 0.838 (26.873) 9.0 -70.929 (68.613) -24.824 (92.450) -0.423 (12.367) 0.509 (11.945) -196.595 (142.673) -53.852 (234.286) 0.169 (29.724) -1.050 (26.876) 19.0
0.8
-395.364 (166.251) -185.900 (228.931) 0.254 (12.735) 0.040 (12.080) -840.543 (281.685) -326.036 (561.663) 2.275 (29.854) 0.901 (28.127)
References
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Appendix: Checking the Validity of the Method by Hsu and Prentice
Investigation #1: Is the distribution of αˆ reasonable?
Figure A.1
Finding: There seems to be a bound on αˆ .
Investigation #2: Whether the above problem is caused by the root-finding procedure?
Figure A.2 plot of α = 9
Finding: The estimating equation has a unique but wrong solution in some situation.
Investigation #3: Whether the plug-in estimators for the nuisance functions are not accurate?
Figure A.3 the marginal survival function and its estimator
Figure A.4 the cumulative hazard function and its estimator
Finding: The plugged-in estimator have reasonable performance only in some region.
.