Skip to main content

Table 1 Frailty variance estimates bias, coverage probability across shared frailty models in the presence of unobserved heterogeneity and non-proportional hazards

From: Comparison of statistical methods for the analysis of recurrent adverse events in the presence of non-proportional hazards and unobserved heterogeneity: a simulation study

Data Generating Mechanisms

ISF-PH Model

LSF-NPHS1 Model

LSF-NPHS3 Model

% gain in precision relative to ISF-PH model

Scenarioa

Sample size

Frailty variance

TDE

Bias (MCSE), Coverage (MCSE), MSE (MCSE)

Bias (MCSE), Coverage (MCSE), MSE (MCSE)

Bias (MCSE), Coverage (MCSE), MSE (MCSE)

LSF-NPHS1% gain in precision (MCSE)

LSF-NPHS3% gain in precision (MCSE)

1

500

0.25

0.03

−0.0389 (0.0064), 96.40 (0.5891), 0.0423 (0.0021)

−0.0556 (0.0060), 95.90 (0.6270), 0.0386 (0.0020)

− 0.0263 (0.0061), 96.20 (0.6046), 0.0374 (0.0019)

14.6985 (0.7269)

11.1644 (1.6768)

2

500

0.50

0.03

−0.0431 (0.0051), 95.00 (0.6892), 0.0278 (0.0012)

−0.1329 (0.0045), 87.10 (1.0600), 0.0375 (0.0015)

−0.1098 (0.0045), 90.20 (0.9402), 0.0323 (0.0014)

30.9883 (0.9472)

27.8901 (1.8680)

3

500

0.75

0.03

− 0.0689 (0.0048), 92.80 (0.8174), 0.0280 (0.0013)

− 0.2151 (0.0041), 61.80 (1.5365), 0.0628 (0.0020)

−0.1939 (0.0041), 69.10 (1.4612), 0.0544 (0.0019)

40.7088 (1.0645)

38.4618 (1.9530)

4

300

0.25

0.03

−0.0363 (0.0079), 97.20 (0.5217), 0.0637 (0.0029)

−0.0542 (0.0073), 97.50 (0.4937), 0.0569 (0.0027)

− 0.0248 (0.0075), 96.70 (0.5649), 0.0561 (0.0027)

12.5407 (1.8701)

12.5407 (1.8701)

5

300

0.50

0.03

−0.0557 (0.0069), 94.90 (0.6957), 0.0504 (0.0025)

−0.1448 (0.0061), 90.60 (0.9228), 0.0575 (0.0027)

− 0.1221 (0.0061), 92.50 (0.8329), 0.0518 (0.0025)

29.3473 (0.9645)

28.3377 (1.8182)

6

300

0.75

0.03

− 0.0674 (0.0062), 94.50 (0.7209), 0.0431 (0.0020)

− 0.2141 (0.0052), 76.60 (1.3388), 0.0729 (0.0026)

− 0.1950 (0.0053), 80.90 (1.2431), 0.0660 (0.0025)

42.0668 (1.1767)

37.7659 (2.0100)

7

500

0.25

− 0.03

− 0.0212 (0.0063), 96.10 (0.6122), 0.0405 (0.0019)

− 0.0694 (0.0060), 95.20 (0.6760), 0.0413 (0.0021)

− 0.0348 (0.0061), 95.90 (0.6270), 0.0384 (0.0020)

9.7913 (0.6757)

7.8448 (1.5502)

8

500

0.50

− 0.03

−0.0317 (0.0051), 95.30 (0.6693), 0.0268 (0.0012)

− 0.1425 (0.0045), 85.60 (1.1102), 0.0405 (0.0016)

− 0.1163 (0.0045), 89.00 (0.9894), 0.0341 (0.0014)

27.3902 (0.9049)

25.1537 (1.7685)

9

500

0.75

−0.03

− 0.0605 (0.0048), 93.30 (0.7906), 0.0269 (0.0013)

− 0.2236 (0.0041), 59.20 (1.5541), 0.0668 (0.0021)

− 0.1998 (0.0041), 67.70(1.4788), 0.0569 (0.0019)

38.1980 (1.0547)

37.1299 (1.8588)

10

300

0.25

−0.03

− 0.0190 (0.0079), 97.00 (0.5394), 0.0621 (0.0027)

− 0.0680 (0.0075), 97.70 (0.4740), 0.0602 (0.0028)

− 0.0335 (0.0075), 97.00 (0.5394), 0.0577 (0.002)

11.1739 (0.7791)

9.1266 (1.7207)

11

300

0.50

−0.03

− 0.0454 (0.0069), 94.80 (0.7021), 0.0493 (0.0024)

− 0.1549 (0.0061), 89.50 (0.9694), 0.0614 (0.0029)

− 0.1286 (0.0061), 92.50 (0.8329), 0.0537 (0.0026)

26.3505 (0.9108)

26.8512 (1.7308)

12

300

0.75

−0.03

−0.0588 (0.0062), 94.80 (0.7021), 0.0418 (0.0019)

− 0.2219 (0.0053), 75.70 (1.3563), 0.0768 (0.0027)

− 0.2005 (0.0053), 79.90 (1.2673), 0.0684 (0.0026)

39.0878 (1.1318)

36.0944 (1.9042)

13

500

0.25

0

−0.0296 (0.0064), 96.50 (0.5812), 0.0412 (0.0020)

− 0.0619 (0.0060), 95.70 (0.6415), 0.0398 (0.0020)

− 0.0300 (0.0061), 95.90 (0.6270), 0.0378 (0.0019)

12.1870 (0.6812)

9.4918 (1.6201)

14

500

0.50

0

−0.0373 (0.0051), 95.00 (0.6892), 0.0272 (0.0012)

− 0.1373 (0.0045), 86.80 (1.0704), 0.0389 (0.0016)

− 0.1127 (0.0045), 89.70 (0.9612), 0.0331 (0.0014)

29.0623 (0.8985)

26.5314 (1.8048)

15

500

0.75

0

−0.0644 (0.0048), 93.00 (0.8068), 0.0274 (0.0013)

−0.2188 (0.0041), 61.30 (1.5402), 0.0646 (0.0020)

−0.1964 (0.0041), 68.80 (1.4651), 0.0554 (0.0019)

39.4216 (1.0400)

37.9288 (1.8940)

16

300

0.25

0

−0.0273 (0.0079), 97.10 (0.5307), 0.0627 (0.0028)

− 0.0605 (0.0074) 97.70 (0.4740), 0.0583 (0.0028)

−0.0283 (0.0075), 96.70 (0.5649), 0.0567 (0.0027)

13.2530 (0.7841)

10.8368 (1.7814)

17

300

0.50

0

−0.0500 (0.0069), 94.80 (0.7021), 0.0495 (0.0024)

−0.1491 (0.0061), 90.20 (0.9402), 0.0591 (0.0028)

− 0.1249 (0.0061), 92.60 (0.8278), 0.0526 (0.0026)

27.7522 (0.9164)

27.0827 (1.7637)

18

300

0.75

0

−0.0627 (0.0062), 94.50 (0.7209) 0.0423 (0.0020)

−0.2174 (0.0052), 76.20 (1.3467), 0.0745 (0.0027)

− 0.1972 (0.0053), 80.60 (1.250) 0.0668 (0.0026)

40.6654 (1.1293)

37.0383 (1.9578)

  1. TDE time dependent effect
  2. aeach scenario is based on 1000 simulations