Background
Methods
Relative survival
Parametric cure models
Flexible parametric cure models
The loss of lifetime function
-
the Nelson et al. [10] relative survival (NRS) model, which is linear on the log cumulative excess hazard scale after the last knot,
-
the Andersson et al. [13] relative survival (ARS) model, which is constant on the log cumulative excess hazard scale after the last knot and thereby incorporates statistical cure, and
-
the flexible mixture cure (FMC) model in (8), which incorporates statistical cure, but is not restricted to a constant log cumulative excess hazard after the last knot.
Implementation
R
. For the purpose of this article, the NRS and ARS models were fitted using the package rstpm2
[17]. Functions for estimating the presented FMC model and computing the loss of lifetime function were assembled in the R
-package cuRe
(see https://github.com/LasseHjort/cuRe). The package also enables estimation of the expected residual lifetime, restricted expected residual lifetime, and restricted loss of lifetime using any of the models considered here. The integrals of the loss of lifetime function were computed numerically by Gauss-Legendre quadrature, while the point-wise variance of the loss of lifetime function was estimated using the delta method and numerical differentiation.Results
Simulation study
Simulation design
-
Draw a general population survival time TE from S∗.
-
Draw a relative survival time, TR from R.
-
Draw a censoring time TC from C.
-
The observed follow-up time is given by T= min(TR,TE,TC) and the event indicator is δ=1[min(TR,TE)≤TC].
Model | Model | Nr. knots | Knot locations |
---|---|---|---|
A | NRS | 6 | 0%, 20%, 40%, 60%, 80%, and 100% quantiles of the uncensored event times. |
B | ARS | 7 | 0%, 20%, 40%, 60%, 80%, and 100% quantiles of the uncensored event times with an additional knot placed at 10 years. |
C | ARS | 7 | 0%, 20%, 40%, 60%, and 80% quantiles of the uncensored event times. The last knot is placed at 80 years and an additional knot is placed at 10 years. |
D | FMC | 5 | 0%, 25%, 50%, 75%, and 100% quantiles of the uncensored event times. |
E | FMC | 5 | First uncensored event time, 0.5, 1, 2, and 5 years. |
Simulation results
Age | Scenario | pi | Model A | Model B | Model C | Model D | Model E |
---|---|---|---|---|---|---|---|
50 | 1 | 0.40 | 8.6(0.9-39.2) | 2.4(0.5-10.7) | 4.1(0.9-19.5) | 2.4(0.2-18.5) | 2.6(0.3-28.6) |
2 | 0.40 | 28.9(5.5-66.9) | 11.7(6.0-23.1) | 12.7(3.7-35.3) | 11.1(0.6-68.0) | 9.3(0.7-52.1) | |
3 | 0.75 | 8.9(0.7-42.7) | 8.9(4.3-15.6) | 4.8(0.9-16.4) | 7.5(0.5-31.5) | 7.2(0.2-23.9) | |
4 | 0.00 | 6.5(0.3-34.5) | 144.8(123.4-171.2) | 37.2(8.9-82.1) | 24.9(0.3-111.8) | 18.1(0.2-104.6) | |
5 | 0.00 | 11.0(0.2-43.3) | 25.4(14.8-35.4) | 13.0(3.2-31.0) | 12.8(0.4-33.8) | 9.6(0.3-30.9) | |
6 | 0.00 | 18.1(0.5-65.4) | 106.6(71.6-127.7) | 49.2(7.9-92.4) | 36.2(0.6-103.7) | 23.9(0.2-89.4) | |
60 | 1 | 0.40 | 6.0(1.3-19.4) | 1.9(0.4-8.0) | 3.1(0.6-13.5) | 2.1(0.2-14.7) | 2.4(0.2-17.6) |
2 | 0.40 | 14.9(2.6-45.4) | 6.4(3.4-14.9) | 7.7(2.0-26.2) | 7.7(0.6-40.2) | 6.6(0.2-42.5) | |
3 | 0.75 | 7.2(0.4-39.6) | 4.2(1.9-10.0) | 4.0(0.4-22.7) | 5.4(0.3-28.2) | 4.7(0.3-19.6) | |
4 | 0.00 | 5.7(0.3-24.1) | 79.5(64.6-93.2) | 21.0(6.1-44.4) | 14.2(0.3-62.4) | 10.2(0.1-49.8) | |
5 | 0.00 | 7.5(0.3-33.4) | 10.7(5.1-18.1) | 5.6(1.5-18.6) | 7.2(0.5-26.0) | 5.0(0.2-17.9) | |
6 | 0.00 | 10.9(0.6-37.3) | 48.2(36.4-61.2) | 18.5(4.1-42.5) | 16.8(1.2-50.9) | 11.2(0.3-45.3) | |
70 | 1 | 0.40 | 3.6(0.9-12.4) | 1.5(0.2-4.8) | 2.2(0.4-7.3) | 1.7(0.1-8.8) | 2.0(0.1-12.9) |
2 | 0.40 | 6.2(1.2-20.8) | 3.4(1.5-8.8) | 4.0(0.9-14.2) | 4.7(0.3-19.4) | 4.3(0.3-19.2) | |
3 | 0.75 | 4.9(0.3-20.6) | 2.4(0.8-7.0) | 3.3(0.3-12.9) | 3.6(0.2-19.0) | 2.8(0.2-11.2) | |
4 | 0.00 | 4.3(0.3-16.1) | 34.9(26.9-44.4) | 9.6(3.5-23.3) | 7.3(0.2-31.3) | 5.7(0.1-27.5) | |
5 | 0.00 | 5.3(0.4-25.2) | 3.9(1.7-8.9) | 3.5(0.6-14.8) | 4.3(0.2-18.5) | 2.9(0.1-10.5) | |
6 | 0.00 | 6.0(0.3-21.7) | 16.5(9.9-23.8) | 6.2(1.8-16.6) | 6.9(0.2-23.7) | 5.1(0.2-17.4) |
Age | Scenario | pi | Model A | Model B | Model C | Model D | Model E |
---|---|---|---|---|---|---|---|
50 | 1 | 0.40 | 6.8(0.6-30.1) | 2.4(0.4-11.3) | 4.0(0.5-21.0) | 2.6(0.3-29.8) | 3.1(0.3-30.5) |
2 | 0.40 | 23.1(5.0-48.1) | 10.2(5.7-18.3) | 7.1(1.8-22.8) | 8.5(0.6-39.3) | 10.3(0.7-43.8) | |
3 | 0.75 | 17.4(1.5-74.8) | 10.2(4.2-19.8) | 7.6(1.0-31.0) | 11.2(0.8-63.1) | 10.0(0.3-56.2) | |
4 | 0.00 | 6.7(0.2-32.6) | 146.5(121.6-166.8) | 39.2(10.4-75.0) | 22.8(0.2-123.1) | 15.4(0.2-95.0) | |
5 | 0.00 | 14.8(0.6-88.3) | 35.9(22.7-45.6) | 18.0(3.2-39.4) | 18.4(0.6-77.5) | 13.7(0.3-40.3) | |
6 | 0.00 | 16.1(0.6-72.9) | 130.5(108.5-153.0) | 56.1(9.8-99.7) | 36.4(0.5-117.8) | 21.3(0.8-100.1) | |
60 | 1 | 0.40 | 4.7(0.9-16.5) | 1.9(0.2-7.5) | 3.0(0.3-12.1) | 2.2(0.2-14.3) | 2.8(0.1-21.5) |
2 | 0.40 | 12.0(2.7-35.4) | 5.5(3.1-10.9) | 4.6(1.0-19.4) | 5.7(0.4-32.1) | 7.2(0.5-35.0) | |
3 | 0.75 | 11.1(1.0-53.0) | 5.2(2.2-11.4) | 6.1(0.7-31.7) | 7.4(0.8-37.6) | 7.5(0.4-31.0) | |
4 | 0.00 | 5.8(0.2-22.2) | 80.3(65.1-96.0) | 21.8(7.3-42.4) | 13.9(0.3-66.4) | 9.5(0.2-49.4) | |
5 | 0.00 | 10.2(0.7-55.2) | 14.8(6.9-22.4) | 7.4(1.6-29.4) | 9.6(0.3-33.6) | 6.8(0.2-22.0) | |
6 | 0.00 | 9.8(0.3-37.7) | 62.2(44.7-77.0) | 24.1(5.9-51.5) | 17.1(0.7-64.5) | 11.4(0.6-54.9) | |
70 | 1 | 0.40 | 3.1(0.6-11.0) | 1.4(0.2-4.8) | 2.2(0.2-7.4) | 1.8(0.2-8.8) | 2.3(0.1-12.7) |
2 | 0.40 | 5.5(1.0-16.7) | 2.6(1.4-6.0) | 2.9(0.4-11.1) | 3.2(0.2-13.6) | 4.2(0.3-14.6) | |
3 | 0.75 | 6.4(0.4-30.7) | 2.8(1.0-8.8) | 4.1(0.3-20.0) | 4.6(0.4-24.6) | 4.5(0.4-17.6) | |
4 | 0.00 | 4.2(0.3-14.9) | 35.4(26.7-43.6) | 10.2(3.9-23.9) | 7.4(0.3-30.9) | 5.4(0.2-24.0) | |
5 | 0.00 | 6.1(0.2-26.8) | 4.9(2.3-10.5) | 3.9(0.6-17.2) | 4.8(0.4-26.0) | 3.6(0.1-13.1) | |
6 | 0.00 | 6.1(0.4-23.2) | 22.4(14.5-30.4) | 8.0(2.4-22.7) | 7.2(0.7-29.7) | 5.2(0.4-25.1) |
Analysis of Danish cancer registry data
Data description
Results
Analysis of Danish lymphoma registry data
Data description
Population-based loss of lifetime
Model | DLBCL | FL | ML | |
---|---|---|---|---|
Median age (range) | 68(18-101) | 63(18-97) | 70(28-99) | |
5-year RS (95% CI) | NRS | 0.66(0.65-0.68) | 0.9(0.88-0.91) | 0.61(0.57-0.65) |
ARS | 0.66(0.65-0.67) | 0.9(0.88-0.91) | 0.61(0.57-0.65) | |
FMC | 0.66(0.64-0.67) | 0.9(0.88-0.91) | 0.61(0.58-0.65) | |
Loss of lifetime (95% CI) | NRS | 7.43(7.06-7.80) | 4.58(3.73-5.42) | 7.66(6.86-8.46) |
ARS | 6.70(6.42-6.98) | 3.57(3.13-4.02) | 6.92(6.26-7.59) | |
FMC | 7.21(6.86-7.55) | 3.97(3.24-4.70) | 7.74(6.95-8.53) |