Accommodating Time-Varying Heterogeneity in Risk Estimation under the Cox Model: A Transfer Learning Approach

Figures & data

Fig. 1 Estimation biases for coefficients β₁, β₂, and β₃ (Panel A), as well as cumulative baseline hazards at times 0.6 and 1.2 (Panel B) over 100 Monte Carlo simulations. $H_1={\widehat{H}}_0(0.6)$ and $H_2={\widehat{H}}_0(1.2)$. The dotted gray line shows the place where bias equals zero.

Fig. 2 Boxplots for absolute personalized prediction error of Trans-Cox and other existing methods in different simulation settings at time 0.6 (APPE1) and 1.2 (APPE2). The results are summarized over 100 Monte Carlo datasets.

Fig. 3 The estimated cumulative baseline hazard (Panel A) and the survival curves (Panel B) for Trans-Cox (green) and Cox-Both (blue) in comparison to the true curves (red) over 100 Monte Carlo experiments. For survival curves, the covariates are fixed at $\mathit{X}=(0.5,1,0.1,0.5,0.5)$.

Fig. 4 Summary of the patients’ characteristics of the MDA and NCDB cohorts.

Table 1 Analysis results of the MDA and NCDB cohorts.

$\widehat{\beta }$
	Trans-Cox	Cox(MDA)	Cox(NCDB)	Cox(Combined) ${}^{\mathrm{d}}a\mathit{gger}$	Cox(Stratified)
Normalized Age	0.091	0.074	0.208	0.197	0.199
Grade:III vs. I/II	0.616	0.388	0.347	0.353	0.348
Race:Black vs. White	0.272	0.163	0.429	0.412	0.416
Race:Other vs. White	−0.128	−0.336	−0.113	−0.107	−0.130
Stage:IIIC vs. IIIB	−0.059	−0.243	0.386	0.368	0.357
Trimodality:Yes vs. No	−0.838	−1.174	−0.589	−0.636	−0.627
SE ${}^{*}$
	Trans-Cox	Cox(MDA)	Cox(NCDB)	Cox(Combined)	Cox(Stratified)
Normalized Age	0.121	0.109	0.021	0.021	0.021
Grade:III vs. I/II	0.197	0.269	0.044	0.043	0.044
Race:Black vs. White	0.265	0.325	0.051	0.051	0.051
Race:Other vs. White	0.261	0.295	0.118	0.109	0.109
Stage:IIIC vs. IIIB	0.179	0.244	0.055	0.053	0.054
Trimodality:Yes vs. No	0.207	0.230	0.069	0.065	0.065
p-value
	Trans-Cox	Cox(MDA)	Cox(NCDB)	Cox(Combined)	Cox(Stratified)
Normalized age	0.453	0.498	¡0.001	¡0.001	¡0.001
Grade:III vs. I/II	0.002	0.150	¡0.001	¡0.001	¡0.001
Race:Black vs. White	0.305	0.616	¡0.001	¡0.001	¡0.001
Race:Other vs. White	0.623	0.295	0.118	0.109	0.19
Stage:IIIC vs. IIIB	0.753	0.318	¡0.001	¡0.001	¡0.001
Trimodality:Yes vs. No	¡0.001	¡0.001	¡0.001	¡0.001	¡0.001

${}^{\ast }$ Boostrap-based standard error estimation with 1000 bootstrap resamplings. ${}^{\mathrm{d}ag}$ Cox(Combined) is the Cox regression model using the combined data of MDA and NCDB.

Fig. 5 The estimated survival curves using different methods for patients with (left panel) or without (right panel) receiving trimodality treatment. Other characteristics were specified as: age 50 years old, grade III, Black race, and stage IIIC.

Fig. 6 Evaluation of the prediction performance of four methods using C-index (Panel A), Uno’s C-index (B), AUC (C), and personalized risk prediction error (D).