A semiparametric Cox-Aalen transformation model with censored data

Abstract

We propose a broad class of so-called Cox-Aalen transformation models that incorporate both multiplicative and additive covariate effects on the baseline hazard function within a transformation. The proposed models provide a highly flexible and versatile class of semiparametric models that include the transformation models and the Cox-Aalen model as special cases. Specifically, it extends the transformation models by allowing potentially time-dependent covariates to work additively on the baseline hazard and extends the Cox-Aalen model through a predetermined transformation function. We propose an estimating equation approach and devise an expectation-solving (ES) algorithm that involves fast and robust calculations. The resulting estimator is shown to be consistent and asymptotically normal via modern empirical process techniques. The ES algorithm yields a computationally simple method for estimating the variance of both parametric and nonparametric estimators. Finally, we demonstrate the performance of our procedures through extensive simulation studies and applications in two randomized, placebo-controlled human immunodeficiency virus (HIV) prevention efficacy trials. The data example shows the utility of the proposed Cox-Aalen transformation models in enhancing statistical power for discovering covariate effects.

Keywords: Cox-Aalen model; ES algorithm; estimating equations; time-dependent covariates; transformation models.

Figure 1.

Estimation results for (a) $A_1\left(t\right)=\log \left(1+t/4\right)$ and (b) $A_2\left(t\right)=0.1t$ in Scenario 1, under the logarithmic transformation $G\left(x\right)=r^{-1}\log \left(1+rx\right)$ with $r=0$. The dashed and solid lines are for data sets with $n=500$ and $n=800$, respectively. Bias, SE, SEE, and CP stand, respectively, for the bias, empirical standard error, mean of the standard error estimator, and empirical coverage probability of the 95% confidence interval.

Figure 2.

Kaplan-Meier plot for four regions in the full cohort. Here, “USAS”, “BP”, “SA” and “Other SSA” represent USA and Switzerland, Brazil and Peru, South Africa and other sub-Saharan African countries, respectively.

Figure 3.

Estimated baseline cumulative hazard function for four regions under the logarithmic transformation $G\left(x\right)=r^{-1}\log \left(1+rx\right)$ with $r=0$. Here, “USAS”, “BP”, “SA” and “Other SSA” represent USA and Switzerland, Brazil and Peru, South Africa and other sub-Saharan African countries, respectively.

Figure 4.

Estimated survival functions by considering different combinations of covariates under the proposed model with the transformation $G\left(x\right)=r^{-1}\log \left(1+rx\right)$ with $r=0$. Here, “Age1”, “Age2”, “Age3” and “Age4” stand for the age goups, [17, 20], [21, 30], [31, 40] and [41, 52], respectively. “USAS”, “BP”, “SA” and “Other SSA” represent USA and Switzerland, Brazil and Peru, South Africa and other sub-Saharan African countries, respectively.