Nonparametric Estimation of Proportional Hazards with Monotone Baseline Hazard and Covariate Effect

Abstract

Order-restricted inference has been applied to survival analysis when its hazard function is known to have a specific shape prior to data analysis. Under the proportional hazards assumption, the partial likelihood approach is commonly used to estimate a covariate effect on the distribution of survival time without specifying its baseline hazard function, but at the same time, the shape information of the baseline hazard function cannot be used in the partial liklelihood estimation procedure. In this paper, we propose a nonparametric full likelihood method for estimating the covariate effect and baseline hazard functions simultaneously under monotone shape restriction. We develop an efficient algorithm using generalized isotonic regression techniques. We extend the algorithm to model with time-dependent covariates. Simulation studies demonstrate that the proposed full likelihood method shows smaller variance than the partial likelihood approach with reduction of bias. Analysis of data from a bone marrow transplantation study illustrates the practical utility of the isotonic methodology in estimating a nonlinear and monotone hazard function.

Appendices

Appendix 1: Proof

Proof Proposition 1

Under censored data with a time-independent covariate (4) are re-formulated to

$$\begin{aligned} \ell (\lambda ,\psi )=-\sum _{i=1}^{n^*} \log \lambda ^*_{(i)} - \sum _{j=1}^{n^*} \psi _{[j]} + \sum _{j=1}^n \sum _{i=1}^n w_{(i)[j]} \lambda _{(i)} e^{\psi _{[j]}}, \end{aligned}$$

(12)

where the parameters for uncensored subjects are included in the all three terms, while the parameters for censored subjects are included only in the last term. Thus, (12) is minimized when the parameters for censored subjects are minimized, which occur among the parameters for uncensored subjects by the order restriction. If there exists an a or b such that $X_{a}<X^*_{(1)}$ or $Z_{b}<Z^*_{(1)}$, (12) is minimized when $\lambda _{a}=0$ or $\psi _{b}=-\infty$, respectively. It shows that (12) is reduced to (9), and the isotonic estimators have jumps only at $X^\star _{(i)}$ and $Z^\star _{[j]}$ for $i=1,\ldots ,n^\star$ and $j=1,\ldots ,n^\star$. □

Proof of Proposition 2

Under censored data with a time-dependent covariate, a negative log likelihood of (10) is formulated to

$$\begin{aligned} -\sum _{s=1}^{n^\star } \log \lambda (X^\star _s) -\sum _{t=1}^{n^\star } \psi (Z^*_t)+\sum _{i=1}^{n} \sum _{j=1}^{n} w_{ij}\lambda (X_{ij})e^{\psi (Z_i(X_{ij}))}, \end{aligned}$$

(13)

where the parameters at $X^\star _s$ and $Z^*_t$, $s=1,\ldots ,n^\star$ and $t=1,\ldots ,n^\star$, are included in all three terms, while the other parameters are included only in the last term. Thus, (13) is minimized when the other parameters are maximized, which occur among the parameters in the first two terms by the order restriction. If there exists an a such that $X_{a}<X^*_{1}$, (12) is minimized when $\lambda (X_{a})=0$. If there exist b and c such that $Z_b(X_{bc})<Z^*_{(1)}$, (12) is minimized when $\psi (Z_b(X_{bc}))=-\infty$. It shows that (12) is reduced to (11), and the isotonic estimators have jumps only at $X^\star _{(i)}$ and $Z^*_{[j]}$ for $i=1,\ldots ,n^\star$ and $j=1,\ldots ,n^\star$. □

Appendix 2: R package isoSurv

In the R package isoSurv, the function named disoph is the main function that includes the methods developed. Here, we explain the usage of the disoph function using a hypothetical example below:

> test1=data.frame(

> time=c(2, 5, 1, 7, 9, 5, 3, 6, 8, 9, 7, 4, 5, 2, 8),

> status=c(0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1),

> z=c(2, 1, 1, 3, 5, 6, 7, 9, 3, 0, 2, 7, 3, 9, 4))

The test1 dataset includes 11 observations with their right-censored time, event status, and covariate values. For the standard Cox proportional hazards model with a linear predictor, the function coxph in the R package survival can be used as

>library(survival)

>fit1=coxph(Surv(time,status)$\sim$z,data=test1)

>print(fit1)

Similarly, the proposed isotonic proportional hazards model with monotone increasing functions $\phi$ and $\lambda _0$ is fitted by using the following R code:

>library(isoSurv)

>fit2=disoph(Surv(time,status)$\sim$iso(z,shape=“inc”),bshape=“inc”,data=test1)

>print(fit2$iso.bh)

>print(fit2$iso.cov)

>plot(fit2)

The print and plot functions printed and visualized estimated BHR and HR, respectively. In the disoph function above, instead of shape=“inc” (or bshape=“inc”), shape=“dec” (or bshape=“inc”) can be used if $\phi$ is monotonically decreasing in z (or $\lambda _0$ is monotonically decreasing in t). Detailed instructions on using the isoSurv package can be found using

> help(disoph)

> example(disoph)