A review of h-likelihood for survival analysis

Abstract

Statistical models with unobservable random variables such as random-effect models have been recently studied for analyzing data of complex types (e.g. longitudinal and time-to-event data) in various areas. The hierarchical likelihood [h-likelihood; Lee and Nelder (Journal of the Royal Statistical Society 58, 619–678, 1996)] provides a unified framework for the inference of such models with unobservable random variables. In this paper, we review the h-likelihood framework for survival analysis. We also demonstrate how to analyze survival data via web-based software Albatross Analytics which was recently developed based on the h-likelihood procedure. Furthermore, we discuss recent extensions of the h-likelihood.

Appendix A: Appendix

Appendix A: Derivation of h-likelihood (1)

Under the assumptions of the conditional independence and non-informative censoring, the h-(log)likelihood based on the (i, j)th observation is defined by a joint density function of $(Y_{ij}, \Delta _{ij}, V_{i})$ as follows.

$$\begin{aligned} h_{ij}= & {} \log f(y_{ij}, \delta _{ij}, v_{i})\\= & {} \log f(y_{ij}, \delta _{ij}|u_{i}) + \log g(v_{i}), \end{aligned}$$

where the first term of $h_{ij}$ can be expressed as

$$\begin{aligned} \log f(y_{ij}, \delta _{ij}|u_{i})= & {} \delta _{ij} \log f(y_{ij}|u_{i}) + (1-\delta _{ij}) \log S(y_{ij}|u_{i})\\= & {} \delta _{ij} \log \lambda (y_{ij}|u_{i}) - \Lambda (y_{ij}|u_{i}). \end{aligned}$$

Here $S(\cdot |u_{i})=\exp \{- \Lambda (\cdot |u_{i})\}$, $\Lambda (\cdot |u_{i})$ and $\lambda (\cdot |u_{i})$ are the conditional survival function, cumulative hazard function and hazard function of $T_{ij}|u_{i}$, respectively. Thus, the h-likelihood (1) of all observations becomes

$$\begin{aligned} h=\sum _{ij} h_{ij}= \sum _{ij}\ell _{1ij}+\sum _{i} \ell _{2i}, \end{aligned}$$

where $\ell _{1ij}= \log f(y_{ij}, \delta _{ij}|u_{i})$ and $\ell _{2i}= \log g(v_{i})$. Note here that for $v_{i}=\nu (u_{i})$ on a strictly monotone function of $u_{i}$,

$$\begin{aligned} \log g(v_{i})= \log g(u_{i}) + \log \biggl | \frac{du_{i}}{d v_{i}} \biggl |. \end{aligned}$$

Note here that the Jacobian is involved when we transform the u-scale to the $\nu$-scale in getting of h-likelihood.

Appendix B: Derivation of the PHL under the joint model

Let $C_{i}$ denote independent censoring time. We assume that given $v_{i}$, $C_{i}$ is independent of $(T_{ik},\Delta _{ik})$ for $k=1,2$. Then the observed event time and event indicator are, respectively, given by

$$\begin{aligned} T_{i}^{*}=\mathrm {min}(T_{i1},T_{i2},C_{i})~\mathrm {and}~\Delta _{ik}=I(T_{i}^{*}=T_{ik}). \end{aligned}$$

Thus, all observable random variables are $(Y_{ij},T_{i}^{*},\Delta _{ik})$ with their observed values $(y_{ij},t_{i}^{*},\delta _{ik})$ $(i=1,\ldots ,q;j=1,\ldots ,n_{i};k=1,2)$. Here the h-likelihood for the general joint models with competing-risk data becomes

$$\begin{aligned} h=\sum _{ij}\ell _{1ij}+\sum _{ik}\ell _{2ik}+\sum _{i}\ell _{3i}, \end{aligned}$$

where $\ell _{1ij}$ is the conditional normal log-likelihood for $y_{ij}$ given $v_{i1}$, and for $k=1,2$

$$\begin{aligned} \ell _{2ik}=\ell _{2ik}(\beta _{k+1},\lambda _{0k};(t_{i}^{*},\delta _{ik})|v_{i})=\delta _{ik}\{\log \lambda _{0k}(t_{i}^{*})+\eta _{2ik}\}-\Lambda _{0k}(t_{i}^{*})\exp (\eta _{2ik}), \end{aligned}$$

where $\eta _{2ik}=x_{i2}^{T}\beta _{k+1}+ v_{i,k+1}$, and $\ell _{3i}$ is the log-likelihood for $v_{i}=(v_{i1}, v_{i2}, v_{i3})^T$ with trivariate normal distribution, given by

$$\begin{aligned} \ell _{3i}(\sigma ;v_{i})=-\frac{1}{2}\log |2\pi \Sigma _{3} |-\frac{1}{2} v_{i}^{T}\Sigma _{3}^{-1}v_{i}. \end{aligned}$$

Following (5) and (6), it can be easily shown that the corresponding PHL $h_{p}$ is given by

$$\begin{aligned} h_{p}=\sum _{ij}\ell _{1ij}+\sum _{ik}\delta _{ik}\eta _{2ik}-\sum _{kr}d_{(kr)}\log \biggl \{\sum _{~i\in R_{(kr)}~}\exp (\eta _{2ik}) \biggl \}+\sum _{i}\ell _{3i}, \end{aligned}$$

where $d_{(kr)}$ is the number of events at time $t_{(kr)}$ and $R_{(kr)}=\{i:t_{i}^{*}\ge t_{(kr)}\}$ is the risk set at $t_{(kr)}$ which is the rth ($r=1,\ldots ,D_{k}$) smallest distinct event time for Type k event among the $t_{i}^{*}$s.