An insurance company’s life table shows information of clients by their age. For each age $i$, it contains

• $n_i$: number of clients
• $y_i$: number of death
• ${\hat{h}}_i=y_i/n_i$: hazard rate
• ${\hat{S}}_i$: survival probability estimate

An example life table

• Also called failure time, survival time, or event time

Probability of dying at age $i$ $$f_i=P(X=i)$$

Probability of surviving past age $i$ $$S_i=\sum _{j\ge i+1}{f}_j=P(X>i)$$

Hazard rate at age $i$: conditional probability $$h_i=\frac{f_i}{S_{i-1}}=P(X=i\mid X\ge i)$$

• Typical frequentist inference: probabilistic results $h_i$ is estimated by the plug-in principle

Probability of surviving past age $j$ given survival past age $i$: $$P(X>j\mid X>i)=\prod _{k=i+1}^{j}P(X>k\mid X\ge k)=\prod _{k=i+1}^j(1-h_k)$$

Probability of survival estimation $${\hat{S}}_j=\prod _{k=i_0}^j(1-{\hat{h}}_k)$$ where $i_0$ is the starting age

Time until event $T$: a continuous positive random variable, with pdf $f\left(t\right)$ and cdf $F\left(t\right)$

Survival function (i.e., reverse cdf) $$S\left(t\right)={\int }_t^{\infty }f\left(x\right)dx=P(T>t)=1-F\left(t\right)$$

• In some other books, hazard rate is denoted as $\lambda \left(t\right)$

Connection between hazard rate $h\left(t\right)$ and survival function $S\left(t\right)$ $$h\left(t\right)=-\frac{\partial \log S\left(t\right)}{\partial t}⟺S\left(t\right)=\exp \{-{\int }_0^{t}h(x\left)dx\right\}$$

Cumulative hazard function $$\Lambda \left(t\right)={\int }_0^{t}h\left(x\right)dx=-\log S\left(t\right)$$

Knowing any of $S\left(t\right)$, $h\left(t\right)$, $\Lambda \left(t\right)$ allows one to derive the other two

• Constant hazard rate: menoryless

• E.g., lost to followup, experiment ended with some patients still alive
• Usually denoted as “number+”

Among the censored data $z_1,\dots ,z_n$, we denote the ordered survival times as $$t_{\left(1\right)}<t_{\left(2\right)}<\dots <t_{\left(n\right)},$$ assuming no ties.

The Kaplan-Meier estimate for survival probability $S_{\left(j\right)}=P(X>t_{\left(j\right)})$ is the life table estimate $${\hat{S}}_{\left(j\right)}=\prod _{k\le j}{\left(\frac{n-k}{n-k+1}\right)}^{d_{\left(k\right)}}$$

Life table curves are nonparametric: no relationship is assumed between the hazard rates $h_i$

Death counts $y_k$ are independent Binomials $$y_k\stackrel{ind}{\sim }\text{B}(n_k,h_k)$$

Logistic regression $$log\left(\frac{h_k}{1-h_k}\right)=\alpha x_k$$

• E.g., cubic regression: $$x_k=(1,k,k^2,k^3{)}^{\prime }$$

• E.g., cubic-linear spline: $$x_k=(1,k,(k-k_0{)}_{-}^2,(k-k_0{)}_{-}^3{)}^{\prime }$$ where $x_{-}=x\cdot 1_{x\le 0}$

Proportional hazards model assumes $$h_i\left(t\right)=h_0\left(t\right)\cdot e^{x_i^{\prime }\beta },$$ where $h_0\left(t\right)$ is a baseline hazard, which we don’t need to specify

Denote ${\theta }_i=e^{x_i^{\prime }\beta }$, then $$S_i\left(t\right)=S_0(t{)}^{{\theta }_i},$$ where $S_0\left(t\right)$ is the baseline survival function

• Larger value of ${\theta }_i$ indicates more quickly declining (i.e., worse) survival curves
• Positive value of the coefficient ${\beta }_j$ indicates increase of the corresponding covariate $x_j$ associating with worse survival curves

Let $J$ be the number of observed deaths, occurring at times $$T_{\left(1\right)}<T_{\left(2\right)}<\dots <T_{\left(J\right)}$$ assuming no ties

Just before time $T_{\left(j\right)}$ there is a risk set of individuals still under observation $$R_j=\{i,t_i\ge T_{\left(j\right)}\}$$

Key results of the proportional hazards model: given one person dies at time $T_{\left(j\right)}$, the probablity it is person $i$, among the set of people at risk, is $$P(i_j=i\mid R_j)=\frac{e^{x_i^{\prime }\beta }}{{\sum }_{k\in R_j}{e}^{x_j^{\prime }\beta }}=\frac{{\theta }_i}{{\sum }_{k\in R_j}{\theta }_j}$$

Estimaiton of $\beta$ is to maximize the partial likelihood $$L\left(\beta \right)=\prod _{j=1}^J\frac{e^{x_{i_j}^{\prime }\beta }}{{\sum }_{k\in R_j}{e}^{x_j^{\prime }\beta }}$$ where individual $i_j$ dies at time $T_{\left(j\right)}$

Semi-parametric: we do not need to specify the baseline $h_0\left(t\right)$, since it is not contained in the objective function