# Survival Analysis

## Life Table and Kaplan-Meier Estimate

### Life table

• 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

Age $$n$$ $$y$$ $$\hat{h}$$ $$\hat{S}$$
34 120 0 0.000 1.000
35 71 1 0.014 0.986
36 125 0 0.000 0.986

### Discrete survival analysis: notations

• A client’s lifetime (time until event): random variable $$X$$
• 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 $$i$$ $S_i = \sum_{j \geq i + 1} f_j = P(X > i)$

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

### Life table estimations

• Hazard rate estimation: binomial proportions $\hat{h}_i = \frac{y_i}{n_i}$
• Typical frequentist inference: probabilistic results $$h_i$$ is estimated by the plug-in principle
• Probability of surviving past age $$j$$ given survival past age $$i$$: $P(X > j \mid X > i) = \prod_{k = i+1}^j P(X > k \mid X \geq k) = \prod_{k = i+1}^j (1 - h_k)$

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

### Continuous survival analysis: notations

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

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

• Hazard rate, also called hazard function $h(t) = \frac{f(t)}{S(t)} = \lim_{\Delta t \rightarrow 0} \frac{P(t < T \leq t + \Delta t \mid T > t)}{\Delta t}$
• In some other books, hazard rate is denoted as $$\lambda(t)$$

### Hazard rate and cumulative hazard function

• Connection between hazard rate $$h(t)$$ and survival function $$S(t)$$ $h(t) = -\frac{\partial \log S(t)}{\partial t} \quad \Longleftrightarrow \quad S(t) = \exp\left\{ -\int_0^t h(x)dx \right\}$

• Cumulative hazard function $\Lambda(t) = \int_0^t h(x) dx = -\log S(t)$

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

• Example: exponential distributed $$T$$ $f(t) = \lambda e^{- \lambda t} \quad \Longrightarrow \quad S(t) = e^{-\lambda t}, \quad h(t) = \lambda$
• Constant hazard rate: menoryless

### Censored data

• Censored data: survival times known only to exceed the reported value
• E.g., lost to followup, experiment ended with some patients still alive
• Usually denoted as “number+”
• Observation $$z_i$$ for censored data: $z = (t_i, d_i),$ where $$t_i$$ is the survival time, and $$d_i$$ is the indicator $d_i = \begin{cases} 1 & \text{if death observed}\\ 0 & \text{if death not observed} \end{cases}$

### Kaplan-Meier estimate

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

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

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

### A parametric approach

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

• Logistic regression $log\left( \frac{h_k}{1-h_k} \right) = \boldsymbol\alpha \mathbf{x}_k$

• E.g., cubic regression: $x_k = (1, k, k^2, k^3)'$

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

## Cox’s Proportional Hazards Model

### Cox’s proportional hazards model

• Proportional hazards model assumes $h_i(t) = h_0(t) \cdot e^{\mathbf{x}_i' \boldsymbol\beta},$ where $$h_0(t)$$ is a baseline hazard, which we don’t need to specify

• Denote $$\theta_i = e^{\mathbf{x}_i' \boldsymbol\beta}$$, then $S_i(t) = S_0(t)^{\theta_i},$ where $$S_0(t)$$ 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

### Proportional hazards model: key results

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

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

• Key results of the proportional hazards model: given one person dies at time $$T_{(j)}$$, the probablity it is person $$i$$, among the set of people at risk, is $P(i_j = i \mid R_j) = \frac{e^{\mathbf{x}_i' \boldsymbol\beta}} {\sum_{k \in R_j} e^{\mathbf{x}_j' \boldsymbol\beta}} = \frac{\theta_i}{\sum_{k \in R_j} \theta_j}$

### Parameter estimation: based on the partial likelihood

• Estimaiton of $$\boldsymbol\beta$$ is to maximize the partial likelihood $L(\boldsymbol\beta) = \prod_{j=1}^J \frac{e^{\mathbf{x}_{i_j}' \boldsymbol\beta}} {\sum_{k \in R_j} e^{\mathbf{x}_j' \boldsymbol\beta}}$ where individual $$i_j$$ dies at time $$T_{(j)}$$

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

### References

• Efron, Bradley and Hastie, Trevor (2016), Computer Age Statistical Inference. Cambridge University Press