# Survival analysis and reliability

Hazard rates, proportional hazard regression, life testing, mean time to failure

March 12, 2019 — March 7, 2022

## 1 Estimating survival rates

Here’s the set-up: looking at a data set of individuals’ lifespans, you would like to infer the distributions—Analysing when people die, or things break, etc. The statistical problem of estimating how long people’s lives are is complicated somewhat by the particular structure of the data — loosely, “every person dies at most one time,” and there are certain characteristic difficulties that arise, such as right-censorship. (If you are looking at data from an experiment and not all your subjects have died yet, they presumably die later, but you don’t know when.)

Handily, the tools one invents to solve this kind of problem end up being useful to solve other problems, such as point process inference, and look not so far from the densities/intensities relations.

So let’s say you have a random variable \(X\) of positive support, according to which the lifetime of your people (components, machines, whatever) is distributed, which possesses a pdf \(f_X(t)\) and cdf \(F_X(T)\).

We define several useful functions:

- The survival function (which is also the right tail CDF)
- \(S(t):=1-F(t)\)
- the hazard function
- \(\lambda(t):=f(t)/S(t)\)
- the cumulative hazard function
- \(\Lambda(t) :=\int_0^t\lambda(s) \textrm{d} s.\)

Why? Because it happens to come out nicely if we do that, and these functions acquire intuitive interpretations once we squint at them a bit. The survival function is the probability of an individual surviving to time \(t\) etc. The hazard function will turn out to be the rate of deaths at time \(t\) *given that one has not yet occurred.*

Using the chain rule, we can find the following useful relation:

\[S(t)=\exp[-\Lambda (t)]={\frac {f(t)}{\lambda (t)}}\]

The hazard function can be pretty much any non-negative function of non-negative support (or more generally, a Schwartz distribution, but let’s ignore that possibility for the moment.)

### 1.1 Life table method

Over intervals of time \([t,u]\) we define the cumulative hazard increment

\[ H(t,u) :=\int_t^u h (s) \textrm{d} s = H(u)-H(t) \]

and the survival increment

\[ \chi(t,u) :=\frac{\chi(u)}{\chi(t)} \]

The following relations are useful

\[ \chi(t)=\exp[-H (t)]={\frac {f(t)}{h (t)}}. \]

and

\[ \chi(t,u)=\frac{\exp[-H (u)]}{\exp[-H (t)]}=\exp[H (t)-H (u)]=\exp[-H (t,u)] \]

and so

\[-\log\chi(t,u)=H (t,u).\]

We estimate hazard via the *life table* method. Given a time interval \([t_{i}, t_{i+1})\) and survival counts \(N(t_{i})\) and \(N(t_{i+1})\) at, respectively, the beginning and end of that interval, (assuming no immigration) the life table estimate of a survival increment is

\[\hat{\chi}(t_i, t_{i+1}):= \frac{N(t_{i+1})}{N(t_{i})}\]

Plugging this in, we obtain cumulative hazard increment estimates

\[\begin{aligned} \hat{H} (t_i, t_{i+1})&=-\log \hat{\chi}(t_i, t_{i+1})\\ &=\log \frac{ N(t_{i}) }{ N(t_{i+1}) } \end{aligned}\]

From this, we construct further point estimates of \(H\) at \(t\in[0, t_1, t_2,\dots]\) as

\[\hat{H} (t)=\sum_{t_i\leq t}\hat{H}(t_{i},t_{i+1})\] By introducing assumptions on the functional form, we can estimate the entire hazard function. For example, we can take \(h (t)\) to be piecewise constant, so that

\[\begin{aligned} h (t)=\sum_i\mathbb{I}\{t_{i}<t<t_{i+1}\} h_i \end{aligned}\]

This corresponds to the assumption that \(H\) is piecewise linear and continuous; we are constructing a piecewise linear interpolant. Thus, for \(t\in(t_i,t_{i+1}],\) we such an interpolant \(\hat{H}\) for \(t\in[0,t_M]\) by a first-order polynomial spline with knots \(0,t_1,t_2,\dots, t_M\) and values \(\hat{H}(0), \hat{H}(t_1), \hat{H}(t_2) \dots,\hat{H}(t_M).\)

### 1.2 Nelson-Aalen estimates

a.k.a. Empirical Cumulative Hazard Function estimator.

The original Aalen paper on this is notoriously beautiful because of clever construction of a life point process and associated martingale. Clear and worth reading. Spoiler: despite the elegant derivation, the actual estimator is something a high-school student could discover by guessing.

TBC.

## 2 Other reliability stuff

Reliawiki has handy stuff, e.g. comprehensive docs on the Weibull law. It’s in support of some software package they are trying to sell, I think?

We can calculate an “effective age” if we want an intuitive risk measure (Brenner, Gefeller, and Greenland 1993).

## 3 tools

- For python sebp/scikit-survival: Survival analysis built on top of scikit-learn.
- For R Emily Zabor’s tutorial is a goofy intro to the large R survival ecosystem

## 4 Score function versus hazard function

- The score function and log-hazard rates are similar beasts. We can exploit that, e.g. in a Langevin dynamics algorithm? But would we gain anything useful from that?

## 5 Incoming

- Michael Betancourt’s survival analysis intro: Outwit, Outlast, Outmodel

Social Desirability Bias: How Psych Can Salvage Econo-Cynicism

Social desirability bias is the tendency of respondents to answer questions in a manner that will be viewed favourably by others. It can take the form of over-reporting “good behaviour” or under-reporting “bad,” or undesirable behaviour. The tendency poses a serious problem with conducting research with self-reports, especially questionnaires. This bias interferes with the interpretation of average tendencies as well as individual differences.

## 6 References

*The Annals of Statistics*.

*Survival and Event History Analysis: A Process Point of View*. Statistics for Biology and Health.

*Statistical models based on counting processes*. Springer series in statistics.

*Wiley StatsRef: Statistics Reference Online*.

*Wiley StatsRef: Statistics Reference Online*.

*Applied Survival Analysis*.

*Epidemiology (Cambridge, Mass.)*.

*Journal of the Royal Statistical Society: Series B (Methodological)*.

*Analysis of Survival Data.*

*Journal of Chronic Diseases*.

*Wiley StatsRef: Statistics Reference Online*.

*Journal of the American Statistical Association*.

*Gastroenterology & Hepatology*.

*Journal of Machine Learning Research*.

*The Annals of Statistics*.

*International Statistical Review / Revue Internationale de Statistique*.

*Survival Analysis: State of the Art*. Nato Science 211.

*Applied Survival Analysis: Regression Modeling of Time to Event Data*. Wiley Series in Probability and Statistics.

*Applied Survival Analysis: Regression Modeling of Time-to-Event Data*. Wiley Series in Probability and Statistics.

*Wiley StatsRef: Statistics Reference Online*.

*Survival Analysis: A Self-Learning Text*. Statistics for Biology and Health 1.0.

*Journal of the American Statistical Association*.

*Biometrika*.

*Journal of Quality Technology*.

*Technometrics*.

*arXiv:1905.09690 [Cs, Stat]*.

*Applied Survival Analysis*.

*Annual Review of Statistics and Its Application*.

*Journal of the American Statistical Association*.

*Journal of Machine Learning Research*.

*Journal of the American Statistical Association*.

*Journal of Applied Probability*.

*Proceedings of the 10th International Conference on Neural Information Processing Systems*. NIPS’97.

*Journal of Statistical Software*.

*Biometrics*.

*International Journal of Forecasting*.

*Statistics in Medicine*.