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Survival Models

Time-to-event models, unified by one idea — censoring as a probability mass over the region an event time is known to lie in. The arc starts with Weibull proportional hazards (parametric, right-censored, fit by MLE and Metropolis), extends it two ways, and generalises the censoring. The frailty model adds a per-subject random multiplier on the hazard — the hierarchical/random-effects extension, literally the survival likelihood × the hierarchical-Poisson sampler. The Cox-via-Poisson bridge fits the semiparametric Cox model as an ordinary Poisson GLM on expanded person-time (the piecewise-exponential = Poisson identity), making the recurring “survival is a count model” theme explicit. And interval-censored survival completes the censoring tour — one likelihood S(l) − S(u) that covers right, left, interval, and exact observations at once. Each is built from scratch and cross-checked across PyMC and R.

What time-to-event data is, and why censoring is unavoidable. The outcome here is how long something takes — until a patient relapses, a shock absorber fails, a tumour appears. What makes that its own subject, rather than ordinary regression on a number, is that you cannot wait forever. Studies end, patients move away, vehicles are taken off the road; and when follow-up stops, some subjects simply have not had the event yet. Their observation is censored: what you know is that the event time is greater than the number you wrote down, not what it is.

Both obvious responses are wrong, and wrong in opposite directions. Discard the censored subjects and you have thrown away precisely the ones who lasted longest — in the leukaemia trial below, the 12 of 21 treated patients still in remission when the study ended are the evidence that the drug worked, so deleting them would delete the finding. Treat still in remission at 60 weeks as though it meant relapsed at 60 weeks and you have invented events that never happened, biasing the other way. Neither is a small approximation: in the reliability example 61% of the sample is censored, and in the mouse study every single observation is.

The way out is to be honest about what each subject actually tells you. One who relapsed in week 8 contributes the probability of an event at week 8. One still event-free at week 60 contributes the probability of surviving past 60 — a whole tail of the distribution rather than a point. Both are perfectly good factors in a likelihood; they are simply different shapes. That single idea — a censored observation contributes the probability mass over the region its event time is known to lie in — is what the whole section rests on, and the interval-censored page at the end shows one formula covering every variety of it at once.

The hazard, which is what every result here is measured in. The natural way to describe a lifetime is the chance of surviving past each moment — the survival function S(t), which starts at 1 and falls. But survival models are built on its derivative in disguise, the hazard: the rate at which the event strikes at time t, among those who have made it that far. That last clause is the whole point. The number of relapses in week 30 depends on how many patients are still in remission at week 30, which is a shrinking and continually changing pool; dividing by it strips that out and leaves a quantity that means the same thing at every time. So a hazard of 0.05 per week says that of the patients still event-free, about 5% relapse in the coming week — whether there are 40 of them left or 4. Hazard and survival carry the same information: accumulate the hazard over time and exponentiate the negative, and you get S(t) back.

Hazard ratios, and the assumption behind them. Every headline number in this section is a hazard ratio — the factor by which a covariate multiplies that rate. A treatment with a hazard ratio of 0.18 leaves patients relapsing at roughly a fifth the rate of controls at any given moment. What makes a single such number meaningful is the proportional-hazards assumption, which names two of the pages below: the covariate multiplies the hazard by the same factor at all times, so the two groups’ hazard curves are vertical shifts of one another on a log scale and never cross. It is a real assumption and it can fail — a surgery that kills some patients immediately but protects the survivors has a hazard ratio above 1 early and below 1 later, and no single number describes it. Where it holds, one coefficient summarises a whole curve; that economy is why the assumption is made so often, and why it is worth remembering that it was made.

Parametric, semiparametric, nonparametric. The three pages that fit covariates differ chiefly in how much they are willing to assume about the baseline hazard — the shape of risk over time before any covariate acts. The Weibull model is parametric: the baseline is a specific curve governed by one shape parameter, which buys precision and extrapolation beyond the observed range at the cost of being wrong if the curve is wrong. The Cox model is semiparametric: the covariate effects are a parametric multiplier, but the baseline is left entirely unrestricted, so nothing at all is assumed about how risk evolves. And the Kaplan–Meier curve is nonparametric — no model, just the observed survival fraction stepping down at each event and pausing at each censored observation, which is why it appears on nearly every figure here as the thing a fitted model has to reproduce.