Discrete-Time Survival Models With Random Effects

From event times to interval risks

When follow-up is recorded at scheduled visits, the exact event time may not be known. Instead, the data may indicate that a person experienced the event between two visits, or remained event-free through a visit. A discrete-time survival model describes these outcomes directly as a sequence of event indicators. In an NLME model, subject-specific random effects can account for persistent differences in event risk among individuals.

The key modeling step is to connect the chance of an event in an interval to an underlying hazard rate. If the hazard is constant within an interval, the probability of an event is not the rate itself: it is one minus the probability of surviving the interval. This relationship leads to the complementary log-log link.

Each subject contributes intervals only until the event or the end of observed follow-up. After an event, that subject is no longer at risk for that event and contributes no further interval indicators. Right-censored subjects contribute their event-free intervals up to censoring. If censoring occurs partway through an interval and the censoring time is known, the exposure interval should end at that time rather than treating the person as observed event-free through the scheduled interval end.

Deriving the complementary log-log link

Let \(\lambda_{ij}(t\mid b_i)\) denote the event hazard for subject \(i\) at time \(t\), conditional on their random effects \(b_i\) and covariate history. The hazard is an instantaneous event rate among those still at risk. Suppose it is constant over interval \(j\), with value \(\lambda_{ij}\), and that the subject is observed over the full interval. The conditional probability of surviving the interval is then the exponential of minus the cumulative hazard over that interval.

For a log-linear hazard model, write \(\log(\lambda_{ij})=\eta_{ij}\), where a typical predictor is \(\eta_{ij}=\alpha_j+x_{ij}^{T}\beta+z_{ij}^{T}b_i\). Here \(\alpha_j\) is an interval-specific baseline log hazard, \(x_{ij}\) and \(z_{ij}\) are covariate vectors, \(\beta\) is a vector of fixed effects, and \(b_i\) is a subject-level random effect. Taking the complementary log-log transformation gives the model’s link directly.

Thus the complementary log-log (cloglog) link is \(g(p)=\log\{-\log(1-p)\}\), and \(\log(\Delta_{ij})\) is an offset when interval lengths vary. The offset has coefficient fixed at one; it accounts for the time at risk and is not an estimated covariate effect. If all intervals have the same length, its value can be absorbed into the interval-specific baseline terms.

This derivation requires a constant hazard within each modeled interval, conditional on covariates and random effects. More generally, if the hazard varies within an interval, the event probability is \(1-\exp\{-\int_{t_{i,j-1}}^{t_{ij}}\lambda_i(t\mid b_i)\,dt\}\). A single constant-rate expression is justified only when that integrated hazard is represented appropriately. Covariates that change within a wide interval may require splitting the interval or calculating the integrated hazard across their changes.

Bernoulli likelihood and random effects

Conditional on \(b_i\), each at-risk interval has a Bernoulli likelihood contribution. For observed intervals \(j=1,\ldots,J_i\), the subject’s conditional likelihood is the product of event probabilities for intervals with an event and survival probabilities for event-free intervals. In a single-event model, at most one observed interval has \(Y_{ij}=1\); all subsequent intervals are absent.

The random effects are usually assigned a distribution such as \(b_i\sim N(0,\Omega)\), where \(\Omega\) is their covariance matrix. The likelihood for the observed data from subject \(i\) is obtained by integrating the conditional likelihood over that distribution:

This integration matters: the conditional model describes risk for a given random effect, while the integrated likelihood describes the data under the assumed distribution of subject effects. The subject effects are shared across that subject’s intervals, so the interval observations are generally dependent after the random effects are integrated out, even when their conditional likelihood factors as a product.

Interpreting conditional and population risks

A coefficient in the linear predictor changes the log hazard, conditional on the random effects and other covariates. For example, a one-unit increase in a covariate with coefficient \(\beta_k\) multiplies the conditional hazard by \(\exp(\beta_k)\), if the model includes that covariate linearly and the other terms are held fixed. It does not generally multiply the interval risk by that factor, because risk is a nonlinear function of both hazard and interval length.

An individual’s interval risk is \(p_{ij}(b_i)\). A population-average risk instead averages that probability over the random-effect distribution. In general, the population-average risk is not the conditional risk evaluated at \(b_i=0\): nonlinear averaging does not commute with transformation. For a particular population prediction, the average also depends on which covariates and interval length are held fixed.

Assumptions, diagnostics, and alternatives

The interval likelihood relies on a defined at-risk set, correctly ordered follow-up intervals, and event indicators that represent whether an event occurred in each observed interval. The event process must be modeled conditional on the included covariates and random effects. Censoring should be independent of the event process conditional on the modeled information, or otherwise handled through an appropriate likelihood. A shared random effect also assumes that the chosen distribution captures relevant persistent subject-level heterogeneity.

Useful checks are specific to these assumptions and the interval representation. Compare observed and model-predicted event proportions across time intervals and across important covariate groups. Inspect whether residual discrepancies concentrate in particular intervals, which may signal an inadequate baseline pattern or an interval length too coarse to represent changing risk. Check whether the assumed random-effect distribution and fitted variability are plausible; a model with negligible estimated variability may not support strong claims about subject heterogeneity. Also verify that interval lengths and offsets are correct, especially when follow-up schedules differ.

The main link alternative is the logit link, which models the log odds of an event in an interval. It can be useful as a direct discrete-time probability model, but its coefficients describe conditional odds ratios rather than hazard ratios, and it does not arise exactly from a piecewise-constant continuous-time hazard. When risks are small, logit and cloglog probabilities may be numerically similar; that approximation does not make their interpretations identical. A continuous-time survival model is another alternative when event times and a continuous-time hazard are central. The interval model is especially natural when event occurrence is only known between scheduled observations, but it does not recover the precise event time within an interval.

A further limitation is that the cloglog derivation does not remove the need to choose a useful time partition. Very wide intervals may conceal important within-interval changes; very fine intervals may create sparse event indicators and many baseline parameters. The piecewise-constant hazard is an assumption on each interval, not a claim that the underlying biological event process literally changes abruptly at every boundary.