Time-Rescaling Diagnostics for Event Histories

Why Rescale Event Times?

A fitted intensity model describes how quickly events are expected to occur as time passes and the observed history changes. A useful diagnostic asks whether the event times, measured on the model’s own cumulative-intensity scale, look like events from a process with a constant rate of one. If the model captures the timing adequately, long periods of high fitted intensity should count as more exposure on this scale than equally long periods of low intensity.

This is a diagnostic, not a way to prove that a model is correct. It assesses the fitted timing mechanism against a clear reference pattern and can reveal systematic departures such as gaps that are too short, clustering, or unmodelled time trends. It complements martingale residuals: rescaling focuses on the spacing and distribution of event times rather than on accumulated signed residuals.

Definition and Conditions

Let \(N(t)\) count events observed by time \(t\), and let \(\lambda(t\mid\mathcal{F}_{t-})\) be its conditional intensity given the observed history immediately before \(t\). The intensity may depend on time-varying covariates and past events, provided those inputs are known just before they affect the event rate. Its integrated intensity, or compensator, is

If observation starts at \(L\), use exposure accumulated from \(L\), or from the previous event for a gap calculation. For event times \(T_1<T_2<\cdots\), define the transformed event times and gaps by

where \(T_0=0\) when observation begins at time zero. With delayed entry at \(L\), the first gap is \(\int_L^{T_1}\lambda(s\mid\mathcal{F}_{s-})\,ds\).

The key conditions are that the intensity is the correct conditional intensity for the observed history, event times are represented by a simple continuous-time process (without simultaneous jumps), and the cumulative intensity accounts for the actual observation window and risk set. A zero fitted intensity contributes no transformed exposure. If a subject leaves observation or experiences a terminal event, rescaling stops there; it does not create exposure after follow-up ends.

Why the Exponential Reference Appears

The result follows by expressing the conditional probability of no event in terms of integrated intensity. Given the history at a starting time, let \(T\) be the next event time. Under the conditional-intensity model, the probability of surviving without an event to time \(t\) is the exponential of minus the accumulated intensity. Write \(x\) for accumulated intensity since the start. At the time when that accumulation reaches \(x\), the probability of no event yet is \(e^{-x}\). This is exactly the survival function of an Exponential random variable with rate 1.

Applying the same argument after each event, conditional on the updated history, gives unit-rate exponential gaps. Their conditional distribution does not depend on the past, so the gaps are independent under the model. Adding these gaps gives the Poisson-process representation: the \(j\)th transformed event time \(U_j\) has a Gamma distribution with shape \(j\) and rate 1, and the number of transformed events over an exposure interval of length \(u\) has a Poisson distribution with mean \(u\). These statements concern the true conditional intensity; using estimated intensities introduces an important qualification discussed below.

Practical Diagnostic Patterns

If the model is adequate and its parameters were known, transformed completed gaps should show the rate-1 exponential distribution: mean 1, variance 1, and survival probability \(e^{-x}\) beyond \(x\). Transformed event times should follow the corresponding Gamma distributions, and transformed counts over exposure \(u\) should have mean and variance \(u\). These are reference patterns, not guarantees that any finite data set will match them exactly.

Useful graphical checks include an exponential quantile plot of the ordered transformed gaps, a plot of cumulative transformed counts against transformed exposure, and checks of transformed counts across comparable subjects or time windows. For an exponential quantile plot, the theoretical quantile at probability \(p\) is \(-\log(1-p)\). A roughly straight pattern supports the marginal gap distribution, while curvature or isolated departures may indicate systematic lack of fit or unusual observations. A cumulative-count plot can reveal intervals with too many or too few events relative to accumulated fitted exposure.

These checks are complementary. A gap distribution can look plausible even when the model misses a pattern in event timing relative to covariates, while an aggregate count check may hide local clustering. Examine diagnostics against scientifically relevant features, such as calendar time or covariate values, without treating every visual irregularity as proof of misspecification.

Competing Event Types and Event Marks

For multiple event types that can occur at most one at a time, let \(\lambda_k(t\mid\mathcal{F}_{t-})\) be the cause-specific intensity for type \(k\). The total intensity while the subject remains at risk is \(\lambda_{\mathrm{tot}}(t)=\sum_k\lambda_k(t)\). Rescaling the time to the first event using the integrated total intensity assesses the timing of any event. Conditional on an event at time \(t\), the model assigns type \(k\) probability \(\lambda_k(t)/\lambda_{\mathrm{tot}}(t)\), when the total intensity is positive. Thus timing and type are distinct diagnostic components: plausible total event timing does not by itself establish that the model assigns event types correctly. After a terminal competing event, follow-up for that subject ends.

Fitted Intensities, Censoring, and Failure Modes

In practice, the intensity is usually estimated from the same data being checked. Substituting estimated parameters into \(\widehat{\Lambda}(t)\) does not make the transformed gaps exactly independent Exponential\((1)\) variables in finite samples. Parameter fitting can pull diagnostics toward a good fit, and uncertainty from estimation affects reference bands and formal goodness-of-fit tests. Treat exponential and Poisson patterns as calibration targets, not as exact finite-sample laws for plug-in residuals. When a formal test or uncertainty band is needed, its construction should reflect parameter estimation and the study’s risk sets and censoring; a naive reference distribution can be misleading.

Censoring must also be represented correctly. If observation ends before the next event, the final transformed gap is right-censored at the integrated intensity accumulated by the end of observation. It is incorrect to include that exposure as though an event occurred there, or to omit it from a count-versus-exposure diagnostic. Delayed entry similarly means that exposure starts when the subject enters observation, not at an unobserved earlier time.

Common warning signs include systematic trends in transformed gaps, persistent clustering, a transformed-count curve that repeatedly rises faster or slower than its reference, or departures concentrated in particular covariate or calendar-time ranges. Such patterns can point to a misspecified baseline intensity, omitted time dependence, an inadequate history effect, an incorrectly handled risk period, or an error in the fitted intensity calculation. Diagnostics identify tension with the model; they do not uniquely identify its cause.

Time rescaling is most informative when the event process is continuous and the conditional intensity is well defined. Simultaneous recorded events require special treatment rather than a routine application of the simple-process result. In addition, multiple rescaled event-type streams need not be independent merely because each stream is assessed against a unit-rate reference; shared histories and estimation can link their diagnostics. The central practical advantage is a common scale for checking timing across histories with changing fitted rates. Its limitation is equally important: a good-looking transformed pattern does not establish causal interpretation, prove model correctness, or compensate for an incorrectly specified observation process.