The Continuous-Time Limit of Pooled Logistic Models

Why Interval Width Matters

A pooled logistic model represents the probability of an event in an interval among individuals who are still at risk at its start. A continuous-time hazard, by contrast, describes an instantaneous event rate. These are related quantities, but they are not identical: a probability accumulates over time, and the odds of that probability are not generally the instantaneous hazard. The connection becomes close when intervals are short enough that events are rare within each interval.

The aim here is to state that connection precisely. We derive the exact interval relationship, show the limiting contrast, and calculate how large the discrepancy can be when intervals are not short. The statements apply to regime-specific hazards, including hazards targeted by a correctly identified marginal structural model (MSM). They do not by themselves establish that a fitted weighted model identifies those hazards; that interpretation still depends on the causal identification assumptions and appropriate weights.

From Interval Event Probability to Cumulative Hazard

This relationship assumes a well-defined, nonnegative, locally integrable hazard and the usual continuous-time survival model on the interval. It follows from the conditional survival probability \(\exp\{-H_a(t,\Delta)\}\). In particular, the interval probability depends on the hazard over the whole interval, not just its value at the beginning or end.

For \(0<p_a<1\), the interval log odds have an exact expression:

The last equality is exact: \(1-p_a=\exp(-H_a)\), so \(p_a/(1-p_a)=\exp(H_a)-1\). It is not generally true that interval log odds equal the log cumulative hazard, let alone the log instantaneous hazard. For small \(H\), however, \(\exp(H)-1\) is close to \(H\), which is the source of the rare-event approximation.

The Shrinking-Interval Result

Proof. Define \(g(x)=\log\{(\exp(x)-1)/x\}\) for \(x>0\). Using the exact log-odds identity, for either regime \(a\),

Continuity at \(t\) gives

Since \(\lambda_a(t)>0\), this also implies \(H_a(t,\Delta)\to0\) and \[ \frac{H_1(t,\Delta)}{H_0(t,\Delta)} \longrightarrow \frac{\lambda_1(t)}{\lambda_0(t)}. \] Furthermore, \((\exp(x)-1)/x\to1\) as \(x\to0\), so \(g(x)\to0\). Subtracting the two log-odds expressions therefore gives

This proves the result. \(\square\)

The positivity condition matters: if a regime’s instantaneous hazard is zero at \(t\), its log hazard ratio may be undefined, and the stated finite limit need not hold. Continuity is a convenient sufficient condition; the key local property is that the interval-average hazards converge to finite, positive values at \(t\).

Size of the Finite-Interval Correction

A Taylor expansion makes the approximation more informative than the limit alone. Near zero,

Equivalently, \(g(x)=x/2+x^2/24+O(x^4)\). Thus the difference between the interval log-odds contrast and the log cumulative-hazard ratio is \(g(H_1)-g(H_0)\). When both cumulative hazards are small, its leading term is \((H_1-H_0)/2\). If the hazards are constant over an interval, \(H_a=\lambda_a\Delta\), so the correction tends to zero with \(\Delta\). At a non-negligible interval width, the correction can matter, particularly when the cumulative hazards differ substantially.

Reading Pooled-Logistic Contrasts Carefully

In a pooled logistic MSM, rows represent person-intervals and the event indicator records whether an event occurred during that interval. A fitted treatment coefficient is a log-odds contrast under the model’s specification, on the model’s chosen scale and with its other terms held as specified. The shrinking-interval result explains when this contrast can approximate a log hazard ratio: the interval must be short, the interval hazards must be small, and the fitted model must appropriately represent the regime-specific interval probabilities.

The derivation also distinguishes two sources of error. First, even if the interval probabilities are known exactly, finite interval widths create a logistic-versus-hazard correction. Second, a fitted contrast can differ from the true interval log-odds contrast because of model misspecification, inadequate representation of time, or unstable estimation. Shorter intervals address the first issue, but do not automatically fix the second or ensure valid causal identification.

When the hazard varies within an interval, the relevant cumulative hazard is an integral. Dividing it by \(\Delta\) gives the interval-average hazard. Only as the interval shrinks, under suitable local regularity, does that average approach the instantaneous hazard at the interval’s time point. Therefore, coarse intervals can obscure changing effects or yield an interval contrast that does not represent the local effect of interest.

Diagnostics, Alternatives, and Limitations

• Inspect event frequencies by interval. Frequent events within intervals warn that the rare-event approximation may be poor. The relevant probability is conditional on being at risk at the interval start.
• Check sensitivity to interval width. If the scientific time scale supports it, refit with a finer partition and compare the estimated contrasts. Large changes can indicate a meaningful width effect or other model changes; they are not proof of one specific cause.
• Represent time flexibly enough. A single treatment coefficient assumes a constant contrast on the model’s log-odds scale unless interactions or time-varying terms are included. If effects vary over time, a single number may conceal that variation.
• Keep the estimand explicit. A logistic coefficient is an interval log-odds contrast. Calling it a hazard ratio requires the short-interval approximation and an appropriate model; calling it a causal effect additionally requires the causal identification assumptions.

The complementary log-log link is a standard alternative for modeling interval probabilities through cumulative hazards under a piecewise-constant hazard representation. It is useful when the target parameterization is directly hazard-based, but choosing a link does not remove the need to define intervals, model time, or justify causal identification. The continuous-time limit derived here is a mathematical approximation, not a claim that every pooled logistic fit is equivalent to a continuous-time survival model.

A common mistake is to say, “The event probability is small, so the logistic coefficient equals the hazard ratio.” The correction is twofold: coefficients are on a log scale, and the exact link is \(\log\{\exp(H)-1\}\), not \(\log(\lambda)\). Only after taking a contrast, imposing the stated regularity conditions, and shrinking the interval does the contrast converge to the log hazard ratio. With wide intervals, changing hazards, or poor model fit, the approximation may be inadequate.