Interval Width Effects in Pooled Logistic Survival MSMs

Why Interval Width Matters

A pooled logistic survival MSM models whether an event occurs within each interval among individuals still at risk at its start. Its treatment coefficient therefore describes an interval odds ratio, not automatically a continuous-time hazard ratio. The distinction matters whenever interval widths differ, or when the same follow-up is divided into different numbers of intervals: even if the underlying treatment effect on the hazard is constant, the interval odds ratio can change with interval width.

We derive this dependence for a simple continuous-time model, then examine what it implies for model specification and interpretation. The calculations also show why a coefficient can change when intervals are regrouped even though the underlying event process and cumulative risks have not changed.

From a Continuous-Time Hazard to an Interval Odds Ratio

Let interval \(j\) run from \(c_{j-1}\) to \(c_j\), with width \(\Delta_j=c_j-c_{j-1}\). Let \(Y_{ij}=1\) indicate that individual \(i\) experiences the event in that interval, conditional on being event-free and at risk at \(c_{j-1}\). For treatment level \(a\), write the cumulative hazard over the interval as \(H_{aj}=\int_{c_{j-1}}^{c_j}\lambda_a(t)\,dt\). Assume a continuous event time, no competing event, and a hazard \(\lambda_a(t)\) that correctly describes the event process under treatment level \(a\). The event probability in the interval is \(p_{aj}=1-\exp(-H_{aj})\).

The hazard-to-risk relationship gives a direct expression for these odds. Since \(1-p_{aj}=\exp(-H_{aj})\), the odds equal \(\{\;1-\exp(-H_{aj})\;\}/\exp(-H_{aj})=\exp(H_{aj})-1\). Thus the interval odds depend on the cumulative hazard in that interval, and so generally depend on its width.

Suppose treatment has a constant proportional effect on the continuous-time hazard: \(\lambda_1(t)=r\lambda_0(t)\), where \(r>0\) is the hazard ratio. Then \(H_{1j}=rH_{0j}\), and \(\operatorname{OR}_j=\{\exp(rH_{0j})-1\}/\{\exp(H_{0j})-1\}\). This is not generally \(r\). The interval odds ratio is a function of the baseline cumulative hazard in the interval, which usually grows as the interval becomes wider.

To verify the small-interval limit, put \(x=H_{0j}\). The Taylor expansion \(\exp(x)-1=x+O(x^2)\) gives \(\{\exp(rx)-1\}/\{\exp(x)-1\}=\{rx+O(x^2)\}/\{x+O(x^2)\}\), which tends to \(r\) as \(x\) decreases to zero. For the strict comparison at positive \(x\), the function \(f(s)=\exp(sx)-1\) is strictly convex, with \(f(0)=0\). If \(r>1\), strict convexity implies \(f(1)<f(r)/r\), so \(f(r)/f(1)>r\). If \(0<r<1\), it implies \(f(r)<rf(1)\), so \(f(r)/f(1)<r\). These are the claimed comparisons. \(\square\)

What a Pooled Logistic MSM Estimates

With baseline treatment \(A_i\), a pooled logistic MSM with interval-specific intercepts and a common treatment coefficient has the form \(\operatorname{logit}P(Y_{ij}=1\mid R_{ij}=1,A_i)=\alpha_j+\beta A_i\), where \(R_{ij}=1\) indicates that individual \(i\) is at risk at the start of interval \(j\). In an MSM, treatment and, when needed, censoring weights are used to represent the target pseudo-population. A causal interpretation requires the relevant consistency, exchangeability, and positivity assumptions, adequate handling of censoring, and appropriate weights; the weighted outcome model must also represent the target interval risks adequately.

Under a continuous-time proportional-hazards process, the interval-specific log odds ratio implied by the process is \(\beta_j=\log\{\{\exp(rH_{0j})-1\}/\{\exp(H_{0j})-1\}\}\). A common \(\beta\) is therefore generally not the exact interval-specific effect when the \(H_{0j}\) differ. If that common-effect model is fitted anyway, its estimate is a compromise determined by the interval-specific risks, risk sets, weights, and the fitted model. It is not, in general, a fixed transformation of the continuous-time hazard ratio.

Worked Consequences of Unequal and Split Intervals

Model Checks, Alternatives, and Limitations

When intervals are unequal, or when baseline event rates vary substantially over time, inspect whether the model’s treatment effect is plausibly constant on the interval log-odds scale. A useful diagnostic is to fit or examine treatment-by-interval terms and compare the interval-specific estimates with their uncertainty. Large systematic differences, especially differences associated with interval width or baseline event frequency, can indicate that a common pooled-logistic coefficient is an inadequate summary. Also check whether conclusions change when a scientifically defensible alternative interval partition is used. This is a sensitivity check, not a guarantee that one partition is correct.

The main alternatives answer somewhat different questions. A continuous-time Cox model targets a hazard-ratio description on the continuous-time scale and does not define its treatment coefficient through chosen event intervals. A pooled logistic model is useful when the discrete interval risk is itself the target or when the analysis is naturally organized in discrete time. In either case, adequate weighting, risk-set construction, and uncertainty estimation remain necessary. The introductory material on the continuous-time limit of pooled logistic models explains why short intervals can make the logistic and hazard comparisons numerically close; that approximation does not make them identical for finite intervals.

Interval width is not merely a formatting choice if the fitted coefficient is interpreted as an interval odds ratio. Coarsening combines event-free time and events into wider intervals, while fine partitioning creates smaller interval risks and odds ratios closer to hazard ratios. Both choices can be reasonable, but the interpretation must match the interval definition. Unequal intervals deserve particular care: a common treatment coefficient may obscure genuine width-related differences, while a single reported coefficient can be sensitive to how the follow-up has been partitioned.