Pooled Logistic Models for Discrete-Time Survival MSMs

Why Pool Person-Intervals?

In discrete-time survival data, follow-up is divided into intervals, and each person contributes an event indicator for each interval in which they are still at risk. A pooled logistic model uses these person-interval records to describe how the event hazard varies over time and across treatment regimes. “Pooled” means that the records are analyzed together in one regression, rather than fitting a separate regression for every interval.

For a marginal structural model (MSM), the target is a marginal hazard under a specified treatment regime, not a hazard conditional on adjusting for the time-varying confounders that motivated weighting. The outcome regression is therefore written in terms of regime features and time. Weights, estimated from the treatment and censoring processes, are used when fitting the model; this tutorial concerns the outcome-model specification and its interpretation, not how those weights are constructed.

Definition: Let $t=1,\ldots,K$ index discrete follow-up intervals, and let $T^g$ be the event time that would be observed under treatment regime $g$. The regime-specific discrete-time marginal hazard is $$ h^g(t)=P(T^g \text{ occurs in interval }t \mid T^g \text{ has not occurred before interval }t). $$ A pooled logistic MSM specifies the log odds of this hazard as a function of time and features of $g$: $$ \operatorname{logit}\{h^g(t)\}=\alpha(t)+\boldsymbol{\beta}^{\mathsf T}\mathbf{q}(g,t), $$ where $\operatorname{logit}(p)=\log\{p/(1-p)\}$, $\alpha(t)$ is a prespecified time component, and $\mathbf{q}(g,t)$ contains treatment-regime features.

In the observed data, define $R_{it}=1$ when person $i$ is event-free and under observation at the start of interval $t$. For each such person-interval, $Y_{it}=1$ if the event occurs during that interval and $Y_{it}=0$ otherwise. A person stops contributing event-risk records after an event; intervals after censoring are not treated as observed event-free intervals. Thus each included record has one binary outcome, and the model’s probability refers to an event in that interval among those still at risk at its start.

The regime feature $\mathbf{q}(g,t)$ must encode the treatment contrast the analysis intends to estimate. It might represent treatment assigned in the current interval, treatment received so far, or another prespecified summary of the treatment history. A summary that omits consequential earlier exposure cannot generally represent a regime effect that depends on that exposure. The time component and treatment features together must be rich enough for the scientific contrast; their form is a modeling choice, not a consequence of weighting.

What the Coefficients Mean

Suppose, for example, the model uses a binary treatment feature $A$, with $A=1$ for the regime feature being contrasted and $A=0$ for its reference: $$ \operatorname{logit}\{h^g(t)\}=\alpha(t)+\beta A. $$ The treatment coefficient $\beta$ compares the interval event odds for the two feature values at the same time. It is not, in general, a risk ratio or a difference in event probabilities.

Result: Under the specified logistic model, changing a binary treatment feature from 0 to 1 multiplies the interval event odds by $\exp(\beta)$, holding time and all other model features fixed.

Write the odds at feature value 0.
They are $\exp\{\alpha(t)\}$, because odds equal $\exp\{\operatorname{logit}(h)\}$.

Write the odds at feature value 1.
They are $\exp\{\alpha(t)+\beta\}$.

Take the ratio.
$\exp\{\alpha(t)+\beta\}/\exp\{\alpha(t)\}=\exp(\beta)$. Thus the odds ratio is constant over time if the model contains no treatment-by-time interaction. $\square$

For a model with a treatment-by-time interaction, the contrast can vary by interval. For example, if $\operatorname{logit}\{h^g(t)\}=\alpha(t)+\beta A+\delta A s(t)$, where $s(t)$ is a specified time function, then the treatment odds ratio at interval $t$ is $\exp\{\beta+\delta s(t)\}$. This follows by subtracting the log odds at $A=0$ from those at $A=1$, then exponentiating. The coefficient $\beta$ alone is the contrast only where $s(t)=0$.

The odds-scale interpretation follows exactly from the model; a risk-scale interpretation does not. If the reference hazard is $p_0$ and the odds ratio is $r$, then the other hazard is $p_1=rp_0/(1-p_0+rp_0)$. It equals $rp_0$ only in special cases or as a rare-event approximation. This distinction matters even though the model is marginal: “marginal” identifies the target hazard, not the scale of the regression coefficient.

Worked Specifications

Worked Example: A Constant Treatment Contrast

Suppose the fitted model is $\operatorname{logit}\{h^g(t)\}=\alpha(t)+0.693A$, where $A=1$ and $A=0$ denote the two regime features. At a particular interval, suppose the reference hazard is $0.05$. Find the treatment-feature hazard and interpret the coefficient.

The reference odds are $0.05/(1-0.05)=0.05/0.95=0.05263$. Since $\exp(0.693)$ is approximately $2$, the treatment-feature odds are $2(0.05263)=0.10526$. Converting those odds back to a probability gives $0.10526/(1+0.10526)=0.09524$.

The estimated interval hazard is therefore about $0.095$, not $0.10$. The coefficient says the interval event odds are doubled; it does not say that the event probability is doubled. Here the probability ratio is approximately $0.09524/0.05=1.90$.

Worked Example: Specifying a Two-Interval Regime Contrast

Consider two intervals and two sustained regimes: treatment in both intervals ($g_1$) and no treatment in either interval ($g_0$). Suppose the intended model has interval-specific baseline hazards and a common regime contrast: $\operatorname{logit}\{h^g(t)\}=\alpha(t)+\beta I(g=g_1)$. At interval 1, let the estimated reference hazard be $0.10$, and let $\beta=\log(1.5)$. What does the model imply?

The interval-1 reference odds are $0.10/0.90=1/9$. Under $g_1$, the odds are multiplied by $1.5$, giving $1.5/9=1/6$. The corresponding hazard is $(1/6)/(1+1/6)=1/7$, approximately $0.143$. Thus the model estimates a 1.5-fold contrast in interval event odds at interval 1.

The same coefficient gives a 1.5-fold odds contrast in interval 2 because the specification has no treatment-by-time interaction. It does not require the two regime-specific hazards to be equal across intervals: $\alpha(1)$ and $\alpha(2)$ may differ. If the scientific question allows the regime contrast itself to change over time, this common-contrast specification would need an interaction feature.

Worked Example: Choosing a Treatment-History Feature

Suppose an event hazard in interval 2 may depend on whether treatment was received in interval 1, even if treatment is absent in interval 2. Compare a model using only current treatment $A_{i2}$ with one that also uses prior treatment $A_{i1}$.

The current-treatment-only model might be $\operatorname{logit}\{h^g(2)\}=\alpha(2)+\beta A_{i2}$. It assigns the same interval-2 log odds to histories $(A_{i1},A_{i2})=(0,0)$ and $(1,0)$, because both have $A_{i2}=0$. That restriction rules out any remaining association of the interval-1 treatment feature with the interval-2 hazard.

A model allowing that contrast is $\operatorname{logit}\{h^g(2)\}=\alpha(2)+\beta_1 A_{i1}+\beta_2 A_{i2}$. For histories $(0,0)$ and $(1,0)$, the fitted log-odds difference is $\beta_1$, so their interval-2 hazard odds ratio is $\exp(\beta_1)$. This calculation shows why the history features must reflect the treatment regimes and carryover patterns of interest; it does not establish that either model is scientifically correct.

Assumptions, Diagnostics, and Alternatives

A fitted coefficient has a causal MSM interpretation only under the causal assumptions needed for the weighted analysis. In particular, treatment strategies must be well defined and consistency must connect observed outcomes to the relevant potential outcomes. Sequential exchangeability requires that, conditional on the measured history used to construct the treatment weights, treatment assignment is independent of future potential outcomes. Positivity requires nonzero probability of following the relevant treatment choices for histories in the target population. Censoring must also be handled under appropriate assumptions if censoring weights are used. Finally, the specified marginal hazard model must adequately represent the target contrasts.

These assumptions are distinct from the logistic functional-form assumption. A correctly written logistic equation cannot repair unmeasured confounding, poor overlap, or incorrectly timed weights. Conversely, valid weights do not guarantee that a chosen time component or set of regime features captures the marginal hazards adequately.

Check the person-interval records. Verify that each at-risk interval contributes exactly one binary outcome, that event intervals are coded as events, and that no post-event or unobserved post-censoring intervals are coded as event-free.
Check treatment timing and history. Ensure each feature refers to treatment assigned or received at the intended time. Compare the feature definition with the actual regime contrast; current treatment alone may omit relevant prior treatment.
Inspect weights and overlap. Extreme weights or very few observations supporting a regime contrast can make a weighted fit unstable. A large nominal sample does not guarantee substantial information in the weighted data.
Assess model fit over time and treatment features. Systematic discrepancies between observed weighted interval event patterns and fitted hazards may indicate an inadequate time component or missing treatment interactions. This is a model diagnostic, not a test of unmeasured confounding.

A pooled logistic MSM is an alternative to a weighted Cox MSM when time is naturally represented by intervals and the target is an interval-specific marginal hazard. The discrete-time specification directly models the probability of an event within each interval. Separate logistic models for each interval are another possibility, but they do not pool information through shared treatment parameters unless constraints are imposed. The pooled model is compact, but its implications depend on the chosen time and treatment features. It is also tied to the interval definition: changing interval widths changes the event probabilities being modeled.

Common Pitfalls

Common mistake: “The treatment coefficient is $\log(1.5)$, so treatment increases the event risk by 50%.”

Correction: In a logistic MSM, $\exp(\beta)=1.5$ is a ratio of interval event odds, holding the other model features fixed. If the reference hazard is $0.10$, the implied treated hazard is $1.5(0.10/0.90)/[1+1.5(0.10/0.90)]$, approximately $0.143$, not $0.15$. The risk ratio is about $1.43$, not $1.5$.

A second error is to add time-varying confounders to the outcome regression automatically because they predict the event. In a weighted MSM, those variables are used in the weighting strategy to address treatment assignment; conditioning on them in the outcome model changes the model’s target and can undermine the intended marginal interpretation. Any additional outcome-model terms should have a clear role and should be justified for the estimand and analysis.

Check Your Understanding

Answer each question, then compare your reasoning with the solutions.

In a pooled logistic MSM, what does the binary outcome $Y_{it}=1$ represent?
A model has a binary regime feature with coefficient $\log(2)$. State the model-implied contrast.
At a reference hazard of $0.20$, a treatment feature has an odds ratio of $2$. What hazard does the model imply for that feature value?
Why might a model using only current treatment be inadequate when earlier treatment can affect the present interval’s event hazard?

Solutions. (1) It indicates that the event occurred during interval $t$, for a person who was at risk at that interval’s start. (2) The interval event odds are multiplied by 2, holding time and other features fixed; this is not necessarily a doubling of risk. (3) Reference odds are $0.20/0.80=0.25$. Doubled odds are $0.50$, which correspond to hazard $0.50/(1+0.50)=1/3$, approximately $0.333$. (4) Histories with the same current treatment but different earlier treatment would be forced to have the same fitted hazard, so the model would exclude a prior-treatment contrast.