Pooled Logistic and Cox Models Compared

Two Models, Two Time Representations

Weighted pooled logistic and weighted Cox models can analyze the same weighted longitudinal survival data, but they do not automatically estimate the same parameter. The pooled logistic model describes the probability of an event over a specified interval among those event-free at its start. The Cox model describes an instantaneous event rate among those still at risk at time \(t\). Their coefficients can be close when intervals are short and event probabilities are small, but that approximation is not an identity.

Here, \(i\) indexes individuals and \(j\) indexes intervals \((t_{j-1},t_j]\), with width \(\Delta_j=t_j-t_{j-1}\). Let \(Y_{ij}=1\) indicate that individual \(i\) is at risk at the start of interval \(j\), and let \(D_{ij}=1\) indicate an event during that interval. For clarity, formulas use a binary treatment \(A_i\); a longitudinal treatment history or regime can replace \(A_i\) with a vector of treatment terms. Let \(W_{ij}\) denote the appropriate treatment and censoring weight for that individual and risk interval.

In both models, weighting aims to create a pseudo-population in which treatment assignment is independent of measured treatment-confounding history, subject to consistency, sequential exchangeability, positivity, correct weight construction, and appropriate handling of censoring. A causal interpretation also requires the marginal structural model to be adequate for the target contrasts. The regression formula alone does not establish these conditions.

What Each Coefficient Represents

In the pooled logistic model, \(\exp(\beta_L)\) is the treatment odds ratio for an event during a particular interval, conditional on being at risk at the interval's start, when the model has a common treatment coefficient. In the Cox model, \(\exp(\beta_C)\) is the hazard ratio comparing treatment values at an instant, under the proportional-hazards specification. A hazard is a rate, not an interval probability or an odds.

Both are marginal structural contrasts when the outcome model is fit using appropriate weights and does not condition on post-treatment confounders as ordinary covariates. However, marginal hazards condition on survival up to the time being compared. The surviving groups under two regimes can differ as time passes, so a marginal hazard ratio need not be constant even when an individual-level effect is constant. The earlier material on marginal hazard ratios and noncollapsibility is relevant to that interpretation.

The weighted pooled logistic score for a model with covariate vector \(X_{ij}\) and fitted probability \(p_{ij}\) has the form \[ \sum_i\sum_j W_{ij}Y_{ij}X_{ij}(D_{ij}-p_{ij})=0. \] A weighted Cox score, written for a time-dependent covariate \(A_i(t)\), has the form \[ \sum_i\int W_i(t)\{A_i(t)-\bar A_W(\beta,t)\}\,dN_i(t)=0, \] where \(N_i(t)\) jumps when an event occurs and \[ \bar A_W(\beta,t)= \frac{\sum_i Y_i(t)W_i(t)A_i(t)\exp\{\beta A_i(t)\}} {\sum_i Y_i(t)W_i(t)\exp\{\beta A_i(t)\}}. \] The pooled logistic score compares observed events with fitted interval probabilities over person-intervals. The Cox score compares event covariates with weighted covariate averages in event-time risk sets. These are different estimating equations, even when they use the same individuals and weights.

Exact Relation Within an Interval

A proportional-hazards model gives an exact expression for the probability of an event in an interval. Suppose, for the moment, that the treatment is fixed over the interval and that the hazard under treatment \(a\) is \(\lambda^a(t)=\lambda_0(t)\exp(\beta_Ca)\). Define the baseline cumulative hazard increment \[ \Lambda_{0j}=\int_{t_{j-1}}^{t_j}\lambda_0(u)\,du. \] Under this model, the event probability conditional on being event-free at \(t_{j-1}\) and the corresponding odds are as follows.

Derivation. For a subject event-free at the interval start, survival through the interval is the exponential of minus its cumulative hazard, so \(1-p_j(a)=\exp\{-\Lambda_{0j}\exp(\beta_Ca)\}\). Therefore \[ \frac{p_j(a)}{1-p_j(a)} = \frac{1-\exp\{-\Lambda_{0j}\exp(\beta_Ca)\}} {\exp\{-\Lambda_{0j}\exp(\beta_Ca)\}} = \exp\{\Lambda_{0j}\exp(\beta_Ca)\}-1. \] Taking the ratio of these odds at \(a=1\) and \(a=0\) gives the stated expression. Unless the increments are small enough for the approximation below, the ratio varies with \(j\). \(\square\)

When \(x\) is near zero, \(\exp(x)-1\) is approximately \(x\). Thus, if the baseline increments are small, the interval odds ratio is approximately \[ \frac{\Lambda_{0j}\exp(\beta_C)}{\Lambda_{0j}}=\exp(\beta_C). \] This explains why the coefficients can be similar in short intervals with few events; it does not make the two models or estimands identical.

How Fitted Survival Contrasts Compare

A pooled logistic fit yields fitted interval hazards \(p_j(a)\); its regime-specific survival estimate through interval \(k\) is the product of the probabilities of surviving each interval. A Cox fit instead uses a baseline cumulative hazard estimate together with its treatment coefficient. The following identity explains when these survival predictions agree.

Proof. By the stated interval probability, \(1-p_j(a)=\exp\{-\Delta\Lambda_{0j}\exp(\beta a)\}\). Multiplying over intervals and using the rule that a product of exponentials is the exponential of the sum gives \[ \prod_{j=1}^k\{1-p_j(a)\} = \prod_{j=1}^k\exp\{-\Delta\Lambda_{0j}\exp(\beta a)\} = \exp\left\{-\exp(\beta a)\sum_{j=1}^k\Delta\Lambda_{0j}\right\}. \] The sum of baseline increments is the cumulative baseline hazard through \(t_k\); this is the Cox survival formula under the specified proportional-hazards model. \(\square\)

More generally, the pooled logistic survival product remains the appropriate reconstruction from its fitted discrete hazards, whether or not those hazards have a proportional-hazards representation. The equality above is conditional on the interval probabilities having the stated form. If the pooled logistic model instead imposes a common treatment odds ratio, its fitted probabilities need not satisfy that form.

Fitting and Diagnostics in Practice

For a meaningful comparison, use the same treatment history or regime contrast, the same target population, and weights appropriate to the relevant risk intervals and risk sets. The pooled logistic analysis uses person-interval records and specifies how time enters the event-probability model. The Cox analysis uses event times and weighted risk sets. Differences between estimates can therefore reflect both the different time representations and differences in model fit, not simply numerical error.

Useful checks include examining weighted event counts and risk-set sizes over time, checking whether a pooled logistic model's treatment contrast changes when the interval baseline is made more flexible, and comparing fitted regime-specific survival curves over times represented in both analyses. Large changes across plausible time specifications, unstable estimates in sparse intervals, or material disagreement in predicted survival are reasons to investigate rather than to select whichever coefficient looks preferable. Robust variance estimation addresses dependence or weighting-related uncertainty under its assumptions; it does not make a misspecified mean model correct.

A common mistake is to report \(\exp(\hat\beta_L)\) as the hazard ratio because the model was fit to survival data. That is incorrect: it is an interval event odds ratio under the pooled logistic model. Another mistake is to presume that two coefficients should match because the data and weights match. The corrections are to label each contrast with its time scale and interpretation, then compare regime-specific survival or risk over a stated horizon when an absolute contrast is of interest.

The Cox model naturally uses event-time risk sets, while pooled logistic regression accommodates an explicitly discrete time grid and interval-specific event probabilities. Neither dominates in every setting. The Cox contrast relies on a proportional-hazards representation if summarized by one constant coefficient; a pooled logistic contrast relies on its chosen discrete-time link and time specification. Both can be affected by weight instability, sparse events, model misspecification, and violations of causal identification assumptions. Their comparison is informative, but it is not a substitute for checking those assumptions.