Why Fitted Odds Are Not Discrete-Time Risks

What a Pooled Logistic Fit Predicts

In a discrete-time survival analysis, each interval contributes an event indicator for individuals who are event-free at its start. A pooled logistic model uses these records to model the conditional probability of an event in the interval. The word logistic refers to the link: the model is linear in the log odds, not in the risk itself. Confusing these quantities changes predicted probabilities, sometimes substantially.

Consider a single event type and intervals indexed by \(t\). Let \(R_t=1\) indicate that a person is at risk at the start of interval \(t\), and let \(Y_t=1\) indicate that the event occurs during that interval. For a specified treatment regime \(a\), define the interval risk among those still at risk as

This is a conditional risk: it is the probability of an event during the interval, given survival to its start. It is not the unconditional probability of having had the event by the end of the interval. A pooled logistic model represents the log odds of this conditional event as a linear predictor \(\eta_t(a)\):

In a weighted marginal structural model, the weights and model specification determine the estimand and the fitted linear predictor. They do not change the meaning of the logistic link. Under the assumptions required for the causal interpretation of the weighted model, the fitted predictor describes the log odds in the target pseudo-population. In all cases, converting the fitted log odds to a probability is a mathematical step, not an additional causal assumption.

Converting Fitted Odds to Interval Risks

The conversion follows directly from the definition. If \(o=p/(1-p)\), then \(o(1-p)=p\). Rearranging gives \(o=p(1+o)\), hence \(p=o/(1+o)\). Since \(\eta=\log(o)\), substituting \(o=\exp(\eta)\) gives the familiar inverse-logit formula. This conversion applies at each time interval and for each treatment regime or covariate pattern for which a prediction is required.

This result is an identity, not an approximation. It holds for any value of \(\eta\), regardless of whether the model is correctly specified. Correct specification matters for whether the fitted quantity estimates the intended population quantity; it does not alter the conversion from the model's linear predictor to its implied probability.

Why Odds Used as Risks Overstate the Event Probability

For any positive odds \(o\), the risk is \(p=o/(1+o)\). Subtracting the risk from the odds gives an exact expression for the error made by reporting \(o\) as if it were a probability.

To verify the result, subtract \(p=o/(1+o)\) from \(o\):

For the relative error, divide this difference by the true risk \(p\):

For example, odds of \(0.10\) overstate the risk by 10% relative to the risk; odds of \(1\) overstate it by 100%. When the risk is small, odds and risk are numerically close, which can make the mistake appear harmless. But closeness is an approximation that depends on the event being rare, not an identity. At \(p=0.5\), the odds are \(1\), twice the risk.

From Interval Risks to Survival and Cumulative Risk

A fitted interval risk is conditional on still being event-free at the interval's start. For consecutive intervals, the probability of surviving through all of them is the product of the corresponding conditional probabilities of surviving each interval. The conversion from odds to risks must therefore happen before combining intervals.

For the derivation, write \(S_K(a)=P(R_{K+1}=1\mid a)\), the probability of remaining event-free through interval \(K\). By the multiplication rule,

For each interval, no event occurs with probability \(1-p_t(a)\), so \(P(R_{t+1}=1\mid R_t=1,a)=1-p_t(a)\). Substitution yields the survival product. Since survival through \(K\) intervals and an event by the end of interval \(K\) are complementary events, cumulative event risk is one minus that product. This derivation assumes a single event type and intervals defined so that remaining at risk means no event has yet occurred.

Odds Ratios Do Not Supply Absolute Risks

A fitted treatment coefficient in a logistic model is often exponentiated and described as an odds ratio. An odds ratio compares odds between model conditions; it is not itself a risk, a risk ratio, or a risk difference. To obtain absolute risks, one needs the relevant fitted linear predictors, or equivalently the baseline odds together with the odds ratio. This distinction matters especially when reporting regime-specific predictions or absolute treatment contrasts.

Diagnostics, Alternatives, and Common Pitfalls

Several checks help prevent an interpretation error. First, label model output according to what it is: a linear predictor, a log odds, an odds value, or a risk. Second, verify that every reported interval probability lies between 0 and 1; an odds value need not. Third, when presenting cumulative risks, confirm that each interval input is a converted conditional risk, rather than an odds value or an unconditional probability. For a weighted causal interpretation, these checks do not replace the assumptions and diagnostics concerning the weights and model specification.

The main alternative is to model interval risks directly using a model with a probability-scale mean structure. That approach still requires a link or other method to keep predictions in the interval from 0 to 1, and it does not make the logistic model's output a risk without conversion. The logistic link is useful because it maps any finite linear predictor to a valid probability and gives coefficients a log-odds interpretation. The cost is that those coefficients are not direct absolute-risk contrasts.

• Wrong: “The fitted odds are \(0.25\), so the event probability is 25%.” Correction: Convert \(0.25\) to \(0.25/1.25=20\%\).
• Wrong: “The treatment odds ratio is 2, so treatment doubles the interval risk.” Correction: An odds ratio of 2 doubles the odds; calculate both risks using their fitted odds before comparing risks.
• Wrong: “A fitted odds above 1 shows the model is invalid.” Correction: Odds can exceed 1. The corresponding risk remains below 1 after applying \(o/(1+o)\).
• Wrong: “Odds and risk are interchangeable when interpreting fitted values.” Correction: They are close only when risks are small; the exact conversion is required for probability predictions.

The transformation itself is exact, but a converted prediction can still be misleading if the fitted model, target population, or causal assumptions are inappropriate. It also does not address competing risks: with competing events, the outcome and risk being predicted must be defined for the relevant event process. The practical lesson is narrow but essential: interpret the logistic model on the odds scale, then convert to the probability scale before communicating interval or cumulative risk.