Pooled Logistic Regression for Discrete Censoring Hazards

Why Pool Person-Period Records?

A discrete-time censoring model asks, at each follow-up interval, which individuals who are still being observed become censored before the next interval. Pooled logistic regression expresses this question as a binary-outcome regression on person-period records. Pooling permits a common regression structure across intervals while allowing the baseline hazard to change over time.

The key is to build the data set from the risk set, not by treating every person as observed at every time. A person who has already been censored cannot contribute a later censoring outcome. The resulting fitted probabilities are conditional hazards: they describe censoring in an interval among people still at risk at its start.

Definition: For individual \(i\) and interval \(t\), let \(C_{it}=1\) indicate censoring during interval \(t\), and \(C_{it}=0\) indicate no censoring during that interval. Let \(R_{it}=1\) indicate that the individual is at risk and observed at the start of interval \(t\). Set \(R_{i1}=1\), and, for this censoring process, define \(R_{i,t+1}=R_{it}(1-C_{it})\). Let \(H_{it}\) denote the observed history available before the censoring outcome in interval \(t\). The discrete censoring hazard is \(h_t(H_{it})=P(C_{it}=1\mid R_{it}=1,H_{it})\).

This notation assumes censoring status is defined for an interval only when the individual is at risk at its start. If an event or another process ends eligibility for censoring, the study’s risk-set definition must reflect that process as well. The history \(H_{it}\) must be available before \(C_{it}\) is realized; it cannot include information learned only later.

Model and Estimation

A pooled logistic model specifies the log odds of censoring as a function of time and the available history. One common specification uses interval indicators for the baseline hazard and a shared coefficient vector for history predictors:

\operatorname{logit}\{h_t(H_{it})\} =\alpha_t+\beta^{\mathsf T}g(H_{it}), \qquad \operatorname{logit}(p)=\log\!\left(\frac{p}{1-p}\right).

Here \(\alpha_t\) is an interval-specific intercept, \(g(H_{it})\) is a chosen vector of history variables or transformations, and \(\beta\) is a common coefficient vector. This model allows the baseline hazard to vary freely by interval, while assuming the specified history effects are shared over time. Interactions with time can relax that assumption.

Create one row for each pair \((i,t)\) for which \(R_{it}=1\). The row contains \(C_{it}\), the interval, and \(g(H_{it})\). Fit a binomial regression with a logit link to these rows. Under the specified model, the conditional likelihood for the observed censoring outcomes is

L(\alpha,\beta) = \prod_i\prod_{t:R_{it}=1} h_t(H_{it})^{C_{it}} \{1-h_t(H_{it})\}^{1-C_{it}}.

This likelihood includes no row after an individual is censored. Although the rows are convenient for fitting, rows from the same individual are not independent sampling units. For inference, account for within-individual dependence when estimating uncertainty, for example with a subject-clustered sandwich variance or an appropriate model-based approach.

Result: If the conditional hazard is correctly specified and the usual regularity conditions for logistic maximum likelihood hold, the fitted model estimates the stated conditional censoring probabilities. Relevant conditions include independent sampling across individuals, a fixed finite number of intervals, identifiable model parameters, and no complete or quasi-complete separation. For a conventional large-sample maximum-likelihood approximation, the true parameter must be finite and the information matrix nonsingular.

The likelihood form follows from sequential conditioning. At each interval, among individuals who have reached its risk set, the model assigns probability \(h_t(H_{it})\) to censoring and \(1-h_t(H_{it})\) to remaining uncensored. Multiplying these conditional contributions over the observed at-risk intervals gives the displayed likelihood. This does not assert that all person-period rows are independent; it is a factorization of the sequential conditional model.

Worked Examples

Worked Example: Constructing Risk-Set Rows

Five individuals begin follow-up. One is censored during interval 1. Of the four who remain at risk in interval 2, one is censored during that interval. Fit a time-only pooled logistic model with a separate intercept for each interval.

Risk sets. All five individuals contribute an interval-1 row, with one \(C=1\) and four \(C=0\). Only the four who remain uncensored contribute interval-2 rows; those rows contain one \(C=1\) and three \(C=0\). The first interval’s censored individual has no interval-2 row.

With a separate intercept for each interval, the maximum-likelihood hazard in each interval is its censoring proportion in that interval’s risk set: \( \hat h_1=1/5=0.20\) and \(\hat h_2=1/4=0.25\). To check this, the interval-specific likelihood is proportional to \(p^d(1-p)^{m-d}\), where \(m\) is the risk-set size and \(d\) is the number censored. Its log-likelihood derivative is \(d/p-(m-d)/(1-p)\); setting it to zero gives \(p=d/m\).

For a specified path that remains at risk through both intervals, the model’s sequential no-censoring factors are \(1-\hat h_1=0.80\) and \(1-\hat h_2=0.75\). Their product is \(0.80(0.75)=0.60\). This is a pathwise sequential calculation, not a probability conditional on future histories.

Worked Example: Extracting Hazards From Fitted Coefficients

Suppose a fitted model has two interval indicators and a binary history variable \(X_{it}\), with \(\operatorname{logit}\{\hat h_t(X_{it})\}=\hat\alpha_t+\hat\beta X_{it}\). The fitted coefficients are \(\hat\alpha_1=0\), \(\hat\alpha_2=-\log(3)\), and \(\hat\beta=\log(3)\).

For an individual with \(X=1\) in interval 1, the linear predictor is \(0+\log(3)=\log(3)\). Therefore \(\hat h_1=3/(1+3)=0.75\). If \(X=1\) is also the history available at the start of interval 2, the linear predictor there is \(-\log(3)+\log(3)=0\), so \(\hat h_2=1/(1+1)=0.50\).

Along the specified sequential history with \(X=1\) in both intervals, the estimated no-censoring factors are \(0.25\) and \(0.50\), giving \(0.25(0.50)=0.125\) for the pathwise survival product. If instead \(X=0\) at both starts, the hazards are \(0.50\) and \(1/(1+3)=0.25\), and the corresponding product is \(0.50(0.75)=0.375\). The covariate value used at each step must be one available before that interval’s censoring outcome.

Worked Example: Evaluating the Person-Period Likelihood

Consider the risk-set counts from the first example. Under the time-only model, the interval-1 contribution to the likelihood is \(p_1(1-p_1)^4\), and the interval-2 contribution is \(p_2(1-p_2)^3\). At the maximum-likelihood estimates \(p_1=0.20\) and \(p_2=0.25\), the likelihood is

$$ L=(0.20)(0.80)^4(0.25)(0.75)^3 =0.20(0.4096)(0.25)(0.421875) =0.00864. $$

There are five interval-1 outcomes and four interval-2 outcomes, for nine person-period rows. The total number of censoring outcomes is two. The likelihood uses the interval-specific risk-set denominators: it does not divide the two censoring events by the original five people and call that a common hazard.

Sequential Probability Extraction

For a fitted model, the interval-specific predicted censoring probability for an at-risk record is \(\hat h_{it}=\operatorname{expit}\{\hat\alpha_t+\hat\beta^{\mathsf T}g(H_{it})\}\), where \(\operatorname{expit}(u)=1/(1+\exp(-u))\). The corresponding predicted probability of remaining uncensored through that interval is \(1-\hat h_{it}\).

A product of these no-censoring factors is useful when evaluated sequentially along a specified history path: at each interval, use the history available at that interval’s start, conditional on having remained at risk so far. It is not generally valid to interpret that product as \(P(C_{i1}=0,\ldots,C_{ik}=0\mid H_{i1},\ldots,H_{ik})\). Conditioning on future histories can change earlier conditional probabilities.

Derivation: The chain rule conditions each censoring outcome on the information available before it and on survival in the risk set up to that point. For a sequentially specified history path, the no-censoring contribution at step \(t\) is \(1-h_t(H_{it})\). Multiplying those successive conditional contributions yields the pathwise product \(\prod_{t=1}^k\{1-h_t(H_{it})\}\). This derivation does not condition an earlier outcome on future history values; if histories are random and only baseline information is conditioned on, obtaining a marginal survival probability can also require averaging over the evolving histories.

Diagnostics, Alternatives, and Limitations

Begin by checking that every included row belongs to the risk set and that each person stops contributing rows after censoring. Tabulate risk-set sizes and censoring counts by interval. Sparse late intervals can produce unstable estimates; a time-specific intercept is especially vulnerable when an interval has no censoring events or everyone is censored. Large coefficients, very large standard errors, convergence warnings, or predicted hazards near zero or one can signal separation or sparse data.

Assess whether the chosen time structure and shared history effects are plausible. Compare observed censoring proportions with fitted probabilities across intervals and useful covariate groupings, while recognizing that such checks do not prove the model is correct. Systematic discrepancies suggest that the time baseline, functional form, or time-by-history structure is inadequate. Predictions outside the histories represented in the risk set should be treated cautiously.

Separate logistic regressions by interval allow every coefficient to vary over time, but may be unstable when risk sets are small and do not pool information about shared effects. Pooled logistic regression is more parsimonious when common effects are reasonable; time interactions provide an intermediate option. A discrete-time complementary log-log model or a continuous-time hazard model may be preferable when its assumptions and time scale fit the data better. These approaches do not remove the need to define the risk set correctly.

The fitted hazard is conditional on the history variables and model form actually used. A pooled logistic model cannot by itself repair unmeasured causes of dropout, inadequate history measurement, or positivity problems. Nor does a well-fitting model establish sequential independent censoring; that is a substantive assumption about the censoring process, not a consequence of regression fit.

Common Mistake

Wrong reasoning: “The product of predicted no-censoring probabilities must equal the probability of no censoring conditional on the entire history through the final interval.” This silently conditions early censoring outcomes on future histories.
Correction: Interpret each fitted hazard conditional on the risk set and history available before that interval. A product is a sequential pathwise calculation using those histories in order. Conditioning on the full future history can alter the earlier conditional probabilities, so it does not generally justify the same product.
Another frequent error: Keeping rows after a person has been censored. Those rows represent outcomes the study could not observe and must not enter the risk-set likelihood.

Check Your Understanding

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

Eight people are at risk at the start of interval 1; two are censored. How many interval-2 rows can they contribute, assuming no other process ends follow-up?
In a time-only logistic model, 3 of 12 people in an interval’s risk set are censored. What is the maximum-likelihood hazard for that interval?
A model predicts censoring hazards of 0.10 and 0.20 on a specified sequential history path. What is the pathwise probability of remaining uncensored over both intervals?
Why can that product not automatically be called the probability of remaining uncensored conditional on all histories through interval 2?

Solutions. (1) Six interval-2 rows: the two censored people leave the risk set, while the other six remain. (2) The estimate is \(3/12=0.25\), the censoring proportion among those at risk. (3) The no-censoring factors are \(0.90\) and \(0.80\), so the product is \(0.90(0.80)=0.72\). (4) The second history may contain information about the earlier censoring outcome. The product follows sequential conditioning on information available at each interval, not conditioning an earlier outcome on future history.