Direct Multinomial Landmark Models for State Occupation

Why Model State Occupation Directly?

At a landmark time, a clinician may want to know the probability that a person will be in each possible state one year later, given the state the person occupies now and covariates measured at the landmark. A direct multinomial model treats the state at that fixed future horizon as a single categorical outcome. It therefore targets the desired horizon-specific probabilities without separately modeling the sequence or timing of transitions between the landmark and horizon.

The prediction is conditional: it applies to people eligible for prediction at the landmark, with the stated current state and covariate values. It is not a probability of a particular transition path. The earlier material on conditioning on the current state and using landmark covariates provides the prediction setting; here the focus is specifying and interpreting the direct regression model.

The time origin and horizon must be specified. For example, “one year after the landmark” is different from “one year after diagnosis” unless the landmark is diagnosis. The target population is also defined by eligibility at the landmark; it is not automatically the original cohort.

Multinomial Logistic Specification

Choose one future state \(K\) as the reference category. Let \(z_i(t)\) be the model covariate vector, which can include indicators for the landmark state, measured covariates, and prespecified interactions. For each non-reference future state \(k\), multinomial logistic regression specifies the log odds of state \(k\) rather than the reference state as a linear function of those predictors:

The landmark state can enter as a categorical predictor, typically through indicator variables. If the association between a covariate and future state is expected to differ by landmark state, an interaction can represent that difference. The specification should be chosen before interpreting a single fitted coefficient: omitting an important interaction imposes the same log-odds association across the affected landmark states.

The state probabilities follow from the logits by the softmax transformation:

These fitted probabilities are nonnegative and sum to one. A coefficient \(\beta_{kj}\) is a change in the log odds of future state \(k\) versus the reference state for a one-unit increase in predictor \(j\), holding the other modeled predictors fixed. Its exponential is a ratio of those category odds, not a risk ratio and not, by itself, a change in the probability of state \(k\).

Estimation and Conditions for Interpretation

With one fully observed horizon outcome per landmark row, maximum likelihood estimates the coefficients by maximizing the multinomial likelihood. If \(y_{ik}\) is 1 when row \(i\)'s outcome is state \(k\) and 0 otherwise, the likelihood contribution is the fitted probability for the observed state. The log likelihood is:

This likelihood presumes the recorded horizon state is known. If a person is lost to follow-up or censored before \(t+h\), their state at the horizon is generally unknown; assigning their last observed state as the outcome is not justified. One valid simple setting is complete ascertainment of the horizon outcome for every included row. With incomplete follow-up, an analysis may instead use inverse-probability-of-censoring weights when censoring is conditionally independent of the unobserved horizon state given the variables in the censoring model and the probability of remaining observed is positive for relevant covariate patterns. The censoring model and weighting procedure must then be appropriate; the ordinary unweighted likelihood of observed outcomes alone does not automatically solve the problem.

Interpretation also depends on adequate support: the data need observations across the landmark-state and covariate combinations for which predictions are requested. The multinomial logit functional form must be adequate, and all predictors used for prediction must be measured by the landmark. In repeated landmark data, rows from the same person can be dependent. This dependence does not change the definition of the conditional prediction, but it must be addressed in uncertainty estimates, for example with subject-clustered inference or resampling people rather than treating their rows as independent.

Turning Fitted Coefficients Into Predictions

Coefficients are useful for describing relative category odds, but prediction requires calculating all category probabilities for the covariate profile of interest. Because the probabilities share a denominator, a change in one predictor can affect several state probabilities at once.

Repeated Landmarks, Diagnostics, and Limitations

A person can contribute multiple landmark rows, each with a current state, covariates measured at that time, and a corresponding fixed-horizon outcome. The fitted model then represents the conditional prediction for a landmark row. If the intended target is a particular population of landmark occasions, ensure that the sampling and analysis weights do not unintentionally give people with more eligible landmarks disproportionate influence. For standard errors, account for within-person dependence; treating all landmark rows as independent generally understates uncertainty when rows are correlated.

Useful checks include verifying that the outcome categories and their coding are consistent, confirming that every prediction uses only information available at the landmark, and examining whether the data support the requested state-and-covariate profiles. Sparse categories, separation, or very large fitted coefficients can indicate unstable estimation. Compare predicted probabilities with observed horizon outcomes across meaningful groups and inspect whether discrepancies are systematic; formal calibration measures are a separate topic. Also check censoring patterns: a high rate of unknown horizon outcomes or differences in follow-up across covariate groups can make complete-case predictions misleading.

A direct multinomial model and a model built from transition processes answer related but distinct modeling questions. The direct model targets state occupation at the selected horizon, and need not impose a particular transition structure between now and then. It also does not, by itself, provide transition times, intermediate state probabilities, or a coherent prediction for every possible horizon. Those limitations matter when the scientific question concerns the path as well as the endpoint.

Short Justification of the Probability Formula

The multinomial model specifies log odds relative to the reference category. Exponentiating each non-reference equation gives \(p_k/p_K=\exp(\eta_k)\). Hence \(p_k=p_K\exp(\eta_k)\) for each \(k<K\). Since all category probabilities sum to one, \(p_K+\sum_{k=1}^{K-1}p_K\exp(\eta_k)=1\). Solving gives \(p_K=1/\{1+\sum_{k=1}^{K-1}\exp(\eta_k)\}\), and substituting this expression into \(p_k=p_K\exp(\eta_k)\) gives the softmax formula. This derivation also shows why the fitted probabilities are positive and sum to one.