Conditional logistic model for longitudinal binary data ($\mathrm{logit}P(Y_t=1\mid {\mathcal{H}}_{t-1},x)={\eta }_t$)

A logistic regression can specify each binary response conditionally on recorded past outcomes and baseline covariates. A chain rule for probabilities gives the path likelihood as the product of the conditional Bernoulli distribution masses. This permits dependence within a person's history without treating their outcomes as unconditionally independent. The model must specify the initial history and which past features enter its conditional mean.

The accumulated number of previous successes may enter the conditional mean in a history-dependent logistic regression. Its coefficient exponentiates to the conditional success odds ratio per previous success. For a specified future path, update this cumulative predictor after each outcome before multiplying the conditional probabilities.

A previous outcome may affect a subsequent outcome after controlling both measured covariates and persistent unobserved heterogeneity. This is true state dependence. An observed lagged-outcome association alone does not establish it, because omitted individual propensities can create apparent persistence.