Binary and Categorical Covariate Transitions

Why Model Covariate Transitions?

In the parametric g-formula, a time-varying covariate must be generated according to its distribution given the history that precedes it. When the covariate is binary or takes a small number of categories, logistic and multinomial regression provide familiar ways to specify that conditional distribution. The model’s predictions are probabilities for the next covariate value, not predictions of treatment assignment.

Use the following timing convention. At time \(t\), \(L_t\) is observed after treatment \(A_{t-1}\) and before treatment \(A_t\). Write the information available before \(L_t\) as \(H_t^L=(V,\bar L_{t-1},\bar A_{t-1})\), where \(V\) denotes baseline covariates, \(\bar L_{t-1}=(L_0,\ldots,L_{t-1})\), and \(\bar A_{t-1}=(A_0,\ldots,A_{t-1})\). Thus the transition of interest is \(P(L_t\mid H_t^L)\). A specified treatment regime supplies the treatment values in this history; as established in the earlier material on why g-computation does not model treatment assignment, the transition model is not a model for how treatment was allocated.

The model must use variables that are available before the covariate being generated. If \(L_t\) is measured after \(A_{t-1}\), its model can include \(A_{t-1}\), but it cannot condition on a future treatment \(A_t\) or on a future covariate \(L_{t+1}\). This ordering is substantive: the transition probability is meant to describe how the covariate evolves given the history that has already occurred.

Binary Covariate Transitions With Logistic Regression

For a binary covariate, a standard model is logistic regression. Let \(x_t(H_t^L)\) be a specified vector of history features, including an intercept, and let \(\beta_t\) be its coefficient vector. A separate model can be fitted at each time, or a pooled model can include time indicators and shared coefficients.

A coefficient describes a change in conditional log odds for a one-unit change in the corresponding feature, holding the other model features fixed. For example, a coefficient of \(0.8\) for a binary treatment indicator means that the conditional odds are multiplied by \(\exp(0.8)\), about \(2.23\), when that indicator changes from zero to one. It does not mean the probability rises by 0.8; the probability change depends on the other history values.

Categorical Transitions With Multinomial Logistic Regression

For a covariate with \(J\) unordered categories, a reference-category multinomial logistic model specifies \(J-1\) log odds. Choose category \(0\) as the reference and define a linear predictor \(\eta_{tj}(H_t^L)\) for each \(j=1,\ldots,J-1\). Set \(\eta_{t0}(H_t^L)=0\). The corresponding probabilities are

Each non-reference linear predictor is commonly modeled as \(x_t(H_t^L)^\top\beta_{tj}\). Its coefficients describe changes in the log odds of category \(j\) relative to category \(0\). The resulting probabilities are a single distribution over all categories, not a collection of separately fitted binary probabilities that might fail to sum to one.

Fitting and Using the Transition Distributions

For observed individuals \(i=1,\ldots,n\), let \(l_{it}\) be the observed value of \(L_t\), and let \(h_{it}^L\) be that individual’s history before \(L_t\). Under independent sampling of individuals and a correctly specified collection of conditional transition models, the conditional likelihood contribution for the transitions is

The log-likelihood is the sum of the logarithms of the conditional probabilities. These expressions concern the transition distributions conditional on the observed histories; they do not add a treatment-assignment model. In practice, models may be fitted separately by time or pooled across times with time represented in the model. The choice depends on the number of observations, the amount of history information, and whether sharing parameters across times is defensible.

Once coefficients are estimated, substitute the histories of interest into the fitted model to obtain conditional probabilities. For a binary covariate, a draw can be generated using a uniform random variable \(U\sim\operatorname{Uniform}(0,1)\): set \(L_t=1\) if \(U\leq\hat p_t(H_t^L)\), and \(L_t=0\) otherwise. For a categorical covariate, form cumulative probabilities in category order and select the category whose interval contains \(U\). The next transition uses the newly generated history, not the original observed history.

Proof. For a path \((l_1,\ldots,l_K)\) under the fixed regime, the chain rule for conditional probabilities gives \(P(L_1=l_1,\ldots,L_K=l_K)=\prod_{t=1}^{K}P(L_t=l_t\mid L_1=l_1,\ldots,L_{t-1}=l_{t-1})\), with the regime’s treatment values included in the conditioning history at each time. The transition-model assumption identifies each factor with the model probability evaluated at the history along this path. Sequential drawing assigns precisely that product probability to the path: at each draw, conditional on the previous draws, the probability of the required next category is the corresponding transition probability. Multiplying these conditional probabilities yields the stated path probability. Finally, summing the product over all paths gives one by successively summing the last transition probabilities, then the preceding ones, because each conditional distribution sums to one. Thus the sequentially generated paths have the joint distribution specified by the transitions. \(\square\)

Diagnostics, Alternatives, and Limits

Transition probabilities are only useful if the fitted models represent the observed conditional distributions adequately and provide support for the histories at which predictions are needed. Examine fitted probabilities by time and category, and compare predicted transition frequencies with observed frequencies across meaningful history groups. Very sparse groups, probabilities close to zero or one, and categories that are never observed for some histories can signal instability or lack of support. A model that predicts a category as essentially impossible when it is observed often, or predicts it often when it is absent, warrants investigation.

Calibration should be assessed for the transitions themselves, conditional on relevant history features; good overall prediction can conceal poor fit in particular histories. Separation in a binary logistic model or sparse category-history combinations in a multinomial model can yield unstable or extreme estimates. Also check whether the histories used for prediction resemble those represented in the data. A fitted model can return probabilities for an unsupported history, but this is extrapolation, not evidence that the transition distribution is known there.

A saturated table of observed transition frequencies is a direct alternative when the history has only a few well-populated combinations. It avoids imposing a particular regression form but becomes unreliable or undefined in sparse combinations. Logistic and multinomial regression share information through a chosen parametric form and can be more stable, at the cost of depending on that form. Neither approach removes the need for adequate data support. The model choice also does not replace the causal identification assumptions, including consistency and sequential exchangeability.

A further distinction is important when several covariates are measured at the same time. Modeling each covariate separately as if they were conditionally independent can give the wrong joint distribution if they are associated given the history. One option is a joint categorical model when the joint state space is manageable; another is to specify a conditional ordering among the covariates and model each in turn. The chosen factorization must represent the intended joint transition distribution.

The main limitation is that predicted transitions inherit the model’s assumptions. A correctly normalized probability distribution is not necessarily a well-fitting one. Model diagnostics can reveal important failures but cannot prove that all conditional distributions are correctly specified. In particular, a transition model cannot repair a lack of observed support for a regime-generated history.