Covariate-Specific Support and Positivity

Why Examine Treatment Support Within Covariate Strata?

A treatment can be common overall yet rarely observed for people with particular covariate profiles. That distinction matters in a marginal structural model: estimating a contrast for a treatment option in a region where that option is uncommon requires information that the observed data may provide only weakly. An overall treatment percentage can conceal the problem because it averages across profiles with very different treatment patterns.

This tutorial focuses on discrete treatment options and covariate-stratified frequencies. For a longitudinal decision at time \(t\), let \(H_t\) denote the measured covariate and treatment history available just before that decision, and let \(A_t\) be the treatment assigned. The same ideas apply to a single treatment decision by taking \(H_t\) to be the baseline covariates. The earlier material on structural versus practical positivity violations and sparse treatment histories provides useful context; here the emphasis is on locating possible support problems in covariate space.

Support is therefore conditional, not simply a property of a treatment group’s overall size. A profile with a very small \(q_t(a\mid h)\) raises a practical concern even if the probability is technically positive. A profile with \(q_t(a\mid h)=0\) represents a structural lack of support for that option there. In finite data, however, observing no one on an option does not by itself establish that its population probability is zero.

Stratified Frequencies and Probability Summaries

Choose a set of covariate strata \(s=1,\ldots,S\), defined using information available before the treatment decision. For example, strata might be combinations of age group and disease severity. Let \(n_s\) be the number of individuals in stratum \(s\), and let \(n_{sa}\) be the number receiving treatment option \(a\). The empirical within-stratum frequency is

For a treatment with several options, calculate a frequency for every option in every stratum. Each row should sum to one, apart from rounding. A useful display reports \(n_s\), the option-specific counts, and the corresponding frequencies together: a frequency of 0.05 based on one of 20 people is less precise than 0.05 based on 1,000 of 20,000.

Probability summaries can help scan many strata. For each option, one can report the minimum and lower quantiles of \(\widehat q_s(a)\), the number or proportion of strata below a chosen screening value, and the sample sizes in those strata. It can also be informative to report the fraction of the target sample lying in flagged strata. These are descriptive summaries, not universal pass/fail tests. A threshold such as 0.05 is a screening choice; its relevance depends on the analysis, sample size, and consequences of limited support.

What a Zero Count Can—and Cannot—Tell You

The distinction between population support and observed support is essential. Suppose a fixed stratum contains \(n_s=n\) independent individuals with the same treatment probability \(q=P(A=a\mid H\in s)\). The number \(N_a\) receiving option \(a\) has a binomial distribution. Even if \(q>0\), a finite sample can contain no recipients of that option.

To verify the result, each of the \(n\) individuals independently fails to receive option \(a\) with probability \(1-q\). The event \(N_a=0\) requires all \(n\) failures, so its probability is the product \((1-q)\cdots(1-q)=(1-q)^n\). Thus, the chance of missing a genuinely available option can be substantial when the stratum is small or the option is rare.

The frequency estimator itself has a straightforward sampling justification. For a fixed stratum with \(n\) independent observations and constant probability \(q\), \(\widehat q=N_a/n\) is unbiased because \(E[N_a]=nq\). Its variance is \(q(1-q)/n\). Consequently, for any \(\epsilon>0\), Chebyshev’s inequality gives

As \(n\) grows, the final bound converges to zero. Thus, within this fixed-stratum, independent-sampling setup, the empirical frequency converges in probability to the population probability. The guarantee is about increasing sample size within the stratum; it does not make a small observed cell precise, and it does not automatically apply to a growing number of increasingly fine strata.

Why Marginal Summaries Can Mislead

An overall frequency is a sample-size-weighted average of the stratum frequencies. If \(n=\sum_s n_s\), then

The identity follows by substituting \(n_{sa}=n_s\widehat q_s(a)\) into the numerator and dividing by \(n\). It explains how a large stratum with a high frequency can dominate the overall summary and conceal a small stratum with a low frequency. It also means that the overall frequency may change when the covariate distribution of the sample changes, even if the within-stratum frequencies remain the same.

A comparison of treatment groups’ marginal covariate distributions or a marginal propensity-score plot can be useful, but neither replaces checking treatment availability within relevant covariate regions. The earlier material on weight distributions and propensity score overlap plots for longitudinal histories addresses related diagnostics. Extreme or highly variable weights can be consistent with poor support, but weight summaries alone do not identify every covariate region in which an option is rare. Conversely, a small stratum frequency is a support warning, not a complete assessment of a fitted weighting analysis.

Practical Choices, Diagnostics, and Limitations

The definition of strata affects what the table can reveal. Strata should be based on covariates measured before the treatment decision and should correspond to profiles relevant to the target analysis. Very broad strata can average over important differences; very narrow strata can have tiny sample sizes or empty cells. With many covariates, a full cross-classification may become too sparse to interpret. Prespecified, clinically meaningful groupings can help, but they do not eliminate the need to report cell sizes or examine whether conclusions depend on the grouping.

A concise diagnostic report can include the following for every treatment option:

• the number of observations and option-specific counts in each stratum;
• the smallest and lower-quantile empirical frequencies, paired with their stratum sizes;
• the number and sample proportion of observations in strata below a stated screening value; and
• any empty cells, together with an explicit note that an empty sample cell is not proof of zero population probability.

Investigate whether a flagged cell reflects a data or coding error, a very small stratum, a genuine prescribing pattern, or an overly fine stratification. If the pattern changes sharply under reasonable alternative groupings, that instability is itself informative. Such checks help locate where treatment comparisons depend on thin data, but they cannot establish that an unobserved option would be impossible in the population.

Covariate-specific support summaries are diagnostic rather than a cure for limited data. They do not create observations for rare treatment–covariate combinations, and they do not by themselves justify changing the target population or treatment contrast. Their value is to show where the data are informative and where a causal comparison may rely heavily on a small number of observations.