Conditional Treatment Support and Positivity

Why Overall Treatment Frequency Is Not Enough

At a treatment decision, a treatment level may be common in the study overall but unavailable for some observed histories. For example, a treatment might be assigned frequently to patients with mild disease and never assigned to patients with severe disease. The overall frequency alone cannot show whether both treatment options are supported within the histories relevant to a comparison.

The earlier material on positivity at each treatment decision motivates checking treatment availability conditional on observed history. Here the emphasis is on expressing that availability as a conditional support set, and on separating support for a target treatment level from marginal treatment prevalence. We consider a discrete treatment at a decision time \(t\). Let \(H_t\) denote the observed history just before that decision, and let \(A_t\) be the treatment assigned then. For example, \(H_t\) can contain baseline covariates, prior treatments, and covariates measured before time \(t\).

For a target level \(a^\star\), conditional positivity at a particular relevant history means \(P(A_t=a^\star\mid H_t=h)>0\). If the target is to assign \(a^\star\) at every history in a set \(\mathcal{H}^\star\), the corresponding requirement is that this conditional probability be positive for almost every history in that set under the history distribution relevant to the requirement. The precise choice of relevant histories matters: positivity is not a claim about every imaginable history, including histories that cannot occur.

This is a support condition, not a claim that treatment is equally likely at all histories. A probability of \(0.02\) is positive, although it signals limited representation; a probability of zero means the treatment level is absent at that history. The distinction between zero and small positive probabilities is important for interpretation and estimation.

What Marginal Prevalence Establishes

Write \(p_a(h)=P(A_t=a\mid H_t=h)\). Marginal prevalence averages these conditional probabilities over the observed distribution of histories. The averaging identity is useful, but it does not establish positivity at every history.

Proof. Let \(I_a=I(A_t=a)\), the indicator that treatment level \(a\) was assigned. By the definition of conditional probability for a discrete treatment, \(E[I_a\mid H_t]=p_a(H_t)\). Taking expectations and applying the law of iterated expectations gives

The random variable \(p_a(H_t)\) is nonnegative. If \(P\{p_a(H_t)>0\}=0\), then it is zero almost surely, so its expectation is zero. Conversely, suppose \(P\{p_a(H_t)>0\}>0\). Since $$ \{p_a(H_t)>0\}=\bigcup_{k=1}^{\infty}\{p_a(H_t)\geq 1/k\}, $$ at least one set on the right has positive probability. For that \(k\), $$ E[p_a(H_t)]\geq \frac{1}{k}P\{p_a(H_t)\geq 1/k\}>0. $$ This proves the stated equivalence. It only guarantees positivity on some non-negligible set of histories, not across all histories of interest. \(\square\)

Support for a Target Level, History by History

A target treatment level must be compared with the conditional support at the histories where it is intended to be used. For a fixed target level \(a^\star\), one can examine \(p_{a^\star}(h)\) across those histories. If the target changes with history, write it as \(g(h)\); the relevant conditional probability is then \(P(A_t=g(h)\mid H_t=h)\). This notation describes the support check without changing the central point: the probability must be evaluated conditional on the history, not averaged over the study.

There is also a useful identity for a history-dependent target. Let \(M=I\{A_t=g(H_t)\}\). Conditional on \(H_t=h\), the target level \(g(h)\) is fixed, so \(E[M\mid H_t=h]=P(A_t=g(h)\mid H_t=h)\). Taking expectations yields the following result.

Derivation. The indicator \(M\) is measurable, and conditioning on \(H_t\) fixes \(g(H_t)\). Therefore, $$ E[M\mid H_t] =P\{A_t=g(H_t)\mid H_t\}. $$ The law of iterated expectations gives \(P\{A_t=g(H_t)\}=E[M]=E\{E[M\mid H_t]\}\), which is the identity. As in the preceding result, an average of nonnegative conditional probabilities can be positive even when some of them are zero. \(\square\)

How to Check Support and Recognize Trouble

A practical assessment starts by specifying the treatment level of interest and the histories at which it is required. For categorical histories, cross-tabulations can reveal zero cells directly. With many covariates, conditional probabilities may instead be estimated, but an estimated probability is evidence about support, not a definition of it. A fitted value above zero does not prove that a population conditional probability is positive, especially when the model extrapolates into sparse data.

Common warning signs include zero treatment counts within a covariate stratum, extremely imbalanced counts, and large changes in estimated conditional probabilities under modest model changes. Sparse strata also make estimates uncertain: an observed zero in a finite sample does not by itself prove the population probability is exactly zero. Conversely, a smooth fitted model that predicts nonzero probabilities in empty strata cannot establish empirical representation there.

Common Mistake, Alternatives, and Limits

Common mistake: “Treatment 1 was given to 30% of the sample, so treatment 1 is supported for every patient history.” This reasoning replaces a conditional requirement with a marginal average. The correction is to inspect \(P(A_t=1\mid H_t=h)\) at each relevant history; the same marginal percentage can arise from widespread availability or from treatment concentrated in only a subset of histories.

The discussion here is limited to conditional support for discrete treatment levels at an observed decision history. For continuous treatment, exact treatment values generally have probability zero, so support requires a different formulation. Also, checking one decision’s conditional support does not by itself settle support for a multi-decision treatment strategy. These are separate questions; the present check concerns whether the target level is available conditional on the history at the decision being assessed.

The practical importance is direct: when the target level is absent for a relevant history, the observed data do not provide treatment comparisons within that history. A marginal frequency, a model-based extrapolation, or a large total sample cannot turn that absence into conditional support. When support is positive but weak, estimates may still be highly sensitive to a small number of observations, so both the conditional probabilities and the counts behind them should be reported.