Positivity Limits and Parametric Extrapolation

Why Support Matters for Parametric g-Computation

Parametric g-computation uses fitted outcome and covariate-transition models to predict outcomes under specified treatment regimes. Those models can return predictions for histories not observed in the data. This computational ability is not, by itself, evidence that those predictions are identified by the data. When a treatment value or history is absent from the relevant observed population, the model must extend its fitted relationships beyond the empirical support. The resulting estimate may be precise numerically while remaining highly dependent on assumptions that the data cannot check.

The issue is not limited to treatment assignment. A regime may create covariate histories rarely or never seen in the observed data, and transition models may then be asked to predict the next covariate distribution in those histories. Positivity is the condition that connects the histories required by an intervention to histories for which the observed data provide information.

In a longitudinal analysis, positivity must hold at each treatment decision, not merely at baseline or for the marginal treatment frequency. For example, both treatments may be common overall while treatment \(A_t=1\) is never observed among patients with a particular earlier treatment and covariate history. A regime that assigns \(A_t=1\) in that history has a support problem.

With continuous treatment, the statement needs additional care. An observed treatment distribution with a density does not assign positive probability to an exact dose. A density condition can describe local support, but it does not automatically identify a conditional mean at one exact dose: a conditional expectation is defined only up to sets of treatment values having probability zero. Point-dose identification generally requires an additional regularity condition, such as continuity of the response surface at the target dose. The earlier material on Continuous Treatment Levels and Parametric G-Formula develops the continuous-treatment setup; here the emphasis is on what limited support permits us to interpret.

Support, Identification, and Extrapolation

For a categorical regime, the parametric g-formula averages conditional outcome and covariate-transition distributions over the histories that the regime generates. Under consistency, sequential exchangeability, and positivity, those conditional distributions can be learned from the corresponding observed histories. The following result states the support implication precisely.

The first assertion follows by iterated conditioning. At each decision, exchangeability allows the conditional counterfactual distribution under the assigned treatment to be represented by the corresponding observed conditional distribution; consistency equates the observed outcome for that treatment with its counterfactual outcome. Positivity ensures that this observed conditional distribution is available at almost every history to which the regime assigns positive probability. Applying these equalities successively through the covariate transitions and final outcome yields the g-formula integral. If positivity fails on a positive-probability set, the observed data place no restriction on the treatment-specific conditional outcome distribution there. The next example shows why that gap cannot generally be filled by data alone. \(\square\)

This example also illustrates why “the software fit a model” is not an answer to a support problem. A parametric form can impose relationships across cells, but the resulting prediction inherits those restrictions. If the unobserved cell is given a prediction by an additive model, the additivity assumption supplies information that the data did not provide. A more flexible model may expose the lack of information by leaving that cell’s prediction unstable or undetermined; it does not create support.

Continuous Treatment: Local Support Is Not the Same as Point Identification

For a continuous treatment \(A\), positivity is often described by requiring the conditional density of treatment to be positive around the target dose at relevant histories. This is useful for assessing whether nearby doses are represented. However, it is not the same as the discrete condition \(P(A=a\mid H=h)>0\), since that probability is zero for any single dose under a continuous distribution. Moreover, if the target dose lies outside the observed support, nearby observations do not directly identify its response.

Longitudinal Support Can Fail Along a Path

A longitudinal regime is a sequence of treatment decisions. Even if each prescribed treatment occurs in the data somewhere, the complete histories leading to those decisions may not. The g-formula needs transition and outcome distributions at histories generated by the intervention, so a lack of support at a later decision can affect the regime-specific mean.

What to Check and What the Estimate Means

Support should be assessed against the intervention and target population, not only by looking at overall treatment frequencies. Useful checks include tabulating treatment frequencies across important history strata, plotting continuous treatment distributions within those strata, and identifying histories that the regime generates but the observed data scarcely represent. For continuous treatment, inspect whether the target dose is inside the observed range at relevant histories, and distinguish a thinly represented neighborhood from a dose outside the range.

Sparse observations are a practical warning even when formal positivity holds. Estimates can then be driven by a small number of observations or by the chosen functional form. Exact zero support is more fundamental: no amount of increasing the number of observations from the same data-generating process fills a cell that has probability zero. A model can conceal this distinction by returning a smooth prediction in both situations.

A common mistake is to argue, “The parametric g-formula predicts every regime, so positivity is unnecessary.” The correction is that prediction and identification are different. Parametric restrictions may yield a number in an unsupported region, but the number is identified only if those restrictions or other assumptions justify extending the observed relationships there. That justification is not supplied merely by successful model fitting.

The main alternatives address the target or the assumptions, not the missing information by magic. One can restrict the target population or regime to histories with support, accepting that the estimand changes. One can consider a stochastic intervention that assigns treatment values represented in the observed data, rather than forcing an unsupported value; its support must still be checked. Inverse-probability weighting and other g-methods also depend on support, although the way the problem appears in their calculations differs. Parametric extrapolation is therefore not a way to evade positivity.

Interpretation should state where the estimate relies on extrapolation. If only a small share of the regime-generated population falls outside well-supported regions, the estimate may still be useful, but its uncertainty about model assumptions is not captured simply by a narrow simulation or sampling interval. If substantial population mass is unsupported, a single marginal estimate can obscure that its key contribution is assumption-driven. The appropriate response is to clarify the estimand, report the unsupported regions, and avoid presenting the model prediction as an empirical comparison.