Positivity Assumption

The positivity assumption requires that every group of similar people must have had some chance of receiving each treatment option being compared. Without this overlap, we're trying to compare treatments that were never actually both available to the same types of people—like asking how a medicine works for patients who were never allowed to take it. This doesn't mean everyone did receive each option, just that it was possible for each group. For example, if a hospital only performs surgery on patients under 65, we can't honestly estimate how surgery would work for older patients because we have no data from that group. Check this: When comparing groups, ask whether similar people could have realistically ended up in either option.

The positivity assumption ensures that for every combination of background characteristics (like age, health status, or location), there was a non-zero chance of receiving each treatment. When this fails, we have a violation called 'nonoverlap'—regions in the data where only one treatment group appears. This makes causal claims about those regions impossible to support from the data alone.

In healthcare, this often happens due to clinical guidelines: patients with severe kidney disease might never be prescribed a particular drug because it's contraindicated for them. We can estimate effects for patients who could have received either treatment, but not for those who were structurally excluded from one option.

In education policy, if a new teaching method is only piloted in well-funded schools, we can't generalize its effects to underfunded schools that never had the opportunity to try it. The violation is 'structural' when certain groups are systematically excluded, versus 'practical' when they simply didn't appear in our sample by chance.

Researchers address positivity violations through 'trimming' (excluding groups without overlap), 'overlap weighting' (giving more influence to groups with good comparability), or acknowledging that estimates apply only to the 'overlap population' where both treatments were actually possible. Transparent reporting requires clearly stating which subgroups remain in the analysis.

The positivity assumption is formalized as requiring that the conditional probability of receiving each treatment level is strictly greater than zero for all covariate values: P(A = a | X) > 0 for all treatment levels a and covariate combinations X. This is sometimes called 'overlap' because the treatment and control groups must share regions of the covariate space. Violations are distinguished as structural (theoretical impossibility, such as clinical contraindications) versus practical (random absence in finite samples).

Positivity cannot be tested definitively because we never observe the true probability of treatment—only whether treatment occurred in our sample. What appears to be a structural violation might reflect limited sample variation rather than true impossibility. Additionally, different statistical models for estimating propensity scores can yield different overlap assessments, making positivity partly model-dependent. The assumption also becomes increasingly difficult to satisfy as we condition on more covariates, since each additional variable creates more covariate combinations that may lack overlap—a challenge known as the 'curse of dimensionality.'