Individuals differ not only in their background characteristics, but also in how they respond to a particular treatment, intervention, or stimulation. In particular, treatment effects may vary systematically by the propensity for treatment. In this paper, we discuss a practical approach to studying heterogeneous treatment effects as a function of the treatment propensity, under the same assumption commonly underlying regression analysis: ignorability. We describe one parametric method and two non-parametric methods for estimating interactions between treatment and the propensity for treatment. For the first method, we begin by estimating propensity scores for the probability of treatment given a set of observed covariates for each unit and construct balanced propensity score strata; we then estimate propensity score stratum-specific average treatment effects and evaluate a trend across them. For the second method, we match control units to treated units based on the propensity score and transform the data into treatment-control comparisons at the most elementary level at which such comparisons can be constructed; we then estimate treatment effects as a function of the propensity score by fitting a non-parametric model as a smoothing device. For the third method, we first estimate non-parametric regressions of the outcome variable as a function of the propensity score separately for treated units and for control units and then take the difference between the two non-parametric regressions. We illustrate the application of these methods with an empirical example of the effects of college attendance on womens fertility.
This research output has an Altmetric Attention Score of 82. This is our high-level measure of the quality and quantity of online attention that it has received. This Attention Score, as well as the ranking and number of research outputs shown below, was calculated when the research output was last mentioned on 25 August 2026.
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