Assessment of the predictive power of a causal variable: An application to the Head Start impact study

In a study attempting to estimate a causal effect of a causal variable, an assessment of the predictive power of the causal variable can shed light on the heterogeneity around its average effect. Using data from the Head Start Impact Study, a randomized controlled trial of the Head Start, a nation-wide early childhood education program in the United States, we provide a parallel comparison between measures of average effect and predictive power of the Head Start on five cognitive outcomes. We observed that one year of the Head Start increased scores for all five outcomes, with effect sizes ranging from 0.12 to 0.19 standard deviations. Percent variation explained by the Head Start ranged from 0.56 to 1.62%. For binary versions of the outcomes, the overall pattern remained; the Head Start on average improved the outcomes by meaningful magnitudes. In contrast, in a fully adjusted model, the Head Start only improved area under the curve (AUC) by less than 1% and its influence on the variance of predicted probabilities was negligible. The Head-Start-only model only achieved AUC ranging from 50.22 to 55.24%. Negligible predictive power despite the significant average effect suggests that the heterogeneity in effects may be large. The average effect estimates may not generalize well to different populations or different Head Start program settings. Assessment of the predictive power of a causal variable in randomized data should be a routine practice as it can provide helpful information on the causal effect and especially its heterogeneity.