Quantifying Sensitivity to Selection on Unobserved Covariates: Recasting the Coefficient of Proportionality Within a Correlational Framework

Sensitivity analyses can inform evidence-based policy by quantifying the hypothetical conditions necessary to change an inference. Perhaps the most prevalent index used for sensitivity analyses is Oster’s Coefficient of Proportionality (COP) which expresses how strong selection on unobserved covariates would have to be relative to selection on observed covariates to nullify an estimated effect. But Oster has been critiqued based on its two-stage conceptualization of the COP and its estimation of the COP based on coefficient stability across estimated models. In this article, we reconceptualize the COP as a function of unobserved covariates’ correlations with the focal predictor (e.g., treatment) and with the outcome. Our correlation-based approach addresses the critiques of Oster while preserving the comparison of selection on unobserved covariates to selection on observed covariates. As importantly, our expressions do not depend on analysts’ subjective choices of covariates to include in a baseline model, are adapted to a threshold for inference based on statistical significance, and can be directly calculated from conventionally reported quantities (e.g., estimated effect, standard error) through the Konfound packages in R or Stata or the R-shiny app https://konfound-project.shinyapps.io/konfound-it/ . Thus, for most published studies in the social sciences our correlation-based COP index can be easily applied and intuitively interpreted.