The probability of a robust inference for internal validity

Tenglong Li*, Ken Frank

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


The internal validity of observational study is often subject to debate. In this study, we define the counterfactuals as the unobserved sample and intend to quantify its relationship with the null hypothesis statistical testing (NHST). We propose the probability of a robust inference for internal validity, that is, the PIV, as a robustness index of causal inference. Formally, the PIV is the probability of rejecting the null hypothesis again based on both the observed sample and the counterfactuals, provided the same null hypothesis has already been rejected based on the observed sample. Under either frequentist or Bayesian framework, one can bound the PIV of an inference based on his bounded belief about the counterfactuals, which is often needed when the unconfoundedness assumption is dubious. The PIV is equivalent to statistical power when the NHST is thought to be based on both the observed sample and the counterfactuals. We summarize the process of evaluating internal validity with the PIV into a six-step procedure and illustrate it with an empirical example.

Original languageEnglish
Pages (from-to)1947-1968
Number of pages22
JournalSociological Methods and Research
Issue number4
Publication statusPublished - Nov 2022
Externally publishedYes


  • Bayesian statistics
  • causal inference
  • hypothesis testing
  • internal validity
  • observational study
  • robustness indices
  • sensitivity analysis


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