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The internal validity of observational study is often subject to debate. Inthis study, we define the counterfactuals as the unobserved sample and intendto quantify its relationship with the null hypothesis statistical testing(NHST). We propose the probability of a causal inference is robust for internalvalidity, i.e., the PIV, as a robustness index of causal inference. Formally,the PIV is the probability of rejecting the null hypothesis again based on boththe observed sample and the counterfactuals, provided the same null hypothesishas already been rejected based on the observed sample. Under eitherfrequentist or Bayesian framework, one can bound the PIV of an inference basedon his bounded belief about the counterfactuals, which is often needed when theunconfoundedness assumption is dubious. The PIV is equivalent to statisticalpower when the NHST is thought to be based on both the observed sample and thecounterfactuals. We summarize the process of evaluating internal validity withthe PIV into an eight-step procedure and illustrate it with an empiricalexample (i.e., Hong and Raudenbush (2005)).

On the probability of a causal inference is robust for internal validity