Finite Population Inference for Causal Parameters
Loh, Wen Wei
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Randomized experiments are often employed to determine whether a treatment X has a causal effect on an outcome Y. Under the Neyman-Rubin causal model with binary X and Y, each patient is characterized by two binary potential outcomes, leading to four possible response types. In a finite population, the set of individuals of each response type is regarded as fixed over hypothetical rerandomizations, so that individuals are sampled without replacement. The resulting observed-data likelihood, which we term the Neyman-Rubin-Copas (NRC) likelihood, is a convolution of multivariate hypergeometric probabilities. We will derive results for the NRC likelihood that may be used to facilitate calculation of the generalized likelihood ratio (GLR) in more complicated finite population settings. A key finding is that the maximum likelihood under the `Neyman' null (where the population average causal effect is zero) is always attained by the population in which the `Fisher' null holds (where the individual causal effect is zero). Next we consider the setting where treatment X is no longer randomized, but there is an instrumental variable Z that is randomized. For example, patients in a randomized controlled trial may choose not to adhere to their randomly assigned treatment Z, possibly due to side-effects. In such randomized experiments with noncompliance, scientific interest is often in testing whether the treatment exposure X has an effect on the final outcome Y, among the subset of 'Compliers' who take the treatment only if assigned to do so and would not if assigned not to do so. We propose a finite population significance test of the `Fisher' null hypothesis among the principal stratum of 'Compliers', using the GLR test statistic under an extended Neyman-Rubin-Copas likelihood that accounts for the noncompliance. New methods that improve the computational efficiency when evaluating the exact p-values are described. We then extend the randomization-based significance tests using the GLR to construct an exact confidence interval for the Complier Average Causal Effect (CACE). The procedure is illustrated with a small toy example. Finally, we propose a GLR test statistic for a significance test of the ‘Fisher’ null under the noncompliance setting where we allow for a direct effect of Z on Y .
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