J D Angrist, G W Imbens, and D B Rubin. Identi cation of causal e ects using instrumental variables. Journal of the American Statistical Association An Algorithm to Compute the Likelihood Ratio Test Statistic of the Sharp Null Hypothesis for Compliers Wen Wei Loh wloh@u.washington.edu 0 Thomas S. Richardson thomasr@u.washington.edu 0 Department of Statistics, University of Washington , USA 1996 91 434

In a randomized experiment with noncompliance, scienti c interest is often in testing whether the treatment exposure X has an e ect on the nal outcome Y [2, 1]. We have proposed a nite-population signi cance test of the sharp null hypothesis that X has no e ect on Y , within the principal stratum of compliers, using a generalized likelihood ratio test [4]. As both the null and alternative hypotheses are composite hypotheses (each comprising a di erent set of distributions), computing the value of the generalized likelihood ratio test statistic [6] requires two maximizations: one where we assume that the sharp null hypothesis holds, and another without making such an assumption. In our work [4], we have assumed that there are no Always Takers, such that the nuisance parameter is a bivariate parameter describing the total number of Never Takers with observed outcomes y = 0 and y = 1. Extending the approach to the more general case in which there are also Always Takers would require a nuisance parameter of higher dimension that describes the total number of Always Takers with observed outcomes y = 0 and y = 1 as well. This increases the size of the nuisance parameter space and the computational e ort needed to nd the likelihood ratio test statistic. We present a new algorithm that extends [5] to solve the corresponding integer programs in the general case where there are Always Takers. The procedure for the nite-population signi cance test may be illustrated using a toy example from [3].