A note on exact confidence interval for causal effects on a binary outcome in randomized trials

Sharp nonparametric bounds and randomization inference for treatment effects on an ordinal outcome

In clinical research, investigators are interested in inferring the average causal effect of a treatment. However, the causal parameter that can be used to derive the average causal effect is not well defined for ordinal outcomes. Although some definitions have been proposed, they are limited in that they are not identical to the well-defined causal risk for a binary outcome, which is the simplest ordinal outcome. In this paper, we propose the use of a causal parameter for an ordinal outcome, defined as the proportion that a potential outcome under one treatment condition would not be smaller than that under the other condition. For a binary outcome, this proportion is identical to the causal risk. Unfortunately, the proposed causal parameter cannot be identified, even under randomization. Therefore, we present a numerical method to calculate the sharp nonparametric bounds within a sample, reflecting the impact of confounding. When the assumption of independent potential outcomes is included, the causal parameter can be identified when randomization is in play. Then, we present exact tests and the associated confidence intervals for the relative treatment effect using the randomization-based approach, which are an extension of the existing methods for a binary outcome. Our methodologies are illustrated using data from an emetic prevention clinical trial.

Stratified exact tests for the weak causal null hypothesis in randomized trials with a binary outcome

Fisher's exact test is commonly used to compare two groups when the outcome is binary in randomized trials. In the context of causal inference, this test explores the sharp causal null hypothesis (i.e. the causal effect of treatment is the same for all subjects), but not the weak causal null hypothesis (i.e. the causal risks are the same in the two groups). Therefore, in general, rejection of the null hypothesis by Fisher's exact test does not mean that the causal risk difference is not zero. Recently, Chiba (Journal of Biometrics and Biostatistics 2015; 6: 244) developed a new exact test for the weak causal null hypothesis when the outcome is binary in randomized trials; the new test is not based on any large sample theory and does not require any assumption. In this paper, we extend the new test; we create a version of the test applicable to a stratified analysis. The stratified exact test that we propose is general in nature and can be used in several approaches toward the estimation of treatment effects after adjusting for stratification factors. The stratified Fisher's exact test of Jung (Biometrical Journal 2014; 56: 129-140) tests the sharp causal null hypothesis. This test applies a crude estimator of the treatment effect and can be regarded as a special case of our proposed exact test. Our proposed stratified exact test can be straightforwardly extended to analysis of noninferiority trials and to construct the associated confidence interval.

Response to comment on 'Randomization inference for treatment effects on a binary outcome'

We thank Professor Yasutaka Chiba [1 ] for commenting on Rigdon and Hudgens (RH) [2 ]. Chiba [1 ] described a certain exact confidence interval reported in RH as “somewhat unnatural.” Chiba also presented an alternative approach to constructing confidence intervals [3 ]. In this response, we (i) provide a simple explanation why the confidence interval in RH appeared “unnatural,” and (ii) explain the relationship between the RH [2 ] and Chiba [3 ] confidence intervals. Essentially the two approaches are equivalent, except RH entails inverting one two-sided test whereas Chiba inverts two one-sided tests. We present a more computationally efficient method (RLH) for computing the RH intervals based on Chiba’s principal stratification formulation of the problem. We also propose a third method based on Blaker [4 ] which inverts a single two-sided test but forms a confidence interval that is at least as narrow as inverting two one-sided tests. Simulation results show the RLH intervals tend to be as narrow or narrower than the Chiba and Blaker intervals on average.