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  1. Scholkopf, Bernhard ; Uhler, Caroline ; Zhang, Kun (Ed.)
    In order to test if a treatment is perceptibly different from a placebo in a randomized experiment with covariates, classical nonparametric tests based on ranks of observations/residuals have been employed (eg: by Rosenbaum), with finite-sample valid inference enabled via permutations. This paper proposes a different principle on which to base inference: if — with access to all covariates and outcomes, but without access to any treatment assignments — one can form a ranking of the subjects that is sufficiently nonrandom (eg: mostly treated followed by mostly control), then we can confidently conclude that there must be a treatment effect. Based on a more nuanced, quantifiable, version of this principle, we design an interactive test called i-bet: the analyst forms a single permutation of the subjects one element at a time, and at each step the analyst bets toy money on whether that subject was actually treated or not, and learns the truth immediately after. The wealth process forms a real-valued measure of evidence against the global causal null, and we may reject the null at level if the wealth ever crosses 1= . Apart from providing a fresh “game-theoretic” principle on which to base the causal conclusion, the i-bet has other statistical and computational benefits, for example (A) allowing a human to adaptively design the test statistic based on increasing amounts of data being revealed (along with any working causal models and prior knowledge), and (B) not requiring permutation resampling, instead noting that under the null, the wealth forms a nonnegative martingale, and the type-1 error control of the aforementioned decision rule follows from a tight inequality by Ville. Further, if the null is not rejected, new subjects can later be added and the test can be simply continued, without any corrections (unlike with permutation p-values). Numerical experiments demonstrate good power under various heterogeneous treatment effects. We first describe i-bet test for two-sample comparisons with unpaired data, and then adapt it to paired data, multi-sample comparison, and sequential settings; these may be viewed as interactive martingale variants of the Wilcoxon, Kruskal-Wallis, and Friedman tests. 
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  2. null (Ed.)
    We propose a method for multiple hypothesis testing with familywise error rate (FWER) control, called the i-FWER test. Most testing methods are predefined algorithms that do not allow modifications after observing the data. However, in practice, analysts tend to choose a promising algorithm after observing the data; unfortunately, this violates the validity of the conclusion. The i-FWER test allows much flexibility: a human (or a computer program acting on the human’s behalf) may adaptively guide the algorithm in a data-dependent manner. We prove that our test controls FWER if the analysts adhere to a particular protocol of masking and unmasking. We demonstrate via numerical experiments the power of our test under structured non-nulls, and then explore new forms of masking. 
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  3. null (Ed.)