Rerandomization Inference on Regression and Shift Effects: Computationally Feasible Methods

An idea due to Quade (1973) is used to develop a method that reduces the computational load for rerandomization inferences in certain situations, and, for experiments carried out with a sufficiently small reference set of randomizations, makes computations entirely feasible. Consider an experimental design for N units for which an experimental plan E is to be chosen randomly from a set of K plans. Plan E will assign treatment dosage dE,u to unit u with postulated additive effect δdE,u reflected in the observation zu . Rerandomization inferences are to be carried out by means of “pivotal” statistics te,E (z) = [bE (z) - be (z)]/[1 - be (d E )], where z and d E are observation and dosage vectors and be ([mdot]) is the regression (slope) onto d e , the dosages under plan e (∈ ). (a) The P value of a test of the hypotheses δ = Δ versus δ > Δ is essentially the proportion of te,E (z) values, e ∈ , that are at most Δ. (b) If t (l),E (z) is the lth largest of the te,E (z)'s, e ∈ , then a (L - l)/K confidence interval is t (l),E (z) ≤ δ ≤ t (L),E (z). (c) The power of the α-level test of δ = Δ is F(δ - Δ), where F is the left-continuous empirical cdf of -t (l),E (ζ), E ∈ —l being the largest integer ≤ αK and ζ the vector of N potential observations without any treatment effects. The computations for (a) and (b) are O(K) and for (c) O(K 2), and K may be as small as 1,000 (or less). This article formulates the rerandomization regression model and develops the pivotal method. The special case of the two-sample shift problem is also treated. The merits of the randomization model, in contrast to a random sampling model, are stressed.