Rerandomization Inference on Regression and Shift Effects: Computationally Feasible Methods

Abstract

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 d E,u to unit u with postulated additive effect δd E,u reflected in the observation z u . Rerandomization inferences are to be carried out by means of “pivotal” statistics t e,E ( z ) = [ b E ( z ) - b e ( z )]/[1 - b e ( d E )], where z and d E are observation and dosage vectors and b e ([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 t e,E ( z ) values, e ∈ , that are at most Δ. (b) If t ( l ), E ( z ) is the l th largest of the t e,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.