The seminal work of Morgan & Rubin (2012) considers rerandomization for all the units at one time.In practice, however, experimenters may have to rerandomize units sequentially. For example, a clinician studying a rare disease may be unable to wait to perform an experiment until all the experimental units are recruited. Our work offers a mathematical framework for sequential rerandomization designs, where the experimental units are enrolled in groups. We formulate an adaptive rerandomization procedure for balancing treatment/control assignments over some continuous or binary covariates, using Mahalanobis distance as the imbalance measure. We prove in our key result that given the same number of rerandomizations, in expected value, under certain mild assumptions, sequential rerandomization achieves better covariate balance than rerandomization at one time.

A combination of three psychiatric screening tests was used to uncover the extent of affective-psychotic symptoms, the indications of "chronic brain syndrome," and the perceptual limitations among two selected populations of elderly persons. It was found that 25 percent of 48 residents in a home for the aged, and 9 percent of 45 members in a social club, had four or more affective-psychotic com-

In this paper, we resolve a longstanding open statistical problem. The problem is to mathematically prove Yule's 1926 empirical finding of "nonsense correlation" ([15]). We do so by analytically determining the second moment of the empirical correlation coefficient

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