A note on generalized Genome Scan Meta-Analysis statistics

We provide an Edgeworth approximation to the null distribution of the weighted GSMA statistic, and, we examine the limiting distribution of the GSMA statistics under the order statistic formulation, and quantify the relevance of the pairwise correlations of the GSMA statistics across different bins on this limiting distribution. We also remark on aggregate criteria and multiple testing for determining significance of GSMA results.Theoretical considerations detailed herein can lead to clarification and simplification of testing criteria for generalizations of the GSMA statistic.Wise, Lanchbury and Lewis [1] introduced a rank-based statistical technique for meta-analysis of genome scans, the Genome Scan Meta-Analysis (GSMA) method, and derived its exact null distribution using a clever inclusion/exclusion argument. Koziol and Feng [2] provided an alternative derivation of the null distribution of the GSMA statistic via a combinatoric approach involving probability generating functions, and suggested an Edgeworth series approximation to its exact null distribution that improves upon the Wise [1] normal approximation.Levinson [3] described two generalizations to the GSMA statistic: (i) a weighted version of the GSMA statistic, so that different studies could be ascribed different weights for analysis; and (ii) an order statistic approach, reflecting the fact that a GSMA statistic can be computed for each chromosomal region or bin across the various genome scan studies. Wise [1] had suggested that each chromosomal region (bin) be about 30 cM, leading to a total of about n = 120 bins spanning the entire genome, and correspondingly 120 GSMA statistics. Wise [1] and Koziol and Feng [2] had investigated the marginal distribution of any of these (exchangeable) GSMA statistics, whereas under the order statistic formulation of Levinson [3], the joint distribution of the entire set of GSMA statistics is taken into account. In this note, we consider both generalizations in turn.