Expected monotonicity--a desirable property for evidence measures?

We consider here the principle of 'evidential consistency' - that as one gathers more data, any well-behaved evidence measure should, in some sense, approach the true answer. Evidential consistency is essential for the genome-scan design (GWAS or linkage), where one selects the most promising locus(i) for follow-up, expecting that new data will increase evidence for the correct hypothesis. Earlier work [Vieland, Hum Hered 2006;61:144-156] showed that many popular statistics do not satisfy this principle; Vieland concluded that the problem stems from fundamental difficulties in how we measure evidence and argued for determining criteria to evaluate evidence measures. Here, we investigate in detail one proposed consistency criterion - expected monotonicity (ExpM) - for a simple statistical model (binomial) and four likelihood ratio (LR)-based evidence measures. We show that, with one limited exception, none of these measures displays ExpM; what they do display is sometimes counterintuitive. We conclude that ExpM is not a reasonable requirement for evidence measures; moreover, no requirement based on expected values seems feasible. We demonstrate certain desirable properties of the simple LR and demonstrate a connection between the simple and integrated LRs. We also consider an alternative version of consistency, which is satisfied by certain forms of the integrated LR and posterior probability of linkage.