Evidence for Alternative Hypotheses

Abstract

Most researchers want evidence for the direction of an effect, not evidence against a point null hypothesis. Such evidence is ideally on a scale that is easily interpretable, with an accompanying standard error. Further, the evidence from identical experiments should be repeatable, and evidence from independent experiments should be easily combined, such as required in meta-analysis. Such a measure of evidence exists and has been shown to be closely related to the Kullback-Leibler symmetrized distance between null and alternative hypotheses for exponential families. Here we provide more examples of the latter phenomenon, for distributions lying outside the class of exponential families, including the non-central chi-squared family with unknown non-centrality parameter.

Indicators of Evidence for Bioequivalence

“…For exponential families, Reference [10] show that the expected evidence of the variance stabilized statistic T is approximately equal to the signed square root of the Kullback-Leibler symmetrized divergence. Examples not from exponential families are in [11][12][13]. These results gives support to calling T the "evidence for the alternative hypothesis".…”

“…Further, at perfect equivalence λ = 0 it has E[T] ≈ K λ 0 (0) for K λ 0 given by (12). These claims are made based on simulation studies, using R scripts in Appendix B.…”

“…The evidence in S ∼ χ 2 ν (λ) for equivalence λ < λ 0 was derived in Section 3.1.The practical problem is to choose λ 0 . Once λ 0 is chosen one can compute the maximum evidence in the experiment; it is by (12) …”

“…Let f λ be the noncentral χ 2 density with ν degrees of freedom, so that by (12), n1 <-10 x1bar <-99.8120 s1 <-7.5639 n2 <-12 x2bar <-99.2903 s2 <-5.9968 n3 <-13 x3bar <-100.0024 s3 <-10.4809 n4 <-15 x4bar <-98.6407 s4 <-4.5309 N <-n1+n2+n3+n4 ## xbarbar <-(n1*x1bar+n2*x2bar+n3*x3bar+n4*x4bar)/N ## 99.3849 K <-4 nbar <-N/K ssqW <-(n1-1)*s1^2+(n2-1)*s2^2+(n3-1)*s3^2+(n4-1)*s4^2 ssqB <-n1*(x1bar-xbarbar)^2+n2*(x2bar-xbarbar)^2+n3*(x3bar-xbarbar)^2 ssqB <-ssqB+n4*(x4bar-xbarbar)^2…”

Evidence for goodness of fit in Karl Pearson chi-squared statistics

“…In Section 1.3 the evidence against the desired model in the chi-squared statistic and the calibration scale for this evidence is described, based on earlier work by Kulinskaya et al (2008), Morgenthaler & Staudte (2012, 2013, 2016. It is then shown that this evidence is essentially an estimate of the square root of the symmetrized Kullback & Leibler (1951) divergence between the null and alternative chi-square models.…”