Advantages of Variance Stabilization

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.

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“…It is also true for the non-central t, see [9], and the non-central chi-square models, see Section 3.…”

Section: Example 1 Normal Modelmentioning

confidence: 99%

“…Our purpose here is to explain in more detail why we advocate this particular definition. Connections with other measures of evidence, such as the p-value and Bayes factor, are given in [9].…”

Section: Desirable Properties Of Statistical Evidencementioning

confidence: 99%

Evidence for Alternative Hypotheses

Morgenthaler

Staudte

2013

Robustness and Complex Data Structures

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“…As seen earlier in this paper, the Key plays an important role in estimation by confidence intervals. Another advantage of variance stabilized statistics is that they can be readily combined in a meta-analysis of effects from multiple studies, as shown in (Kulinskaya et al 2008;Malloy et al 2013;and Morgenthaler and Staudte 2012).…”

Section: Choosing Rmentioning

confidence: 97%

Inference for the Standardized Median

Staudte

2013

Contemporary Developments in Statistical Theory

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We carry out inference for the median, divided by a fixed interquantile range (IQR). It is a standardized effect or 'effect size' defined by 3 quantiles. Applied statisticians sometimes prefer effect sizes to raw effects, such as the median, because it is scalefree. Inference regarding the median itself has already been thoroughly investigated by a number of authors, including (McKean and Schrader (1984) and Sheather and McKean 1987). More generally, the problem of Studentizing a quantile estimator has been studied by (Bloch and Gastwirth 1968;Hall andSheather 1988 andSiddiqui 1960), amongst others.We are given a location-scale family F α,β (x) = F ((x − α)/β) where α, β > 0 are unknown. Assume F = F 0,1 has a continuous derivative f which is symmetric about 0 and is positive over a possibly infinite symmetric interval containing 0. Denote the quantile function of F by G = F −1 , its value at any 0 < r < 1 by x r = G(r), and its derivative at r by g(r) = {f (x r )} −1 . The rth quantile of F α,β is related to x r of F by α + βx r , and g α,β (r) = β g(r).For 0 < r < 0.5, a value to be chosen later, let the rth IQR of F α,β be β IQR r , where IQR r = x 1−r − x r . Also define the rth standardized median of F α,β by:th order statistic of a sample of size n from F . Let M = X ([n/2]) be the sample median, which is consistent for x 0.5 and for a fixed 0 < r < 1/2 define the rth sample IQR by, which is a consistent estimator of x 1−r − x r . We want to estimate the rth standardized median defined in (22.1) byδ r = M/R r . In the next Sect. 22.2 we derive a variance stabilizing transformation (VST) ofδ r . This leads to confidence intervals for δ r , whose effectiveness of coverage is evaluated R. G. Staudte ( )

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“…The first row in Table 1 gives a much more reasonable estimate of "evidence for the alternative" together with an easily understood standard error. The compatibility of this calibration scale for evidence with Bayesian calibration scales for p-values and Bayes factors is discussed in [10] (Section 4.3). Table 1.…”

Section: Properties Of Evidence In One-sided Z-testsmentioning

confidence: 99%

“…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].…”

Section: Background and Summarymentioning

confidence: 99%

Indicators of Evidence for Bioequivalence

Morgenthaler

Staudte

2016

Entropy

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Some equivalence tests are based on two one-sided tests, where in many applications the test statistics are approximately normal. We define and find evidence for equivalence in Z-tests and then one-and two-sample binomial tests as well as for t-tests. Multivariate equivalence tests are typically based on statistics with non-central chi-squared or non-central F distributions in which the non-centrality parameter λ is a measure of heterogeneity of several groups. Classical tests of the null λ ≥ λ 0 versus the equivalence alternative λ < λ 0 are available, but simple formulae for power functions are not. In these tests, the equivalence limit λ 0 is typically chosen by context. We provide extensions of classical variance stabilizing transformations for the non-central chi-squared and F distributions that are easy to implement and which lead to indicators of evidence for equivalence. Approximate power functions are also obtained via simple expressions for the expected evidence in these equivalence tests.

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Advantages of Variance Stabilization

Cited by 21 publications

References 28 publications

Evidence for Alternative Hypotheses

Evidence for Alternative Hypotheses

Inference for the Standardized Median

Indicators of Evidence for Bioequivalence

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