A Note on the Transformation of Chi-Squared Variables to Normality

Hawkins, Douglas M.; Wixley, R. A. J.

doi:10.1080/00031305.1986.10475420

Cited by 52 publications

(39 citation statements)

References 5 publications

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“…We first calculated for each gene set its SUMSTAT score, which is the sum of selection scores of genes in the set of interest (Tintle et al 2009; Daub et al 2013). As selection score we took the fourth-root of the ΔlnL g values (called ΔlnL4 hereafter) to ensure that the distribution of nonzero ΔlnLs is approximately Normal (Hawkins and Wixley 1986; Roux et al 2014). This procedure also prevents extreme scoring genes from getting too much weight in the test, which would otherwise result in significant pathways mostly due to a few outlier genes.…”

Section: Methodsmentioning

confidence: 99%

Detection of Pathways Affected by Positive Selection in Primate Lineages Ancestral to Humans

Daub

Moretti

Davydov

et al. 2017

Molecular Biology and Evolution

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Gene set enrichment approaches have been increasingly successful in finding signals of recent polygenic selection in the human genome. In this study, we aim at detecting biological pathways affected by positive selection in more ancient human evolutionary history. Focusing on four branches of the primate tree that lead to modern humans, we tested all available protein coding gene trees of the Primates clade for signals of adaptation in these branches, using the likelihood-based branch site test of positive selection. The results of these locus-specific tests were then used as input for a gene set enrichment test, where whole pathways are globally scored for a signal of positive selection, instead of focusing only on outlier “significant” genes. We identified signals of positive selection in several pathways that are mainly involved in immune response, sensory perception, metabolism, and energy production. These pathway-level results are highly significant, even though there is no functional enrichment when only focusing on top scoring genes. Interestingly, several gene sets are found significant at multiple levels in the phylogeny, but different genes are responsible for the selection signal in the different branches. This suggests that the same function has been optimized in different ways at different times in primate evolution.

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Section: Methodsmentioning

confidence: 99%

Detection of Pathways Affected by Positive Selection in Primate Lineages Ancestral to Humans

Daub

Moretti

Davydov

et al. 2017

Molecular Biology and Evolution

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“…This observation was evaluated using a rmANOVA. The dependent measure was the Chi-square values characterizing the quality of the function fit, transformed by applying an exponent of 0.25 to approximate a normal distribution (Hawkins and Wixley 1986). The independent variables were dataset (novel-stimulus, mean of five repeated-stimulus runs, all repeated-stimulus runs pooled), and masker (multi-talker babble, speech-shaped noise, AM noise).…”

Section: Resultsmentioning

confidence: 99%

Masked Sentence Recognition Assessed at Ascending Target-to-Masker Ratios

Buss

Calandruccio

Hall³

2015

Ear &Amp; Hearing

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Objectives Masked sentence recognition is typically evaluated by presenting a novel stimulus on each trial. As a consequence, experiments calling for replicate estimates in multiple conditions require large corpora of stimuli. The present study evaluated the consequences of repeating sentence-plus-masker pairs at ascending target-to-masker ratios (TMRs). The hypothesis was that performance on each trial would be consistent with the cues available to the listener at the associated TMR, resulting in similar estimates of threshold and slope for procedures using novel vs. repeated sentences within an ascending-TMR block of trials. Design A group of 37 normal-hearing young adults participated. Each listener was tested in the presence of one of three maskers: a multi-talker babble, a speech-shaped noise, or an amplitude-modulated speech-shaped noise. There were two data collection procedures, both proceeding in blocks of trials with ascending TMRs. The novel-stimulus procedure used five lists of AzBio sentences, one presented at each of five TMRs, with a novel sentence and masker sample on each trial. The repeated-stimulus procedure used a single list of AzBio sentences, with each sentence presented at multiple TMRs, progressing from low to high; each sentence was paired with a single masker sample, such that only the TMR changed within blocks of repeated stimuli. Listeners completed one run with the novel-stimulus procedure and five runs with the repeated-stimulus procedure. The resulting values of percent correct at each TMR were fitted with a logit function to estimate threshold and psychometric function slope. Results The novel-stimulus and repeated-stimulus procedures resulted in generally similar data patterns. After controlling for effects related to the order in which listeners completed the six data collection runs, mean thresholds were slightly higher (<0.5 dB) for the repeated-stimulus procedure than the novel-stimulus procedure in all three maskers. Function slopes for the multi-talker babble and amplitude modulated noise maskers were slightly shallower using the repeated-stimulus than the novel-stimulus procedure, but slopes were comparable for the speech-shaped noise. The quality of psychometric function fits was significantly better for the repeated- than the novel-stimulus procedure, even when comparing a single run of the repeated-stimulus procedure (using one list) to a run of the novel-stimulus procedure (using five lists). Conclusions Repeating sentences at ascending TMRs is an efficient method for estimating thresholds and psychometric function slopes, both in terms of the number of sentences and the number of trials.

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“…3.2, for a justification of the choice λ = 1 3 ). Hawkins and Wixley (1986) have argued that the approximation can be improved for smaller values of a by using λ = 1 4 . In what follows, we compare the foregoing two methods for approximating gamma quantiles.…”

Section: Normal Approximations and The Proposed Methodsmentioning

confidence: 98%

“…In particular, the associate editors' comments resulted in the clarification of several ideas, and the associate editor also drew the author's attention to the work of Hernandez and Johnson (1980) and Hawkins and Wixley (1986). The research of Krishnamoorthy and Mathew was supported by National Institute of Occupational Safety and Health grant R01-OH03628.…”

Section: Acknowledgmentsmentioning

confidence: 97%

“…Specifically, the Wilson-Hilferty (WH) approximation states that if X follows a two-parameter gamma distribution, then the distribution of X 1/3 can be approximated by a normal distribution. We investigated the accuracy of this approximation and made a comparison with a more recent approximation due to Hawkins and Wixley (1986), which states that X 1/4 can be approximated by a normal distribution. Specifically, we compared the quantiles of a gamma distribution with those obtained using the WH and Hawkins-Wixley (HW) approximations.…”

Section: Introductionmentioning

confidence: 98%

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Normal-Based Methods for a Gamma Distribution

2008

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In this article we propose inferential procedures for a gamma distribution using the Wilson-Hilferty (WH) normal approximation. Specifically, using the result that the cube root of a gamma random variable is approximately normally distributed, we propose normal-based approaches for a gamma distribution for (a) constructing prediction limits, one-sided tolerance limits, and tolerance intervals; (b) for obtaining upper prediction limits for at least l of m observations from a gamma distribution at each of r locations; and (c) assessing the reliability of a stress-strength model involving two independent gamma random variables. For each problem, a normal-based approximate procedure is outlined, and its applicability and validity for a gamma distribution are studied using Monte Carlo simulation. Our investigation shows that the approximate procedures are very satisfactory for all of these problems. For each problem considered, the results are illustrated using practical examples. Our overall conclusion is that the WH normal approximation provides a simple, easy-to-use unified approach for addressing various problems for the gamma distribution.

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A Note on the Transformation of Chi-Squared Variables to Normality

Cited by 52 publications

References 5 publications

Detection of Pathways Affected by Positive Selection in Primate Lineages Ancestral to Humans

Detection of Pathways Affected by Positive Selection in Primate Lineages Ancestral to Humans

Masked Sentence Recognition Assessed at Ascending Target-to-Masker Ratios

Normal-Based Methods for a Gamma Distribution

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