Simulating multivariate <i>g</i>‐and‐<i>h</i> distributions

Kowalchuk, Rhonda K.; Headrick, Todd C.

doi:10.1348/000711009x423067

Cited by 28 publications

(34 citation statements)

References 33 publications

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“…Many non-normal distributions were investigated via g-and-h distributions (see Headrick, Kowalchuk, & Sheng, 2008;Hoaglin, 1983;1985;Kowalchuk & Headrick, 2010;Tukey, 1960). These distributions with their values for skewness and kurtosis are enumerated in Table 1.…”

Section: Methodsmentioning

confidence: 99%

“…These equations generate symmetric (g = 0) and asymmetric distributions (g ≠ 0), respectively. As Kowalchuk and Headrick (2010) noted, "The parameter ± g controls the skew of a distribution in terms of both direction and magnitude. The parameter h controls the tail weight or elongation of a distribution and is positively related with kurtosis" (p. 63).…”

Section: Methodsmentioning

confidence: 99%

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Generalized Linear Model Analyses for Treatment Group Equality when Data are Non-Normal

Kesleman

Othman

Wilcox

2016

J. Mod. App. Stat. Meth.

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One of the validity conditions of classical test statistics (e.g., Student's t-test, the ANOVA and MANOVA F-tests) is that data be normally distributed in the populations. When this and/or other derivational assumptions do not hold, the classical test statistic can be prone to too many Type I errors (i.e., falsely rejecting too often) and/or have low power (i.e., failing to reject when the null hypothesis is false) to detect treatment effects when they are present. However, alternative procedures are available for assessing equality of treatment group effects when data are non-normal. For example, researchers can use robust estimators instead of the usual least squares estimators to test that treatment effects are equivalent across groups. As well, recent advances in statistical methodology allow researchers to test for equality of treatment group effects by assuming other distributional shapes for the data. One class of such analyses is generalized linear model techniques. On the other hand, researchers can adopt sequential analyses where they first assess the normality assumption and then depending on the result determine the type of analysis that should be adopted. The purpose of the present study was to compare the above approaches for assessing equality of treatment group effects in the presence of non-normal data. Simulation results which were based on various non-normal distributions and the values of group variances and sample sizes revealed that sequential analysis coupled with a generalized linear model solution were just as prone to inflated or depressed rates of Type I error as the classical ANOVA F-test. Keywords:Tests for equality of treatment effects, non-normal data, multi-group problem, goodness-of-fit statistic, skewed and kurtotic data, 5-point Likert data, familywise Type I error control KESELMAN ET AL. 33

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Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

Generalized Linear Model Analyses for Treatment Group Equality when Data are Non-Normal

Kesleman

Othman

Wilcox

2016

J. Mod. App. Stat. Meth.

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“…As Kowalchuk and Headrick (29) note "The parameter ± g controls the skew of a distribution in terms of both direction and magnitude. The parameter h controls the tail weight or elongation of a distribution and is positively related with kurtosis."…”

Section: Methodsmentioning

confidence: 99%

Testing for normality in the multi-group problem: is this a good practice?

Keselman¹

2014

CDERM

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SummaryOne of the validity conditions of classical test statistics (e.g., the ANOVA F-test) is that data be normally distributed in the populations. When this derivational assumption does not hold classical test statistics can be prone to falsely rejecting too often and/or fail to reject when the null hypothesis is false. Thus, many authors have recommended that researchers routinely check that the assumption of normality is satisfied. The findings regarding the power of normality tests are mixed. That is, some authors believe that tests for normality lack power to detect non-normality in the data; others however, have found that some tests for non-normality are quite effective in detecting non-normal data. The purpose of our study was to re-examine this question but in the multi-group problem. We found that two of the goodness-of-fit statistics, the Anderson-Darling and Cramer-von Mises tests were quite effective in detecting non-normality over a large number of non-normal data conditions with sample sizes that were not unreasonably large, particularly if a wiser choice is made regarding the level of significance (e.g., α = 15 or α = 20). We also demonstrate how overall Type I error control can be maintained over the multiple tests for normality with a sequentially-rejective Bonferroni procedure.

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“…Schoder, et al (2006 investigated a normal distribution with a single outlier, a normal distribution with 10% outliers, skewed lognormal distributions with varying skewness, and an ordinal 5-point Likert scale with varying multinomial probabilities (common they state in dermatological investigations). Many non-normal distributions were investigated via g-and-h distributions (See Headrick, Kowalchuk, & Sheng, 2008;Hoaglin, 1983;1985;Kowalchuk & Headrick, 2010;Tukey, 1960). These distributions with their values for skewness and kurtosis are enumerated in Table 1.…”

Section: Methodsmentioning

confidence: 99%

“…As Kowalchuk and Headrick (2010) noted "The parameter ±g controls the skew of a distribution in terms of both direction and magnitude. The parameter h controls the tail weight or elongation of a distribution and is positively related with kurtosis."…”

Section: Methodsmentioning

confidence: 99%