Studies in the Theory of Sampling

Pepper, Joseph

doi:10.2307/2332559

Cited by 2 publications

(2 citation statements)

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“…To account for dependence of variables in the linear regression context, the variance of skewness difference (σγX−γY2=σγX2+σγY2−2cov(γX,γY)) can be used as a basis for the denominator, where σγX2 and σγY2 are the variances of γ X and γ Y , and cov (γ X , γ Y ) is the covariance of γ X and γ Y . The variances of standard normally distributed quantities equal one and the correlation between third central moments of two variables can be approximated by the cube of the correlation between X and Y , ργXγY=cov(γX,γY)/(σγXσγY)≈ρXY3 (see Pearson & Young, 1918; Pepper, 1929; Rider, 1929; Wishart, 1928, for detailed discussions of sampling moments of product moment coefficients and sampling moments of moments). Thus, σγX−γY2=σγX2+σγY2−2ρ…”

Section: Direction Dependence In Mediation Analysis: the Regression R...mentioning

confidence: 99%

Direction of effects in mediation analysis.

Wiedermann¹,

Eye²

2015

Psychological Methods

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Data collected in the social sciences are rarely normally distributed. The linear regression methods that are usually employed to test mediation hypotheses consider moments no higher than second order. Recently discussed methods of direction dependence do consider higher moments. After a review of commonly used methods for mediation analysis, the present article demonstrates that these methods do not allow one to make decisions about competing mediation models, that is, models in which the reverse flow of causality is considered. Then, direction of dependence methodology is introduced which allows one to evaluate hypotheses of direction of effects, and extend its application to mediation analysis. Significance tests for statistical inference on direction of effects are proposed and discussed. Results of a Monte-Carlo simulation of the performance of the tests under various data scenarios are presented. An empirical example from research on intimate partner violence is given. Finally, possible limitations of these methods are addressed, issues of implicit assumptions concerning the origin of observed skewness are discussed, and the new methodology is embedded into the larger framework of causal inference.

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Section: Direction Dependence In Mediation Analysis: the Regression R...mentioning

confidence: 99%

Direction of effects in mediation analysis.

Wiedermann¹,

Eye²

2015

Psychological Methods

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show abstract

“…Thus, we need to account for the dependency between ε and ε′ as well. Early work by Pearson and Filon (), Pearson and Young (), Wishart (), Pepper (), and Rider () examined the sampling moments of moments of variables and the sampling moments of product moment coefficients. From these studies, it is known that the correlation of the first moments of two variables from a normal population (i.e., μ X and μ Y ) and the correlation of the second moments of two variables from a normal population (i.e.,

σ_{X}^{2}

and

σ_{Y}^{2}

) can be expressed as

{italicρ}_{{italicμ}_{X} {italicμ}_{Y}} = {italicρ}_{X Y}

and

{italicρ}_{{italicσ}_{X}^{2} {italicσ}_{Y}^{2}} = {italicρ}_{X Y}^{2}

, respectively.…”

Section: Statistical Inference On γε and γBold-italicε′mentioning

confidence: 99%

Significance tests to determine the direction of effects in linear regression models

Wiedermann

Hagmann

Eye

2014

Brit J Math & Statis

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Previous studies have discussed asymmetric interpretations of the Pearson correlation coefficient and have shown that higher moments can be used to decide on the direction of dependence in the bivariate linear regression setting. The current study extends this approach by illustrating that the third moment of regression residuals may also be used to derive conclusions concerning the direction of effects. Assuming non-normally distributed variables, it is shown that the distribution of residuals of the correctly specified regression model (e.g., Y is regressed on X) is more symmetric than the distribution of residuals of the competing model (i.e., X is regressed on Y). Based on this result, 4 one-sample tests are discussed which can be used to decide which variable is more likely to be the response and which one is more likely to be the explanatory variable. A fifth significance test is proposed based on the differences of skewness estimates, which leads to a more direct test of a hypothesis that is compatible with direction of dependence. A Monte Carlo simulation study was performed to examine the behaviour of the procedures under various degrees of associations, sample sizes, and distributional properties of the underlying population. An empirical example is given which illustrates the application of the tests in practice.

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Studies in the Theory of Sampling

Cited by 2 publications

References 0 publications

Direction of effects in mediation analysis.

Direction of effects in mediation analysis.

Significance tests to determine the direction of effects in linear regression models

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