On the Distribution of Products of Random Variables

Łomnicki, Z. A.

doi:10.1111/j.2517-6161.1967.tb00713.x

Cited by 78 publications

(49 citation statements)

References 4 publications

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“…Whereas the ab point estimate of the indirect effect and the standard error associated with the z test have been found to have minimal bias in single level and multilevel simulation studies (Krull & MacKinnon, 1999;MacKinnon et al, 2002), the ab product does not generally follow a normal distribution (Bollen & Stine, 1990;Lomnicki, 1967;Springer & Thompson, 1966), which is an assumption associated with the z test. As noted in MacKinnon et al (2004), when Paths a and b both have a mean of zero, the sampling distribution of the ab product is symmetric but has higher kurtosis than a normal distribution.…”

Section: Single Sample Methodsmentioning

confidence: 96%

A Comparison of Single Sample and Bootstrap Methods to Assess Mediation in Cluster Randomized Trials

Pituch

Stapleton

Kang

2006

Multivariate Behavioral Research

117

109

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A Monte Carlo study examined the statistical performance of single sample and bootstrap methods that can be used to test and form confidence interval estimates of indirect effects in two cluster randomized experimental designs. The designs were similar in that they featured random assignment of clusters to one of two treatment conditions and included a single intervening variable and outcome, but they differed in whether the mediator was measured at the participant or site level. A bias-corrected bootstrap had the best statistical performance for each design and was closely followed by the empirical-Mtest, either of which is recommended for testing and estimating indirect effects in multilevel designs. In addition, consistent with previous research, the commonly used z test had relatively poor performance.

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Section: Single Sample Methodsmentioning

confidence: 96%

A Comparison of Single Sample and Bootstrap Methods to Assess Mediation in Cluster Randomized Trials

Pituch

Stapleton

Kang

2006

Multivariate Behavioral Research

117

109

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“…The distribution of the mean seed weight per pod, corresponding to the product of the former two traits, did not follow a normal distribution. With the exception of a very large sample, it is known that the product of two normal distributions does not follow a normal distribution but a Log-normal distribution or a related distribution (Lomnicki 1967). The distribution of the mean seed weight per pod fitted well to a generalized extreme value (GEV) distribution (Fig.…”

Section: Variability and Heritability Of The Traits Studiedmentioning

confidence: 96%

Relations between several traits linked to sexual plant reproduction in Theobroma cacao L.: number of ovules per ovary, number of seeds per pod, and seed weight

Cilas

Machado²,

Motamayor³

2009

Tree Genetics & Genomes

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Tropical perennial plants cultivated outside their place of origin call for major germplasm characterization and prebreeding work. For cacao breeding, the selection process has to optimize the weight of cacao beans produced per tree by increasing the number of pods produced, the number of seeds per pod (NoSP), and the mean weight of seeds produced (SW). NoSP and SW were studied over nine consecutive years in a collection of more than 200 clones. The number of ovules per ovary (NoOV) was also examined in a subsample of this collection in order to evaluate the ability to predict NoSP. NoSP and SW showed a relatively normal distribution with heritabilities of 0.29 and 0.51, respectively. The composite trait "mean seed weight per pod" showed a generalized extreme value distribution of intermediate heritability (0.43). NoSP and SW were found to be genetically independent. NoOV was highly heritable (h 2 =0.810), but it did not provide good prediction of NoSP. The largest seeds were observed in genotypes derived from crosses between cultivated cacao trees, which had been formerly domesticated. Domestication therefore favored seed size, while for the other traits examined, similar variation was found between wild and domesticated populations.

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“…In general, the product of two random variables is not normally distributed, even if both components are normally distributed (Lomnicki, 1967; Springer & Thompson, 1966). Therefore, there are concerns over the accuracy of normal-theory tests and CIs for the ab product because of expected departures from the normality assumption.…”

Section: Tests Of Mediationmentioning

confidence: 99%

Tests of Mediation: Paradoxical Decline in Statistical Power as a Function of Mediator Collinearity

Beasley

2013

The Journal of Experimental Education

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Increasing the correlation between the independent variable and the mediator (a coefficient) increases the effect size (ab) for mediation analysis; however, increasing a by definition increases collinearity in mediation models. As a result, the standard error of product tests increase. The variance inflation due to increases in a at some point outweighs the increase of the effect size (ab) and results in a loss of statistical power. This phenomenon also occurs with nonparametric bootstrapping approaches because the variance of the bootstrap distribution of ab approximates the variance expected from normal theory. Both variances increase dramatically when a exceeds the b coefficient, thus explaining the power decline with increases in a. Implications for statistical analysis and applied researchers are discussed.

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On the Distribution of Products of Random Variables

Cited by 78 publications

References 4 publications

A Comparison of Single Sample and Bootstrap Methods to Assess Mediation in Cluster Randomized Trials

A Comparison of Single Sample and Bootstrap Methods to Assess Mediation in Cluster Randomized Trials

Relations between several traits linked to sexual plant reproduction in Theobroma cacao L.: number of ovules per ovary, number of seeds per pod, and seed weight

Tests of Mediation: Paradoxical Decline in Statistical Power as a Function of Mediator Collinearity

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