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. 2012;47(1):61-87.
doi: 10.1080/00273171.2012.640596.

Explanation of Two Anomalous Results in Statistical Mediation Analysis

Matthew S Fritz  1 , Aaron B Taylor, David P Mackinnon
Affiliations

Explanation of Two Anomalous Results in Statistical Mediation Analysis

Matthew S Fritz et al. Multivariate Behav Res. 2012.

Abstract

Previous studies of different methods of testing mediation models have consistently found two anomalous results. The first result is elevated Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap tests not found in nonresampling tests or in resampling tests that did not include a bias correction. This is of special concern as the bias-corrected bootstrap is often recommended and used due to its higher statistical power compared with other tests. The second result is statistical power reaching an asymptote far below 1.0 and in some conditions even declining slightly as the size of the relationship between X and M, a, increased. Two computer simulations were conducted to examine these findings in greater detail. Results from the first simulation found that the increased Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap are a function of an interaction between the size of the individual paths making up the mediated effect and the sample size, such that elevated Type I error rates occur when the sample size is small and the effect size of the nonzero path is medium or larger. Results from the second simulation found that stagnation and decreases in statistical power as a function of the effect size of the a path occurred primarily when the path between M and Y, b, was small. Two empirical mediation examples are provided using data from a steroid prevention and health promotion program aimed at high school football players (Athletes Training and Learning to Avoid Steroids; Goldberg et al., 1996), one to illustrate a possible Type I error for the bias-corrected bootstrap test and a second to illustrate a loss in power related to the size of a. Implications of these findings are discussed.

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Figures

FIGURE 1
FIGURE 1
The single mediator model.
FIGURE 2
FIGURE 2
Type I error rate as a function of effect size of a and b, when the other parameter is equal to zero, collapsing across test and sample size.
FIGURE 3
FIGURE 3
Type I error rate as a function of sample size and effect size of a, collapsed across number of bootstrap samples. Note. b = 0.
FIGURE 4
FIGURE 4
Actual and ideal power curves for the numerical integration test as a function of a for condition in which b = .20, c′ = .50, and n = 160.
FIGURE 5
FIGURE 5
Power stagnation as a function of b and sample size. The figure is truncated at b = .40 because power stagnation was very near zero for all methods beyond this value.
FIGURE 6
FIGURE 6
Power stagnation as a function of b and test of mediation. The Baron and Kenny (1986) test has results separated by size of c′ because it was the only test affected by size of c′. The joint significance test and percentile bootstrap are omitted because their power stagnation values were nearly identical to those of the numerical integration test. The figure is truncated at b = .40 because power stagnation was very near zero for all methods beyond this value.
FIGURE 7
FIGURE 7
Power stagnation as a function of average power. Average power is the proportion of the area in a power curve figure falling below the power curve, which is also the average level of power across the power curve.
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