69 and the approximation to this component is
1
2 21 ( )ˆ 2 .
2tr Σ β H = mx ym x (4.27) Substituting these results into Δbias yields the appropriate asymptotic bias approximation
1
2 2 2 2 2 2
1 ( )ˆ 1 ( )ˆ
2 4
.
yy
ym x mx mx ym x mx ym x
tr tr
= −
= + −
Σ β H Σ β H
(4.28)
The second bias approximation in Equation 4.28 is multiplied by ½ because the first approximation does not include approximations of second-order terms (i.e., mx2 ym x2 ), so adjusting by Equation 4.27 yields an over-correction.
Although unbiasedness is a desirable property for estimators, it is possible for other important properties of these estimators to be deficient, such that a biased estimator of the same parameter is more useful in practice. Particularly notable for estimators are consistency (i.e., the estimator converges to the parameter as sample size increases), and variability. For example, an unbiased estimator with high sampling variability can be substantially less useful than a biased estimator that is more efficient.
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importance for researchers to know if there are certain conditions under which the estimators would not be expected to yield accurate estimates.
4.4.1 Simulation Design
The generating model for this simulation was a parallel mediation model with a single predictor x, a single outcome y, and two mediators m1 and m2 (Figure 2). Variables in this simulation were considered standard normal in the population. As described in Section 4.2.1, effect sizes can be estimated for three indirect effects: 1) the specific indirect effect of x on y through m1 ( m x1 ym1x), 2) the specific indirect effect of x on y through m2 ( m x2 ym2x), and 3) and the total indirect effect of x on y through m1 and m2 ( m x1 ym1x + m x2 ym2x). Because of the symmetry in the magnitudes of the specific effects, effect sizes for this simulation will be evaluated for the total indirect and one specific indirect effect, m x1 ym1x.
4.4.2 Simulation Conditions
Parameter values for the paths were varied among .15, .39, and .59, magnitudes for small, medium, and large standardized coefficients common in applied research. Because of the large number of parameter combinations possible in this model, some generating parameters were constrained to be equal (m x2 =ym2x) and some were fixed to zero, including the direct effect and the residual correlation of the mediators. Parameter values consistent with the null
hypothesis of no indirect effect were not considered in this simulation because zero is on the boundary of the parameter space for . In addition, the properties of estimators typically evaluated under the null hypothesis (i.e., Type I error rate, power) are not of interest for estimators because ˆ and were not intended to be used for null hypothesis significance
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testing. Sample size was varied among 50, 100, 250, and 500. This yields a total of 3x3x3x4 = 108 total conditions.
Because the sampling distributions of ˆ and are unknown and assumed to be non- normally distributed, nonparametric bootstrapping was used to construct 95% CIs. Two
bootstrap CI methods were evaluated in this simulation: a) percentile and b) BCa (Section 2.3).
Although percentile bootstrap CIs performed satisfactorily in terms of coverage and balance for
estimates from a three-variable mediation model (Lachowicz et al., 2018), BCa CIs are often recommended for estimators with non-normal, heavy-tailed sampling distributions, which is characteristic of the distributions of both ˆ and . It is expected that BCa CIs will outperform percentile CIs in terms of coverage and balance, particularly in conditions with small effect magnitudes.
The simulation was conducted in R (version 3.4.1; R Core Team, 2017). 1,000
replications per condition is sufficient to obtain accurate estimates of bias for point estimates and coverage for CIs. For each replication, 1,000 bootstrap resamples are used to construct 95% CIs using the boot package (Canty & Ripley, 2017). The point estimators will be evaluated in terms of bias, overall accuracy, and relative efficiency, and CIs will be evaluated in terms of coverage, coverage balance, and CI width.
4.4.3 Evaluation criteria
Bias was evaluated using percent relative bias, defined as the difference between the expected value of the estimator and the population value, divided by the population value
[ ]ˆ
( )ˆ .
rel
bias E
= − (4.29)
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The metric of percent relative bias is often more interpretable than is raw bias, although it is possible for trivially small raw bias to appear quite large in terms of percent relative bias when the population parameter is close to zero. Researchers often use a criterion of 5% for acceptable relative bias (Boomsma, 2013), which I will also use in this study. It was hypothesized that the unadjusted estimator will exhibit positive bias, with the largest biases in conditions with smallest sample sizes and smallest effects. It was hypothesized that the adjusted estimator will exhibit acceptable bias (< 5%) in all conditions.
Accuracy was evaluated in terms of mean square error (MSE). MSE is defined as
ˆ 2 ˆ
MSE=E bias[ ( )] +var( ). (4.30) It follows that for an unbiased estimator, MSE is equivalent to the estimator variance, and MSE will favor estimators with less variability. However, it possible there are circumstances where a biased estimator can have less variance than an unbiased estimator, such that the biased estimator returns more accurate estimates. To further examine the variability of the estimators, relative efficiency (RE) was evaluated by the ratio of the empirical sampling variances
1 2
var( )ˆ ˆ . var( )
RE
= (4.31)
1
RE corresponds to the sampling variance of ˆ2 ˆ1, RE 1 to the sampling variance of
2 1
ˆ ˆ
, and RE =1 to equal sampling variances. RE was defined as the ratio of the sampling variance of to ˆ for this simulation. It is not clear if there are conditions where the unadjusted
ˆ would be more accurate or have greater sampling variability than the bias-adjusted , so these questions will be addressed empirically.
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CIs were evaluated using average CI width, coverage, and balance of coverage. Average CI width was used to evaluate the precision of the estimates, where smaller widths represent estimates with greater precision. It was unclear which estimator will have smaller average CI widths; this will be evaluated empirically. Coverage is defined as the proportion of CIs that contain the true population value. It is expected that the coverage probabilities of proper CIs are equal to one minus the nominal alpha level. The nominal alpha level for CIs in this study is .05, and coverage will be evaluated according to Bradley’s criteria (.925 - .975; Bradley, 1978). It is expected that 95% CIs will have coverage probabilities acceptably close to the nominal .95 level.
In addition to achieving nominal coverage, it is assumed for proper CIs that the proportion of times the population value is greater than the upper CI limit and less than the lower CI limit are equal (i.e., 2.5% for 95% CIs). However, it is possible for CIs to achieve the nominal alpha and be imbalanced in the proportion of misses above and below the confidence limits, which results in biased estimates of Type 1 error rates and power. It is expected that the proportion of misses above and below the 95% CI limits are equal.
Finally, it is unclear how or if the sampling properties of the estimators of the effect size for the total indirect effect would differ as compared to the specific indirect effect, so this question will be addressed empirically.
4.5 Simulation Results