Sample Size for AIPE on a Mediated (Indirect) Effect $ab$

Determines the sample size needed for the confidence interval on a mediated effect $ab$ (the product of the $X \to M$ and $M \to Y$ coefficients in a simple three-variable mediation model) to have a desired full width. Two methods are available. "closed_form" (the default) plans for the symmetric Wald interval built on the delta method standard error of the product (Sobel, 1982) and answers instantly. "monte_carlo" plans for the Monte Carlo confidence interval (MacKinnon, Lockwood, & Williams, 2004; Tofighi & MacKinnon, 2011), the interval mediation_mbco reports under ci_method = "monte_carlo", and it plans by a priori Monte Carlo simulation: at each candidate sample size the mediation model is fit to G simulated data sets and the realized interval widths are recorded. Because the Monte Carlo interval respects the skewness of the sampling distribution of a product, and because the simulation measures the widths that fitted models actually deliver, method = "monte_carlo" is the recommended way to settle on the final sample size; the closed form is its fast first approximation.

Usage

ss_aipe_indirect_effect(
  a,
  b,
  width,
  method = c("closed_form", "monte_carlo"),
  conf_level = 0.95,
  n_max = 10000L,
  B = 5000L,
  G = 1000L,
  seed = NULL
)

Arguments

a: Anticipated population coefficient for $X \to M$, on the standardized scale. Numeric scalar in $(-1, 1)$.
b: Anticipated population coefficient for $M \to Y$ controlling for $X$, standardized. Numeric scalar in $(-1, 1)$.
width: Desired full width of the confidence interval on $ab$.
method: One of "closed_form" (default) or "monte_carlo"; see Details.
conf_level: Desired confidence level (default 0.95).
n_max: Upper bound on the search; default 10000.
B: Number of Monte Carlo draws forming the interval within each simulated study when method = "monte_carlo"; default 5000. This is the same B the analysis-stage interval uses (see mediation_mbco).
G: Number of simulated studies per candidate sample size when method = "monte_carlo"; default 1000. The mean simulated width carries a simulation error of about its standard deviation over $\sqrt{G}$; raise G for a sharper answer.
seed: Optional integer seed for the Monte Carlo method, used locally (the caller's random number generator state is restored on exit). Default NULL leaves the random number generator state alone.

Value

A data.frame with rows for the recommended sample size, the expected CI width at that size, and the inputs echoed back. Under method = "closed_form" the expected width is the delta method width evaluated at the returned sample size; under method = "monte_carlo" it is the mean simulated width there. The method is carried on the returned object as the ci_method attribute.

Details

The mediation model. The simple mediator model is $$M = \alpha_1 + a X + \varepsilon_M,$$ $$Y = \alpha_2 + c' X + b M + \varepsilon_Y,$$ with the indirect (mediated) effect of $X$ on $Y$ through $M$ equal to $ab$ (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002). Both methods plan on the standardized scale with no direct effect: the planning population takes $X$, $M$, and $Y$ with unit variances and $c' = 0$, so a and b are the standardized paths.

The closed form. Under the planning population the sampling variance of $\hat a$ is $(1 - a^2)/(n - 2)$. In the equation for $Y$ the mediator is regressed alongside $X$, with which it is correlated at $a$, so the sampling variance of $\hat b$ carries the variance inflation factor $1/(1 - a^2)$: $$\mathrm{Var}(\hat b) \;=\; \frac{1 - b^2}{(n - 3)(1 - a^2)}.$$ The estimators come from two separate equations and are uncorrelated, so the delta method (Sobel, 1982) standard error of the product is $$\mathrm{SE}(\hat a \hat b) \;=\; \sqrt{\,a^2 \mathrm{Var}(\hat b) + b^2 \mathrm{Var}(\hat a)\,},$$ and the closed form returns the smallest $n$ at which the Wald width $2 z_{1 - \alpha/2}\, \mathrm{SE}(\hat a \hat b)$ is at or below width. Two approximations remain. The Wald interval is symmetric while the sampling distribution of a product is skewed, so the Wald interval is not the interval an indirect effect should be reported with (Tofighi & Kelley, 2020). And the closed form evaluates the standard error at the planning values, while a fitted model evaluates it at the estimates, which leaves a discrepancy of a percent or two in realized width at moderate sample sizes. Both are reasons to treat the closed form as the first approximation and to verify the final plan with method = "monte_carlo", which measures the realized widths directly.

Planning for the Monte Carlo interval. With method = "monte_carlo", each candidate $n$ is evaluated by a priori Monte Carlo simulation (Muthén & Muthén, 2002; Schoemann, Boulton, & Short, 2017): G data sets of size $n$ are drawn from the planning population, the two mediation regressions are fit to each, and the Monte Carlo interval is formed by drawing B pairs $(\tilde a, \tilde b)$ from normal distributions centered at the estimates with the estimated standard errors, multiplying, and reading off the empirical $(\alpha/2, 1 - \alpha/2)$ quantiles (MacKinnon, Lockwood, & Williams, 2004). Since $\hat a$ and $\hat b$ are uncorrelated here, the independent draws realize the joint normal approximation of the estimates, the same construction mediation_mbco uses for its Monte Carlo interval. The necessary sample size is the smallest $n$ whose mean simulated width is at or below width; the search starts from the closed-form answer, brackets the crossing geometrically, and bisects. A planning call fits the mediation model several thousand times and takes a few seconds, which is why the Monte Carlo example below is shown rather than run. The necessary sample size inherits the simulation error of the mean widths; raising G narrows it, and supplying seed makes a plan reproducible.

Relation to the MBCO procedure. The model-based constrained optimization (MBCO) likelihood ratio test of Tofighi and Kelley (2020), implemented in mediation_mbco, is the recommended test of a mediation effect, and the intervals that suit an indirect effect are the profile likelihood interval and the Monte Carlo interval, both of which accommodate the skewness of the product. This planner targets the Monte Carlo interval. Planning for the profile likelihood interval would require inverting a pair of constrained optimizations in every simulated study (two constrained OpenMx fits per interval, times G, times every candidate sample size), while the Monte Carlo interval costs B products of normal draws per study and is the inexpensive interval that also accommodates the skewness, the one Tofighi and Kelley (2020) report for their memory example. A study planned with method = "monte_carlo" and analyzed with mediation_mbco(ci_method = "monte_carlo") is therefore planned and analyzed on the same interval.

Beyond the simple model. The planning population assumes standardized observed variables, one mediator, no covariates, and no direct effect. With a nonzero direct effect the residual variance of $Y$ is $1 - b^2 - c'^2 - 2abc'$ rather than $1 - b^2$, so assuming $c' = 0$ errs toward a larger sample whenever $c'(c' + 2ab) > 0$ (consistent mediation) and toward a smaller one otherwise. When the direct effect, covariates, several mediators, or latent variables matter to the design, plan by simulation from the full model with ss_aipe_composite_sem, labeling the paths and defining the indirect effect via ab := a*b; its intervals are the Wald intervals of the fitted model, the same target as the closed form here. ss_aipe_indirect_effect_sensitivity quantifies what a plan from this page delivers when the population paths differ from the planning values.

References

Fritz, M. S., & MacKinnon, D. P. (2007). Required sample size to detect the mediated effect. Psychological Science, 18(3), 233–239. doi:10.1111/j.1467-9280.2007.01882.x

Lachowicz, M. J., Preacher, K. J., & Kelley, K. (2018). A novel measure of effect size for mediation analysis. Psychological Methods, 23, 244–261. doi:10.1037/met0000165

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological Methods, 7(1), 83–104. doi:10.1037/1082-989X.7.1.83

MacKinnon, D. P., Lockwood, C. M., & Williams, J. (2004). Confidence limits for the indirect effect: Distribution of the product and resampling methods. Multivariate Behavioral Research, 39(1), 99–128. doi:10.1207/s15327906mbr3901_4

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge.

Muthén, L. K., & Muthén, B. O. (2002). How to use a Monte Carlo study to decide on sample size and determine power. Structural Equation Modeling, 9(4), 599–620. doi:10.1207/S15328007SEM0904_8

Preacher, K. J., & Kelley, K. (2011). Effect size measures for mediation models: Quantitative strategies for communicating indirect effects. Psychological Methods, 16(2), 93–115. doi:10.1037/a0022658

Schoemann, A. M., Boulton, A. J., & Short, S. D. (2017). Determining power and sample size for simple and complex mediation models. Social Psychological and Personality Science, 8(4), 379–386. doi:10.1177/1948550617715068

Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312.

Tofighi, D., & Kelley, K. (2020). Improved inference in mediation analysis: Introducing the model-based constrained optimization procedure. Psychological Methods, 25, 496–515. doi:10.1037/met0000259

Tofighi, D., & MacKinnon, D. P. (2011). RMediation: An R package for mediation analysis confidence intervals. Behavior Research Methods, 43(3), 692–700. doi:10.3758/s13428-011-0076-x

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Plan n so the 95% CI on ab has full width <= 0.20, with
#        anticipated standardized a = 0.40 and b = 0.40. The closed
#        form answers instantly:
ss_aipe_indirect_effect(a = 0.40, b = 0.40, width = 0.20)
#>  term           value
#>  necessary_N    116  
#>  expected_width 0.2  
#>  a              0.4  
#>  b              0.4  
#>  ab             0.16 
#>  width_target   0.2  
#>  conf_level     0.95 
#> 
#> Confidence level: 95%

# 2. The recommended plan targets the Monte Carlo interval directly:
#        every candidate sample size fits the mediation model to G
#        simulated data sets and measures the realized widths. The
#        call takes a few seconds, so it is shown here rather than
#        run. It returns a slightly larger sample size than the
#        closed form because the interval it plans for is a little
#        wider than the Wald interval:
# ss_aipe_indirect_effect(a = 0.40, b = 0.40, width = 0.20,
#                         method = "monte_carlo", seed = 113)

Sample Size for AIPE on a Mediated (Indirect) Effect \(ab\)