Mediation Analysis With Bootstrap Confidence Intervals

Estimates the simple mediation model, a predictor $X$ affecting an outcome $Y$ directly and through a mediator $M$, and reports the indirect, direct, and total effects with confidence intervals in one tidy table. The indirect effect $a b$ is the product of the $X \to M$ path and the $M \to Y$ path (holding $X$), and its sampling distribution is skewed, which is why the default interval is the percentile bootstrap rather than a normal approximation; the bias-corrected and accelerated (BCa) bootstrap, the Monte Carlo (parametric simulation) interval, and the Sobel normal-theory interval are available for comparison. This is the analysis counterpart of the planning functions ss_aipe_indirect_effect and ss_power_indirect_effect.

Usage

mediate(
  data,
  x,
  m,
  y,
  covariates = NULL,
  ci_method = c("boot_percentile", "boot_bca", "monte_carlo", "sobel"),
  B = 2000,
  conf_level = 0.95,
  seed = NULL
)

Arguments

data: A data.frame containing the variables.
x, m, y: Names (single character strings) of the predictor, the mediator, and the outcome columns in data.
covariates: Optional character vector of covariate column names, entered in both the mediator and the outcome models.
ci_method: Confidence interval method for the indirect effect: "boot_percentile" (default), "boot_bca", "monte_carlo" (simulate $a$ and $b$ from their joint normal approximation; MacKinnon, Lockwood, & Williams, 2004), or "sobel" (the first-order normal-theory interval; reported for comparison, not recommended for inference). Direct and total effects always carry their ordinary t-based intervals.
B: Number of bootstrap or Monte Carlo replications. Defaults to 2000; published analyses often use 5000 or more, and the BCa interval in particular rewards a large B (see Details).
conf_level: Confidence level. Defaults to 0.95.
seed: Optional integer seed for the resampling, 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 (class dmar_tbl) with rows indirect_effect (with its ci_method interval), direct_effect, total_effect, the paths a and b, their standard errors (se_indirect per Sobel, se_a, se_b), the interval limits (indirect_lower / indirect_upper, direct_lower / direct_upper, total_lower / total_upper), proportion_mediated ($ab/c$; NA when the total effect is near zero, where the ratio is unstable), N, and B. The interval method is recorded in the "ci_method" attribute and the confidence level in "conf_level".

Details

The model is the standard pair of regressions $$M = i_M + a X + \mathbf{g}'\mathbf{C} + e_M, \qquad Y = i_Y + c' X + b M + \mathbf{h}'\mathbf{C} + e_Y,$$ with $\mathbf{C}$ the optional covariates. The indirect effect is $a b$, the direct effect $c'$, and the total effect $c = c' + a b$ (an identity in linear models with the same cases, which the implementation exploits as an internal consistency check).

Bootstrap intervals resample cases (rows) with replacement B times, refitting both regressions in each resample (Efron & Tibshirani, 1993). "boot_percentile" takes the interval limits from the empirical quantiles of the bootstrapped $a b$ estimates; it is not forced to be symmetric about the estimate, which is the point for a skewed sampling distribution. "boot_bca" (the bias-corrected and accelerated interval) additionally adjusts the two quantile positions for median bias, estimated from the bootstrap distribution, and for the rate at which the variance of the estimator changes with the parameter, the acceleration, estimated by the jackknife (which adds N extra pairs of fits); the adjustments make it second-order accurate where the percentile interval is first-order accurate (DiCiccio & Efron, 1996). Because the adjusted quantile positions sit farther into the tails of the bootstrap distribution, the BCa interval benefits more than the percentile interval does from a B well above the default. Each resample refits the two regressions by least squares, a closed-form fit with no iterative estimation, so in ordinary data all B replications enter the interval. A degenerate resample (one whose refit is rank deficient, possible with a near-constant predictor) returns no indirect effect; such replications are dropped with a warning stating how many, and the interval is computed from the replications that returned a value. The Sobel standard error is computed by var_indirect_effect. The proportion mediated is one of the effect size measures for mediation models surveyed by Preacher and Kelley (2011); its instability when the total effect is small is why it is reported as NA near a zero total effect. Listwise deletion is applied to the analysis variables; for full information maximum likelihood under missingness, fit the model in lavaan (see mlmr for the package's FIML front end philosophy).

Mediation language implies causal structure: with observational data the estimates are conditional associations, and the causal reading requires the usual no-unmeasured-confounding assumptions for both the $X \to M$ and $M \to Y$ links (MacKinnon, 2008). The function computes; the design earns the interpretation.

References

DiCiccio, T. J., & Efron, B. (1996). Bootstrap confidence intervals. Statistical Science, 11(3), 189–228.

Efron, B., & Tibshirani, R. J. (1993). An introduction to the bootstrap. New York, NY: Chapman & Hall/CRC.

MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Erlbaum.

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

Preacher, K. J., & Hayes, A. F. (2008). Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models. Behavior Research Methods, 40(3), 879–891. doi:10.3758/BRM.40.3.879

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

Author

Ken Kelley kkelley@nd.edu

Examples

# Simulated mediation: X raises M (a = .5), M raises Y (b = .4), and a
# little direct effect remains (c' = .2).
set.seed(113)
n <- 200
x <- rnorm(n)
m <- 0.5 * x + rnorm(n, 0, sqrt(1 - 0.25))
y <- 0.2 * x + 0.4 * m + rnorm(n, 0, 0.8)
d <- data.frame(x = x, m = m, y = y)

# The Sobel interval is closed form, so it is the call that runs here.
# It assumes the product ab is normally distributed, which is why it is
# reported for comparison rather than used for inference. The B row of
# the result is NA because no replications are drawn.
mediate(d, x = "x", m = "m", y = "y", ci_method = "sobel")
#>  term                value 
#>  indirect_effect     0.182 
#>  indirect_lower      0.105 
#>  indirect_upper      0.259 
#>  direct_effect       0.194 
#>  direct_lower        0.0634
#>  direct_upper        0.324 
#>  total_effect        0.375 
#>  total_lower         0.25  
#>  total_upper         0.501 
#>  a                   0.449 
#>  b                   0.404 
#>  se_a                0.0615
#>  se_b                0.0678
#>  se_indirect         0.0393
#>  proportion_mediated 0.484 
#>  N                   200   
#>  B                   <NA>  
#> 
#> Confidence level: 95%

# The percentile bootstrap is the default and is what a reported
# analysis would use. It is not run here because it refits both
# regressions in each of the B resamples; the call is:
# mediate(d, x = "x", m = "m", y = "y", seed = 113)

# The Monte Carlo interval needs no refitting, so it stays quick even
# for a large B, but it still draws B replications. Also not run here:
# mediate(d, x = "x", m = "m", y = "y", ci_method = "monte_carlo",
#         B = 10000, seed = 113)