Confidence limits for a product, without pretending it is normal

The product of two normal estimates is not normal: it is skewed and sharply peaked, so the usual estimate-plus-or-minus-Sobel interval is mis-centred and too short, and the deficit is worst exactly where mediation studies live – small a or small b. Taking the quantiles of the product distribution itself fixes the shape.

Usage

MedCI(a, b, sa, sb, n_sim = 20000, level = 0.95)

Arguments

a, b

Path coefficients.

sa, sb

Their standard errors, strictly positive.

n_sim

Number of deterministic draws.

level

Confidence level.

Value

List with estimate, ci_lo, ci_hi, se_mc, sobel_se, sobel_lo, sobel_hi, asymmetry, n_sim.

Details

Formula: draw a* ~ N(ahat, sa^2) and b* ~ N(bhat, sb^2) independently, form a* b*, and read off its alpha/2 and 1 - alpha/2 quantiles – MacKinnon, Lockwood and Williams (2004) Section 3. The draws are a two-dimensional Halton sequence (van der Corput in bases 2 and 3, pushed through AS 241), so the interval is the same number every time and in both language arms.

References

MacKinnon, D. P., Lockwood, C. M. and Williams, J. (2004). Multivariate Behavioral Research 39(1):99-128. doi:10.1207/s15327906mbr3901_4 .

Examples

MedCI(a = 0.3, b = 0.4, sa = 0.1, sb = 0.12, n_sim = 5000)[c("estimate", "ci_lo", "ci_hi", "sobel_lo", "sobel_hi")]
#> $estimate
#> [1] 0.12
#> 
#> $ci_lo
#> [1] 0.02909682
#> 
#> $ci_hi
#> [1] 0.2426789
#> 
#> $sobel_lo
#> [1] 0.01452553
#> 
#> $sobel_hi
#> [1] 0.2254745
#>