Variance of the Mediated (Indirect) Effect $ab$ — var_indirect

Computes the asymptotic variance of the product of two regression coefficients $\hat a \hat b$ (the mediated/indirect effect in a simple three-variable mediator model: $X \to M \to Y$) under four competing formulas: Sobel (1982) first-order, Aroian (1947) / Goodman (1960) second-order, and the full second-order delta method with optional cross-product covariance. All four are reported in a single output so the user can see the relative contributions of the higher-order terms.

Usage

var_indirect_effect(a, b, var_a, var_b, cov_ab = 0)

Arguments

a, b: Anticipated population (or estimated) regression coefficients for $X \to M$ and $M \to Y$ (typically on the standardized scale).
var_a, var_b: Variances (squared standard errors) of $\hat a$ and $\hat b$. For standardized regression with no covariates, $\mathrm{Var}(\hat a) \approx (1 - a^2)/(n - 2)$ and $\mathrm{Var}(\hat b) \approx (1 - b^2)/\{(n - 3)(1 - a^2)\}$, the standardized-model variance accounting for the correlation the $a$ path induces between the predictors of $Y$.
cov_ab: Optional covariance between $\hat a$ and $\hat b$. Defaults to 0 (the assumption underlying the standard Sobel formula). In practice the two estimators are nearly uncorrelated when the controls in the $Y$-on-$M$ regression are uncorrelated with the $M$-on-$X$ regression's predictors.

Value

A data.frame with rows for the four variance formulas; columns are term and value.

Details

Sobel (1982) first-order. The delta method variance of $\hat a \hat b$ under independent $\hat a$, $\hat b$ is $$\mathrm{Var}_{\mathrm{Sobel}}(\hat a \hat b) \;=\; a^2 \mathrm{Var}(\hat b) + b^2 \mathrm{Var}(\hat a).$$ This is the most cited form and is the variance used by the standard Sobel z-test (Sobel, 1982).

Aroian (1947). Aroian retains the second-order term: $$\mathrm{Var}_{\mathrm{Aroian}}(\hat a \hat b) \;=\; a^2 \mathrm{Var}(\hat b) + b^2 \mathrm{Var}(\hat a) + \mathrm{Var}(\hat a)\,\mathrm{Var}(\hat b).$$ Aroian shows this is exact under joint normality of the two independent estimators.

Goodman (1960). Goodman's "unbiased" variance subtracts the second-order term instead of adding it: $$\mathrm{Var}_{\mathrm{Goodman}}(\hat a \hat b) \;=\; a^2 \mathrm{Var}(\hat b) + b^2 \mathrm{Var}(\hat a) - \mathrm{Var}(\hat a)\,\mathrm{Var}(\hat b).$$ For small variances the three forms agree to leading order; they diverge for noisy $\hat a$, $\hat b$.

Second-order delta method (with covariance). When $\hat a$ and $\hat b$ share variability (e.g., they are both estimated from the same regression of $Y$ on $X$ and $M$), the cross-product covariance term enters: $$\mathrm{Var}(\hat a \hat b) \;\approx\; a^2 \mathrm{Var}(\hat b) + b^2 \mathrm{Var}(\hat a) + 2 a b \cdot \mathrm{Cov}(\hat a, \hat b).$$ MacKinnon et al.\ (2002) show this matters in models with covariates that simultaneously load on $M$ and $Y$.

Connection to ss_aipe_indirect_effect. The Sobel (delta method) variance is what ss_aipe_indirect_effect builds on under method = "closed_form" for AIPE planning; this function makes the alternative formulas available for explicit comparison.

References

Aroian, L. A. (1947). The probability function of the product of two normally distributed variables. The Annals of Mathematical Statistics, 18(2), 265–271.

Goodman, L. A. (1960). On the exact variance of products. Journal of the American Statistical Association, 55(292), 708–713.

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

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

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

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

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. a = 0.40, b = 0.40, var_a = 0.02, var_b = 0.02, no covariance:
var_indirect_effect(a = 0.40, b = 0.40, var_a = 0.02, var_b = 0.02)
#>  term                   value 
#>  var_sobel              0.0064
#>  var_aroian             0.0068
#>  var_goodman            0.006 
#>  var_delta_second_order 0.0064

# 2. With a positive covariance between a-hat and b-hat:
var_indirect_effect(a = 0.40, b = 0.40,
                     var_a = 0.02, var_b = 0.02, cov_ab = 0.005)
#>  term                   value 
#>  var_sobel              0.0064
#>  var_aroian             0.0068
#>  var_goodman            0.006 
#>  var_delta_second_order 0.008

Variance of the Mediated (Indirect) Effect \(ab\)