Indirect Effects

What the page reads

Three parts of the output. The block headed TOTAL, TOTAL INDIRECT, SPECIFIC INDIRECT, AND DIRECT EFFECTS, which MODEL INDIRECT produces, gives every effect the run was asked for, and for a specific indirect effect it also gives the chain of variables it runs through, printed from the outcome back to the predictor. MODEL RESULTS gives the coefficient of every link of that chain with its standard error. TECHNICAL 1 OUTPUT numbers the free parameters and TECHNICAL 3 OUTPUT gives their covariance matrix, from which the covariance between two links of the same chain is taken.

Why the interval is built again

An indirect effect is a product of coefficients. The standard error beside it in the output is the delta-method standard error (Sobel, 1982), and the interval that follows from it is symmetric about the estimate. The sampling distribution of a product of two normal variables is not normal: it is typically skewed, especially when the component coefficients are small relative to their standard errors. A symmetric interval can therefore be poorly positioned and have coverage that differs from its nominal level (MacKinnon, Lockwood, Hoffman, West, & Sheets, 2002). The bootstrap is one answer and Mplus provides it with BOOTSTRAP and CINTERVAL(BOOTSTRAP). The interval below is the other: it is built from the joint distribution of the coefficients that make the product, which is what the run already estimated.

The interval from the joint distribution of the coefficients

Preacher and Selig (2012) obtain it as follows. For an indirect effect that runs through two links with coefficients a and b, values are drawn repeatedly from

Here the means are the estimates the output gives, the variances are the squares of their standard errors, and the covariance is the one TECH3 gives for those two parameters. The product $a^{*}{b}^{*}$ is formed at every draw, and the confidence limits are the percentiles of the values so obtained. The authors note that in the three-variable model the covariance is often replaced by zero; the page uses the value from the file when it is there and says which of the two it used. A chain of more than two links is treated the same way, with as many coefficients as the chain has and the product of all of them at every draw.

The draws are made with a generator whose seed is fixed, so the same file gives the same interval every time it is read. With 20 000 draws the limits are stable to about the third decimal of a coefficient of ordinary size; the number of draws can be raised above.

The interval from the standard error

Beside it the page reports the symmetric interval, which is the estimate plus and minus the standard error times the normal quantile. It is the interval that follows from what the output prints, and it is shown so that the two can be compared: when they agree the product is close enough to normal for either to serve, and when they do not it is the asymmetric one that describes the sampling distribution of the product.

The standard error is the one the output gives, which for a two-link chain is the delta-method standard error of Sobel (1982), $\mathrm{SE}=\sqrt{{\hat{b}}^2{\mathrm{SE}}_a^2+{\hat{a}}^2{\mathrm{SE}}_b^2}$. The page recomputes that quantity from the two coefficients and compares it with the one in the file, and reports the comparison: they agree when the two coefficients are uncorrelated, and differ by the term in their covariance when they are not.

The share of the total effect

Where the file reports the total effect as well, the page gives the ratio of the indirect effect to it. The ratio is worth reading only when the two have the same sign and the total is not near zero: a ratio formed on a total that is small, or on effects of opposite sign, takes values outside the unit interval and changes wildly from sample to sample, and it is reported with that warning attached (MacKinnon, 2008). No value of the ratio is taken as a threshold.

Two indirect effects against each other

Two specific indirect effects of the same model are compared by their difference, and the interval for the difference is built in the same way as the interval for each of them: at every draw the two products are formed from the same draw of the coefficients they share, the difference is taken, and the limits are the percentiles of the differences. Drawing them together is what makes the comparison right when the two chains have a link in common, since the two products are then not independent.

The completely standardized effect

When the run contains the STDYX block, the same effects appear there in the completely standardized metric, and the page reports those estimates beside the unstandardized ones. Its symmetric interval can be formed from the STDYX standard error printed by Mplus. The page does not construct a Monte Carlo interval by drawing the standardized path coefficients independently: TECH3 is the covariance matrix of the free, unstandardized model parameters, not the covariance matrix of the derived STDYX coefficients. A correct Monte Carlo interval on the standardized metric would require propagating each draw of the unstandardized parameter vector through the standardization transformation, including the variances that enter that transformation.

Unsupported inputs

A file with no MODEL INDIRECT block, since there is then no effect to work on. A chain whose links cannot all be found in MODEL RESULTS, which happens when a link is a fixed parameter or belongs to a part of the model printed under a heading the page does not read; the effect is then listed with the interval from the standard error alone and the reason is given. A run whose messages put it among those whose results are not to be interpreted.

Self-test

Append ?selftest=1 to check output parsing, delta-method standard errors, Monte Carlo intervals, percentile calculations, chain reconstruction, and unsupported inputs.

• MacKinnon, D. P. (2008). Introduction to statistical mediation analysis. Lawrence Erlbaum. https://doi.org/10.4324/9780203809556
• 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. https://doi.org/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. https://doi.org/10.1207/s15327906mbr3901_4
• Preacher, K. J., & Selig, J. P. (2012). Advantages of Monte Carlo confidence intervals for indirect effects. Communication Methods and Measures, 6(2), 77–98. https://doi.org/10.1080/19312458.2012.679848
• Sobel, M. E. (1982). Asymptotic confidence intervals for indirect effects in structural equation models. Sociological Methodology, 13, 290–312. https://doi.org/10.2307/270723
• Muthén, L. K., & Muthén, B. O. (1998–2017). Mplus user's guide (8th ed.). Muthén & Muthén. https://www.statmodel.com/ugexcerpts.shtml