Advantages of Monte Carlo Confidence Intervals for Indirect Effects

Abstract

Monte Carlo simulation is a useful but underutilized method of constructing confidence intervals for indirect effects in mediation analysis. The Monte Carlo confidence interval method has several distinct advantages over rival methods. Its performance is comparable to other widely accepted methods of interval construction, it can be used when only summary data are available, it can be used in situations where rival methods (e.g., bootstrapping and distribution of the product methods) are difficult or impossible, and it is not as computer-intensive as some other methods. In this study we discuss Monte Carlo confidence intervals for indirect effects, report the results of a simulation study comparing their performance to that of competing methods, demonstrate the method in applied examples, and discuss several software options for implementation in applied settings.In its simplest form, mediation occurs when the effect of an independent variable (X) on a dependent variable (Y) is transmitted via a mediator variable (M) (see Figure 1).1 This mediation effect is also commonly referred to as the indirect effect of X on Y through M. Mediation models permit researchers to test simple hypotheses about how causal processes may occur and form the building blocks of more complicated structural models. Mediation models often involve parsing the total effect (c) of X on Y into a direct effect (c ) and an indirect effect (a × b, or simply ab). These coefficients can be derived from fitting the following simultaneous equations to sample data using linear regression or path analysis, with straightforward extensions to latent variable models:Figure 1 depicts these effects using simple path diagrams of the effect of X on Y both without and with M included in the model. In mediation analysis, attention focuses mostly on the indirect 1 In order for the results of a mediation analysis to have a causal interpretation, several additional assumptions must be met.

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“…As discussed previously, this normal-theory based approach of the statistical inference could be inaccurate since it relies on the unrealistic assumption that the sampling distribution of indirect effect terms are normal (Hayes, 2013;Preacher & Selig, 2012). Instead, scholars increasingly rely on the method that empirically generates a distribution of the indirect effect in question from cases bootstrap or from a large number of random draws, known as the Monte Carlo method (Bauer et al, 2006;MacKinnon et al, 2004;Preacher & Selig, 2012).…”

“…Most of the multilevel programs or the covariance structure programs provide an option to display a variance-covariance matrix, and particularly, some covariance structure modeling programs such as Mplus implement this normal-theory based SE derivation by default. Yet just as the Sobel test, the normal-theory based approach is relying on the unrealistic assumption that the sampling distribution of the average indirect effect follows a normal distribution, therefore many researchers have warned against its use for hypothesis testing (MacKinnon, Lockwood, & Williams, 2004;Hayes & Scharkow, 2013;Preacher & Selig, 2012).…”

“…Instead, scholars increasingly rely on the method that empirically generates a distribution of the indirect effect in question from cases bootstrap or from a large number of random draws, known as the Monte Carlo method (Bauer et al, 2006;MacKinnon et al, 2004;Preacher & Selig, 2012). Within the multilevel context, the MC method has a unique advantage over other bootstrapping-based approaches, where choice of exact resampling method is ambiguous due to the nested data structure.…”

“…Another approach has been suggested, which is superior to the other alternative and therefore a recommended practice, is using either cases bootstrap (Efron & Tibshirani, 1994) or Monte Carlo (MC) simulations (Preacher & Selig, 2012). The cases bootstrap requires no a priori assumption of its distributional form regarding indirect effect, but it empirically simulates its sampling distribution from repeatedly drawing model estimates over a large number of replications (say, k).…”

“…The cases bootstrap requires no a priori assumption of its distributional form regarding indirect effect, but it empirically simulates its sampling distribution from repeatedly drawing model estimates over a large number of replications (say, k). In contrast, MC methods derive sampling distributions of the indirect effect by simulating k values of its constituent components from a multivariate normal distribution, as described in Preacher and Selig (2012). 4 Once the distribution of indirect effects are estimated from either the cases bootstrap or from MC method, one could obtain CIs for the average indirect effect by taking the value corresponding to, for instance, 2.5 th and 97.5 th percentiles to form a 95% confidence interval.…”