Matthew S. Fritz

Required Sample Size to Detect the Mediated Effect

Mediation models are widely used, and there are many tests of the mediated effect. One of the most common questions that researchers have when planning mediation studies is, “How many subjects do I need to achieve adequate power when testing for mediation?” This article presents the necessary sample sizes for six of the most common and the most recommended tests of mediation for various combinations of parameters, to provide a guide for researchers when designing studies or applying for grants.

Distribution of the product confidence limits for the indirect effect: program PRODCLIN.

This article describes a program, PRODCLIN (distribution of the PRODuct Confidence Limits for INdirect effects), written for SAS, SPSS, and R, that computes confidence limits for the product of two normal random variables. The program is important because it can be used to obtain more accurate confidence limits for the indirect effect, as demonstrated in several recent articles (MacKinnon, Lockwood, & Williams, 2004; Pituch, Whittaker, & Stapleton, 2005). Tests of the significance of and confidence limits for indirect effects based on the distribution of the product method have more accurate Type I error rates and more power than other, more commonly used tests. Values for the two paths involved in the indirect effect and their standard errors are entered in the PRODCLIN program, and distribution of the product confidence limits are computed. Several examples are used to illustrate the PRODCLIN program. The PRODCLIN programs in rich text format may be downloaded from www.psychonomic.org/archive.

Explanation of Two Anomalous Results in Statistical Mediation Analysis

Previous studies of different methods of testing mediation models have consistently found two anomalous results. The first result is elevated Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap tests not found in nonresampling tests or in resampling tests that did not include a bias correction. This is of special concern as the bias-corrected bootstrap is often recommended and used due to its higher statistical power compared with other tests. The second result is statistical power reaching an asymptote far below 1.0 and in some conditions even declining slightly as the size of the relationship between X and M, a, increased. Two computer simulations were conducted to examine these findings in greater detail. Results from the first simulation found that the increased Type I error rates for the bias-corrected and accelerated bias-corrected bootstrap are a function of an interaction between the size of the individual paths making up the mediated effect and the sample size, such that elevated Type I error rates occur when the sample size is small and the effect size of the nonzero path is medium or larger. Results from the second simulation found that stagnation and decreases in statistical power as a function of the effect size of the a path occurred primarily when the path between M and Y, b, was small. Two empirical mediation examples are provided using data from a steroid prevention and health promotion program aimed at high school football players (Athletes Training and Learning to Avoid Steroids; Goldberg et al., 1996), one to illustrate a possible Type I error for the bias-corrected bootstrap test and a second to illustrate a loss in power related to the size of a. Implications of these findings are discussed.

A graphical representation of the mediated effect

Mediation analysis is widely used in the social sciences. Despite the popularity of mediation models, few researchers have used graphical methods, other than structural path diagrams, to represent their models. Plots of the mediated effect can help a researcher better understand the results of the analysis and convey these results to others. This article presents a method for creating and interpreting plots of the mediated effect for a variety of mediation models, including models with (1) a dichotomous independent variable, (2) a continuous independent variable, and (3) an interaction between an independent variable and the mediating variable. An empirical example is then presented to illustrate these plots. Sample code for creating plots of the mediated effect in R and SAS is also included, and may be downloaded from www.psychonomic.org/archive.

The Combined Effects of Measurement Error and Omitting Confounders in the Single-Mediator Model.

ABSTRACT Mediation analysis requires a number of strong assumptions be met in order to make valid causal inferences. Failing to account for violations of these assumptions, such as not modeling measurement error or omitting a common cause of the effects in the model, can bias the parameter estimates of the mediated effect. When the independent variable is perfectly reliable, for example when participants are randomly assigned to levels of treatment, measurement error in the mediator tends to underestimate the mediated effect, while the omission of a confounding variable of the mediator-to-outcome relation tends to overestimate the mediated effect. Violations of these two assumptions often co-occur, however, in which case the mediated effect could be overestimated, underestimated, or even, in very rare circumstances, unbiased. To explore the combined effect of measurement error and omitted confounders in the same model, the effect of each violation on the single-mediator model is first examined individually. Then the combined effect of having measurement error and omitted confounders in the same model is discussed. Throughout, an empirical example is provided to illustrate the effect of violating these assumptions on the mediated effect.

An exponential decay model for mediation

Mediation analysis is often used to investigate mechanisms of change in prevention research. Results finding mediation are strengthened when longitudinal data are used because of the need for temporal precedence. Current longitudinal mediation models have focused mainly on linear change, but many variables in prevention change nonlinearly across time. The most common solution to nonlinearity is to add a quadratic term to the linear model, but this can lead to the use of the quadratic function to explain all nonlinearity, regardless of theory and the characteristics of the variables in the model. The current study describes the problems that arise when quadratic functions are used to describe all nonlinearity and how the use of nonlinear functions, such as exponential decay, address many of these problems. In addition, nonlinear models provide several advantages over polynomial models including usefulness of parameters, parsimony, and generalizability. The effects of using nonlinear functions for mediation analysis are then discussed and a nonlinear growth curve model for mediation is presented. An empirical example using data from a randomized intervention study is then provided to illustrate the estimation and interpretation of the model. Implications, limitations, and future directions are also discussed.

Increasing Statistical Power in Mediation Models Without Increasing Sample Size

Inadequate statistical power to detect treatment effects in health research is a problem that is compounded when testing for mediation. In general, the preferred strategy for increasing power is to increase the sample size, but there are many situations where additional participants cannot be recruited, necessitating the use of other methods to increase statistical power. Many of these other strategies, commonly applied to analysis of variance and multiple regression models, can be applied to mediation models with similar results. Additional predictors or blocking variables will increase or decrease statistical power, however, depending on whether these variables are related to the mediator, the outcome, or both. The effect of these two methods on the power for tests of mediation is illustrated through the use of simulations. Implications for health researchers using these methods are discussed.

Health beliefs as a key determinant of intent to use anabolic-androgenic steroids (AAS) among high-school football players: implications for prevention

Abstract The use of anabolic-androgenic steroids (AAS) is problematic for youth because of negative effects such as reduced fertility, increased aggression and exposure to toxic chemicals. An effective programme for addressing this problem is Adolescents Training and Learning to Avoid Steroids (ATLAS). This secondary analysis expands prior research by identifying prominent mechanisms of change and highlighting key longitudinal processes that contributed to the success of ATLAS. The current sample consists of high-school football players (N = 1.068; Mage = 15.25) who began ATLAS in grades nine through eleven and participated in booster sessions for two years post-baseline. Knowledge of AAS effects, belief in media ads, reasons not to use AAS, perceived severity of and susceptibility to AAS effects and ability to resist drug offers were critical mediators of the relations between ATLAS and outcomes. Modern applications of the ATLAS programme are also discussed.

Review of Doing Statistical Mediation & Moderation, by Paul E. Jose

As noted by Paul Jose in Doing Statistical Mediation & Moderation, although there are many articles on mediation and moderation, most do not focus on the actual procedures for estimating these models. Jose uses a wealth of empirical examples to illustrate the nuts and bolts of estimating mediation and moderation models while also focusing on the interpretation of the results in everyday language.