Mediation

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Conducting mediational analysis is easier than you think.


Take-home message

►The details are provided below, but the take-home message is that the Baron & Kenny method is the one most often used but has some limitations, the Sobel test is more accurate but has low statistical power, and Bootstrapping is the preferred method because it's the only test that doesn't violate assumptions of normality (and it's recommended for small sample sizes). The same applies to Multiple mediation and Reverse mediation.

Whenever possible the links to helpful articles and websites about mediation are posted below, including a link to the Preacher website and Hayes website which has macros for SPSS and SAS that do everything you need because they provide output simultaneously for the (1) Baron & Kenny method, (2) sobel method, and (3) bootstrapping method. Macros exist for both simple mediation (click here for macro) and multiple mediation (click here for macro).


Contents

What is mediation?


Baron & Kenny

The causal steps methods developed by (Baron & Kenny, 1986) is the most commonly used and most frequently cited test of mediation in psychology.
(Baron & Kenny, 1986) defined three conditions necessary for mediation, and the updated version by (Kenny, 1998) describes four steps to infer mediation.
FYI - see the Kenny website for a pictorial representation of paths represented as a, b, c, c'
The four steps involve:
  1. IV predicts DV (so estimate path c).
  2. IV predicts mediator (so estimate path a)
  3. Mediator predicts DV (while controlling for IV) (so estimate path b)
  4. IV does NOT predict DV (while controlling for mediator) (so estimate path c' )
Using regression analysis in SPSS (or related statistical program), you test:
  1. IV is predictor, and DV is criterion variable (want this significant)
  2. IV is predictor, and mediator is criterion variable (want this significant)
  3. IV & mediator as predictor, and DV as criterion variable (want b path to be significant but want c' path n.s.).
  1. Most argue that only Step 2 & 3 are required because the initial correlation between the IV and DV (step 1) is not essential, and the finding of a subsequent n.s. correlation between the IV and DV (step 4) is only necessary for arguing complete mediation (see below).
  2. Complete (or perfect) mediation occurs when path c' decreases to zero. Partial mediation occurs when c' decreases to nontrivial amount but not to zero. See (Shrout & Bolger, 2002) page 432 for the four reasons why partial mediation may occur.
Over the years the (Baron & Kenny, 1986) method has been criticized for low power, Type I error, not being able to address suppression effects, and not addressing the central question of whether the indirect effect is significantly different from zero and in the expected direction (see (MacKinnnon, Lockwood, Hoffman, & West, 2002), (Preacher & Hayes, 2004), (Shrout & Bolger, 2002) for more detailed information.)

Sobel Test

The Sobel test determines the significance of the indirect effect of the mediator by testing the hypothesis of no difference between the total effect (path c) and the direct effect (path c' ). The indirect effect of the mediator is the product of path ab which is equivalent to (c - c' ).
The Sobel test is superior to the Baron & Kenny method in terms of all the limitations of the B&K method discussed above (e.g., power, Type I error, suppression effects, addressing the significance of the indirect effect).
  1. Determine the standard error of the indirect effect.
  2. Divide the ab path by the standard error of the indirect effect.
  3. The ratio is compared to critial value from standard normal distribution for a given alpha level (i.e., treated as Z-test).
  1. There is an easy-to-use Sobel test calculator posted online by Kristopher J. Preacher. Just run regression analyses in SPSS (or similar statistical program) and input the requested numbers into the online calculator, OR
  2. Preacher and Hayes also offer a macro that calculates the Sobel test directly within SPSS and SAS.
  3. There is also a command to do this in Stata called sgmediation.
The assumption for conducting the Sobel test (like most tests in psychology) is that the sampling distribution is normal. Hundreds of articles in statistical journals have shown that assumptions of normality are usually violated, especially in small samples, leading to reduced ability to detect true relationships amongst variables (see (Wilcox, 1998), (Wilcox, 2003), and (Wilcox, 2005) for more information).

Bootstrapping

Bootstrapping is a way to overcome the limitations of statistical methods that make assumptions about the shape of sampling distributions, such as normality. It is becoming the preferred method for analyzing data.
See (Shrout & Bolger, 2002) for details, but basically bootstrapping involves repeatedly randomly sampling observations with replacement from the data set and computing the statistic of interest in each resample. Over many bootstrap resamples, an empirical approximation of the sampling distribution of the statistic can be generated and used for hypothesis testing.
Preacher and Hayes offer a macro that calculates bootstrapping directly within SPSS and SAS.
The Stata command sgmediation also offers bootstrapping and has a simple syntax, e.g., "sgmediation Y, mv(M) iv(X) bootstrap reps(5000)"
The syntax in Mplus for a simple mediation model with 5000 bootstrapped samples is shown below. The important bits are bolded.

TITLE:

data from Preacher and Hayes figure 2
satis: satisfaction
therapy: therapy
0: standard
1: cognitive
attrib: attributional positivity

DATA:

File is figure2data.dat ;

VARIABLE:

Names are satis therapy attrib;

ANALYSIS:

Bootstrap = 5000;

MODEL:

attrib ON therapy ;
satis ON attrib therapy;

MODEL INDIRECT:

satis IND attrib therapy;

OUTPUT:

cinterval (bcbootstrap);

SEM

Structural Equation Modeling is a statistical technique for simultaneously analyzing the relationships among multiple IVs and DVs by building and testing different models of the data. There are various programs for conducting SEM including EQS, AMOS, and LISREL.
SEM offers many of the same tests listed above (Sobel test, bootstrapping), but can also control for measurement error and provides greater flexibility in model design.
SEM tests significants of indirect effects similar to the Sobel test.
The details of conducting SEM is beyond the scope of this page, and there are many helpful books/articles on how to analyze data using SEM.
See also informative websites such as the David Kenny website on SEM, and the Ed Rigdon website on SEM.
(Shrout & Bolger, 2002) provide syntax that enables EQS and AMOS to conduct tests of indirect effects with bootstrapping.

Multiple Mediation

Moderated Mediation

Moderated mediation refers to testing mediational models for different groups (e.g., control group versus condition group, males versus females, etc) or different levels of a continuous moderator variable. If the magnitude of the indirect effect changes significantly across values of a moderator, that's moderated mediation.
The article by Muller, Judd & Yzerbyt (2005) provides an in-depth explanation as does Preacher, Rucker, and Hayes (2007). You can find the Muller et al. paper here and the Preacher et al. paper here

Summary...




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