Mediation analysis tests a hypothetical causal chain where one variable X affects a second variable M and, in turn, that variable affects a third variable Y. Mediators describe the how or why of a relationship between two other variables. Mediators describe the process through which an effect occurs. This is also sometimes called an indirect effect. We therefore have the following properties:
Perfect mediation occurs when the effect of X on Y decreases to 0 with M in the model.
Partial mediation occurs when the effect of X on Y decreases by a nontrivial amount (the actual amount is up for debate) with M in the model.
Mediation was previously considered statistically significant if a and b were significant and c’ was less than c. One calculated c-c’ (difference method) and regarded a mediation as statistically significant if c was significant and c’ was not. The mediation was thus validated by a logical conclusion without considering a*b (the indirect effect) specifically.
However, this approach might not be suitable for the following cases:
In order to be able to speak of mediation, the following must be statistically reliable: ab not equal 0 (i.e. a not equal to 0 and b not equal to 0!).
To determine the significance of the indirect effect, the indirect effect is backed up statistically using the so-called Sobel test. Due to unrealistic requirements, however, it is no longer recommended.
Instead, a bootstrap method (also “resampling method”) is preferred. Empirical sample is considered as a pseudo-population, and a large number of random samples (each with replacement) are drawn from this pseudo-population (1000 or more). The indirect effect is calculated in each case. The result is a distribution of the indirect effects (which are normally distributed!)
We will cover two ways to conduct mediation analysis:
This method includes 4-steps:
This package uses the more recent bootstrapping method of Preacher & Hayes (2004) to address the power limitations of the Sobel test. This method computes the point estimate of the indirect effect (ab) over a large number of random sample (typically 1000) so it does not assume that the data are normally distributed and is especially more suitable for small sample sizes than the Barron & Kenny method.
This method includes 2-steps:
The mediate function gives us our Average Causal Mediation Effects (ACME), our Average Direct Effects (ADE), our combined indirect and direct effects (Total Effect), and the ratio of these estimates (Prop. Mediated). The ACME here is the indirect effect of M (total effect - direct effect) and thus this value tells us if our mediation effect is significant.
In general, complete mediation implies that these step 1-2-3-4 (in the 4-steps from Baron & Kenny) be met. However, in practice, some researchers also consider that the essential steps in establishing mediation are steps 2 (X on M) and 3 (Z on Y, controlling for X).
Furthermore, if c′ (direct effect of X on Y after controlling for M) had the opposite sign to that of ab (indirect effect of X on Y), then there would still be mediation (even if step 1 could not be met). In this case the mediator acts like a suppressor variable (see MacKinnon, Fairchild & Fritz, 2007).
For instance, we could have the following model: participation in strikes (or social demonstrations) and satisfaction with government mediated by media consumption. Presumably, we have the following relations:
Thus, we obtain a positive indirect effect, and the total effect of participation in demonstration(s) on satisfaction with government is likely to be very small because the direct and indirect effects will tend to cancel each other out.
Given the above example, it is recommended to perform a single test of ab (rather than two separated tests of a and b). The test was first proposed by Sobel (1982). The Sobel test uses the following standard error estimate of ab:
\[ \sqrt{b^2*s^2_{a} + a^2*s^2_{b}} \]
The test of the indirect effect is given by dividing ab by the above standard error estimate and treating the ratio as a Z test:
\[ z = \frac{ab}{\sqrt{b^2*s^2_{a} + a^2*s^2_{b}}} \].
If a z-score is larger than 1.96 in absolute value, the mediation effect is significant at the .05 level.
The Sobel test is easy to conduct but presumes that a and b are independent (which may not always be true). Furthermore, it assumes that ab is normally distributed (which might not work well for small sample sizes).
One solution is to use the bootstrap method (see Bollen & Stine, 1990). This method has no distribution assumption on the indirect effect ab, but rather approximates the distribution of ab using its bootstrap distribution.
Using the original data set (Sample size = n) as the population, the model draws a bootstrap sample of n individuals with paired (Y, X, M) scores randomly from the data set with replacement. From the bootstrap sample, estimate ab based on a set of regression models. The steps are repeated for a total of N times (N=number of bootstraps).
One speaks of total mediation when the direct effect disappears as a result of mediation: c != 0, and thus ab = c.
One speaks of partial mediation when the direct effect does not disappear as a result of the mediation, but a residue remains: c!= 0 (i.e. if ab != c).
In the social science context, mediations are mostly partial, because one process rarely explains the full influence of X on Y.
The difference method (c-c’) originally assumed that c must be significant (that is, that the X must have a total effect on the Y, which is to be explained by mediation).
However, mediation can also exist if c = 0. This is possible because there may be several significant mediator effects that cancel each other out (but not all may have been measured).
The targeted examination of ab thus provides a more precise picture of individual significant mediation processes, even if there is no total effect.
The R package that handles measurement and analysis models most easily is lavaan. For the calculation, the model formula and the data are passed to the sem() function.
The result is a complex object with many methods of its own. The most important are: summary(), fitMeasures(), parameterEstimates(), and inspect().
A mediation analysis is a regression analysis in which the influence of an independent variable (X) on a dependent variable Y is fully or partially explained by a mediator (M).
The indirect effect via the mediator is formed by multiplying two effects: effect of X on M and effect of M on Y. The significance of the indirect effect is determined by bootstrapping.
Significant mediation occurs when the confidence interval for the indirect effect is not 0.
The size of the effect can be described by the unstandardized or standardized coefficients.
A mediation can exist even if there is no significant total effect.
See the lecture slides on mediation analysis:
You can also download the PDF of the slides here: 
| True | False | Statement |
|---|---|---|
| Perfect mediation occurs when the relationship between the predictor (X) and the outcome (Y) is completely wiped out when the mediator (M) is included in the model. | ||
| To measure the effect size of mediation, we rely on the ratio of the indirect to total effect. | ||
| The Sobel test works well for small sample sizes. | ||
| To test for a mediation effect, we assess the size of the indirect effect and ensure that the confidence interval does not contains 0. |
The following article relies explains mediation (and moderation) analysis with an illustrative example:
Igartua, J. J., & Hayes, A. F. (2021). Mediation, moderation, and conditional process analysis: Concepts, computations, and some common confusions. The Spanish Journal of Psychology, 24, e49. Available here.
Please reflect on the following questions:
The following article relies on moderated mediation as a method of analysis:
Matthes, J. (2013). Do hostile opinion environments harm political participation? The moderating role of generalized social trust. International Journal of Public Opinion Research, 25(1), 23-42. Available here.
Please reflect on the following questions:
You can download the PDF of the exercises here:
In this exercise, we will use the data “protest.sav” (Hayes, 2022) which can be downloaded 
or here under “data files and code”. Especially, we will focus on the following variables:
Hypothesis: If the lawyer protests, her reaction will be perceived as more appropriate, and therefore the lawyer will be evaluated more favorably.
Start by drawing the regression equations.
The regression equations go as:
\[ Y_i = \beta_0 + \beta_1*Protest_i + \beta_2*Respappr_i + \epsilon_i = c' + b \] \[ M_i = \beta_0 + \beta_1*Protest_i + \epsilon_i = a \] \[ Y_i = \beta_0 + \beta_1*Protest_i + \epsilon_i = c = c' + ab \]
Now, we want to calculate the mediation model and to answer the following questions:
Start by loading and selecting the data:
# load the data
library(foreign)
db <- read.spss(file=paste0(getwd(),
"/data/protest.sav"),
use.value.labels = F,
to.data.frame = T)
# get the data
sel <- db |>
dplyr::select(protest, respappr, liking) |>
stats::na.omit()Now, define the model in laavan and give the results for the c path:
# define the model
modell.c = "liking ~ protest"
# get the complete output
fit.c = lavaan::sem(modell.c, data=sel)
lavaan::parameterestimates(fit.c, standardized=T)[1:8]
## lhs op rhs est se z pvalue ci.lower
## 1 liking ~ protest 0.479 0.193 2.478 0.013 0.100
## 2 liking ~~ liking 1.044 0.130 8.031 0.000 0.789
## 3 protest ~~ protest 0.217 0.000 NA NA 0.217
lavaan::inspect(fit.c,"r2")
## liking
## 0.045The C path is significant and explains 4.5% of the variance of liking. Linear regression makes sense for this data.
Now get the model for the c’ path. The output is a little better understandable if you take the indirect and the overall effect on Y with the names of the paths:
# define the model
modell.mediation = "
## direct effect
liking ~ cp*protest
## mediation path
respappr ~ a*protest
liking ~ b*respappr
## indirect effect (a*b)
ab := a*b
## total effect (c+a*b)
total := cp+a*b
"
# get the complete output
fit.med = lavaan::sem(modell.mediation, data=sel)
lavaan::parameterestimates(fit.med, standardized=T)[1:8]
## lhs op rhs label est se z pvalue
## 1 liking ~ protest cp -0.101 0.198 -0.508 0.611
## 2 respappr ~ protest a 1.440 0.220 6.544 0.000
## 3 liking ~ respappr b 0.402 0.069 5.857 0.000
## 4 liking ~~ liking 0.824 0.103 8.031 0.000
## 5 respappr ~~ respappr 1.354 0.169 8.031 0.000
## 6 protest ~~ protest 0.217 0.000 NA NA
## 7 ab := a*b ab 0.579 0.133 4.364 0.000
## 8 total := cp+a*b total 0.479 0.193 2.478 0.013
lavaan::inspect(fit.med,"r2")
## liking respappr
## 0.246 0.249
lavaan::fitMeasures(fit.med)[c("chisq","df","aic","cfi","rmsea")]
## chisq df aic cfi rmsea
## 0.000 0.000 756.355 1.000 0.000The c’ path is not significant, but both a and b are significant and the variance explained by liking is significantly higher (24.6%) than in the last model.
Give the full interpretation of the model:
In this exercise, we will use the data “protest.sav” (Hayes, 2022) which can be downloaded here under “data files and code”. Especially, we will focus on the following variables:
We want to test the assumption that if the lawyer protests, her response will be judged more appropriate by women who perceive sexism as a problem (moderator: dichotomous variable “sexism”), and therefore the lawyer will be judged better.
We want to test the following hypothesis: If the lawyer protests against gender discrimination, her response is perceived as more appropriate and therefore the lawyer is judged better.
Start by drawing the regression equations.
The regression equations go as:
\[ Y_i = \beta_0 + \beta_1*Protest_i + \beta_2*Respappr_i + \epsilon_i = c' + b \] \[ M_i = \beta_0 + \beta_1*Protest_i + \beta_2*Sexism_i + \beta_3*(Protest_i*Sexism_i) + \epsilon_i = a \] \[ Y_i = \beta_0 + \beta_1*Protest_i + \epsilon_i = c = c' + ab \]
Now, we want to calculate the mediation model and to answer to following questions:
Start by loading and selecting the data, and also construct the interaction variable:
# load the data
library(foreign)
db <- read.spss(file=paste0(getwd(),
"/data/protest.sav"),
use.value.labels = F,
to.data.frame = T)
# get the data
sel <- db |>
dplyr::select(protest, respappr, liking, sexism) |>
stats::na.omit()
# construct the interaction variable
sel$protest.sexism = sel$protest*sel$sexismNow, define the model in laavan:
# define the model
modell.mod = "
liking ~ cp*protest ## direct effect
respappr ~ a*protest + sexism + protest.sexism ## moderation/mediation paths
liking ~ b*respappr
protest ~~ sexism ## covariances
protest ~~ protest.sexism
sexism ~~ protest.sexism
respappr ~1 ## Intercepts
liking ~1
ab := a*b ## indirect effect
total := cp+a*b ## total effect
"
# get the complete output
fit.mod = lavaan::sem(modell.mod, data=sel)
lavaan::parameterestimates(fit.mod, standardized=T)[1:8]
## lhs op rhs label est se z pvalue
## 1 liking ~ protest cp -0.101 0.198 -0.508 0.611
## 2 respappr ~ protest a -2.687 1.429 -1.880 0.060
## 3 respappr ~ sexism -0.529 0.232 -2.278 0.023
## 4 respappr ~ protest.sexism 0.810 0.278 2.919 0.004
## 5 liking ~ respappr b 0.402 0.069 5.857 0.000
## 6 protest ~~ sexism 0.015 0.032 0.456 0.648
## 7 protest ~~ protest.sexism 1.114 0.141 7.886 0.000
## 8 sexism ~~ protest.sexism 0.501 0.176 2.846 0.004
## 9 respappr ~1 6.567 1.191 5.516 0.000
## 10 liking ~1 3.747 0.302 12.400 0.000
## 11 liking ~~ liking 0.824 0.103 8.031 0.000
## 12 respappr ~~ respappr 1.269 0.158 8.031 0.000
## 13 protest ~~ protest 0.217 0.027 8.031 0.000
## 14 sexism ~~ sexism 0.610 0.076 8.031 0.000
## 15 protest.sexism ~~ protest.sexism 6.151 0.766 8.031 0.000
## 16 protest ~1 0.682 0.041 16.640 0.000
## 17 sexism ~1 5.117 0.069 74.441 0.000
## 18 protest.sexism ~1 3.505 0.218 16.053 0.000
## 19 ab := a*b ab -1.081 0.604 -1.790 0.073
## 20 total := cp+a*b total -1.182 0.664 -1.781 0.075
lavaan::inspect(fit.mod,"r2")
## liking respappr
## 0.246 0.296What do the results mean in terms of content?
The overall model for the mediator (respappr) significantly explains 29.6% of the mediator’s variance.
There is a significant conditional effect of the independent variable on the mediator when the moderator has a value of 0 (= with a moderate level of sexism, the lawyer’s protest leads to her reaction being perceived as more appropriate).
The a-path from the independent variable to the mediator is significantly moderated by the sexism attitude. This moderation alone explains 4.8% (0.296-0.246=0.05) of the variance of the mediator.
The extent to which protesting influences the perceived appropriateness of the reaction thus varies depending on the subjects’ sexism attitude.
However, the strong correlations between the independent variables are a problem for the model! The standardized coefficients are outside the natural limits of -1 to +1. The model represents the data poorly. Moderation is better studied independently of mediation.
Finally, we want to know whether there is a significant moderated mediation effect? If so, how can this be described and interpreted in terms of content?
Interaction is significant: The moderator has an influence on the a-path.
Path ab is not significant: There is not an indirect effect for all values of the moderator.
The data are not suitable for this evaluation and the estimators cannot be fully trusted.