Emma Mills
Watch the lecture on mediation theory here
Watch the mediation demonstration here
Please download the Slides. In this zipped folder is the slides that accompany the lecture and the week 14 lab task question sheet.
The folder is a zipped folder called Wk14_Lecture_Lab_R_Upload_Materials.zip. It can be downloaded here This has the model lecture script and the .sav file that was demonstrated in the lecture. It also has the basic layout of a .Rmd file and the data for the Week 14 lab task.
We are working with the Coffee_Mediation_1.csv from the folder listed above. Upload it to the R server, (preferably before you come to the lab) and complete the Mediation_Lab_Script.Rmd file using that data. There is no model output as a guide this week. The lecture demonstration video and the model script is your guide.
The link to this week’s multiple choice quiz, WBA 4, is here
A t test tests for significant difference between two - and only two - mean values. ANOVA extends the t test functionality by exploring differences between several means either within one factor that has two or more levels or across multiple factors that can have more than two levels. ANOVA’s (and the regression method’s) ability to manage more than two variables within an analysis, give us the capacity to explore whether predictor variables effect each other and the outcome variable at the same time: we can test for interactions.
Measures of association follow a similar pattern: Correlation in its simplest form measures the direction and a degree of strength in a relationship between two variables.
Simple regression nominates one of those two variables as a predictor of the other. Indeed, if you use standardised values for predictor and outcome variables, the simple regression will return the Pearson’s r correlation coefficient as the simple regression coefficient estimate.
Multiple regression methods extend the correlation test functionality to many more predictor variables, each with the capcacity for multiple levels within a predictor.
Once we have multiple variables, we can take stock of the relationships not just between X and Y but also between the predictor variables. In week 13, we explored interactions in the context of effects on Y for a value for one predictor variable changing as the levels of a second predictor variable changed - an interaction. As the levels of one predictor changes, it moderates the effect of another predictor on Y.
Predictor variable relationships are not always those of interactions / moderations, however. There are other types of relationships that occur between predictor variables. The focus of this week’s lecture is that of the mediated relationship.
Mediation helps answer the question: how does a predictor variable effect an outcome variable, in the presence of a mediating variable? Look at the path diagram below.
There are still two predictors of Y, denoted by the two arrow heads that point to Y. If this were an interaction effect relationship then M would have an arrow connecting onto the line between X and Y. Instead, for a mediated relationship, the line between X and Y is both direct from X to Y on the bottom part of the diagram, but also goes through the predictor M, on the top of the diagram. One expression of X’s relationship with Y is direct, unmediated. An alternative expression of X’s relationship with Y is indirect, is mediated, by the predictor variable M. Some of X’s influence of effect passes through or is accounted for by M.
A mediation model is defined by four paths. These paths are labelled on the connecting arrows of the diagram, the lie between the variables in the rectangular boxes.
c: The path of total effect - perhaps you will recognise that this pathway is the arrow between what is otherwise a simple regression model diagram (see above and directly below).
c': (read Path c-Prime) The path of direct effecta: The path of X predicting Mb: The path of M predicting YTogether, path a and b are referred to as the path of indirect effect. See the diagram below.
Pause for a moment and look at the mediated model path diagram above. If we start at X and trace along to Y, then the single-arrow line denotes a simple regression model. Go back to X and now trace the single arrow line to M. This also denotes a simple regression model. Now start at M and follow the single-arrow line to Y. We have yet another diagram for a simple regression model.
However, X predicts both M and Y in the model below, so if we want to be sure that we are isolating M’s effect on Y, we have to partial out X’s influence on Y at the same time. We do this by putting X as a predictor of Y in the same time and in the same model for when we use M as a predictor of Y. Hence, the single-arrow line from M to Y, is a multiple regression because we have two predictors - our primary predictor of M and (in the context of this model) our secondary predictor of X, whose effect on Y we are seeking to control.
Now think about this:
To find the effect for the indirect paths a and b, we multiply them together. If we then add them to the effect of the direct path, c', then we should have the same magnitude or value of the effect as that of the total effect path, c.
\[ indirect\ effect + direct\ effect = total\ effect \]
or
\[ ab + c' = c \]
There are many types of mediation models. Here we will concentrate on only two: fully or partially mediated models.
When all of the effect passes through the indirect path, or through the mediating variable M, then we say we have Full Mediation. We can recognise this because the direct path, or c' is reduced to such a small value that it is no longer significantly different from zero, i.e. the regression coefficient in the multiple regression model of Y = M + X is non-significant
\[ Full\ Mediation: ab + (c' = 0) = c \]
When some of the effect passes through the indirect path, or through the mediating variable M, and a significant amount still passes through the direct path, then we say we have Partial Mediation.
\[ Partial\ Mediation = (ab > 0) + (c' > 0) = c \]
All of the usual assumptions for linear regression models apply to mediation models - because they are, after all only a series of regression models. However, we are beginning to claim a causal relationship and so there are some extra considerations. As an explanatory process, X can be said to be ‘causally’ related to Y when: + X is associated with Y + X precedes changes in Y + No other unmeasured variables are related to X and also affect Y
There are also some conditions that are specific for claiming an effect is a mediated relationship:
X need not be a significant predictor of YM must not be a primary predictor variable for YM must not be any of the study conditionsM must be singificantly predicted by XM must reduce or eradicate the impact of X on YX should / could precede M in timeM should significantly predict Y but Y could also significantly predict MM and Y could be correlated if they are both causally related to X.While this is known as a four step approach, it involves making three regression models:
X and outcome Ya; a simple regression model involving predictor X and outcome Ma and direct effect c'; a multiple regression model involving predictor X and M on outcome YHere are the steps are linear regression equations:
\[ Total\ effect: = Y = b_0 + b_1 * X + e \]
\[ indirect\ effect\ a = M = b_0 + b_1 * X + e \]
\[ indirect\ effect\ b = Y = b_0 + b_1 * X + b_2 * M + e \]
This occurs within model 3. For this step, look at the regression coefficient for the X predictor. This is the direct effect coefficient, or the coefficient for c' (c-prime):
X coefficient in the third model is significant, and M is significant, then you have a partial mediation effect;X coefficient in the third model is non-significant, and M is significant, then you have a full mediation effectDue to concerns for low power in mediation analysis methods, the dynamic duo above constructed a bootstrapped method to harness the power of machine learning and automate a resampling technique, just so we could have greater confidence in our results. The mediation package (Tingley et al., 2013) has prepared the process for R users. In order to use the automated process, the researcher has to write the three models as above and then write a further line of code, telling the bootstrapping algorithm what models to use. The only process that is new in this method is the line of code for the bootstrapping part. The three models are simple and multiple regression models.
If at this point, you have not watched the week 14 lecture demonstration video, please watch it now here.
Read this carefully because it sounds repetitive and as if I may have made an error, but it is the correct method:
First, as with all results, report the method that you used and your predictors, outcome variables and any transformations that you made to inform your modelling process.
Then, report the indirect effect with its confidence intervals. This is generally the effect that has the most power. Then report the direct effect with its confidence intervals. At this point you may want to claim either a full or partially mediated effect, however be careful. In a context of low statistical power, Type II errors can occur for the direct path coefficient. Finally report the total effect coefficient with its confidence intervals and discuss how the additional assumptions of mediation analysis are met.
As with all results, interpretation is largely driven by theory. A partial mediation may be illuminating an otherwise concealed relationship and so inform some theory development. The size of an indirect effect may indicate the strenght of a mediating relationship which could prompt more attention and greater dedication of research resource in future studies. It is your job as the researcher to make these statements, in the context of both theory and prior research.
The following is the script from the lecture video
In this demonstration, we will model a mediation analysis.
Is the quality of Supervision related to Dissertation Performance? What, if any, mediating effect may the self-esteem of a student have on this relationship?
We are going to test for the mediator variable of self esteem.
data <- c(0, "a", 0,
0, 0, 0,
"b", "`c'`", 0)
M <- matrix (nrow=3, ncol=3, byrow = TRUE, data=data)
plot<- plotmat (M, pos=c(1,2),
name= c( "Self-esteem",
"Supervision",
"Dissertation \nPerformance"),
box.type = "rect",
box.size = 0.12,
box.prop=0.5,
curve=0)
First, if you haven’t already, download the data file from here and upload to the R server. You may need to change the upload codes below to suit your directory structure.
d_full <- read_sav("data/Wk4/mediation exercise 2 data.sav")
head(d_full)# A tibble: 6 × 3
supervision self_esteem dissertation_performance
<dbl> <dbl> <dbl>
1 2.67 3.14 3
2 2.67 3.57 3.56
3 3.33 3 3.41
4 3.33 2.57 1
5 2.67 4.71 2.26
6 2.67 3.29 2.41Early in the construction of the script, I noticed that there were different levels of missingness across the models - this means that coefficients are estimated on different datasets, so we are introducing a potential source of systematic error if we do not correct for this.
Checking for missingness:
summary(d_full) supervision self_esteem dissertation_performance
Min. :1.667 Min. :1.571 Min. :1.000
1st Qu.:3.333 1st Qu.:3.286 1st Qu.:2.380
Median :4.000 Median :3.857 Median :3.037
Mean :3.786 Mean :3.864 Mean :3.028
3rd Qu.:4.333 3rd Qu.:4.500 3rd Qu.:3.593
Max. :5.000 Max. :5.714 Max. :5.815
NA's :2 NA's :3 For the purposes of this analysis, I will remove the observations (rows) with NA values. This is not the best way of working with missingness, but for the purposes of the demonstration it is ok.
d <- na.omit(d_full) # 4 observations removed
summary(d) # no NA values listed supervision self_esteem dissertation_performance
Min. :1.667 Min. :1.571 Min. :1.000
1st Qu.:3.333 1st Qu.:3.286 1st Qu.:2.370
Median :4.000 Median :3.857 Median :3.074
Mean :3.791 Mean :3.853 Mean :3.030
3rd Qu.:4.333 3rd Qu.:4.429 3rd Qu.:3.593
Max. :5.000 Max. :5.714 Max. :5.815 I am going to copy and rename the variables to save on typing. You don’t have to do this of course, and it may help you if you don’t.
X = supervision Y = dissertation_performance M = self_esteem
d <- d %>%
mutate(X = supervision,
Y = dissertation_performance,
M = self_esteem)Four steps but three independent linear regression models (as listed above):
X on Y (total effect = pathway c)X on M (indirect effect pathway a)Y as the outcome variable andM (indirect effect pathway b)X (for direct effect estimation = pathway c')When running your models, you need to assign them to objects in the environment to then be able to use them in a call to the mediation package when you run your bootstrapping process. Take note of how I call my models below to help you do this.
c pathway of the total effect\[ Y = b_0 + b_1 * X + e \]
via a simple regression model:
(fit_total <- summary(lm(Y ~ X, d)))
Call:
lm(formula = Y ~ X, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.83011 -0.56623 -0.04308 0.51478 2.25553
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.3718 0.4679 2.931 0.004470 **
X 0.4375 0.1209 3.620 0.000532 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8361 on 75 degrees of freedom
Multiple R-squared: 0.1488, Adjusted R-squared: 0.1374
F-statistic: 13.11 on 1 and 75 DF, p-value: 0.0005321The total effect of our predictor on our outcome is significant. In other words, supervision on dissertation performance is significant (p < .001). A change in one unit of supervision is associated with an increase in dissertation performance of 0.44.
Check the assumptions at this point.
There is an effect that can be tested for mediation.
a pathway of the indirect effect\[ M = b_0 + b_1 * X + e \]
a second simple regression model, using X as a predictor but this time, M (self esteem here) is our outcome variable:
(fit_indirecta <- summary(lm(M ~ X, d)))
Call:
lm(formula = M ~ X, data = d)
Residuals:
Min 1Q Median 3Q Max
-2.34553 -0.48838 0.02337 0.61288 1.49353
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 2.7018 0.4406 6.132 3.74e-08 ***
X 0.3038 0.1138 2.670 0.0093 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.7872 on 75 degrees of freedom
Multiple R-squared: 0.08679, Adjusted R-squared: 0.07461
F-statistic: 7.128 on 1 and 75 DF, p-value: 0.0093The indirect effect pathway a (X on M) is also significant (p = .009). A change in one unit of supervision is associated with an increase in self-esteem of 0.30.
So now we know that X and M share some variance - they are correlated. We have met one of the assumptions that we need to be able to perform a mediation analysis.
b pathway of the indirect effect & the c' pathway of the direct effect\[ Y = b_0 + b_1 * X + b_2 * M + e \]
A multiple regression model, with Y (dissertation performance) as our outcome variable, and X (supervision) and M (self-esteem) as predictors. Remember that this model is controlling for the effect of X on Y, because interpreting one predictor in a multiple regression model always assumes that the effect of the other predictors are already taken care of, or controlled for.
(fit_indirectb <- summary(lm(Y ~ X + M, d)))
Call:
lm(formula = Y ~ X + M, data = d)
Residuals:
Min 1Q Median 3Q Max
-1.39749 -0.48798 0.02245 0.44603 1.29565
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.46168 0.44402 -1.040 0.3018
X 0.23135 0.09794 2.362 0.0208 *
M 0.67861 0.09497 7.145 5.26e-10 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.6475 on 74 degrees of freedom
Multiple R-squared: 0.4963, Adjusted R-squared: 0.4827
F-statistic: 36.46 on 2 and 74 DF, p-value: 9.564e-12The indirect effect pathway b (M on Y), while controlling for X is also significant (p < .001). A change in one unit of self-esteem is associated with an increase in dissertation-performance of 0.68 of a unit. The predictor (X) relationship with the outcome (Y and pathway c' - the direct effect) remains significant also (p < .021) , but reduced relative to the step 1 model coefficient (step 1 \(b\) = 0.44, step 4 \(b\) = 0.23).
Since the direct pathway c' is significant, we can say that we have a partially mediated effect of self-esteem on the relationship between supervision and dissertation performance. Supervision predicts self esteem and dissertation performance, while self esteem also predicts dissertation performance.
If instead the X coefficient in the model above had been > .05 i.e. no significant, we could have claimed a full or complete mediation of self esteem on the relationship between supervision and dissertation performance
mediation packageThe mediation package is called by the library() function loaded at the top of the document.
a and b (fit_indirecta and fitindirectb here),boot argument to T for TRUE, to be able to generate confidence intervals on our co-efficients.results <- mediate(fit_indirecta,
fit_indirectb,
treat = 'X',
mediator = 'M',
boot = T,
dropobs = TRUE)Running nonparametric bootstrapsummary(results)
Causal Mediation Analysis
Nonparametric Bootstrap Confidence Intervals with the Percentile Method
Estimate 95% CI Lower 95% CI Upper p-value
ACME 0.2062 0.0625 0.37 0.006 **
ADE 0.2313 0.0423 0.41 0.018 *
Total Effect 0.4375 0.2071 0.65 <2e-16 ***
Prop. Mediated 0.4712 0.1785 0.88 0.006 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Sample Size Used: 77
Simulations: 1000 a and pathway b from fit indirecta (X = 0.30) and fit_indirectb (M = 0.68).c'. This is the X coefficient in our fit_indirectbX in model fit_total.X on Y that goes through M. We divide ACME (or ab) by the total effect (c).plot(results)
and to use our diagram from the top of the document:
We are going to test for the mediator variable of self esteem.
data <- c(0, 0.30, 0,
0, 0, 0,
0.68, 0.23, 0)
M<- matrix (nrow=3, ncol=3, byrow = TRUE, data=data)
plot<- plotmat (M, pos=c(1,2),
name= c( "Self-esteem",
"Supervision",
"Dissertation \nPerformance"),
box.type = "rect",
box.size = 0.12,
box.prop=0.5,
curve=0)
(Remember that these data are not standardised so we cannot compare between them for strength of relationships!)
The effect of supervision on dissertation performance was partially mediated via self-esteem. The effect of supervision on dissertation performance and the effect of self-esteem on dissertation performance were independently significant predictors. The indirect effect equals (.3)*(0.68) = .0.21. We tested the significance of this indirect effect using bootstrapping procedures. We computed the average indirect effect over 1,000 bootstrapped samples with 95% confidence intervals (bootstrapped indirect effect = 0.21 95% CI [0.06, 0.38]). Since the confidence intervals do not cross zero, we infer statistical significance.
Unlike weeks 12 and 13, week 14’s lab task requires you to run a mediation analysis and answer some questions. The staff will have the answers in the lab, so you can check them either as you go along or once you have finished. The lecture demonstration and model script is there to guide you.
The mediation analysis for the lab task involves exploring how a property of Alertness mediates the effect of Coffee intake on Problem Solving.
You can download the data and a starter .Rmd file in a zipped folder that’s needed for this lab from here. This is the same folder as the one at the top of the webpage. The Task and Question Sheet is in the other zipped folder, located here. This is the same Slides folder that is also at the top of the webpage. You will need to store this elsewhere on your computer
In your working with the data, get to know the variables and think how you would describe them in your measures sub-section of your methods section of a report.
Y)Step 1: Total effect: Run the code and write a short paragraph of results
Step 2: Indirect effect path a: Run the code and write a short paragraph of results.
Step 3 & 4: Indirect effect path b and direct effect path: Run the code and write a short paragraph of results.
X predicting the mediator M?M on outcome Y?X on Y?Run the bootstrap method Report the following values to 2 dp.
Draw the mediation analysis diagram with the correct values on the pathways.
Report the bootstrap analysis and the results.