Multiple_Regression_Mediation

Mediation analysis

In this demonstration, we will model a mediation analysis. The total effect of an unmediated relationship is below in pathway c.

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( "M",
                         "Supervision", 
                         "Dissertation \nPerformance"), 
                box.type = "rect", 
                box.size = 0.12, 
                box.prop=0.5,  
                curve=0)

We are going to test for the mediator variable of self esteem. This needs the prime mark added to the direct pathway “c” text!

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)

Read in the data

d_full <- read_sav("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                     2.41

Early 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

X = supervision Y = dissertation_performance M = self_esteem

d <- d %>% 
  mutate(X = supervision,
         Y = dissertation_performance,
         M = self_esteem)

Longhand - Steps of Baron & Kenny (1986)

Four independent linear regression models

  1. The effect of X on Y (total effect = pathway c)
  2. The effect of X on M (indirect effect pathway a)
  3. The effect of M on Y (indirect effect pathway b) while controlling for X
  4. The effect of X and M on Y (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.

Step 1: Test the total effect - pathway c

\[ Y = b_0 + b_1 * X + e \]

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: 0.0005321

The 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.

There is an effect that can be tested for mediation.

Step 2: Test the 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: 0.0093

The 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.

Step 3 & 4: Test the b pathway of the indirect effect & the direct effect (pathway c’)

\[ Y = b_0 + b_1 * X + b_2 * M + e \] 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-12

The 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 < .02 , 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

Using the power of R and the mediation package

The mediation package is called by the library() function loaded at the top of the document.

  • It takes the models for pathways a and b (fit_indirecta and fitindirectb here),
  • It needs us to tell it the name of the predictor or treatment variable and the name of the mediator variable as labelled in the models
  • and we set the 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 bootstrap
summary(results)

Causal Mediation Analysis 

Nonparametric Bootstrap Confidence Intervals with the Percentile Method

               Estimate 95% CI Lower 95% CI Upper p-value    
ACME           0.206160     0.057689     0.375392   0.008 ** 
ADE            0.231347     0.037914     0.406644   0.016 *  
Total Effect   0.437508     0.214473     0.642776  <2e-16 ***
Prop. Mediated 0.471215     0.184343     0.874170   0.008 ** 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Sample Size Used: 77 


Simulations: 1000
  • ACME stands for average causal mediation effects and is the product of pathway a and pathway b from fit indirecta (X = 0.3037992) and fit_indirectb (M = 0.6786069).
  • ADE stands for average direct effects or pathway c’. This is the X coefficient in our fit_indirectb
  • Total Effect does what it says on the tin. It is the sum of the direct and indirect effect, ACME + ADE, and also calculated as X in model fit_total.
  • Prop. Mediated is the proportion of the effect of X on Y that goes through M. We divide ACME (or ab) by the total effect (c).
plot(results)

Reporting the mediation analysis

and to use our diagram from the top of the document:

We are going to test for the mediator variable of self esteem. (This needs the prime mark added to the direct pathway “c” text!)

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.

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