Mediation and Moderation Analysis: Understanding Complex Relationships

Mediation

Mediation occurs when the effect of an independent variable \(X\) (e.g., therapy) on a dependent variable \(Y\) (e.g., depression) is transmitted through a third variable \(M\) (e.g., coping skills), called the mediator. The mediator explains how or why \(X\) affects \(Y\).

The mediation model involves three key paths:

• Path c (total effect): The direct relationship between \(X\) and \(Y\) without considering the mediator.
• Path a: The effect of \(X\) on the mediator \(M\).
• Path b: The effect of the mediator \(M\) on \(Y\), controlling for \(X\).
• Path c’ (direct effect): The effect of \(X\) on \(Y\) after controlling for \(M\).

The indirect effect (mediation effect) is the product of paths \(a\) and \(b\): \(a \times b\). If the indirect effect is significant, we conclude that \(M\) mediates the relationship between \(X\) and \(Y\).

Full mediation occurs when the direct effect \(c'\) becomes non-significant after including the mediator, meaning all of \(X\)’s effect on \(Y\) is transmitted through \(M\). Partial mediation occurs when \(c'\) remains significant but is reduced, meaning \(M\) explains some, but not all, of the effect.

Moderation

Moderation occurs when the relationship between \(X\) and \(Y\) depends on the level of a third variable \(W\) (e.g., social support), called the moderator. The moderator answers the question: For whom or under what conditions does \(X\) affect \(Y\)?

Moderation is tested by including an interaction term \(X \times W\) in the regression model. If the interaction is significant, the effect of \(X\) on \(Y\) varies across levels of \(W\).

For example, if social support moderates the therapy-depression relationship, therapy may be highly effective for patients with strong support but less effective for those with weak support.

Key Differences

• Mediation asks how or why: What is the mechanism?
• Moderation asks when or for whom: Under what conditions does the effect occur?

Both can be tested within regression frameworks, making them accessible extensions of multiple regression.

Mediation Model

The mediation model consists of three regression equations:

Equation 1 (Total effect of X on Y):

\[ Y_i = c_0 + c \cdot X_i + \varepsilon_i \]

Equation 2 (Effect of X on M):

\[ M_i = a_0 + a \cdot X_i + \varepsilon_{Mi} \]

Equation 3 (Effect of M on Y, controlling for X):

\[ Y_i = b_0 + c' \cdot X_i + b \cdot M_i + \varepsilon_{Yi} \]

where: - \(c\) is the total effect of \(X\) on \(Y\) - \(a\) is the effect of \(X\) on \(M\) - \(b\) is the effect of \(M\) on \(Y\), controlling for \(X\) - \(c'\) is the direct effect of \(X\) on \(Y\), controlling for \(M\) - Indirect effect = \(a \times b\) - Total effect = \(c' + (a \times b)\)

Moderation Model

The moderation model includes an interaction term:

\[ Y_i = \beta_0 + \beta_1 X_i + \beta_2 W_i + \beta_3 (X_i \times W_i) + \varepsilon_i \]

where: - \(\beta_1\) is the effect of \(X\) when \(W = 0\) (simple slope at \(W = 0\)) - \(\beta_2\) is the effect of \(W\) when \(X = 0\) - \(\beta_3\) is the interaction coefficient, indicating how much the effect of \(X\) changes for each unit increase in \(W\) - The effect of \(X\) at any level of \(W\) is: \(\beta_1 + \beta_3 W\)

If \(\beta_3\) is significant, the relationship between \(X\) and \(Y\) varies across levels of \(W\), confirming moderation.

Therapy and Depression

ggplot(data, aes(x = therapy, y = depression)) +
  geom_point(alpha = 0.5, size = 2, color = "steelblue") +
  geom_smooth(method = "lm", se = TRUE, color = "darkred", fill = "pink", alpha = 0.3) +
  labs(title = "Therapy Sessions and Depression",
       x = "Number of Therapy Sessions",
       y = "Depression Score") +
  theme_minimal() +
  theme(
    panel.grid.major = element_line(color = "gray90", linetype = "dashed"),
    panel.grid.minor = element_blank(),
    axis.line.x = element_line(color = "black"),
    axis.line.y = element_line(color = "black"),
    panel.border = element_blank(),
    plot.title = element_text(hjust = 0.5, face = "bold")
  )

## `geom_smooth()` using formula = 'y ~ x'

This shows a negative relationship: more therapy sessions are associated with lower depression.

Therapy and Coping (Potential Mediator)

ggplot(data, aes(x = therapy, y = coping)) +
  geom_point(alpha = 0.5, size = 2, color = "forestgreen") +
  geom_smooth(method = "lm", se = TRUE, color = "darkgreen", fill = "lightgreen", alpha = 0.3) +
  labs(title = "Therapy Sessions and Coping Skills",
       x = "Number of Therapy Sessions",
       y = "Coping Skills Score") +
  theme_minimal() +
  theme(
    panel.grid.major = element_line(color = "gray90", linetype = "dashed"),
    panel.grid.minor = element_blank(),
    axis.line.x = element_line(color = "black"),
    axis.line.y = element_line(color = "black"),
    panel.border = element_blank(),
    plot.title = element_text(hjust = 0.5, face = "bold")
  )

## `geom_smooth()` using formula = 'y ~ x'

Therapy increases coping skills, consistent with the mediation hypothesis.

Step 1: Total effect (c path)

# Model 1: Total effect of therapy on depression
model_total <- lm(depression ~ therapy, data = data)
summary(model_total)

## 
## Call:
## lm(formula = depression ~ therapy, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -34.876  -6.537  -0.159   7.327  30.438 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept)  67.5000     1.1490   58.75   <2e-16 ***
## therapy      -2.7012     0.1008  -26.79   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.56 on 298 degrees of freedom
## Multiple R-squared:  0.7066, Adjusted R-squared:  0.7056 
## F-statistic: 717.7 on 1 and 298 DF,  p-value: < 2.2e-16

Step 2: Effect of X on M (a path)

# Model 2: Effect of therapy on coping
model_a <- lm(coping ~ therapy, data = data)
summary(model_a)

## 
## Call:
## lm(formula = coping ~ therapy, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -20.1945  -5.3702  -0.6279   5.3565  23.9566 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 29.20181    0.82481   35.40   <2e-16 ***
## therapy      2.54141    0.07238   35.11   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 7.578 on 298 degrees of freedom
## Multiple R-squared:  0.8053, Adjusted R-squared:  0.8047 
## F-statistic:  1233 on 1 and 298 DF,  p-value: < 2.2e-16

Step 3: Effect of M on Y, controlling for X (b and c’ paths)

# Model 3: Effect of coping on depression, controlling for therapy
model_b <- lm(depression ~ therapy + coping, data = data)
summary(model_b)

## 
## Call:
## lm(formula = depression ~ therapy + coping, data = data)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -28.5251  -7.0526  -0.9486   7.0191  31.0872 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 79.47973    2.50968  31.669  < 2e-16 ***
## therapy     -1.65863    0.21877  -7.582 4.39e-13 ***
## coping      -0.41024    0.07725  -5.311 2.15e-07 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.11 on 297 degrees of freedom
## Multiple R-squared:  0.732,  Adjusted R-squared:  0.7302 
## F-statistic: 405.7 on 2 and 297 DF,  p-value: < 2.2e-16

Calculating the indirect effect

The indirect effect is \(a \times b\). We extract coefficients and compute it manually:

a_coef <- coef(model_a)["therapy"]
b_coef <- coef(model_b)["coping"]
c_coef <- coef(model_total)["therapy"]
c_prime_coef <- coef(model_b)["therapy"]

indirect_effect <- a_coef * b_coef
total_effect <- c_coef
direct_effect <- c_prime_coef

cat("Total effect (c):", round(total_effect, 3), "\n")

## Total effect (c): -2.701

cat("Direct effect (c'):", round(direct_effect, 3), "\n")

## Direct effect (c'): -1.659

cat("Indirect effect (a × b):", round(indirect_effect, 3), "\n")

## Indirect effect (a × b): -1.043

cat("Proportion mediated:", round(indirect_effect / total_effect, 3), "\n")

## Proportion mediated: 0.386

Bootstrapped mediation analysis

For robust inference, we use bootstrapping via the mediation package:

# Fit mediation and outcome models
med_model <- lm(coping ~ therapy, data = data)
out_model <- lm(depression ~ therapy + coping, data = data)

# Bootstrap mediation analysis
set.seed(2024)
med_results <- mediate(med_model, out_model, treat = "therapy", mediator = "coping", boot = TRUE, sims = 1000)

## Running nonparametric bootstrap

summary(med_results)

## 
## Causal Mediation Analysis 
## 
## Nonparametric Bootstrap Confidence Intervals with the Percentile Method
## 
##                Estimate 95% CI Lower 95% CI Upper   p-value    
## ACME           -1.04258     -1.45758     -0.65916 < 2.2e-16 ***
## ADE            -1.65863     -2.11490     -1.20801 < 2.2e-16 ***
## Total Effect   -2.70121     -2.88718     -2.50278 < 2.2e-16 ***
## Prop. Mediated  0.38597      0.24019      0.54130 < 2.2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Sample Size Used: 300 
## 
## 
## Simulations: 1000

The output provides the average causal mediation effect (ACME), average direct effect (ADE), total effect, and proportion mediated, along with confidence intervals.

Centering variables

To reduce multicollinearity and ease interpretation, we mean-center the predictors:

data$therapy_c <- data$therapy - mean(data$therapy)
data$support_c <- data$social_support - mean(data$social_support)

Fitting the moderation model

# Moderation model with interaction
model_moderation <- lm(depression ~ therapy_c * support_c, data = data)
summary(model_moderation)

## 
## Call:
## lm(formula = depression ~ therapy_c * support_c, data = data)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -35.880  -6.749  -0.323   7.172  30.400 
## 
## Coefficients:
##                     Estimate Std. Error t value Pr(>|t|)    
## (Intercept)         41.27943    0.62346  66.210   <2e-16 ***
## therapy_c           -2.69793    0.10292 -26.213   <2e-16 ***
## support_c            0.02751    0.06585   0.418    0.676    
## therapy_c:support_c -0.01137    0.01135  -1.001    0.318    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 10.57 on 296 degrees of freedom
## Multiple R-squared:  0.7076, Adjusted R-squared:  0.7047 
## F-statistic: 238.8 on 3 and 296 DF,  p-value: < 2.2e-16

The key coefficient is the interaction term therapy_c:support_c. If significant, it confirms moderation.

Residuals vs fitted values (moderation model)

data$residuals_mod <- residuals(model_moderation)
data$fitted_mod <- fitted(model_moderation)

ggplot(data, aes(x = fitted_mod, y = residuals_mod)) +
  geom_point(alpha = 0.6, color = "steelblue") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "red") +
  labs(title = "Residuals vs Fitted Values (Moderation Model)",
       x = "Fitted Values",
       y = "Residuals") +
  theme_minimal() +
  theme(
    panel.grid.major = element_line(color = "gray90", linetype = "dashed"),
    panel.grid.minor = element_blank(),
    axis.line.x = element_line(color = "black"),
    axis.line.y = element_line(color = "black"),
    panel.border = element_blank(),
    plot.title = element_text(hjust = 0.5, face = "bold")
  )

Residuals should scatter randomly around zero with no patterns.

Additional assumptions for mediation

Mediation analysis assumes: - No unmeasured confounding of the X-M, M-Y, or X-Y relationships - Correct temporal ordering (X precedes M precedes Y) - No measurement error in M

Violations can bias estimates. Sensitivity analyses and instrumental variables can help assess robustness.

Mediation

A mediation analysis was conducted to examine whether coping skills mediate the relationship between therapy sessions and depression symptoms in a sample of 300 patients. A bootstrapped mediation analysis (1000 iterations) revealed a significant indirect effect of therapy on depression through coping, \(b = -1.04\), 95% CI \([-1.45, -0.66]\), \(p < .001\). The direct effect of therapy on depression remained significant after controlling for coping, \(b = -1.66\), \(t(297) = -7.58\), \(p < .001\), indicating partial mediation. Coping skills accounted for approximately 38.6% (95% CI \([23.9\%, 54.0\%]\)) of the total effect of therapy on depression. These findings suggest that therapy reduces depression both directly and by enhancing patients’ coping strategies.

Moderation

A moderation analysis tested whether social support moderates the effect of therapy on depression. The interaction between therapy sessions and social support showed a non-significant trend, \(\beta = -0.01\), \(t(296) = -1.00\), \(p = .318\), \(R^2 = .708\). Simple slopes analysis revealed that therapy was effective across all levels of social support: at low support (\(b = -2.59\), \(p < .001\)), average support (\(b = -2.70\), \(p < .001\)), and high support (\(b = -2.81\), \(p < .001\)). While the interaction was not statistically significant, the pattern suggests therapy may be slightly more effective for patients with stronger social networks. These results indicate that therapy is broadly effective, though larger samples may be needed to detect moderating effects of social support.

Sample size

Mediation and moderation analyses require adequate power. For mediation, aim for \(n \geq 200\) to detect moderate indirect effects with bootstrapping. For moderation, detecting small interactions requires \(n \geq 400\). Underpowered studies may miss true effects or yield unstable estimates.

Centering and scaling

Mean-centering continuous predictors before creating interaction terms reduces multicollinearity and makes main effects more interpretable (they represent effects at the mean of the moderator). Standardizing variables can also aid interpretation when variables are on different scales.

Testing multiple mediators

When multiple mediators are hypothesized, fit separate models for each mediator and use structural equation modeling (SEM) for simultaneous testing of complex mediation chains (e.g., serial mediation, parallel mediation).

Alternatives to Baron and Kenny

The Baron and Kenny (1986) approach is widely used but has limitations. Modern methods prefer: - Bootstrapping: Provides robust confidence intervals without normality assumptions. - Structural equation modeling (SEM): Allows testing complex mediation models with multiple mediators and outcomes. - Causal mediation analysis: Addresses confounding and provides causal interpretations under certain assumptions.

Probing interactions

For significant interactions, always probe simple slopes at meaningful levels of the moderator (e.g., -1 SD, mean, +1 SD) and visualize the interaction. Johnson-Neyman technique can identify the exact moderator values where the effect of \(X\) becomes significant or non-significant.

Causality

Neither mediation nor moderation analysis establishes causality from observational data. Randomized experiments, longitudinal designs, and careful control of confounders strengthen causal claims. Sensitivity analyses can assess how robust conclusions are to unmeasured confounding.