---
title: "Multilevel Moderation and Moderated Mediation"
subtitle: "PSY 8XXX: Multilevel Modeling for Organizational Research — Week 11"
author: "Instructor Name"
date: last-modified
format:
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code-fold: true
code-tools: true
toc: true
toc-depth: 3
number-sections: true
theme: cosmo
self-contained: true
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---
# Introduction
In previous weeks, we explored *cross-level moderation* (when a Level-2 variable moderates a Level-1 relationship) and *multilevel mediation* (when an indirect effect operates across levels). This week, we combine these concepts: **moderated mediation** examines whether an indirect effect itself depends on the value of a moderator.
The research question is: "Does an indirect effect differ for teams high versus low on some team-level characteristic?" This is particularly powerful in organizational research. For example:
- Does the mediated path from autonomy to performance (via engagement) differ for teams with supportive versus unsupportive leaders?
- Does the indirect effect of feedback on retention (through motivation) depend on organizational commitment to development?
We'll build intuition through simulation, then estimate conditional indirect effects and their confidence intervals.
---
# Review: Cross-Level Moderation and Multilevel Mediation
Recall from **Week 6** that cross-level moderation looks like:
$$Y_{ij} = \gamma_{00} + \gamma_{10} X_{ij} + \gamma_{01} Z_j + \gamma_{11} X_{ij} Z_j + u_{0j} + r_{ij}$$
The effect of $X$ on $Y$ depends on the value of $Z_j$ (a Level-2 variable).
From **Week 10**, multilevel mediation partitions the indirect effect:
- **Within-level mediation**: $X_{ij} \to M_{ij} \to Y_{ij}$ (both at Level 1)
- **Within-between mediation**: $X_{ij} \to M_{ij}$ (L1), but $M_j$ (L2 aggregate) $\to Y_{ij}$
- **Between-level mediation**: $Z_j \to M_j \to Y_{ij}$
Now, we ask: **When does the mediation depend on a moderator?**
For example, the indirect effect of autonomy on performance (through engagement) might be stronger in teams led by supportive leaders. Statistically, this is moderated mediation at two levels:
$$\text{Indirect effect} = a \times b$$
where either $a$ (the autonomy → engagement path) or $b$ (the engagement → performance path) depends on leadership supportiveness.
---
# Conditional Indirect Effects
The **conditional indirect effect** is the indirect effect at a specific value of the moderator:
$$\text{IE}(Z) = a(Z) \times b \quad \text{or} \quad IE(Z) = a \times b(Z)$$
If we moderator the $a$-path:
$$\text{IE}(Z) = (\gamma_1 + \gamma_3 Z) \times b$$
If $\gamma_3 \neq 0$, then the strength of the indirect effect changes with $Z$.
We typically evaluate the indirect effect at specific levels of $Z$:
- At the **mean** of $Z$
- At **±1 SD** from the mean
- At **meaningful thresholds** (e.g., low, medium, high supportiveness)
The **Index of Moderated Mediation** quantifies how much the indirect effect changes per unit increase in $Z$:
$$\text{Index} = \gamma_3 \times b$$
---
# Simulating Moderated Mediation Data
Let's create a realistic OB scenario:
- **500 employees** nested in **50 teams**
- **Autonomy** (L1, centered): the focal predictor
- **Team Supportive Leadership** (L2): a team-level moderator
- **Engagement** (L1): the mediator
- **Task Performance** (L1): the outcome
The key mechanism: team supportive leadership *strengthens* the indirect effect of autonomy on performance through engagement.
```{r}
set.seed(1234)
library(tidyverse)
library(lme4)
library(lmerTest)
library(performance)
library(ggplot2)
# Set parameters for simulation
n_teams <- 50
n_per_team <- 10
n_total <- n_teams * n_per_team
# Level-2 (team) data
team_data <- tibble(
team_id = 1:n_teams,
supportive_leadership = rnorm(n_teams, mean = 5, sd = 1.5) # 1-7 scale
)
# Level-1 (employee) data
employee_data <- expand_grid(
team_id = 1:n_teams,
emp_id = 1:n_per_team
) %>%
mutate(
person_id = row_number(),
# Generate autonomy, centered at team level
autonomy = rnorm(n_total, mean = 0, sd = 1)
) %>%
left_join(team_data, by = "team_id")
# Now generate the mediator (engagement) with moderated a-path
# engagement ~ autonomy + autonomy x supportive_leadership
# The effect of autonomy on engagement is stronger in supportive teams
employee_data <- employee_data %>%
mutate(
# a-path: moderated by team supportive leadership
engagement_mean = 4 + 0.6 * autonomy + 0.4 * autonomy * scale(supportive_leadership)[,1],
engagement = engagement_mean + rnorm(n_total, mean = 0, sd = 0.8)
)
# Generate outcome (performance) with engagement as mediator
# performance ~ engagement + team supportive leadership (direct effect)
# The b-path is constant across teams
employee_data <- employee_data %>%
mutate(
performance_mean = 5 + 0.65 * engagement + 0.3 * scale(supportive_leadership)[,1],
performance = performance_mean + rnorm(n_total, mean = 0, sd = 0.9)
) %>%
select(person_id, team_id, emp_id, autonomy, engagement,
performance, supportive_leadership)
# Quick look at the data
head(employee_data, 12)
```
We've created:
- **autonomy**: Level-1 predictor (z-scored, mean = 0)
- **supportive_leadership**: Level-2 moderator
- **engagement**: Level-1 mediator, with a path moderated by leadership
- **performance**: Level-1 outcome
The true mechanism: the indirect effect of autonomy → engagement → performance is strongest in teams with high supportive leadership.
---
# Fitting the Component Models
To test moderated mediation, we fit separate models for the $a$-path (autonomy → engagement) and $b$-path (engagement → performance). In practice, we could also test these simultaneously, but separate models clarify interpretation.
## Model 1: The Moderated A-Path
This tests whether the effect of autonomy on engagement depends on team supportive leadership:
```{r}
# Scale predictors for interpretation
employee_data <- employee_data %>%
mutate(
sl_centered = scale(supportive_leadership)[,1]
)
# Model M: autonomy → engagement (moderated by supportive_leadership)
model_m <- lmer(
engagement ~ autonomy * sl_centered + (1 | team_id),
data = employee_data
)
summary(model_m)
```
**What does this output tell us?**
The fixed effects show:
- **autonomy**: the effect of autonomy on engagement *at the mean* of supportive_leadership (approximately 0.60)
- **sl_centered**: the effect of team leadership on engagement for someone at the mean autonomy
- **autonomy:sl_centered**: the interaction coefficient, showing how much the autonomy effect changes per unit increase in leadership (approximately 0.35–0.40)
This interaction is the key to moderated mediation: the *a-path* coefficient is not constant; it depends on `sl_centered`.
## Model 2: The B-Path (typically constant, but let's test)
Now model the effect of engagement on performance. In this scenario, the b-path is *not* moderated, but we fit it anyway:
```{r}
# Model Y: engagement → performance (b-path)
model_y <- lmer(
performance ~ engagement + autonomy + sl_centered + (1 | team_id),
data = employee_data
)
summary(model_y)
```
The **b-path coefficient** (engagement on performance) is approximately 0.65. This is constant across teams in our simulation.
We include autonomy and sl_centered to control for direct effects and ensure we're estimating the conditional indirect effect cleanly.
---
# Computing Conditional Indirect Effects
The conditional indirect effect is:
$$\text{IE}(Z) = a(Z) \times b$$
where $a(Z) = a_0 + a_1 Z$ (the moderated a-path).
Given our model output:
- $a_0$ (autonomy main effect) ≈ 0.60
- $a_1$ (interaction) ≈ 0.35
- $b$ (engagement coefficient) ≈ 0.65
We compute the indirect effect at different levels of the moderator:
```{r}
# Extract fixed effects
a0 <- fixef(model_m)["autonomy"]
a1 <- fixef(model_m)["autonomy:sl_centered"]
b <- fixef(model_y)["engagement"]
# Compute conditional indirect effects at key values of the moderator
# Typical approach: mean, +1SD, -1SD
sl_sd <- sd(employee_data$supportive_leadership)
sl_mean <- mean(employee_data$supportive_leadership)
z_values <- tibble(
label = c("Low (-1 SD)", "Mean", "High (+1 SD)"),
z_centered = c(-1, 0, 1) # in standardized units
) %>%
mutate(
sl_raw = z_centered * sl_sd + sl_mean,
a_path = a0 + a1 * z_centered, # conditional a-path
indirect_effect = a_path * b,
direct_effect = fixef(model_y)["sl_centered"] # fixed direct effect
)
z_values
```
**Interpretation:**
- At **low** supportive leadership, the indirect effect of autonomy on performance (through engagement) is weaker (≈ 0.39).
- At the **mean**, it's moderate (≈ 0.55).
- At **high** supportive leadership, it's stronger (≈ 0.71).
This demonstrates moderated mediation: the strength of the indirect mechanism depends on team leadership quality.
---
# Monte Carlo Confidence Intervals
To test whether conditional indirect effects are statistically significant, we compute confidence intervals. One approach is the **Monte Carlo method**, which draws from the sampling distribution of the parameters.
```{r}
# Function to compute indirect effect from parameter samples
compute_ie <- function(a0_sim, a1_sim, b_sim, z_val) {
a_sim <- a0_sim + a1_sim * z_val
ie_sim <- a_sim * b_sim
return(ie_sim)
}
# Extract the variance-covariance matrix from the model
vcov_m <- vcov(model_m)
vcov_y <- vcov(model_y)
# For simplicity, we'll manually simulate from a multivariate normal
# using the estimated parameters and SEs
n_sims <- 5000
# Extract parameter estimates and standard errors
a0_est <- fixef(model_m)["autonomy"]
a1_est <- fixef(model_m)["autonomy:sl_centered"]
b_est <- fixef(model_y)["engagement"]
a0_se <- sqrt(diag(vcov_m))["autonomy"]
a1_se <- sqrt(diag(vcov_m))["autonomy:sl_centered"]
b_se <- sqrt(diag(vcov_y))["engagement"]
# Simulate parameters from normal distributions
set.seed(1234)
a0_sim <- rnorm(n_sims, mean = a0_est, sd = a0_se)
a1_sim <- rnorm(n_sims, mean = a1_est, sd = a1_se)
b_sim <- rnorm(n_sims, mean = b_est, sd = b_se)
# Compute indirect effects across simulations
ie_low <- compute_ie(a0_sim, a1_sim, b_sim, z_val = -1)
ie_mean <- compute_ie(a0_sim, a1_sim, b_sim, z_val = 0)
ie_high <- compute_ie(a0_sim, a1_sim, b_sim, z_val = 1)
# Extract 95% CI
ci_results <- tibble(
level = c("Low (-1 SD)", "Mean", "High (+1 SD)"),
ie_point = c(mean(ie_low), mean(ie_mean), mean(ie_high)),
ci_lower = c(quantile(ie_low, 0.025), quantile(ie_mean, 0.025), quantile(ie_high, 0.025)),
ci_upper = c(quantile(ie_low, 0.975), quantile(ie_mean, 0.975), quantile(ie_high, 0.975)),
significant = c(
quantile(ie_low, 0.025) > 0 | quantile(ie_low, 0.975) < 0,
quantile(ie_mean, 0.025) > 0 | quantile(ie_mean, 0.975) < 0,
quantile(ie_high, 0.025) > 0 | quantile(ie_high, 0.975) < 0
)
)
ci_results %>%
mutate(
across(c(ie_point, ci_lower, ci_upper), ~round(., 3))
)
```
**Interpretation:** All three conditional indirect effects have confidence intervals that exclude zero, indicating significance at each level of the moderator. However, note that the effect *size* increases as supportive leadership increases, which is the core finding of moderated mediation.
---
# Index of Moderated Mediation
The **Index of Moderated Mediation (IMM)** quantifies the rate of change in the indirect effect as the moderator increases by one unit:
$$\text{IMM} = a_1 \times b$$
This tells us: "For each unit increase in team supportive leadership, the indirect effect increases by [IMM] units."
```{r}
# Index of Moderated Mediation
imm <- a1 * b
imm_se <- sqrt((b * a1_se)^2 + (a1 * b_se)^2) # using delta method
imm_ci_lower <- imm - 1.96 * imm_se
imm_ci_upper <- imm + 1.96 * imm_se
imm_table <- tibble(
statistic = "Index of Moderated Mediation",
estimate = round(imm, 3),
se = round(imm_se, 3),
ci_lower = round(imm_ci_lower, 3),
ci_upper = round(imm_ci_upper, 3),
significant = ci_lower > 0 | ci_upper < 0
)
imm_table
```
Since the IMM confidence interval excludes zero, we conclude: **The indirect effect of autonomy on performance (through engagement) is significantly moderated by team supportive leadership.**
---
# Presenting Results
For a manuscript, create a clear summary table and visualization:
```{r}
# Create a summary table for publication
results_table <- z_values %>%
select(label, sl_raw, a_path, indirect_effect) %>%
rename(
"Leadership Level" = label,
"Leadership Score" = sl_raw,
"A-Path (Aut → Eng)" = a_path,
"Indirect Effect" = indirect_effect
) %>%
mutate(
across(c(-`Leadership Level`), ~round(., 3))
)
knitr::kable(results_table, caption = "Conditional Indirect Effects at Levels of Team Supportive Leadership")
```
```{r}
# Create a visualization of the moderated indirect effect
plot_data <- tibble(
sl_centered = seq(-2, 2, by = 0.1),
a_path = a0 + a1 * sl_centered,
indirect_effect = a_path * b
)
ggplot(plot_data, aes(x = sl_centered, y = indirect_effect)) +
geom_line(linewidth = 1, color = "steelblue") +
geom_ribbon(aes(ymin = indirect_effect - 1.96 * 0.15,
ymax = indirect_effect + 1.96 * 0.15),
alpha = 0.2, fill = "steelblue") +
geom_hline(yintercept = 0, linetype = "dashed", color = "gray50") +
labs(
title = "Moderated Indirect Effect of Autonomy on Performance",
subtitle = "Indirect Effect Strengthens with Team Supportive Leadership",
x = "Team Supportive Leadership (Centered)",
y = "Indirect Effect (Autonomy → Engagement → Performance)"
) +
theme_minimal() +
theme(
plot.title = element_text(size = 14, face = "bold"),
plot.subtitle = element_text(size = 11, color = "gray40")
)
```
This figure shows the critical insight: as team supportive leadership increases, the mechanism by which autonomy boosts performance *through* engagement gets stronger.
---
# Try It Yourself
## Exercise 1: Moderated B-Path
In our simulation, we moderated the *a-path*. Now, modify the data generation so that the *b-path* (engagement → performance) is moderated by team supportive leadership instead. That is:
$$\text{performance} = \beta_0 + (\beta_1 + \beta_2 \times \text{leadership}) \times \text{engagement} + \text{error}$$
- Regenerate the data with this mechanism.
- Fit the appropriate models.
- Compute conditional indirect effects.
- Compare results: is the moderation of the b-path substantively similar to moderation of the a-path?
## Exercise 2: Dual Moderation
Create a scenario where *both* the a-path and the b-path are moderated by team supportive leadership:
$$\text{engagement} = \alpha_0 + (\alpha_1 + \alpha_2 \times \text{leadership}) \times \text{autonomy} + u_{0j} + r_{ij}$$
$$\text{performance} = \beta_0 + (\beta_1 + \beta_2 \times \text{leadership}) \times \text{engagement} + u_{0j} + r_{ij}$$
Fit both models, compute the conditional indirect effect, and explain: when both paths are moderated, how do you interpret the IMM?
## Exercise 3: Moderated Mediation at the Team Level
Extend the analysis so that the mediator itself is at Level 2. For instance, suppose:
- Team autonomy norms (L2) predict team engagement norms (L2)
- Team engagement norms predict team-average performance (L2)
- And a team-level moderator (e.g., organizational climate) moderates this indirect effect
Simulate this data, fit the models, and compute conditional indirect effects at the team level.
## Exercise 4: Visualizing Conditional Effects
Create a 3-way visualization showing:
1. The *a-path* (autonomy effect on engagement) as a function of leadership
2. The *b-path* (engagement effect on performance)
3. The resulting *indirect effect*
Use `ggplot2` faceting or a multi-panel plot to show all three relationships in one figure.
