The Hayes Process Models provide a powerful framework for analyzing complex relationships between variables in social, behavioral, and other scientific research fields. Developed by Andrew F. Hayes, this approach offers a set of statistical tools that allow researchers to explore and test relationships that go beyond simple linear correlations. These models focus on mediation, moderation, and conditional process modeling, offering a clear method for investigating how and when effects occur.
Mediation
Mediation analysis helps identify why or how a certain effect occurs. It examines whether the relationship between an independent variable (X) and a dependent variable (Y) is explained, at least partially, by a third variable called the mediator (M). In simpler terms, mediation investigates if X affects Y through M, by decomposing the total effect of X on Y into direct and indirect effects.
Example: A researcher might hypothesize that the impact of study habits (X) on academic performance (Y) is mediated by motivation (M). The mediation model would test whether better study habits increase motivation, which in turn improves academic performance.
Moderation
Moderation analysis looks at when or under what conditions a relationship between X and Y holds. It introduces a moderator variable (W), which influences the strength or direction of the effect that X has on Y. The goal is to determine if the effect of X on Y changes depending on the level of W.
Example: A company might test whether the relationship between employee training (X) and job performance (Y) depends on the years of experience (W) the employees have. If training improves job performance more for experienced workers than for beginners, experience moderates the training-performance relationship.
Conditional Process Modeling
Conditional process modeling combines mediation and moderation, exploring both how and when an effect occurs. It investigates whether the indirect effect of X on Y through M is conditional upon the level of a moderator W, allowing researchers to examine more nuanced relationships.
Example: A researcher could examine whether the indirect effect of parental involvement (X) on a child’s academic success (Y) through school engagement (M) depends on the socio-economic status (W) of the family. In this case, socio-economic status moderates the indirect effect of parental involvement on academic success.
Hayes’ models have become increasingly popular because they provide a comprehensive and structured way to test hypotheses involving complex variable relationships. They allow researchers to:
Analyze causal mechanisms and understand the process through which an independent variable influences an outcome.
Determine under what conditions an effect holds, identifying critical contextual or situational factors.
Move beyond traditional statistical approaches, which often assume linear, direct relationships, by modeling indirect, conditional, or interactive effects.
The statistical foundation for these models is rooted in ordinary least squares (OLS) regression and bootstrapping methods, which allow for robust hypothesis testing even when assumptions such as normality are violated. Hayes’ popular PROCESS macro, which integrates into software like SPSS, SAS, and R, simplifies the estimation of mediation and moderation models, making it accessible for researchers with varying levels of statistical expertise.
Hayes developed over 90 distinct models for mediation, moderation, and moderated mediation. Each model is designed for different types of hypotheses:
Model 4 is widely used for simple mediation.
Model 1 tests for simple moderation.
Model 7 is a common choice for moderated mediation (also known as conditional process models).
These models provide flexible and easily interpretable ways to test intricate theoretical relationships.
Mediation analysis is a statistical technique used to explore the underlying process or mechanism through which an independent variable (X) influences a dependent variable (Y) through a third variable, called the mediator (M). This chapter will provide an in-depth look at the theory behind mediation, introduce you to the necessary statistical steps, and show how to conduct mediation analysis using R with practical examples.
In many research scenarios, a simple direct relationship between an independent variable and a dependent variable does not fully explain the phenomenon being studied. Mediation analysis is useful when we suspect that this relationship is at least partly driven by an intermediate variable, called the mediator. The goal is to assess both the direct effect of X on Y and the indirect effect, which is the portion of the effect that passes through M.
Path Diagram for Mediation
A typical mediation model is illustrated with three paths:
Path a: The effect of the independent variable (X) on the mediator (M).
Path b: The effect of the mediator (M) on the dependent variable (Y), while controlling for X.
Path c’: The direct effect of X on Y, while controlling for the mediator (M).
Indirect effect: The product of paths a and b, representing the portion of the effect of X on Y that occurs through M.
Mediation Formula
The relationship between the variables is expressed using the following equations:
\[ M = aX + e_1 \] \[ Y= bM + c'X + e_2 \]
Where:
a is the regression coefficient for the effect of X on M.
b is the regression coefficient for the effect of M on Y (controlling for X).
c’ is the direct effect of X on Y after accounting for the mediator.
The total effect of X on Y is the sum of:
The direct effect of X on Y (c’).
The indirect effect, which is mediated through M.
Mathematically:
\[ c=c'+(a \cdot b) \]
Where:
c: Total effect of X on Y.
c’: Direct effect of X on Y (not through M ).
\(a \cdot b\): Indirect effect of X on Y via M.
Mediation analysis generally follows three main steps:
Testing the effect of X on M (path a):
Testing the effect of M on Y while controlling for X (path b):
Testing the direct effect of X on Y (path c’):
Bootstrapping for Mediation
Since the product of coefficients (a * b) often has a non-normal distribution, bootstrapping is commonly used to obtain more accurate confidence intervals for the indirect effect. Bootstrapping repeatedly samples the data with replacement to estimate the distribution of the indirect effect, which allows for more robust statistical inferences.
To illustrate mediation analysis, we will use Hayes’ Model 4—a simple mediation model. Below is an example of how to run a mediation analysis in R using a fictional dataset where the independent variable (X) is hours studied, the mediator (M) is motivation, and the dependent variable (Y) is test score.
Data Simulation Example
# Simulating the dataset
set.seed(123)
n <- 200
X <- rnorm(n, mean = 10, sd = 2) # Hours studied
M <- 0.5 * X + rnorm(n) # Motivation (mediator)
Y <- 0.6 * M + 0.3 * X + rnorm(n) # Test score (dependent variable)
data <- data.frame(X = X, M = M, Y = Y)
head(data) X M Y
1 8.879049 6.638335 6.573159
2 9.539645 6.082235 5.342583
3 13.117417 6.293563 7.076615
4 10.141017 5.613702 6.381685
5 10.258575 4.714948 6.577237
6 13.430130 6.238818 6.121783Running Mediation Analysis in R
We will use the mediation package in R, which provides functions for conducting mediation analysis. First, we fit two linear models: one for predicting the mediator (M) and one for predicting the outcome (Y).
# Load the mediation package
# install.packages("mediation")
library(mediation)Loading required package: MASSLoading required package: MatrixLoading required package: mvtnormLoading required package: sandwichmediation: Causal Mediation Analysis
Version: 4.5.0# Step 1: Fit the model for the mediator
med_model <- lm(M ~ X, data = data)
# Step 2: Fit the model for the outcome
out_model <- lm(Y ~ M + X, data = data)
# Step 3: Conduct mediation analysis
mediation_analysis <- mediate(med_model,
out_model,
treat = "X",
mediator = "M",
boot = TRUE,
sims = 500)Running nonparametric bootstrapsummary(mediation_analysis)
Causal Mediation Analysis
Nonparametric Bootstrap Confidence Intervals with the Percentile Method
Estimate 95% CI Lower 95% CI Upper p-value
ACME 0.267 0.205 0.35 <2e-16 ***
ADE 0.309 0.202 0.41 <2e-16 ***
Total Effect 0.576 0.477 0.66 <2e-16 ***
Prop. Mediated 0.463 0.346 0.61 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Sample Size Used: 200
Simulations: 500 Interpreting the Results
The output of the mediate() function provides several key statistics:
ACME (Average Causal Mediation Effect): This is the estimated indirect effect (a * b). If it is significantly different from zero, we can conclude that mediation exists.
ADE (Average Direct Effect): This represents the direct effect (c’) of X on Y, after accounting for the mediator.
Total Effect: The sum of the direct and indirect effects.
Proportion Mediated: The proportion of the total effect that is mediated by M.
An example interpretation might be:
The indirect effect of hours studied on test scores through motivation is significant, meaning motivation partially mediates the relationship between hours studied and test scores.
The direct effect of hours studied on test scores remains significant even after accounting for motivation, indicating that both direct and mediated effects are important.
While mediation analysis is powerful, it is not without its limitations:
Causality: Mediation assumes a causal order (X → M → Y), but establishing causality requires more than just statistical analysis (e.g., experimental designs or longitudinal data).
Confounding Variables: Unaccounted confounders can bias the results. It’s important to ensure that potential confounders are controlled for in the analysis.
Measurement Error: Inaccurate measurement of X, M, or Y can distort the mediation effect.
Moderation analysis is a statistical technique used to explore how the relationship between an independent variable (X) and a dependent variable (Y) changes depending on the level of a third variable, known as the moderator (W). This technique allows researchers to examine when and under what conditions an effect occurs, rather than simply whether or not it exists.
In many research contexts, the relationship between an independent variable (X) and a dependent variable (Y) is not consistent across all situations or individuals. Instead, this relationship may depend on a third variable, called the moderator (W). Moderation occurs when the effect of X on Y varies depending on the level of W.
Moderation analysis answers the question: “When or under what conditions does X influence Y?” For example, the relationship between study time (X) and test performance (Y) may depend on the student’s motivation (W). If motivation is high, the effect of study time on test performance might be stronger than when motivation is low.
Moderation occurs when the strength or direction of the relationship between two variables (X and Y) is influenced by a third variable (W). Rather than explaining how the effect happens (as in mediation), moderation analysis addresses the question of when or under what conditions an effect is stronger, weaker, or even reversed.
A moderator can be either:
Quantitative (e.g., income, age, temperature) or
Categorical (e.g., gender, region, treatment vs. control).
Example of Moderation
Imagine we are investigating whether the relationship between study hours (X) and test scores (Y) is moderated by sleep quality (W). It is possible that the effect of studying on test scores is stronger when sleep quality is high and weaker when sleep quality is low. This would suggest that sleep quality moderates the relationship between study hours and test scores.
Moderation Formula
The general form of a moderation model can be expressed as:
\[ Y = b_0 + b_1X + b_2W + b_3(X \times W)+e \]
Where:
X is the independent variable,
W is the moderator,
XW is the interaction term between X and W, representing the moderation effect,
b1 represents the effect of X on Y (when W = 0),
b2 represents the effect of W on Y (when X = 0),
b3 represents the effect of the interaction between X and W on Y, quantifying how W moderates the effect of X on Y.
In this model, the key coefficient of interest is b3, which tests whether the interaction between X and W significantly predicts Y. A significant interaction effect (b_3 ≠ 0) indicates the presence of moderation.
The interaction term captures how the effect of X on Y depends on W. Specifically:
\[ \text{Effect of X on Y} = \beta_1 + \beta_3 W \]
When W increases, the slope of (X on Y) changes by \(\beta_3\).
The relationship between X and Y is conditional on the value of W
If \(W=0\): The effect of X on Y is simply \(\beta_1\)
If \(W>0\): The effect of X on Y becomes \(\beta_1+\beta_3W\)
This demonstrates that the magnitude and direction of X’s effect on Y change depending on W.
Path Diagram for Moderation
A typical moderation model can be depicted as follows:
X → Y: The main effect of the independent variable (X) on the dependent variable (Y).
W (Moderator): The moderator affects the strength or direction of the X → Y relationship.
Interaction Effect (X * W): This term captures the effect of the interaction between X and W on Y. If the interaction term is significant, it indicates that W moderates the relationship between X and Y.
To illustrate moderation analysis, we will use Hayes’ Model 1, which tests for a simple moderation effect. In this example, we will examine whether motivation (W) moderates the relationship between hours studied (X) and test score (Y).
Data Simulation Example
# Simulating the dataset
set.seed(123)
n <- 300
X <- rnorm(n, mean = 10, sd = 2) # Hours studied (X)
W <- rnorm(n, mean = 5, sd = 1) # Motivation (Moderator, W)
Y <- 0.4 * X + 0.6 * W + 0.3 * X * W + rnorm(n) # Test score (Y)
data <- data.frame(X = X, W = W, Y = Y)
head(data) X W Y
1 8.879049 4.284758 18.60986
2 9.539645 4.247311 18.49225
3 13.117417 4.061461 23.63328
4 10.141017 3.947487 16.91829
5 10.258575 4.562840 21.67399
6 13.430130 5.331179 29.83955Running Moderation Analysis in R
We can use the lm() function to fit the moderation model, including the interaction term between X and W.
# Step 1: Fit the main effects model
main_effects_model <- lm(Y ~ X + W, data = data)
summary(main_effects_model)
Call:
lm(formula = Y ~ X + W, data = data)
Residuals:
Min 1Q Median 3Q Max
-2.8798 -0.7630 0.0867 0.6989 4.1000
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -14.44599 0.50512 -28.60 <2e-16 ***
X 1.84417 0.03499 52.71 <2e-16 ***
W 3.59820 0.06692 53.77 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.142 on 297 degrees of freedom
Multiple R-squared: 0.9474, Adjusted R-squared: 0.947
F-statistic: 2673 on 2 and 297 DF, p-value: < 2.2e-16# Step 2: Fit the moderation model (including the interaction term)
moderation_model <- lm(Y ~ X * W, data = data)
summary(moderation_model)
Call:
lm(formula = Y ~ X * W, data = data)
Residuals:
Min 1Q Median 3Q Max
-2.52213 -0.72725 0.04238 0.68765 2.63550
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.61640 1.77458 -0.347 0.72857
X 0.48400 0.17158 2.821 0.00511 **
W 0.78321 0.35421 2.211 0.02779 *
X:W 0.27753 0.03441 8.067 1.82e-14 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.036 on 296 degrees of freedom
Multiple R-squared: 0.9568, Adjusted R-squared: 0.9564
F-statistic: 2188 on 3 and 296 DF, p-value: < 2.2e-16Interpreting the Results
In the output of the lm() function, focus on the coefficient for the interaction term (X). If this term is significant, it means that motivation (W) moderates the relationship between hours studied (X) and test score (Y).
Main effects: The coefficients for X and W tell us the effect of hours studied and motivation on test scores when the other variable is held constant.
Interaction effect: A significant interaction term indicates that the relationship between hours studied and test scores changes depending on the level of motivation.
For example:
If the interaction term is positive, the effect of study hours on test scores is stronger when motivation is higher.
If the interaction term is negative, the effect of study hours on test scores is weaker when motivation is higher.
Visualizing the Interaction Effect
To better understand the moderation effect, it’s helpful to plot the interaction between X and W. The interactions package in R provides convenient functions for visualizing interaction effects.
# Load the interactions package
# install.packages("interactions")
library(interactions)
# Plot the interaction effect
interact_plot(moderation_model,
pred = "X",
modx = "W",
plot.points = TRUE)
This plot shows how the relationship between hours studied and test scores varies depending on the level of motivation. For example, you might see that at low levels of motivation, hours studied have little effect on test scores, but at high levels of motivation, hours studied have a strong positive effect on test scores.
Moderation analysis results need to be interpreted carefully to understand the underlying relationships. The significance of the interaction term helps determine whether moderation exists, and the direction of the interaction coefficient provides insight into how the moderator influences the relationship between X and Y.
Significant Interaction: If the interaction term is significant, W moderates the relationship between X and Y. The interaction plot provides a visual representation of this effect.
Non-Significant Interaction: If the interaction term is not significant, this suggests that the relationship between X and Y is consistent across levels of W, meaning no moderation effect exists.
Significant Interaction but Only One Main Effect is Significant: The significant interaction indicates that the effect of X on Y depends on W, even if X or W alone does not have a straightforward effect on Y.
Example,
Predictor (X): Study habits.
Moderator (W): Sleep quality.
Outcome (Y): Academic performance.
If the main effect of W (sleep quality) is significant, but the main effect of X (study habits) is not, and the interaction (X×W) is significant:
Interpretation: Study habits do not independently predict academic performance, but their effect is moderated by sleep quality. For example:
When sleep quality is high, study habits have a stronger positive impact on academic performance.
When sleep quality is low, the relationship between study habits and academic performance might weaken or even reverse.
Significant interaction, no main effects significant: The effect of X on Y exists only through interaction with W.
Simple Slopes Analysis
To further interpret a significant moderation effect, a simple slopes analysis is often conducted. This involves examining the relationship between X and Y at different levels of the moderator (e.g., low, medium, and high levels of W).
The simple slopes analysis can provide insights into:
How the relationship between X and Y changes across levels of W.
Whether the effect of X on Y is significant at different levels of W.
Moderated mediation, also known as conditional process modeling, refers to the integration of both mediation and moderation in a single model. In a moderated mediation model, the indirect effect of an independent variable (X) on a dependent variable (Y) through a mediator (M) is contingent upon the level of a moderator (W). Let me explain the theoretical framework for moderated mediation, guide you through the steps for conducting such an analysis, and demonstrate how to perform this type of analysis in R using Hayes’ Model 7.
Moderated mediation explores both how and when an effect occurs. Specifically, it investigates whether the strength or direction of the indirect effect of X on Y through M depends on a moderator variable (W). This allows researchers to examine whether the mediation process varies across different conditions or groups.
Key Questions Answered by Moderated Mediation:
How does the mediation process work?: Does X affect Y through a mediator M?
When or under what conditions is the mediation stronger or weaker?: Does the indirect effect of X on Y through M vary based on the level of a moderator W?
Moderated mediation is particularly useful when a simple mediation model is not enough to capture the complexity of the relationships among variables. It allows researchers to test whether the mediation pathway is stronger or weaker under certain conditions.
Path Diagram for Moderated Mediation
A typical moderated mediation model can be represented as:
X → M → Y: This is the basic mediation pathway, where X affects Y indirectly through M.
W moderating X → M or W moderating M → Y: The moderator W influences either the path from X to M (i.e., the indirect effect itself) or the path from M to Y.
Thus, W can moderate the relationship between X and M, M and Y, or both.
Moderated Mediation Formula
The statistical model for moderated mediation can be expressed as follows:
\[ M = a_1X + a_2W + a_3(X \times W) + e_1 \]
\[ Y = b_1M + b_2X + e_2 \]
Where:
a₁: The effect of X on M.
a₃: The interaction between X and W, indicating whether W moderates the relationship between X and M.
b₁: The effect of M on Y (mediated effect).
The direct effect of X on Y is independent of mediation: \(\beta_2\)
The indirect effect is conditional on the level of W, meaning that the mediation pathway (X → M → Y) changes depending on the moderator.
\[ \text{Indirect Effect}=(a_1+a_3W) \cdot b_1 \]
Where:
\(a_1\): Effect of X on M (baseline effect).
\(a_3\): Moderation of the X → M path by W.
\(b_1\): Effect of M on Y.
Let’s work through an example of moderated mediation using Hayes’ Model 7, where we examine whether the indirect effect of hours studied (X) on test score (Y) through motivation (M) is moderated by interest in the subject (W).
Data Simulation Example
# Simulating the dataset
set.seed(123)
n <- 200
X <- rnorm(n, mean = 10, sd = 2) # Hours studied (X)
W <- rnorm(n, mean = 5, sd = 1) # Interest in subject (Moderator, W)
M <- 0.5 * X + 0.3 * W + 0.2 * X * W + rnorm(n) # Motivation (Mediating variable)
Y <- 0.6 * M + 0.3 * X + rnorm(n) # Test score (Dependent variable)
data <- data.frame(X = X, W = W, M = M, Y = Y)
head(data, 10) X W M Y
1 8.879049 7.198810 19.30933 15.323324
2 9.539645 6.312413 17.53853 13.357665
3 13.117417 4.734855 19.76623 15.761632
4 10.141017 5.543194 17.94735 12.294647
5 10.258575 4.585660 16.58415 13.818448
6 13.430130 4.523753 18.57256 14.961842
7 10.921832 4.211397 15.57382 11.964096
8 7.469878 4.405383 12.39449 8.265634
9 8.626294 6.650907 17.24415 12.634614
10 9.108676 4.945972 15.27567 11.048945Running Moderated Mediation Analysis in R
We will use the moderndive package to estimate the moderated mediation model in R. In this example, the mediator is motivation, and the moderator is interest in the subject.
# 필요한 패키지 설치 및 로드
library(mediation)
# Step 1: Fit the model for the mediator (X → M)
med_model <- lm(M ~ X * W, data = data)
# Step 2: Fit the model for the outcome (Y ~ M + X * W)
out_model <- lm(Y ~ M + X, data = data)
# Step 3: Conduct the moderated mediation analysis
mod_med_analysis <- mediate(med_model,
out_model,
treat = "X",
mediator = "M",
boot = TRUE,
sims = 500)Running nonparametric bootstrapsummary(mod_med_analysis)
Causal Mediation Analysis
Nonparametric Bootstrap Confidence Intervals with the Percentile Method
Estimate 95% CI Lower 95% CI Upper p-value
ACME 0.940 0.824 1.04 <2e-16 ***
ADE 0.229 0.107 0.35 <2e-16 ***
Total Effect 1.169 1.065 1.26 <2e-16 ***
Prop. Mediated 0.804 0.711 0.90 <2e-16 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Sample Size Used: 200
Simulations: 500 Conditional mediation effect analysis
# W의 값 설정 (평균, 평균 ± 1표준편차)
W_levels <- c(mean(data$W) - sd(data$W),
mean(data$W),
mean(data$W) + sd(data$W))
W_levels[1] 4.046200 5.042121 6.038042# 기존 모델에서 계수 추출
med_coefs <- coef(med_model)
out_coefs <- coef(out_model)
# 간접효과 계산
indirect_effects <- sapply(W_levels, function(w) {
# 간접효과 = (beta_X + beta_X:W * W) * gamma_M
(med_coefs["X"] + med_coefs["X:W"] * w) * out_coefs["M"]
})
# 결과 출력
names(indirect_effects) <- c("Low W", "Mean W", "High W")
print(indirect_effects) Low W Mean W High W
0.8102705 0.9398953 1.0695201 # 결과 데이터프레임 생성
indirect_effects_df <- data.frame(
W_level = c("Low W", "Mean W", "High W"),
Indirect_Effect = indirect_effects
)
indirect_effects_df W_level Indirect_Effect
Low W Low W 0.8102705
Mean W Mean W 0.9398953
High W High W 1.0695201# 부트스트랩 샘플링 함수 정의
bootstrap_indirect_effect <- function(data, W_value, n_boot = 5000) {
boot_results <- replicate(n_boot, {
# 샘플 재추출
sample_data <- data[sample(1:nrow(data), replace = TRUE), ]
# 매개변수 모델
med_model <- lm(M ~ X * W, data = sample_data)
med_coefs <- coef(med_model)
# 결과 변수 모델
out_model <- lm(Y ~ M + X, data = sample_data)
out_coefs <- coef(out_model)
# 간접효과 계산
indirect_effect <- (med_coefs["X"] + med_coefs["X:W"] * W_value) * out_coefs["M"]
return(indirect_effect)
})
# 평균 및 신뢰구간 계산
mean_effect <- mean(boot_results)
ci_lower <- quantile(boot_results, 0.025)
ci_upper <- quantile(boot_results, 0.975)
return(c(mean_effect, ci_lower, ci_upper))
}
# W 값 설정
W_levels <- c(mean(data$W) - sd(data$W), mean(data$W), mean(data$W) + sd(data$W))
W_labels <- c("Low W", "Mean W", "High W")
# 각 W 수준에서 간접효과 계산
indirect_results <- t(sapply(W_levels, function(w) bootstrap_indirect_effect(data, w)))
colnames(indirect_results) <- c("Effect", "CI_Lower", "CI_Upper")
indirect_results <- data.frame(W_Level = W_labels, indirect_results)
indirect_results W_Level Effect CI_Lower CI_Upper
1 Low W 0.8105840 0.7040393 0.9224319
2 Mean W 0.9414754 0.8352290 1.0516849
3 High W 1.0718155 0.9442989 1.1969065library(ggplot2)
# 간접효과 시각화
ggplot(indirect_results, aes(x = W_Level, y = Effect)) +
geom_point(size = 3, color = "blue") + # Point 표시
geom_errorbar(aes(ymin = CI_Lower, ymax = CI_Upper), width = 0.2, color = "blue") + # Error bar 추가
labs(
title = "Conditional Indirect Effects by Moderator Levels (W)",
x = "Moderator Levels (W)",
y = "Indirect Effect"
) +
theme_minimal(base_size = 14) + # Minimal theme
theme(
axis.title = element_text(face = "bold"),
plot.title = element_text(face = "bold", hjust = 0.5)
)