This glossary collects the key concepts, methods and people in this lesson. It can be used as a reference while working through the material or as a review before assessments. Typing in the search box filters the entries.

Lesson 8 · Section 1

What Role Does the Third Variable Play?

A third variable can confound, mediate or collide, or it can modify an association, and each role leads to a different research question and a different analysis.

Directed acyclic graphs

Three DAG positions, and effect modification

Confounder

It causes both the exposure and the outcome: X ← C → Y.

Mediator

It lies on the pathway from the exposure to the outcome: X → M → Y.

Collider

It is caused by both the exposure and the outcome: X → Z ← Y.

Effect modification

Not a DAG position: the size of the X to Y association depends on the level of V.

From role to analysis

Each role leads to a different analysis

Role	Research question	What the analyst does
Confounder	Is X associated with Y once C is held constant?	Adjust for C.
Mediator	How much of the X to Y association runs through M?	Split the association into direct and indirect parts.
Collider	None: it is a consequence of X and Y.	Leave it out and do not select on it.
Effect modifier	Does the X to Y association differ across levels of V?	Fit an interaction or estimate within groups.

Effect decomposition

Total = direct + indirect

Decomposition in a linear model

\[ {\color{#0B7B6B}{c}} = {\color{#1D4ED8}{c'}} + {\color{#CC0033}{a \times b}} \]

c total effect of support on depressive symptoms c′ direct effect, with loneliness held constant a × b indirect effect through loneliness

Path a runs from support to loneliness, and path b runs from loneliness to depressive symptoms with support held constant.

Assumptions

Reading an indirect effect causally

• The exposure comes before the mediator, and the mediator comes before the outcome.
• There is no unmeasured confounding of the exposure and the outcome.
• There is no unmeasured confounding of the exposure and the mediator.
• There is no unmeasured confounding of the mediator and the outcome, which randomizing the exposure does not remove.
• No confounder of the mediator and the outcome is itself caused by the exposure.

Cross-sectional data

Statistical mediation, described carefully

Suitable wording

Loneliness statistically accounted for about 41% of the association between support and depressive symptoms.

Wording to avoid

Social support reduced depression by reducing loneliness.

All three measures come from the 2021 survey wave, so the order of the variables in time cannot be established from these data.

Running example

The CSCS 2021 data

Carry forward

What to take into the next section

• The role of a third variable is set by a DAG drawn before the analysis, and the role decides the analysis.
• A total effect equals a direct effect plus an indirect effect in a linear model fitted to the same people.
• A causal reading of an indirect effect needs temporal order and no unmeasured confounding, so the CSCS data support a statistical decomposition only.

Introduction and Overview

This is the last lesson of HSCI 410. It continues the loneliness thread from Lesson 7, Measurement and Psychometrics, which asked how well loneliness and related constructs are measured. This lesson asks how such measures relate to one another. Most models in the course so far have related one exposure to one outcome, with other variables added as adjustments. Many research questions are about a third variable in its own right: whether it explains part of an association, or whether an association is stronger in some groups than in others. These are questions of mediation and moderation, and the last section of the lesson shows how both can be written as a path model. This first section sets out the roles a third variable can play, defines the effects that mediation analysis estimates, and states the conditions under which those effects can be read causally.

The section begins with the practical problem that motivates the whole lesson and then turns to the causal diagram that classifies each variable in it.

The Running Case: Two Questions from a Community Connector Program

The methods in this lesson are easier to follow when they answer a concrete question. Box 8.1 describes a community connector program whose staff have two questions about social support, loneliness, depressive symptoms and age, and it names the survey that the analyst uses to explore them. The same case and the same survey are used in every section of the lesson.

The first question is a mediation question, because loneliness is proposed as a step on the pathway from support to depressive symptoms. The second is a moderation question, because age is proposed as a variable that changes the size of an association. The same three variables appear in both, yet the analyses differ, because the third variable plays a different role in each. Identifying that role is the first step of the analysis.

Three DAG Positions, and Effect Modification as a Separate Question

A causal diagram is the tool that identifies the role of a third variable. This part recalls the directed acyclic graph from Lesson 1, uses it to define the confounder, the mediator and the collider, and explains why effect modification is a separate question that the diagram does not encode. Box 8.2 recalls the elements of a DAG and poses a retrieval question about colliders.

With an exposure X and an outcome Y in place, a DAG distinguishes three positions for a third variable: confounder, mediator and collider. Effect modification is a separate question about the size of the X to Y effect across levels of a variable V. A standard DAG does not encode it, because an arrow states only that an effect exists, so it is shown by a note on the diagram or by drawing the diagram within each stratum of V. Figure 8.1 shows the three positions and, in panel D, effect modification drawn within strata.

The cards below describe the three positions and effect modification, the research question each implies and what it asks of the analysis.

Table 8.1 brings the four roles together in one place. For each role it gives the position in the DAG, the research question the role implies and what the analysis does with the variable.

Table 8.1. The four roles a third variable can play, the research question each implies and what the analysis does.

Two of these roles are easily confused in practice, because a mediator and a confounder look alike in the data. Box 8.3 explains how the direction of one arrow separates them.

Sort the roles

Each scenario below names an exposure, an outcome and a third variable. Choose the DAG position the third variable takes (confounder, mediator or collider), or choose effect modifier when the scenario is about the size of the association, which a DAG does not show. Feedback appears after each choice.

Interactive 8.1 gives practice in the distinctions drawn in Table 8.1 and Box 8.3. It presents eight scenarios, five built on the running case and three from other areas of public health.

The DAG has now fixed the role of each third variable in the running case. Loneliness is the proposed mediator of the association between support and depressive symptoms, and age is a proposed effect modifier of the association between support and loneliness. The next part takes the mediator and shows how the association it carries can be divided.

Total, Direct and Indirect Effects

Once loneliness has been identified as a proposed mediator, the association between support and depressive symptoms can be divided into parts. The total effect is the association between the exposure and the outcome when the mediator is ignored. The indirect effect is the part of the total that runs through the mediator. The direct effect is the part that remains when the mediator is held constant, which includes every pathway the model does not name. The word "effect" is the conventional statistical name for these quantities. In cross-sectional data such as the CSCS, each one is an association, and it describes a cause only under the assumptions set out later in this section.

Mediation analysis labels the paths with letters. Path a runs from the exposure to the mediator. Path b runs from the mediator to the outcome, with the exposure held constant. Path c is the total effect, and path c′ (read "c prime") is the direct effect.

Figure 8.2 draws these paths for the running case, with the total effect in the upper diagram and its division into direct and indirect parts in the lower one.

When the mediator and the outcome are continuous and every model is a linear regression fitted to the same people, the parts add up exactly.

Equation 8.1 expresses this decomposition in the path labels of Figure 8.2.

The causal-inference literature defines these quantities more generally with counterfactuals: the indirect effect is the change in the outcome that would follow if the mediator shifted from the value it would take at one level of the exposure to the value it would take at another, with the exposure itself held fixed (Robins and Greenland, 1992; Pearl, 2001). These definitions do not depend on a particular model, and they are the basis of the mediate() function used in Section 3. For linear models without interactions, they give the same numbers as a × b and c′.

The decomposition in Equation 8.1 is arithmetic, and it holds in any dataset in which the linear models are fitted to the same people. Whether the indirect part describes a causal pathway is a separate matter, which depends on the conditions set out in the next part.

Conditions for Reading an Indirect Effect Causally

An estimated indirect effect describes a causal pathway only if several conditions hold (VanderWeele, 2015). None of them can be confirmed from the data alone, which is why the DAG and the study design carry so much of the argument.

The accordion below lists the five conditions, with one panel for each condition.

Box 8.4 describes how careful mediation analyses respond to the fourth condition.

The first condition, temporal order, is a matter of study design, and it is a condition that the CSCS data cannot meet. The next part explains why, and what kind of conclusion the data can still support.

Cross-Sectional Data Support Statistical Mediation Only

The CSCS 2021 wave measured social support, loneliness and depressive symptoms in the same survey, at the same time. The first condition, temporal order, therefore cannot be established. Each arrow in the mediation diagram could plausibly run in the other direction. People with depressive symptoms may withdraw from others and feel lonelier, and they may also perceive the support around them as weaker than others would judge it. Maxwell and Cole (2007) showed that mediation estimates from cross-sectional data can differ substantially from the longitudinal processes they are meant to describe.

The analysis in this lesson is still informative. It shows whether the pattern of associations is consistent with the program's explanation, and how large the indirect association is. That information can justify a longitudinal evaluation of the program. It is reported as a statistical decomposition of an association, and causal language is kept for designs in which the conditions above are plausible.

Table 8.2 sets the two kinds of wording side by side, using the CSCS results that Section 3 estimates. The left column describes a statistical decomposition, and the right column claims a causal pathway that the data cannot show.

Table 8.2. Wording suited to a cross-sectional mediation result, set against wording that claims a causal pathway.

The analyses in this lesson are therefore reported as statistical decompositions of associations in the CSCS. The last part of this section loads those data in R and describes the three variables.

Meeting the Data

Activity 8.1 loads the CSCS data and prepares the variables with the same code as the narrated walkthrough Mediation and Moderation, so that every number in this lesson can be reproduced. Figure 8.3 shows the three scores for the 3,110 people with complete data. Support is concentrated between 4 and 6 on its 1 to 7 scale, loneliness spreads across its 3 to 9 range, and depressive symptoms are most often between 0 and 3 on the 0 to 6 PHQ-2 scale.

# Load the CSCS data from GitHub (this takes a few seconds)
github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
dim(data)                     # people (rows) and variables (columns)

# support:    social support (MSPSS), 1 to 7, higher = more support
# loneliness: UCLA 3-item Loneliness Scale, 3 to 9, higher = lonelier
# depression: PHQ-2 depressive symptoms, 0 to 6, higher = more symptoms
med_data <- data.frame(
  support    = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  loneliness = data$LONELY_ucla_loneliness_scale_score,
  depression = data$WELLNESS_phq_score,
  age        = data$DEMO_age)
summary(med_data)

med_data <- na.omit(med_data)   # keep people with no missing values
nrow(med_data)                  # how many people are in the analysis?
round(cor(med_data[, c("support", "loneliness", "depression")]), 2)

This section has identified the role of each third variable in the running case, defined the total, direct and indirect effects, stated the conditions for a causal reading, and prepared the 3,110 complete cases in R. Section 2 takes up the second question of the running case, whether the association between support and loneliness differs by age. The knowledge check that follows covers the material of this section.

Lesson 8 · Section 2

Moderation: When an Association Differs Across Groups

A moderation question asks whether the association between an exposure and an outcome changes across the levels of a third variable.

Moderation

Three different questions about age

Confounding

Is support associated with loneliness once age is held constant? Age is added as a covariate.

Moderation

Does the support slope differ between age groups? An interaction term is added.

Mediation

Does a third variable lie on the pathway from support to the outcome? The association is split.

The interaction model

One model, two lines

Moderation by age group

\[ \begin{aligned} \text{loneliness} = {} & {\color{#6D28D9}{\beta_0}} + {\color{#1D4ED8}{\beta_1}}\,\text{support} + {\color{#C2410C}{\beta_2}}\,\text{older} \\ & + {\color{#CC0033}{\beta_3}}\,(\text{support} \times \text{older}) \end{aligned} \]

β₁ support slope, under 50 β₂ group difference at support = 0 β₃ difference in slopes (older minus younger)

The support slope is β₁ for adults under 50 and β₁ + β₃ for adults aged 50 and over.

CSCS result

A steeper slope for adults aged 50 and over

Centring

Centring moves zero to the middle of the data

Coefficient	Uncentred	Centred	What the centred value describes
Support	−0.1685	−0.5776	Support slope at the mean age (39.5 years)
Age	0.0575	0.0065	Age slope at the mean support score (4.93)
Support × age	−0.0103	−0.0103	Change in the support slope per year of age

Simple slopes

The support slope at ages 30, 50 and 70

Two scales

Additive and multiplicative interaction

Group	Risk, unexposed	Risk, exposed	Risk difference	Risk ratio
Group A	10%	20%	10 points	2.0
Group B	20%	40%	20 points	2.0

The risk ratios are equal (no multiplicative interaction), and the risk differences differ (additive interaction).

Carry forward

What to take into the next section

• Moderation is tested with an interaction term, and adding a covariate does not test it.
• With an interaction in the model, main effects describe slopes at zero on the other variable, so centring and simple slopes make them readable.
• Moderation is reported with the interaction coefficient, the simple slopes and a plot of one line per group.

Introduction

The second question from the community connector program asks whether the association between social support and loneliness is the same for younger and older adults. This is a question of moderation. A moderator is a variable across whose levels the association between an exposure and an outcome differs in size or direction. Epidemiologists usually call the same idea effect modification, a term met in HSCI 341, and psychologists usually call it moderation. Lesson 3 showed the mechanics of adding an interaction term to a regression. This section treats moderation as a research question: what the interaction coefficient means, how to read the other coefficients when an interaction is present, and how to present the result.

The section first separates moderation from confounding and mediation. It then writes the interaction model, fits it to the CSCS data with age in two groups and as a continuous variable, and closes with the scale of an interaction and the reporting of a moderation result.

Moderation as a Research Question

Moderation asks for whom or under what conditions an association is stronger or weaker. Mediation, the topic of Section 3, asks why an association exists. The two can involve the same variables, but they are different questions with different models. A third question, confounding, asks whether an association remains once a common cause is held constant.

Table 8.3 poses one question of each type about age in the running case and shows the model that answers each one in R.

Table 8.3. Three questions about age in the running case, with the type of question, the model in R and what the model gives.

The models in the first two rows of Table 8.3 both contain age, and they differ by one term. Box 8.5 explains why adjusting for age leaves the moderation question untested.

A moderation question therefore needs a model in which the slope of support can change with age. The next part writes that model and explains each of its coefficients.

The Interaction Model

For the running case, the moderator is an indicator called older, equal to 1 for adults aged 50 and over and 0 for adults under 50 (the reference group). The interaction model adds the product of support and the indicator to a model that already contains both.

Equation 8.2 writes the model with four coefficients, and the explanation under it shows how the model reduces to one line for each age group.

In R, support * age_group is shorthand for support + age_group + support:age_group, so the main effects are included automatically. Table 8.4 gives the meaning of each coefficient.

Table 8.4. The four coefficients of Equation 8.2, their names in the R output and their meaning.

With the meaning of each coefficient fixed, the model can be fitted to the CSCS data. The next part does so with age in two groups.

Moderation by Age Group in the CSCS

Activity 8.2 creates the two age groups, fits the models without and with the interaction, compares them, and calculates the slope of support in each group. Of the 3,110 people, 2,275 are under 50 and 835 are aged 50 and over.

github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
med_data <- data.frame(
  support    = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  loneliness = data$LONELY_ucla_loneliness_scale_score,
  depression = data$WELLNESS_phq_score,
  age        = data$DEMO_age)
med_data <- na.omit(med_data)   # the same people in every model
# The moderator: two age groups, with "Under 50" as the reference
med_data$age_group <- NA
med_data$age_group[med_data$age < 50] <- "Under 50"
med_data$age_group[med_data$age >= 50] <- "50 and over"
med_data$age_group <- factor(med_data$age_group,
                             levels = c("Under 50", "50 and over"))
table(med_data$age_group)

# support * age_group is shorthand for
# support + age_group + support:age_group
model_main <- lm(loneliness ~ support + age_group, data = med_data)
model_int  <- lm(loneliness ~ support * age_group, data = med_data)
summary(model_int)

anova(model_main, model_int)   # does the interaction improve the model?

# Simple slopes: the slope of support in each age group
coef(model_int)["support"]                                     # Under 50
coef(model_int)["support"] + coef(model_int)["support:age_group50 and over"]   # 50 and over

# One fitted line for each age group
plot(jitter(med_data$support), jitter(med_data$loneliness),
     xlab = "Social support (1 to 7)", ylab = "UCLA loneliness score (3 to 9)",
     pch = 16, col = rgb(0, 0, 0, 0.10))
abline(lm(loneliness ~ support,
          data = med_data[med_data$age_group == "Under 50", ]),
       col = "steelblue", lwd = 3)
abline(lm(loneliness ~ support,
          data = med_data[med_data$age_group == "50 and over", ]),
       col = "firebrick", lwd = 3)
legend("topright", legend = c("Under 50", "50 and over"),
       col = c("steelblue", "firebrick"), lwd = 3)

Figure 8.4 shows the plot that the last chunk of code in Activity 8.2 draws. The two fitted lines make the interaction visible, because the line for adults aged 50 and over is the steeper of the two.

Reading the main effects when an interaction is present

Once an interaction term is in the model, each main effect describes a slope or a difference at the zero point of the other variable. The support coefficient (−0.50) is the support slope for adults under 50, and it applies to no one else. The age_group50 and over coefficient (1.83) is the difference between the age groups at a support score of 0. The support scale runs from 1 to 7, so a score of 0 is impossible, and this coefficient has no practical meaning on its own. The group difference changes with support: it equals 1.83 − 0.31 × support. At a support score of 1, adults aged 50 and over are predicted to be 1.52 points lonelier, at a score of 5 the difference shrinks to 0.27 points, and the lines cross near a score of 5.9. Reporting the 1.83 as "the effect of being older" would misread the model, which is one of the most common errors with interaction terms.

The four cards below define the terms used in reading an interaction model: the interaction coefficient, the simple slope, the main effect and the comparison of models.

The binary moderator has given one support slope for each age group. The next part keeps age in years and asks how the support slope changes across the whole age range.

Centring a Continuous Moderator

Splitting age at 50 is easy to explain, but it treats a 49-year-old and a 20-year-old as alike, and it discards information. Age can instead enter the model as a continuous moderator, in lm(loneliness ~ support * age). The difficulty is that the main effects still describe slopes at zero on the other variable. The support coefficient becomes the slope of support for a person aged 0, and the age coefficient becomes the slope of age for a person with a support score of 0. Neither person exists in the data.

Centring solves this. A variable is centred by subtracting a chosen value from every observation, most often the mean, so that 0 on the centred variable means "average" (Aiken and West, 1991). In the CSCS sample the mean age is 39.54 years and the mean support score is 4.93. After centring, the support_c coefficient is the support slope at the mean age, and the age_c coefficient is the age slope at the mean support score. Centring at another meaningful value, such as age 65, is also legitimate, as long as the report says which value was used.

Activity 8.3 fits the uncentred and centred models to the CSCS data and computes the simple slopes of support at ages 30, 50 and 70.

github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
med_data <- data.frame(
  support    = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  loneliness = data$LONELY_ucla_loneliness_scale_score,
  depression = data$WELLNESS_phq_score,
  age        = data$DEMO_age)
med_data <- na.omit(med_data)   # the same people in every model
# Centre each continuous variable by subtracting its mean
med_data$age_c     <- med_data$age - mean(med_data$age)
med_data$support_c <- med_data$support - mean(med_data$support)
round(c(mean_age = mean(med_data$age), mean_support = mean(med_data$support)), 2)

model_raw <- lm(loneliness ~ support * age, data = med_data)       # uncentred
model_cen <- lm(loneliness ~ support_c * age_c, data = med_data)   # centred
round(summary(model_raw)$coefficients[, 1:3], 4)   # estimate, SE, t value
round(summary(model_cen)$coefficients[, 1:3], 4)

# Simple slopes of support at ages 30, 50 and 70
b <- coef(model_cen)
ages <- c(30, 50, 70)
slopes <- b["support_c"] + b["support_c:age_c"] * (ages - mean(med_data$age))
names(slopes) <- paste("age", ages)
round(slopes, 3)

Table 8.5 places the coefficients of the two models side by side and gives the meaning of each one in the centred model.

Table 8.5. Coefficients of the uncentred and centred interaction models of loneliness on support and age, from Activity 8.3.

The interaction coefficient and its t value are identical in the two models. Centring changes the zero point of each variable, and therefore what the main effects describe, but it leaves the interaction, the fitted values and the overall fit of the model unchanged. Centring also lowers the correlation between a main effect and its product term, but the support slope at any given age has the same estimate and standard error in both models, and the test of the interaction is unchanged.

Simple slopes for a continuous moderator

With a continuous moderator there is a different support slope at every age. The simple slope at a chosen age comes straight from the centred coefficients.

Equation 8.3 gives the simple slope at any age A, using the coefficients of the centred model in Table 8.5.

Interactive 8.2 applies Equation 8.3 to any age between 20 and 80. Moving the slider shows how the slope of support becomes steeper with age.

Figure 8.5 draws the predicted lines at ages 30, 50 and 70, the three ages whose simple slopes Activity 8.3 prints.

Simple slopes should be computed at values of the moderator that are well represented in the data. Common choices are the mean and one standard deviation above and below it, or meaningful values such as ages 30, 50 and 70. A plot with one line per chosen value is usually the clearest way to present a moderation result.

The interactions so far compare slopes of a continuous outcome, which are differences. The next part sets this additive scale beside the multiplicative scale of a logistic regression.

Additive and Multiplicative Scales

Whether an interaction exists depends on the scale on which the association is measured. In a linear regression of a continuous outcome, the interaction compares slopes, which are differences, so it is an interaction on the additive scale. In a logistic regression, the interaction term compares odds ratios, so it is an interaction on the multiplicative scale. The same data can show an interaction on one scale and none on the other, as the hypothetical example in Table 8.6 shows.

Table 8.6. A hypothetical exposure with the same risk ratio in two groups and different risk differences.

The risk ratio is 2.0 in both groups, so there is no interaction on the multiplicative scale. The risk difference is twice as large in group B, so there is an interaction on the additive scale. For public health decisions the additive scale is often the more relevant one, because it shows in which group an intervention would prevent more cases. For binary outcomes, reports should present the association within each group and state which scale any interaction refers to (Knol and VanderWeele, 2012; VanderWeele and Knol, 2014). The outcomes in this lesson are continuous scores, so every interaction here is on the additive scale.

The remaining step is to report the CSCS moderation result, which the last part of the section describes.

Reporting a Moderation Result

A moderation result is reported with the interaction coefficient and its confidence interval or p-value, the simple slopes, and a plot with one line per group or per chosen value of the moderator. The language describes associations. A suitable sentence for the CSCS result is: "The association between social support and loneliness differed by age group (interaction coefficient −0.31, p < 0.001). Each additional point of support was associated with a loneliness score 0.50 points lower among adults under 50 and 0.81 points lower among adults aged 50 and over." Refitting the model with "50 and over" as the reference group gives that group's slope with its own confidence interval, which can be added to the report.

The narrated walkthrough in the red card below offers a second route through the steps of Activity 8.2.

This section has shown that the association between social support and loneliness is stronger at older ages, whether age is split at 50 or treated as a continuous moderator, and it has set out how such a result is reported on a stated scale. Section 3 turns to the first question of the running case, whether loneliness carries part of the association between support and depressive symptoms. The knowledge check that follows covers the material of this section.

Lesson 8 · Section 3

Mediation: Direct and Indirect Effects

The association between support and depressive symptoms is split into a part that runs through loneliness and a part that does not.

Three regressions

Estimating the paths one at a time

Model in R	Path	Estimate (95% CI)
lm(depression ~ support)	c, total	−0.560 (−0.608 to −0.511)
lm(loneliness ~ support)	a	−0.626 (−0.676 to −0.576)
lm(depression ~ support + loneliness)	b	0.366 (0.334 to 0.397)
	c′, direct	−0.331 (−0.380 to −0.282)

Indirect effect

Two methods, one answer

Product of coefficients and difference method

\[ \begin{gathered} {\color{#CC0033}{a \times b}} = (-0.626)(0.366) = -0.229 \\ {\color{#0B7B6B}{c}} - {\color{#1D4ED8}{c'}} = -0.560 - (-0.331) = -0.229 \end{gathered} \]

a × b product of coefficients c − c′ difference method

41%

of the total association statistically accounted for by loneliness (0.229 ÷ 0.560)

Baron and Kenny

Why the causal steps are no longer recommended

The four steps

X predicts Y; X predicts M; M predicts Y with X held constant; the X coefficient shrinks.

The problems

The steps never test a × b, they have low power, they miss inconsistent mediation, and their full and partial labels depend on sample size.

mediate()

Bootstrap estimates of the effects

Quantity	Estimate	95% bootstrap CI
ACME (indirect)	−0.229	−0.256 to −0.201
ADE (direct)	−0.331	−0.386 to −0.279
Total effect	−0.560	−0.618 to −0.509
Proportion mediated	0.41	0.36 to 0.46

Caution and sensitivity

Proportion mediated and the sensitivity parameter

Reporting

Reporting without causal overstatement

• The report states the DAG, the covariates in each model and the assumptions a causal reading would need.
• It gives the indirect, direct and total effects with bootstrap confidence intervals and the number of resamples.
• It reports the sensitivity analysis and the limits of a cross-sectional design.

Loneliness statistically accounted for about 41% of the association between social support and depressive symptoms (indirect association −0.23, 95% CI −0.26 to −0.20).Suitable wording for the CSCS result

Carry forward

What to take into the next section

• The indirect effect is a × b, which equals c − c′ in linear models fitted to the same people.
• The indirect effect is tested directly with a bootstrap confidence interval.
• A cross-sectional mediation result is reported as a statistical decomposition, with a sensitivity analysis.

Introduction

Section 1 defined the total, direct and indirect effects and the conditions under which an indirect effect can be read causally. This section estimates them for the program's first question: is social support associated with fewer depressive symptoms partly because people with more support are less lonely? The analysis follows the history of the method. It first estimates each path with an ordinary linear regression, as students did in Lesson 3, then explains why the classic four-step approach to testing mediation has been replaced, and finally estimates the indirect effect with a bootstrap confidence interval and a sensitivity analysis using the mediation package (Tingley et al., 2014).

The section builds on the preview of mediation in Lesson 1. Box 8.6 recalls that example and its output, and it poses a retrieval question about the quantity that mediate() reports.

The first step is to estimate the paths of the mediation diagram with three regressions, which the next part does.

Three Regressions, Four Paths

With support as the exposure (X), loneliness as the mediator (M) and depressive symptoms as the outcome (Y), three linear regressions give every path in the mediation diagram. The outcome model includes both the exposure and the mediator, so that path b describes the association between loneliness and depressive symptoms among people with the same level of support.

Table 8.7 lists the three models, the R code for each, the coefficient that is used from it and the path that the coefficient estimates.

Table 8.7. The three regressions of a simple mediation analysis and the paths they estimate.

The models in this lesson have no covariates, so that the numbers match the narrated walkthrough. In a manuscript, the confounders identified in the DAG (for example age, gender and income) would be added to the mediator model and to the outcome model, so that paths a and b are adjusted for the common causes of the exposure and mediator and of the mediator and outcome.

Activity 8.4 fits the three models to the CSCS data and computes the indirect effect by both of the methods described in the next part.

github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
med_data <- data.frame(
  support    = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  loneliness = data$LONELY_ucla_loneliness_scale_score,
  depression = data$WELLNESS_phq_score,
  age        = data$DEMO_age)
med_data <- na.omit(med_data)   # the same people in every model
nrow(med_data)

# Path c, the total effect: support -> depression
model_total <- lm(depression ~ support, data = med_data)
round(cbind(estimate = coef(model_total), confint(model_total)), 3)   # with 95% CIs

# Path a: support -> loneliness
model_a <- lm(loneliness ~ support, data = med_data)
round(cbind(estimate = coef(model_a), confint(model_a)), 3)   # with 95% CIs

# Path b (loneliness -> depression) and path c' (the direct effect)
model_b <- lm(depression ~ support + loneliness, data = med_data)
round(cbind(estimate = coef(model_b), confint(model_b)), 3)   # with 95% CIs

a <- coef(model_a)["support"]
b <- coef(model_b)["loneliness"]
c_total  <- coef(model_total)["support"]
c_direct <- coef(model_b)["support"]
a * b                  # indirect effect: product of coefficients
c_total - c_direct     # indirect effect: difference method

Figure 8.6 places the estimates from Activity 8.4 on the mediation diagram of Figure 8.2.

The three regressions give every path, and the indirect effect follows from them. The next part sets out the two ways of calculating it.

Product-of-Coefficients and Difference Methods

The indirect effect can be calculated in two ways. The product-of-coefficients method multiplies path a by path b. The difference method subtracts the direct effect from the total effect.

Equation 8.4 applies both methods to the CSCS estimates from Activity 8.4.

The product-of-coefficients method has an intuitive reading. A support score one point higher goes with a loneliness score 0.626 points lower (path a), and each point of loneliness goes with 0.366 more PHQ-2 points among people with the same support (path b), so the pathway through loneliness corresponds to 0.626 × 0.366 = 0.229 fewer PHQ-2 points. The two methods agree exactly only when the mediator and outcome models are linear regressions fitted to the same people. When the outcome is binary and modelled with logistic regression, c − c′ and a × b can differ, because odds ratios change when a variable is added to the model even without confounding. The counterfactual approach used by mediate() handles such models consistently (Imai, Keele and Tingley, 2010; VanderWeele, 2015).

Figure 8.7 shows the result of the decomposition as two bars, with the total association above and its direct and indirect parts below.

The product a × b is the quantity of interest in a mediation analysis. The older causal steps, which the next part describes, never test this product.

The Baron and Kenny Causal Steps

For about two decades, most published mediation analyses followed the causal steps of Baron and Kenny (1986). Mediation was claimed when four conditions were met: the exposure predicted the outcome (path c), the exposure predicted the mediator (path a), the mediator predicted the outcome with the exposure held constant (path b), and the exposure coefficient shrank when the mediator was added (c′ smaller than c). If c′ fell to zero, the result was called full mediation, and otherwise partial mediation. The steps are still described in many textbooks, and students will meet them in published papers. Students who took HSCI 341 met these steps in Lesson 1, Section 2 of that course. Methodologists no longer recommend them as a test of mediation, for the reasons below (MacKinnon et al., 2002; Hayes, 2009; Zhao, Lynch and Chen, 2010).

The accordion below gives these reasons, one in each of its five panels.

An early direct test of a × b, the Sobel test, divides the product by an approximate standard error and compares the result with a normal distribution. The sampling distribution of a product of two estimates is usually skewed, so a symmetric normal interval can be misleading. The bootstrap avoids this assumption (Preacher and Hayes, 2008). It draws many new samples of the same size from the data with replacement, refits the models in each one, and takes the 2.5th and 97.5th percentiles of the resulting indirect effects as the 95% confidence interval.

The next part estimates the indirect effect for the running case with a bootstrap confidence interval, using the mediate() function.

Estimating the Indirect Effect with mediate()

The mediate() function in the mediation package takes the mediator model and the outcome model, names the exposure (treat) and the mediator, and returns four quantities. Their names come from the counterfactual definitions in Section 1, which is why the word "causal" appears in them.

The four cards below define the four quantities and give the path, or combination of paths, to which each corresponds in a linear model.

Activity 8.5 estimates the four quantities for the running case with 1,000 bootstrap samples and then runs the sensitivity analysis described in Box 8.4.

# install.packages("mediation")   # run once, if not yet installed
library(mediation)
github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
med_data <- data.frame(
  support    = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  loneliness = data$LONELY_ucla_loneliness_scale_score,
  depression = data$WELLNESS_phq_score,
  age        = data$DEMO_age)
med_data <- na.omit(med_data)   # the same people in every model
model_a <- lm(loneliness ~ support, data = med_data)                # path a
model_b <- lm(depression ~ support + loneliness, data = med_data)   # paths b and c-prime

set.seed(2021)   # makes the bootstrap results repeatable
med_result <- mediate(model_a, model_b,
                      treat = "support", mediator = "loneliness",
                      boot = TRUE, sims = 1000)
summary(med_result)

plot(med_result, xlim = c(-0.7, 0))   # the three effects with 95% CIs

# Sensitivity analysis: how strong would unmeasured confounding of
# loneliness and depression have to be to remove the indirect effect?
sens <- medsens(med_result, rho.by = 0.1, effect.type = "indirect", sims = 100)
summary(sens)

Figure 8.8 shows the plot drawn in Activity 8.5, in which the indirect, direct and total effects each appear with a 95% bootstrap confidence interval.

The proportion mediated, read with caution

The proportion mediated is 0.41 (95% CI 0.36 to 0.46): loneliness statistically accounts for about 41% of the association between support and depressive symptoms. The figure is easy to communicate, and it is also easy to misuse. Because it is a ratio of two estimates, it becomes unstable when the total effect is small, and its interval can then be very wide. When the direct and indirect effects have opposite signs, it can exceed 1 or fall below 0, values that have no sensible reading as a share. In the CSCS the total effect is large and precisely estimated and both parts have the same sign, so the proportion is stable. It still describes how an association divides in these data, and it does not say that 41% of any benefit of support on mental health works through loneliness.

The estimates from mediate() rest on the assumption of no unmeasured mediator-outcome confounding. The next part examines how far the conclusion depends on that assumption.

Sensitivity Analysis for Unmeasured Mediator-Outcome Confounding

Section 1 showed that a causal reading of the indirect effect requires no unmeasured confounding of the mediator and the outcome, a condition that cannot be checked from the data. A history of depression, for example, could make a person both lonelier and more likely to report depressive symptoms now. The medsens() function asks how strong such confounding would need to be to change the conclusion (Imai, Keele and Tingley, 2010). Unmeasured confounding of loneliness and depressive symptoms would make the residuals of the mediator model and of the outcome model correlated. The sensitivity parameter ρ (rho) is that correlation. The analysis assumes ρ = 0, and medsens() recalculates the indirect effect for a range of other values.

Figure 8.9 plots the indirect effect that medsens() recalculates across values of ρ, together with its confidence band.

The indirect effect falls to zero when ρ is about 0.4. An equivalent reading uses the second line of the summary: an unmeasured confounder would need to explain about 40% of the remaining variance in both loneliness and depressive symptoms (0.4 × 0.4 = 0.16) to remove the indirect effect. That is a strong confounder, but a variable such as a history of depression could plausibly be one, so the result shows that the indirect association is reasonably resistant to this kind of confounding without ruling it out. The sensitivity analysis addresses only mediator-outcome confounding. It says nothing about the order of the variables in time, which remains the main limitation of the CSCS data.

With the estimates and their sensitivity to confounding in hand, the remaining task is to report the analysis, which the last part of the section describes.

Reporting a Mediation Analysis

The AGReMA statement (Lee et al., 2021) gives a checklist for reporting mediation analyses of trials and observational studies. Its main elements can be embedded in a few sentences: the assumed causal structure (a DAG), the timing of each measurement, the covariates in the mediator and outcome models, the estimation method and number of bootstrap samples, the indirect, direct and total effects with confidence intervals, a sensitivity analysis, and the limitations of the design.

Box 8.7 shows how several of these elements read in a results paragraph for the CSCS analysis.

The narrated walkthrough in the red card below offers a second route through the regressions of Activity 8.4 and the mediate() analysis of Activity 8.5.

This section has estimated the indirect association of social support with depressive symptoms through loneliness, −0.23 PHQ-2 points per point of support or about 41% of the total, and it has shown that this association would fall to zero if mediator-outcome confounding produced a residual correlation of about 0.4. Section 4 writes the same model as a path model and extends it to a second outcome. The knowledge check that follows covers the material of this section.

Introduction

Sections 2 and 3 fitted one regression at a time. A path model writes all of the regressions in an analysis as one diagram and estimates them together. This has three advantages. Several outcomes and several pathways can be handled at once, the indirect effects can be defined and tested inside the model, and a whole model can be compared with a simpler one that leaves some paths out. This section expresses the mediation model from Section 3 as a path diagram, fits it with the lavaan package (Rosseel, 2012), extends it to two outcomes, compares two versions, and reports the chosen model. It closes by adding a latent variable, which turns the path model into a structural equation model (SEM) and connects this lesson to the confirmatory factor analysis in Lesson 7.

The section starts with the symbols of a path diagram and the lavaan syntax that turns a diagram into a model.

Path Diagrams

Path analysis was introduced by the geneticist Sewall Wright (1921) as a way of tracing correlations through chains of causes. A path diagram uses a small set of symbols, and the same symbols are used in SEM.

Table 8.8 lists the symbols with their meaning and an example of each from this section.

Table 8.8. The symbols of a path diagram.

A variable with no arrows pointing at it is exogenous: its causes lie outside the model, and in this section support is the only exogenous variable. A variable with at least one arrow pointing at it is endogenous, and the model has an equation for it. Loneliness, depressive symptoms and anxiety symptoms are endogenous. Each single-headed arrow in the diagram is a coefficient in one of those equations, so a path model with three endogenous variables is a set of three regressions fitted together.

To estimate these regressions together, the diagram has to be written as a model that R can fit, which is the subject of the next part.

Writing a Path Model in lavaan

The lavaan package describes a model as a block of text inside single quotes, with one equation or relationship on each line. Four operators cover everything in this lesson.

Table 8.9 sets out the four operators, how each is read, an example of each and the element of the path diagram that each one produces.

Table 8.9. The four lavaan operators used in this lesson.

A label is written in front of a predictor with an asterisk, as in loneliness ~ a * support, which names the path from support to loneliness a. Labels make it possible to define the indirect effect (a * b) and the total effect (c + a * b) inside the model, and lavaan then reports their estimates and standard errors with the other parameters. The model is fitted with sem(model, data = ...).

The simple mediation model in lavaan

The first activity rebuilds the mediation model from Section 3 in lavaan. The path model uses people with complete data on four variables (support, loneliness, depressive symptoms and anxiety symptoms), because the second activity adds anxiety as a second outcome. This leaves 3,099 people, eleven fewer than in Sections 2 and 3, so the estimates differ slightly in the third decimal place.

Activity 8.6 fits the model and defines the indirect and total effects from its labelled paths.

# install.packages("lavaan")   # run once, if not yet installed
library(lavaan)
github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
path_data <- data.frame(
  support    = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  loneliness = data$LONELY_ucla_loneliness_scale_score,
  depression = data$WELLNESS_phq_score,
  anxiety    = data$WELLNESS_gad_score)   # GAD-2 anxiety symptoms, 0 to 6
path_data <- na.omit(path_data)          # people with all four variables
nrow(path_data)

model_med <- '
  loneliness ~ a * support
  depression ~ b * loneliness + c * support
  indirect := a * b
  total    := c + a * b
'
fit_med <- sem(model_med, data = path_data)
summary(fit_med)

The simple mediation model has zero degrees of freedom. It estimates five parameters (three paths and two residual variances), and the data supply exactly five pieces of information about the relationships among the three variables, once the variance of support is set aside. A model like this is called saturated or just-identified. It reproduces the observed covariances exactly, so its chi-square is 0 and its fit cannot be tested. Fit becomes testable only when a model leaves some paths out, which is what the two-outcome models do.

A Path Model with Two Outcomes

The CSCS also includes the two-item Generalized Anxiety Disorder scale (GAD-2; Kroenke et al., 2007), scored from 0 to 6. The analyst asks whether loneliness links support to both depressive and anxiety symptoms, and compares two models.

Figure 8.10 draws the two models in one diagram, with the direct paths that only Model 2 contains shown as dashed arrows.

Model 1 contains no direct paths from support to the outcomes. It corresponds to what the Baron and Kenny tradition calls full mediation. Model 2 adds the direct paths and labels every path so that the two indirect effects can be defined. Both models include depression ~~ anxiety, which allows the parts of depressive and anxiety symptoms that loneliness and support leave unexplained to be correlated. Leaving this line out would assume that nothing else links the two outcomes, which is implausible for two measures of distress.

Activity 8.7 fits both models, compares them and reports the chosen model with bootstrap confidence intervals, standardized coefficients and R-squared values.

# Model 1, full mediation: no direct paths from support
model_full <- '
  loneliness ~ support
  depression ~ loneliness
  anxiety    ~ loneliness
  depression ~~ anxiety
'
fit_full <- sem(model_full, data = path_data)
fitMeasures(fit_full, c("chisq", "df", "pvalue", "cfi", "tli", "rmsea", "srmr"))

# Model 2, partial mediation: adds a direct path to each outcome
model_partial <- '
  loneliness ~ a * support
  depression ~ b1 * loneliness + c1 * support
  anxiety    ~ b2 * loneliness + c2 * support
  depression ~~ anxiety
  indirect_dep := a * b1
  indirect_anx := a * b2
'
fit_partial <- sem(model_partial, data = path_data)
fitMeasures(fit_partial, c("chisq", "df", "cfi", "rmsea", "srmr"))

anova(fit_full, fit_partial)   # chi-square difference test, AIC and BIC

set.seed(2021)
fit_boot <- sem(model_partial, data = path_data,
                se = "bootstrap", bootstrap = 1000)
est <- parameterEstimates(fit_boot, boot.ci.type = "perc")
est[est$op %in% c("~", ":="), c("lhs", "op", "rhs", "est", "ci.lower", "ci.upper")]

std <- standardizedSolution(fit_boot)
std[std$lhs != std$rhs, c("lhs", "op", "rhs", "est.std")]   # paths, covariance, indirect
lavInspect(fit_boot, "rsquare")   # variance explained in each outcome

Reading the fit indices

Fit indices describe how well a model with some paths left out reproduces the observed covariances. The guides for good fit in Table 8.10 follow Hu and Bentler (1999), and the description of an RMSEA above 0.10 as poor follows Browne and Cudeck (1993). All of these guides are rules of thumb, and they work best when several indices are read together (Kline, 2016).

Table 8.10. Common fit indices, what each measures, the usual guide for good fit and the value for Model 1.

Model 1 fits poorly. Its RMSEA of 0.166 is well above 0.10, its TLI is 0.827 and its CFI is just below 0.95. Only the SRMR falls inside its guide. Model 2 has 0 degrees of freedom, so it is saturated, and its perfect fit indices (CFI 1, RMSEA 0, SRMR 0) carry no information about whether it is correct.

Comparing nested models

Model 1 is nested in Model 2: fixing the two direct paths of Model 2 at zero gives Model 1. Nested models fitted to the same people can be compared with a chi-square difference test, which anova(fit_full, fit_partial) reports. The difference is 173.76 on 2 degrees of freedom (one for each path that Model 1 removes), with p < 2.2 × 10⁻¹⁶, so removing the direct paths makes the fit much worse. The Akaike information criterion (AIC), which Lesson 4 used to compare count models, agrees: it is 32,864 for Model 2 and 33,033 for Model 1, a difference of about 170 points in favour of Model 2. The BIC (32,918 and 33,076) points the same way.

In words, the data are inconsistent with a model in which support is related to depressive and anxiety symptoms only through loneliness. Section 3 explained that labels such as "full" and "partial" mediation depend mainly on sample size. With more than 3,000 people, a model with no direct paths was always likely to be rejected, so the report describes the size of each direct and indirect path with its confidence interval and does not rest on a label.

Results from the chosen model

The second activity refits Model 2 with 1,000 bootstrap samples and reports percentile confidence intervals for every path and for the two indirect effects. It then reports the standardized coefficients, which express each path in standard deviation units, so that paths measured on different scales can be compared, and the R-squared for each endogenous variable, the proportion of its variance that the model explains.

Table 8.11 collects these results from Activity 8.7 for every path and for the two indirect effects.

Table 8.11. Unstandardized estimates with 95% bootstrap confidence intervals and standardized estimates from Model 2 (n = 3,099).

Both indirect associations are clearly different from zero. The indirect association for depressive symptoms (−0.229) matches the ACME from Section 3, and the indirect association for anxiety symptoms is similar in size (−0.217). The model explains 16% of the variance in loneliness, 26% in depressive symptoms and 20% in anxiety symptoms. The residual correlation of 0.47 shows that depressive and anxiety symptoms remain strongly related after support and loneliness are taken into account, which justifies the covariance in the model.

The comparison favours Model 2, and Table 8.11 gives its estimates. The next part shows how such a model is presented in a report.

Reporting a Path Model

A path model is usually reported with a figure that shows the standardized coefficients, a table of unstandardized estimates with confidence intervals, the fit indices of each model compared, the indirect effects with bootstrap intervals, and the software, estimator and handling of missing data. Figure 8.11 is a reporting figure for Model 2.

A matching results sentence is: "In a path model fitted to 3,099 CSCS participants, the indirect association of social support with depressive symptoms through loneliness was −0.23 (95% bootstrap CI −0.26 to −0.20), and with anxiety symptoms it was −0.22 (95% CI −0.25 to −0.19). A model without direct paths from support to the outcomes fitted poorly (RMSEA 0.17, CFI 0.94) and significantly worse than a model with them (chi-square difference 173.8 on 2 df, p < 0.001; AIC 33,033 compared with 32,864)." The sentence describes associations, as every result from these cross-sectional data must.

The last part of the section asks what changes when one of the observed scores in a path model is replaced by a latent variable.

From Path Analysis to Structural Equation Modelling

Every variable in the path models so far is an observed score, and every score contains measurement error. Lesson 7 showed that measurement error weakens (attenuates) correlations, and that a latent variable measured by several items separates the shared variation in the items from their error. A structural equation model joins the two ideas. Its measurement model is a confirmatory factor analysis, which defines each latent variable by its items with the =~ operator, and its structural model is a set of paths among the latent and observed variables, written with ~ exactly as in this section.

Figure 8.12 shows such a model for the CSCS data, with a latent loneliness factor measured by the three UCLA items in place of the observed loneliness score.

Worked code 8.1 gives the lavaan code and output for the model in Figure 8.12, and the paragraph after it compares the estimates with those of the path model.

library(lavaan)
github_url <- "https://raw.githubusercontent.com/jorgeandr3s/heal/main/cscs/public_data/CSCS2025_full_cleaned_deidentified_data_and_metadata.RData"
load(url(github_url))                                 # creates a data frame called data
data <- data[data$SURVEY_collection_year == 2021, ]   # keep the 2021 wave
sem_data <- data.frame(
  companionship = data$LONELY_ucla_loneliness_scale_companionship_num,
  left_out      = data$LONELY_ucla_loneliness_scale_left_out_num,
  isolated      = data$LONELY_ucla_loneliness_scale_isolated_num,
  support       = data$PSYCH_zimet_multidimensional_social_support_scale_score,
  depression    = data$WELLNESS_phq_score)
sem_data <- na.omit(sem_data)
nrow(sem_data)

model_sem <- '
  # measurement model: a latent loneliness factor with three items
  Lonely =~ companionship + left_out + isolated
  # structural model: the paths between support, Lonely and depression
  Lonely     ~ a * support
  depression ~ b * Lonely + c * support
  indirect := a * b
'
fit_sem <- sem(model_sem, data = sem_data)
fitMeasures(fit_sem, c("chisq", "df", "cfi", "rmsea", "srmr"))
std <- standardizedSolution(fit_sem)
std[std$op %in% c("=~", "~", ":="), c("lhs", "op", "rhs", "est.std", "pvalue")]

The structural paths through loneliness are stronger in the SEM than in the path model with the observed score: −0.46 compared with −0.40 for support to loneliness, and 0.47 compared with 0.38 for loneliness to depressive symptoms. The standardized indirect effect rises from −0.15 to −0.21, and the direct path from support falls from −0.22 to −0.16. The latent variable removes the attenuation caused by measurement error in the three-item score, so more of the association is attributed to the pathway through loneliness. SEM also brings costs. It needs several good items for each latent variable, larger samples and more decisions about the model, and the latent factor is only as meaningful as the measurement model that defines it, which is why Lesson 7 placed so much weight on evidence of validity. The cross-sectional design limits the SEM in the same way as every other model in this lesson.

The narrated walkthrough in the red card below offers a second route through Activities 8.6 and 8.7.

Box 8.8 restates the main steps of the lesson as a sequence to follow when a new research question involves a possible mediator or moderator.

This section has written the mediation model as a path model, extended it to two outcomes, compared nested models, reported the chosen model and added a latent variable to form a structural equation model. The knowledge check that follows covers the material of this section, and the lesson closes with the key takeaways, a reflection and the final assessment.

Bringing It All Together

This lesson took two questions from a community connector program and answered them with the 2021 wave of the Canadian Social Connection Survey. Section 1 used directed acyclic graphs to separate three positions a third variable can take, and it treated effect modification as a separate question that a standard DAG does not encode. A confounder is adjusted for, a mediator is used to split an association into direct and indirect parts, a collider is left out of the model and the sampling, and an effect modifier is examined with an interaction. The section defined the total, direct and indirect effects and set out the conditions for a causal reading of an indirect effect: temporal order and no unmeasured confounding, including confounding of the mediator and the outcome. Because the survey measured every variable at the same time, every mediation result in the lesson was described as a statistical decomposition.

Section 2 found that the association between social support and loneliness was stronger among adults aged 50 and over (slope −0.81) than among younger adults (slope −0.50), and showed how centring and simple slopes make an interaction model readable. Section 3 estimated the paths of the mediation model one regression at a time, showed that a × b and c − c′ give the same indirect effect (−0.229), explained why the Baron and Kenny steps are no longer used as a test, and used mediate() to obtain a bootstrap interval: loneliness statistically accounted for about 41% of the association between support and depressive symptoms (95% CI 36% to 46%), and a sensitivity analysis showed that confounding inducing a residual correlation of about 0.4 would remove it. Section 4 wrote the model in lavaan, extended it to depressive and anxiety symptoms, preferred a model with direct paths on the chi-square difference test and the AIC, reported it with bootstrap intervals and standardized coefficients, and added a latent loneliness factor to show a structural equation model.

The same steps apply to any question about a third variable: draw the DAG and decide the role, fit the model that the role calls for, estimate effects with confidence intervals, check the assumptions that the data cannot confirm, and write the results in language that matches the design. The final assessment asks for these steps to be applied to new examples.

The final assessment covers all four sections. All 15 questions must be answered correctly (100%), and the final reflection completed, to finish the lesson.

Final Knowledge Assessment