Understanding Interaction Terms in Regression

Author

Heeyoung Lee

Published

April 16, 2026

Understanding Interaction Terms in Regression

Suppose you are studying the relationship between education and self-rated health (SRH). You find that education predicts better health — not surprising. But now ask: does the effect of education on health differ by nativity? Or by age? That is, does education matter more for some groups than others?

This is what interaction terms are designed to test. An interaction term captures the idea that the effect of one variable depends on the value of another. Without interactions, regression assumes every predictor has a constant, additive effect regardless of the other variables in the model. With interactions, you relax that assumption and let the effect of $X_1$ vary as a function of $X_2$.

This post walks through three core types of interactions in linear regression:

Type 1 — Continuous × Continuous: Does the effect of income on health vary by age?
Type 2 — Categorical × Continuous: Does the effect of education on health differ by nativity (immigrant vs. U.S.-born)?
Type 3 — Categorical × Categorical: Does the combination of nativity and insurance status predict health beyond their individual effects?

Each section includes the regression formula, full algebraic derivation, a numeric walkthrough, and R code for simulation, tables, and visualization.

Setup: Shared Packages

Code

library(dplyr)
library(ggplot2)
library(tidyr)
library(marginaleffects)
library(stargazer)
library(kableExtra)
library(gridExtra)

set.seed(42)

1. Continuous × Continuous Interaction

1.1 The Research Question

Does the effect of household income on self-rated health (SRH) vary by age?

We might expect that income matters more for health at older ages — younger adults can recover from deprivation more easily, while older adults rely more heavily on income to access healthcare, maintain housing stability, and manage chronic conditions.

1.2 The Regression Model

The baseline (no-interaction) model assumes income has a constant effect on SRH regardless of age:

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Income}_i + \beta_2 \text{Age}_i + \epsilon_i\]

But if the effect of income depends on age, we need a product term:

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Income}_i + \beta_2 \text{Age}_i + \beta_3 (\text{Income}_i \times \text{Age}_i) + \epsilon_i\]

1.3 Algebraic Interpretation

The key insight is to collect terms involving Income:

\[\text{SRH}_i = \beta_0 + \beta_2 \text{Age}_i + \underbrace{(\beta_1 + \beta_3 \text{Age}_i)}_{\text{Marginal effect of Income}} \times \text{Income}_i + \epsilon_i\]

The marginal effect of Income on SRH is no longer a constant $\beta_1$. The $\partial$ symbol means we are asking: if Income increases by one unit while Age stays fixed, how much does SRH change? The answer depends on which Age we fix it at:

\[\frac{\partial \text{SRH}}{\partial \text{Income}} = \beta_1 + \beta_3 \times \text{Age}\]

This means:

At Age = 30: marginal effect of Income $= \beta_1 + 30\beta_3$
At Age = 50: marginal effect of Income $= \beta_1 + 50\beta_3$
At Age = 70: marginal effect of Income $= \beta_1 + 70\beta_3$

If $\beta_3 > 0$, the income-health association strengthens with age. If $\beta_3 < 0$, it weakens.

Symmetrically, the marginal effect of Age also depends on Income:

\[\frac{\partial \text{SRH}}{\partial \text{Age}} = \beta_2 + \beta_3 \times \text{Income}\]

Both variables are simultaneously “moderated” by each other — continuous × continuous interactions are always symmetric in this sense.

Marginal effect: The marginal effect of a variable $X$ is the estimated change in the outcome $Y$ associated with a one-unit increase in $X$, holding all other variables constant. In a simple linear regression without interactions, this is just the slope coefficient $\hat{\beta}$ — a single number that applies uniformly across all observations. In a model with an interaction term, however, the marginal effect of $X_1$ is no longer constant. It depends on the value of the interacting variable $X_2$.

1.4 Step-by-Step Numeric Walkthrough

Suppose a model yields the following estimates:

Parameter	Estimate
$\hat{\beta}_0$ (Intercept)	3.500
$\hat{\beta}_1$ (Income)	0.080
$\hat{\beta}_2$ (Age)	−0.010
$\hat{\beta}_3$ (Income × Age)	0.002

Predicted SRH for two individuals:

Person A: Income = $30k ($= 3$ in $10k units), Age = 30

\[\hat{\text{SRH}}_A = 3.500 + 0.080(3) + (-0.010)(30) + 0.002(3 \times 30)\] \[= 3.500 + 0.240 - 0.300 + 0.180 = \mathbf{3.620}\]

Person B: Income = $30k, Age = 65

\[\hat{\text{SRH}}_B = 3.500 + 0.080(3) + (-0.010)(65) + 0.002(3 \times 65)\] \[= 3.500 + 0.240 - 0.650 + 0.390 = \mathbf{3.480}\]

Marginal effect of a $10k income increase:

At Age = 30: $\quad 0.080 + 0.002 \times 30 = 0.080 + 0.060 = \mathbf{0.140}$

At Age = 65: $\quad 0.080 + 0.002 \times 65 = 0.080 + 0.130 = \mathbf{0.210}$

The same $10k income increase is associated with a 0.14-point SRH improvement at age 30, but a 0.21-point improvement at age 65 — 50% larger. This is the interaction at work: income matters more for health at older ages. This is Case 1 ($\beta_1 > 0$, $\beta_3 > 0$): an amplifying interaction consistent with cumulative advantage over the life course.

What $\beta_3 = 0.002$ means in plain language: For each one-year increase in age, the income slope increases by 0.002 SRH points. Alternatively, for each $10k increase in income, the age slope increases by 0.002 — the interpretation is symmetric.

1.5 Reading the Direction: Four Sign Combinations of $\beta_1$ and $\beta_3$

Because the marginal effect of Income equals $\beta_1 + \beta_3 \times \text{Age}$, the combination of signs on $\beta_1$ and $\beta_3$ — not either coefficient alone — determines the substantive story. There are four possible patterns.

Case 1: $\beta_1 > 0$, $\beta_3 > 0$ — Amplification

Income has a positive effect on SRH, and that effect grows stronger at older ages. The income-health gradient steepens as the moderator increases. This pattern is consistent with cumulative advantage: those with higher incomes benefit progressively more over the life course.

Example: Education (+) on wages interacted with job experience (+). More experience amplifies the return to education.

Case 2: $\beta_1 > 0$, $\beta_3 < 0$ — Buffering / Diminishing Returns

Income has a positive baseline effect, but higher values of Age attenuate it. The association weakens as the moderator increases, and can reach zero or reverse beyond a crossover point. This is the most common pattern when a protective resource matters less in high-risk contexts, or when ceiling effects apply.

Example: Social support (+) on mental health interacted with chronic illness severity (−). Among the most severely ill, the benefit of social support is diminished.

The crossover point — the value of Age at which the income effect equals zero — is:

\[\text{Age}^* = -\frac{\beta_1}{\beta_3}\]

If $\text{Age}^*$ falls within your observed data range, the direction of the income-health association actually reverses for part of your sample. This should be reported and interpreted substantively.

Case 3: $\beta_1 < 0$, $\beta_3 > 0$ — Mitigation

Income has a negative baseline effect (at Age = 0, a counterfactual), but higher Age weakens that harm. A positive $\beta_3$ moves the marginal effect toward zero and potentially into positive territory. This pattern is common when a moderating resource or condition offsets an otherwise harmful exposure.

Example: Poverty (−) on child health outcomes interacted with neighborhood resource availability (+). Strong neighborhood resources reduce the health penalty of poverty.

The same crossover formula applies: $\text{Age}^* = -\beta_1 / \beta_3$.

Case 4: $\beta_1 < 0$, $\beta_3 < 0$ — Compounding / Cumulative Disadvantage

Income has a negative effect that worsens as Age increases. Both main effect and interaction pull in the same harmful direction. This is the “double jeopardy” scenario, consistent with fundamental cause and cumulative disadvantage theories in medical sociology.

Example: Unemployment (−) on mortality risk interacted with minority status (−). The mortality burden of unemployment is larger among racial minority groups.

Summary table:

$\beta_1$	$\beta_3$	Substantive Pattern	Sociological Label
$+$	$+$	Effect grows with moderator	Amplification / Cumulative advantage
$+$	$-$	Effect shrinks with moderator	Buffering / Diminishing returns
$-$	$+$	Harm shrinks with moderator	Mitigation / Protective factor
$-$	$-$	Harm grows with moderator	Compounding / Cumulative disadvantage

Three critical reminders:

$\beta_1$ is conditional, not unconditional. It is the effect of Income only when Age = 0. If zero is not a meaningful or observed value of your moderator, $\beta_1$ alone is uninterpretable. Mean-centering Age makes $\beta_1$ the effect of Income at the sample mean age — far more useful.
The interaction is symmetric. $\beta_3$ equally describes how the effect of Age varies across levels of Income. You can (and often should) interpret it in both directions depending on your theoretical question.
Report and visualize marginal effects across the moderator range. A single interaction coefficient communicates the rate of change in the slope, but a marginal effects plot communicates whether that change is substantively large, whether a crossover occurs, and where the effect is statistically distinguishable from zero.

1.6 Simulating and Visualizing in R

Code

n <- 800

d1 <- data.frame(
  age    = runif(n, 25, 75),
  income = pmax(rnorm(n, mean = 4.5, sd = 2), 0.5)   # income in $10k
) %>%
  mutate(
    srh = 3.5 +
      0.080 * income +
      -0.010 * age +
      0.002 * income * age +          # true interaction
      rnorm(n, 0, 0.5)
  )

# ── Models ─────────────────────────────────────────────────────────────────
m1_no_int <- lm(srh ~ income + age,             data = d1)
m1_int    <- lm(srh ~ income * age,             data = d1)

Code

# ── Regression Table ────────────────────────────────────────────────────────
stargazer(
  m1_no_int, m1_int,
  type          = "html",
  title         = "OLS Regression: Self-Rated Health ~ Income × Age",
  column.labels = c("No Interaction", "With Interaction"),
  covariate.labels = c("Income ($10k)", "Age", "Income × Age"),
  keep.stat     = c("n", "rsq"),
  star.cutoffs  = c(0.05, 0.01, 0.001)
)

**OLS Regression: Self-Rated Health ~ Income × Age**

	Dependent variable:

	srh
	No Interaction	With Interaction
	(1)	(2)

Income (10k)	0.186^***	0.111^***
	(0.009)	(0.030)

Age	0.0001	-0.007^*
	(0.001)	(0.003)

Income × Age		0.002^**
		(0.001)

Constant	2.969^***	3.303^***
	(0.074)	(0.148)


Observations	800	800
R²	0.360	0.365

Note:	p<0.05; p<0.01; p<0.001

Code

# ── Marginal Effects at Representative Age Values ──────────────────────────
age_vals <- c(30, 45, 60, 75)

me1 <- slopes(
  m1_int,
  variables  = "income",
  newdata    = datagrid(age = age_vals, income = mean(d1$income))
)

me1 %>%
  select(age, estimate, conf.low, conf.high) %>%
  rename(Age = age,
         `ME of Income` = estimate,
         `95% CI Low`   = conf.low,
         `95% CI High`  = conf.high) %>%
  kable(digits = 3,
        caption = "Marginal Effect of Income on SRH at Different Ages") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

Marginal Effect of Income on SRH at Different Ages
Age	ME of Income	95% CI Low	95% CI High
30	0.156	0.128	0.184
45	0.179	0.161	0.197
60	0.202	0.181	0.222
75	0.224	0.191	0.258

Code

# ── Plot 1: Predicted SRH across Income at 3 age values ───────────────────
pred1 <- predictions(
  m1_int,
  newdata = datagrid(
    income = seq(0.5, 10, length.out = 50),
    age    = c(30, 50, 70)
  )
)

p1a <- pred1 %>%
  mutate(age_label = paste0("Age = ", age)) %>%
  ggplot(aes(x = income, y = estimate, color = age_label, fill = age_label)) +
  geom_line(size = 1.1) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.15, color = NA) +
  scale_color_manual(values = c("Age = 30" = "#2166ac",
                                 "Age = 50" = "#f4a582",
                                 "Age = 70" = "#b2182b")) +
  scale_fill_manual(values  = c("Age = 30" = "#2166ac",
                                 "Age = 50" = "#f4a582",
                                 "Age = 70" = "#b2182b")) +
  labs(title    = "Predicted SRH by Income at Different Ages",
       subtitle = "Steeper slopes at older ages indicate a positive interaction",
       x = "Income ($10k)", y = "Predicted SRH", color = "Age Group", fill = "Age Group") +
  theme_minimal()

# ── Plot 2: Marginal effect of income across the age range ──────────────────
me1_full <- slopes(
  m1_int,
  variables = "income",
  newdata   = datagrid(age = seq(25, 75, by = 1), income = mean(d1$income))
)

p1b <- me1_full %>%
  ggplot(aes(x = age, y = estimate)) +
  geom_line(size = 1.1, color = "#2166ac") +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.2, fill = "#2166ac") +
  geom_hline(yintercept = 0, linetype = "dashed", color = "grey50") +
  labs(title    = "Marginal Effect of Income on SRH Across Age",
       subtitle = "Effect grows stronger at older ages (positive β₃)",
       x = "Age", y = "Marginal Effect of Income (per $10k)") +
  theme_minimal()

grid.arrange(p1a, p1b, ncol = 1)

1.7 Reading the Results

$\hat{\beta}_1$ (Income): The effect of income on SRH when age = 0 — not directly interpretable but needed for the formula.
$\hat{\beta}_2$ (Age): The effect of age on SRH when income = 0 — same caveat.
$\hat{\beta}_3$ (Income × Age): The change in the income slope for each one-year increase in age (and vice versa). A positive $\hat{\beta}_3$ means income is a stronger health predictor at older ages. Consult the sign combination table in Section 1.3.1 to identify the substantive pattern.

Tip: For continuous × continuous interactions, the intercept and main effect coefficients are no longer interpretable on their own. Always evaluate them at substantively meaningful values of the moderating variable using marginal effects or predicted value plots.

2. Categorical × Continuous Interaction

2.1 The Research Question

Does the effect of education on self-rated health differ between U.S.-born and immigrant adults?

This directly tests effect heterogeneity by nativity. The healthy immigrant effect suggests immigrants may translate educational credentials into health resources differently than U.S.-born adults — due to occupational mismatch, credential devaluation, or structural barriers. If so, each additional year of education would have a different marginal health payoff depending on nativity.

2.2 The Regression Model

Let $\text{Immigrant}_i = 1$ for foreign-born adults, $= 0$ for U.S.-born (the reference group).

Without interaction (parallel slopes assumed):

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Education}_i + \beta_2 \text{Immigrant}_i + \epsilon_i\]

This forces the education slope to be identical for both groups — only the intercepts differ.

With interaction (slopes allowed to differ):

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Education}_i + \beta_2 \text{Immigrant}_i + \beta_3 (\text{Education}_i \times \text{Immigrant}_i) + \epsilon_i\]

2.3 Algebraic Interpretation

Collect terms separately for each group.

For U.S.-born adults ($\text{Immigrant} = 0$):

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Education}_i + \beta_2(0) + \beta_3 \text{Education}_i(0) = \beta_0 + \beta_1 \text{Education}_i\]

Intercept: $\beta_0$
Education slope: $\beta_1$

For immigrant adults ($\text{Immigrant} = 1$):

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Education}_i + \beta_2(1) + \beta_3 \text{Education}_i(1)\] \[= (\beta_0 + \beta_2) + (\beta_1 + \beta_3) \text{Education}_i\]

Intercept: $\beta_0 + \beta_2$
Education slope: $\beta_1 + \beta_3$

This yields a clean interpretation for each coefficient:

Coefficient	What it represents
$\beta_0$	Predicted SRH for U.S.-born at Education = 0 (baseline intercept)
$\beta_1$	Education slope for U.S.-born adults
$\beta_2$	Intercept shift for immigrants vs. U.S.-born (at Education = 0)
$\beta_3$	Difference in education slopes between immigrants and U.S.-born

Because $Z$ (Immigrant) is binary, the slope of Education takes exactly two values: $\beta_1$ for U.S.-born and $\beta_1 + \beta_3$ for immigrants. The same four directional patterns from Section 1.3.1 apply, but now the “moderator” only switches between two discrete states rather than varying continuously.

A negative $\beta_3$ means immigrants get a smaller health return per year of education relative to U.S.-born adults — the education slope is dampened for the comparison group. This is Case 2 ($\beta_1 > 0$, $\beta_3 < 0$): a buffering interaction consistent with credential devaluation. There is no continuous crossover point here, but you can ask whether the immigrant education slope ($\beta_1 + \beta_3$) remains positive, equals zero, or reverses sign — each has a distinct substantive meaning.

2.4 Step-by-Step Numeric Walkthrough

Suppose the model yields:

Parameter	Estimate
$\hat{\beta}_0$ (Intercept)	1.800
$\hat{\beta}_1$ (Education)	0.120
$\hat{\beta}_2$ (Immigrant)	0.400
$\hat{\beta}_3$ (Education × Immigrant)	−0.050

Education slope by group:

U.S.-born: $\hat{\beta}_1 = 0.120$ → each additional year of education adds 0.120 SRH points
Immigrant: $\hat{\beta}_1 + \hat{\beta}_3 = 0.120 + (-0.050) = 0.070$ → each additional year adds only 0.070 SRH points

Predicted SRH for four profiles:

U.S.-born, 12 years of education (high school):

\[\hat{\text{SRH}} = 1.800 + 0.120(12) + 0.400(0) + (-0.050)(12)(0) = 1.800 + 1.440 = \mathbf{3.240}\]

U.S.-born, 16 years of education (BA):

\[\hat{\text{SRH}} = 1.800 + 0.120(16) = 1.800 + 1.920 = \mathbf{3.720}\]

Immigrant, 12 years of education:

\[\hat{\text{SRH}} = 1.800 + 0.120(12) + 0.400(1) + (-0.050)(12)(1) = 1.800 + 1.440 + 0.400 - 0.600 = \mathbf{3.040}\]

Immigrant, 16 years of education:

\[\hat{\text{SRH}} = 1.800 + 0.120(16) + 0.400(1) + (-0.050)(16)(1) = 1.800 + 1.920 + 0.400 - 0.800 = \mathbf{3.320}\]

Summary table of predicted values:

Group	12 Years Educ	16 Years Educ	Gain (BA vs. HS)
U.S.-born	3.240	3.720	+0.480
Immigrant	3.040	3.320	+0.280

The BA advantage in SRH is 0.480 for U.S.-born adults but only 0.280 for immigrants — a difference of 0.200 points, which is exactly $\hat{\beta}_3 \times (16 - 12) = -0.050 \times 4 = -0.200$.

Also notice: at low education (12 years), immigrants actually have lower predicted SRH than U.S.-born (3.040 vs. 3.240), despite $\hat{\beta}_2 = +0.400$. Why? Because the intercept advantage for immigrants (+0.400) is offset by the slope disadvantage across 12 years of education ($-0.050 \times 12 = -0.600$). The intercept alone does not capture the immigrant health advantage — the slopes cross.

2.5 Simulating and Visualizing in R

Code

n2 <- 900

d2 <- data.frame(
  immigrant = c(rep(0, 550), rep(1, 350)),
  education = c(
    pmin(pmax(rnorm(550, mean = 14.0, sd = 2.5), 8), 20),
    pmin(pmax(rnorm(350, mean = 12.5, sd = 3.0), 8), 20)
  )
) %>%
  mutate(
    srh = 1.800 +
      0.120 * education +
      0.400 * immigrant +
      -0.050 * education * immigrant +   # true interaction: smaller slope for immigrants
      rnorm(n(), 0, 0.45),
    nativity = factor(immigrant, labels = c("U.S.-born", "Immigrant"))
  )

m2_no_int <- lm(srh ~ education + nativity,            data = d2)
m2_int    <- lm(srh ~ education * nativity,            data = d2)

Code

stargazer(
  m2_no_int, m2_int,
  type          = "html",
  title         = "OLS Regression: Self-Rated Health ~ Education × Nativity",
  column.labels = c("No Interaction", "With Interaction"),
  covariate.labels = c("Education (years)", "Immigrant", "Education × Immigrant"),
  keep.stat     = c("n", "rsq"),
  star.cutoffs  = c(0.05, 0.01, 0.001)
)

**OLS Regression: Self-Rated Health ~ Education × Nativity**

	Dependent variable:

	srh
	No Interaction	With Interaction
	(1)	(2)

Education (years)	0.087^***	0.117^***
	(0.006)	(0.008)

Immigrant	-0.253^***	0.615^***
	(0.033)	(0.157)

Education × Immigrant		-0.065^***
		(0.012)

Constant	2.242^***	1.816^***
	(0.085)	(0.112)


Observations	900	900
R²	0.284	0.309

Note:	p<0.05; p<0.01; p<0.001

Code

# ── Marginal Effect of Education by Nativity ────────────────────────────────
me2 <- slopes(
  m2_int,
  variables = "education",
  newdata   = datagrid(nativity = c("U.S.-born", "Immigrant"),
                       education = mean(d2$education))
)

me2 %>%
  select(nativity, estimate, conf.low, conf.high) %>%
  rename(Nativity = nativity,
         `ME of Education` = estimate,
         `95% CI Low`      = conf.low,
         `95% CI High`     = conf.high) %>%
  kable(digits = 3,
        caption = "Marginal Effect of Education on SRH by Nativity") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

Marginal Effect of Education on SRH by Nativity
Nativity	ME of Education	95% CI Low	95% CI High
U.S.-born	0.117	0.102	0.132
Immigrant	0.052	0.035	0.068

Code

# ── Plot 1: Predicted SRH across Education by Nativity ─────────────────────
pred2 <- predictions(
  m2_int,
  newdata = datagrid(
    education = seq(8, 20, length.out = 50),
    nativity  = c("U.S.-born", "Immigrant")
  )
)

p2a <- pred2 %>%
  ggplot(aes(x = education, y = estimate, color = nativity, fill = nativity)) +
  geom_line(size = 1.2) +
  geom_ribbon(aes(ymin = conf.low, ymax = conf.high), alpha = 0.15, color = NA) +
  scale_color_manual(values = c("U.S.-born" = "steelblue", "Immigrant" = "tomato")) +
  scale_fill_manual(values  = c("U.S.-born" = "steelblue", "Immigrant" = "tomato")) +
  labs(title    = "Predicted SRH by Education and Nativity",
       subtitle = "Non-parallel slopes indicate an interaction: immigrants get smaller health returns to education",
       x = "Education (years)", y = "Predicted SRH",
       color = "Nativity", fill = "Nativity") +
  theme_minimal()

# ── Plot 2: Marginal effect comparison (bar + CI) ──────────────────────────
p2b <- me2 %>%
  mutate(nativity = as.character(nativity)) %>%
  ggplot(aes(x = nativity, y = estimate, fill = nativity)) +
  geom_col(alpha = 0.85, width = 0.4) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.12, size = 0.8) +
  geom_text(aes(label = sprintf("%.3f", estimate)), vjust = -0.6, fontface = "bold") +
  scale_fill_manual(values = c("U.S.-born" = "steelblue", "Immigrant" = "tomato")) +
  labs(title    = "Marginal Effect of Education on SRH by Nativity",
       subtitle = "Error bars = 95% CI",
       x = NULL, y = "ME of Education (per year)") +
  theme_minimal() +
  theme(legend.position = "none")

grid.arrange(p2a, p2b, ncol = 2)

2.6 Reading the Results

Non-parallel regression lines in the predicted value plot are the visual signature of an interaction. If $\hat{\beta}_3 = 0$, the two lines would be perfectly parallel.
The interaction coefficient ($\hat{\beta}_3$) directly quantifies the difference in slopes — it is the amount by which the education-health slope changes when moving from the reference group (U.S.-born) to the comparison group (Immigrant).
A negative $\hat{\beta}_3$ here is consistent with credential devaluation: immigrants accumulate education at a lower health return, possibly because their degrees translate less directly into health-promoting occupational and economic resources.

Common mistake: Interpreting $\hat{\beta}_2$ (the dummy coefficient for Immigrant) as the average group difference in SRH. When an interaction is present, $\hat{\beta}_2$ is the intercept difference at Education = 0 — a value that may not even exist in your data. Always use marginaleffects or predictions() to compute meaningful group comparisons at realistic covariate values.

3. Categorical × Categorical Interaction

3.1 The Research Question

Does the combination of nativity (immigrant vs. U.S.-born) and health insurance coverage predict SRH beyond their individual effects?

Insurance coverage may matter differently for immigrants and U.S.-born adults. Uninsured U.S.-born adults face access barriers, but uninsured immigrants may face compounded disadvantages — language barriers, fear of immigration enforcement, and limited eligibility for public programs. This would produce a synergistic (or antagonistic) effect that neither variable captures alone.

3.2 The Regression Model

Let $\text{Immigrant}_i \in \{0, 1\}$ and $\text{Uninsured}_i \in \{0, 1\}$.

Without interaction:

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Immigrant}_i + \beta_2 \text{Uninsured}_i + \epsilon_i\]

With interaction:

\[\text{SRH}_i = \beta_0 + \beta_1 \text{Immigrant}_i + \beta_2 \text{Uninsured}_i + \beta_3 (\text{Immigrant}_i \times \text{Uninsured}_i) + \epsilon_i\]

3.3 Algebraic Interpretation

With two binary variables, the model produces exactly four cell means — one for each combination of the two dummies. Substituting the four combinations:

Nativity	Insurance	Predicted SRH
U.S.-born ($= 0$)	Insured ($= 0$)	$\beta_0$
U.S.-born ($= 0$)	Uninsured ($= 1$)	$\beta_0 + \beta_2$
Immigrant ($= 1$)	Insured ($= 0$)	$\beta_0 + \beta_1$
Immigrant ($= 1$)	Uninsured ($= 1$)	$\beta_0 + \beta_1 + \beta_2 + \beta_3$

This gives a completely transparent mapping from coefficients to group means.

What each coefficient represents:

Coefficient	Meaning
$\beta_0$	Mean SRH for U.S.-born, insured adults (the reference cell)
$\beta_1$	SRH gap: immigrant insured vs. U.S.-born insured
$\beta_2$	SRH gap: U.S.-born uninsured vs. U.S.-born insured
$\beta_3$	Interaction: the extra penalty (or benefit) for being both immigrant and uninsured, beyond the sum of $\beta_1$ and $\beta_2$

The additive counterfactual:

Without an interaction, we would predict the immigrant-uninsured mean as:

\[\hat{\text{SRH}}_{\text{immigrant, uninsured}} = \beta_0 + \beta_1 + \beta_2\]

The interaction term $\beta_3$ measures how much the observed mean deviates from this additive prediction. In the categorical × categorical case, the directional logic from Section 1.3.1 translates directly: the sign of $\beta_3$ tells you whether the doubly-classified cell is better or worse than additivity predicts, and the signs of $\beta_1$ and $\beta_2$ tell you whether each individual disadvantage is harmful or beneficial at baseline.

$\beta_1 < 0$, $\beta_2 < 0$, $\beta_3 < 0$: Both main effects are harmful, and they compound each other — classic double jeopardy or cumulative disadvantage.
$\beta_1 < 0$, $\beta_2 < 0$, $\beta_3 > 0$: Both main effects are harmful, but they are less than additive — one disadvantage partially buffers or offsets the other.
$\beta_1 > 0$, $\beta_2 < 0$, $\beta_3 < 0$: One group has a baseline advantage, but that advantage is erased or reversed when the second condition is present.

A negative $\beta_3$ means the two disadvantages compound each other — the health penalty for being uninsured is worse among immigrants than among U.S.-born adults.

3.4 Step-by-Step Numeric Walkthrough

Suppose the model yields:

Parameter	Estimate
$\hat{\beta}_0$	3.800
$\hat{\beta}_1$ (Immigrant)	−0.150
$\hat{\beta}_2$ (Uninsured)	−0.300
$\hat{\beta}_3$ (Immigrant × Uninsured)	−0.350

Four predicted cell means:

U.S.-born, Insured:

\[\hat{\text{SRH}} = 3.800 = \mathbf{3.800}\]

U.S.-born, Uninsured:

\[\hat{\text{SRH}} = 3.800 + (-0.300) = \mathbf{3.500}\]

Immigrant, Insured:

\[\hat{\text{SRH}} = 3.800 + (-0.150) = \mathbf{3.650}\]

Immigrant, Uninsured:

\[\hat{\text{SRH}} = 3.800 + (-0.150) + (-0.300) + (-0.350) = \mathbf{3.000}\]

Verifying the interaction:

The additive prediction for immigrant-uninsured (assuming no interaction) would be:

\[3.800 + (-0.150) + (-0.300) = 3.350\]

The observed (interaction-adjusted) prediction is 3.000. The interaction term $\hat{\beta}_3 = -0.350$ captures this shortfall: immigrant uninsured adults fare 0.350 SRH points worse than we would expect from adding the immigrant and uninsured penalties separately. This is the $\beta_1 < 0$, $\beta_2 < 0$, $\beta_3 < 0$ pattern — a compounding, cumulative disadvantage scenario.

Computing contrasts manually:

Comparison	Difference	Formula
Insurance penalty, U.S.-born	$3.500 - 3.800 = -0.300$	$\hat{\beta}_2$
Insurance penalty, Immigrant	$3.000 - 3.650 = -0.650$	$\hat{\beta}_2 + \hat{\beta}_3$
Immigrant gap, Insured	$3.650 - 3.800 = -0.150$	$\hat{\beta}_1$
Immigrant gap, Uninsured	$3.000 - 3.500 = -0.500$	$\hat{\beta}_1 + \hat{\beta}_3$

The insurance penalty is more than twice as large for immigrants ($-0.650$) as for U.S.-born adults ($-0.300$). This is the negative interaction: being uninsured is far more consequential for health when you are also an immigrant.

3.5 Simulating and Visualizing in R

Code

n3 <- 800

d3 <- data.frame(
  immigrant = c(rep(0, 500), rep(1, 300)),
  uninsured = rbinom(800, 1, prob = 0.30)
) %>%
  mutate(
    # Uninsured rate is higher among immigrants
    uninsured  = ifelse(immigrant == 1,
                        rbinom(n(), 1, prob = 0.45),
                        rbinom(n(), 1, prob = 0.18)),
    srh = 3.800 +
      -0.150 * immigrant +
      -0.300 * uninsured +
      -0.350 * immigrant * uninsured +   # compounding disadvantage
      rnorm(n(), 0, 0.50),
    nativity  = factor(immigrant,  labels = c("U.S.-born", "Immigrant")),
    insurance = factor(uninsured,  labels = c("Insured", "Uninsured"))
  )

m3_no_int <- lm(srh ~ nativity + insurance,          data = d3)
m3_int    <- lm(srh ~ nativity * insurance,          data = d3)

Code

stargazer(
  m3_no_int, m3_int,
  type          = "html",
  title         = "OLS Regression: Self-Rated Health ~ Nativity × Insurance",
  column.labels = c("No Interaction", "With Interaction"),
  covariate.labels = c("Immigrant", "Uninsured", "Immigrant × Uninsured"),
  keep.stat     = c("n", "rsq"),
  star.cutoffs  = c(0.05, 0.01, 0.001)
)

**OLS Regression: Self-Rated Health ~ Nativity × Insurance**

	Dependent variable:

	srh
	No Interaction	With Interaction
	(1)	(2)

Immigrant	-0.267^***	-0.182^***
	(0.040)	(0.048)

Uninsured	-0.469^***	-0.331^***
	(0.043)	(0.062)

Immigrant × Uninsured		-0.266^**
		(0.086)

Constant	3.833^***	3.810^***
	(0.024)	(0.025)


Observations	800	800
R²	0.230	0.240

Note:	p<0.05; p<0.01; p<0.001

Code

# ── Cell Means and Contrasts ────────────────────────────────────────────────
cell_means <- predictions(
  m3_int,
  newdata = datagrid(nativity  = c("U.S.-born", "Immigrant"),
                     insurance = c("Insured",   "Uninsured"))
)

cell_means %>%
  select(nativity, insurance, estimate, conf.low, conf.high) %>%
  rename(Nativity  = nativity,
         Insurance = insurance,
         `Predicted SRH` = estimate,
         `95% CI Low`    = conf.low,
         `95% CI High`   = conf.high) %>%
  arrange(Nativity, Insurance) %>%
  kable(digits = 3,
        caption = "Predicted SRH by Nativity and Insurance Status") %>%
  kable_styling(bootstrap_options = c("striped", "hover", "condensed"))

Predicted SRH by Nativity and Insurance Status
Nativity	Insurance	Predicted SRH	95% CI Low	95% CI High
U.S.-born	Insured	3.810	3.760	3.859
U.S.-born	Uninsured	3.478	3.368	3.588
Immigrant	Insured	3.627	3.547	3.708
Immigrant	Uninsured	3.030	2.946	3.114

Code

# ── Plot 1: Interaction Plot (classic 2x2 cell mean plot) ─────────────────
pred3 <- predictions(
  m3_int,
  newdata = datagrid(
    nativity  = c("U.S.-born", "Immigrant"),
    insurance = c("Insured", "Uninsured")
  )
)

p3a <- pred3 %>%
  ggplot(aes(x = insurance, y = estimate,
             color = nativity, group = nativity)) +
  geom_point(size = 4) +
  geom_line(size = 1.2) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high),
                width = 0.08, size = 0.9) +
  scale_color_manual(values = c("U.S.-born" = "steelblue", "Immigrant" = "tomato")) +
  labs(title    = "Interaction Plot: SRH by Nativity × Insurance Status",
       subtitle = "Non-parallel lines = interaction; immigrants suffer a larger uninsurance penalty",
       x = "Insurance Status", y = "Predicted SRH", color = "Nativity") +
  theme_minimal()

# ── Plot 2: Grouped bar chart of four cell means ───────────────────────────
p3b <- pred3 %>%
  ggplot(aes(x = nativity, y = estimate, fill = insurance)) +
  geom_col(position = "dodge", alpha = 0.85) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high),
                position = position_dodge(0.9), width = 0.15) +
  geom_text(aes(label = sprintf("%.2f", estimate)),
            position = position_dodge(0.9), vjust = -0.6,
            fontface = "bold", size = 3.8) +
  scale_fill_manual(values = c("Insured" = "#4393c3", "Uninsured" = "#d6604d")) +
  labs(title    = "Predicted SRH by Nativity and Insurance Status",
       subtitle = "The uninsurance penalty is substantially larger for immigrants",
       x = "Nativity", y = "Predicted SRH", fill = "Insurance") +
  theme_minimal()

# ── Plot 3: Marginal effects of insurance penalty by nativity ──────────────
me3 <- slopes(
  m3_int,
  variables = "insurance",
  newdata   = datagrid(nativity = c("U.S.-born", "Immigrant"))
)

p3c <- me3 %>%
  mutate(nativity = as.character(nativity)) %>%
  ggplot(aes(x = nativity, y = estimate, fill = nativity)) +
  geom_col(alpha = 0.85, width = 0.4) +
  geom_errorbar(aes(ymin = conf.low, ymax = conf.high), width = 0.12, size = 0.8) +
  geom_text(aes(label = sprintf("%.3f", estimate)), vjust = 1.5,
            fontface = "bold", color = "white", size = 4) +
  geom_hline(yintercept = 0, linetype = "dashed", color = "grey50") +
  scale_fill_manual(values = c("U.S.-born" = "steelblue", "Immigrant" = "tomato")) +
  labs(title    = "Insurance Penalty (Insured → Uninsured) by Nativity",
       subtitle = "Marginal effect of uninsurance on SRH; 95% CI shown",
       x = NULL, y = "Change in Predicted SRH") +
  theme_minimal() +
  theme(legend.position = "none")

grid.arrange(p3a, p3b, p3c, layout_matrix = rbind(c(1, 2), c(3, 3)))

3.6 Reading the Results

Parallel lines in the interaction plot would indicate no interaction: the uninsurance penalty is the same for both groups. Non-parallel lines mean the effect of insurance depends on nativity — and vice versa.
The interaction coefficient $\hat{\beta}_3$ captures the excess penalty (or benefit) for the doubly-classified cell, beyond what additive effects would predict.
The marginal effects panel (Plot 3) is often the most interpretable: it directly shows the insurance penalty separately for each nativity group, making the interaction visible as a difference in bar heights.

4. Centering and Scaling: A Practical Note

For continuous × continuous and categorical × continuous interactions, the main effect coefficients ($\hat{\beta}_1$, $\hat{\beta}_2$) are conditional on the other variable equaling zero. If zero is not a meaningful value for your moderator — e.g., age = 0 — the main effects become uninterpretable.

The solution is mean-centering the continuous variables before computing the product term:

\[(\text{Income} - \bar{\text{Income}}) \times (\text{Age} - \bar{\text{Age}})\]

After centering:

$\hat{\beta}_1$ = effect of income at the mean age
$\hat{\beta}_2$ = effect of age at the mean income
$\hat{\beta}_3$ = unchanged — the interaction coefficient is scale-invariant

Code

d1_c <- d1 %>%
  mutate(
    income_c = income - mean(income),
    age_c    = age    - mean(age)
  )

m1_centered <- lm(srh ~ income_c * age_c, data = d1_c)

stargazer(
  m1_int, m1_centered,
  type          = "html",
  title         = "Effect of Mean-Centering on Interaction Model Coefficients",
  column.labels = c("Uncentered", "Mean-Centered"),
  covariate.labels = c("Income ($10k)", "Age",
                        "Income × Age",
                        "Income (centered)", "Age (centered)",
                        "Income × Age (centered)"),
  keep.stat     = c("n", "rsq"),
  star.cutoffs  = c(0.05, 0.01, 0.001)
)

**Effect of Mean-Centering on Interaction Model Coefficients**

	Dependent variable:

	srh
	Uncentered	Mean-Centered
	(1)	(2)

Income (10k)	0.111^***
	(0.030)

Age	-0.007^*
	(0.003)

Income × Age	0.002^**
	(0.001)

Income (centered)		0.186^***
		(0.009)

Age (centered)		0.0001
		(0.001)

Income × Age (centered)		0.002^**
		(0.001)

Constant	3.303^***	3.802^***
	(0.148)	(0.017)


Observations	800	800
R²	0.365	0.365

Note:	p<0.05; p<0.01; p<0.001

Note that $\hat{\beta}_3$ is identical across the two models. Only the intercept and main effects change — now they are interpretable as effects at the sample mean of the moderator.

5. Testing and Reporting Interactions

5.1 Is the Interaction Statistically Significant?

The standard approach is to compare the model fit with and without the interaction term using an F-test (for OLS) or a likelihood ratio test:

Code

# Continuous x continuous
anova(m1_no_int, m1_int)

Analysis of Variance Table

Model 1: srh ~ income + age
Model 2: srh ~ income * age
  Res.Df    RSS Df Sum of Sq      F   Pr(>F)   
1    797 194.48                                
2    796 192.83  1    1.6508 6.8146 0.009211 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Code

# Categorical x continuous
anova(m2_no_int, m2_int)

Analysis of Variance Table

Model 1: srh ~ education + nativity
Model 2: srh ~ education * nativity
  Res.Df    RSS Df Sum of Sq      F    Pr(>F)    
1    897 192.69                                  
2    896 186.06  1    6.6325 31.941 2.136e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Code

# Categorical x categorical
anova(m3_no_int, m3_int)

Analysis of Variance Table

Model 1: srh ~ nativity + insurance
Model 2: srh ~ nativity * insurance
  Res.Df    RSS Df Sum of Sq      F   Pr(>F)   
1    797 213.17                                
2    796 210.62  1    2.5569 9.6637 0.001946 **
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

5.2 Reporting Best Practices

Always report the full interaction model, not just the term that “tests” the interaction. Suppressing main effects in the presence of interaction terms is misleading.
Use marginal effects tables (from marginaleffects::slopes()) to communicate the substantive size of the interaction at meaningful covariate values.
Visualize: predicted value plots and marginal effects plots communicate interactions far more effectively than a single interaction coefficient.
For categorical × categorical, always present the full 2 × 2 cell mean table so readers can compute all pairwise contrasts.
Interpret the interaction, not just its significance: a significant interaction coefficient of −0.002 may be statistically reliable but substantively trivial; a p = 0.06 interaction of −0.350 may matter a great deal.
Identify the directional pattern explicitly: use the sign combination framework from Section 1.3.1 to label the substantive story (amplification, buffering, mitigation, or compounding) and connect it to your theoretical argument.

6. Summary Comparison of Interaction Types

Feature	Continuous × Continuous	Categorical × Continuous	Categorical × Categorical
Example	Income × Age → SRH	Education × Nativity → SRH	Nativity × Insurance → SRH
What $\beta_3$ means	Change in slope of $X_1$ per unit of $X_2$	Difference in $X_1$ slope across groups	Extra cell mean deviation from additivity
Main effects interpretable?	Only at $X_2 = 0$ (center first)	Only at $X_2 = 0$ reference level	Only for reference cells
Best visualization	Predicted lines at representative values	Non-parallel regression lines by group	Classic 2×2 interaction plot
Centering needed?	Yes (both variables)	For continuous variable	No (binary)
Marginal effect varies by	All values of the moderator (continuous)	Group membership	Group membership

Directional Interpretation: Quick Reference

The sign combination of $\beta_1$ (main effect of $X_1$) and $\beta_3$ (interaction) determines the substantive pattern across all three interaction types. The table below provides a quick reference:

$\beta_1$	$\beta_3$	What happens to the effect of $X_1$ as $X_2$ increases	Sociological label
$+$	$+$	Positive effect grows stronger	Amplification / Cumulative advantage
$+$	$-$	Positive effect weakens; may reverse at $X_2^* = -\beta_1/\beta_3$	Buffering / Diminishing returns
$-$	$+$	Negative effect weakens; may reverse at $X_2^* = -\beta_1/\beta_3$	Mitigation / Protective factor
$-$	$-$	Negative effect grows stronger	Compounding / Cumulative disadvantage

For categorical × categorical interactions, $\beta_3$ is the deviation of the doubly-classified cell from the additive prediction. The signs of all three coefficients ($\beta_1$, $\beta_2$, $\beta_3$) together determine whether the pattern represents compounding disadvantage, buffering, or advantage amplification.

The Unifying Logic

All three interaction types share the same algebraic idea: the slope of $X_1$ is itself a linear function of $X_2$. The differences are cosmetic — whether $X_2$ takes on a continuous range, two discrete values, or two labeled categories determines how the slope variation is parameterized and visualized, not what it fundamentally means.

\[\frac{\partial Y}{\partial X_1} = \beta_1 + \beta_3 X_2\]

Continuous $X_2$: the marginal effect of $X_1$ traces a line across all values of $X_2$
Binary $X_2$: the marginal effect of $X_1$ takes two values — one per group
The difference between those two values is always $\beta_3$

Nativity	Insurance	Predicted SRH
U.S.-born (\(= 0\))	Insured (\(= 0\))	\(\beta_0\)
U.S.-born (\(= 0\))	Uninsured (\(= 1\))	\(\beta_0 + \beta_2\)
Immigrant (\(= 1\))	Insured (\(= 0\))	\(\beta_0 + \beta_1\)
Immigrant (\(= 1\))	Uninsured (\(= 1\))	\(\beta_0 + \beta_1 + \beta_2 + \beta_3\)

Comparison	Difference	Formula
Insurance penalty, U.S.-born	\(3.500 - 3.800 = -0.300\)	\(\hat{\beta}_2\)
Insurance penalty, Immigrant	\(3.000 - 3.650 = -0.650\)	\(\hat{\beta}_2 + \hat{\beta}_3\)
Immigrant gap, Insured	\(3.650 - 3.800 = -0.150\)	\(\hat{\beta}_1\)
Immigrant gap, Uninsured	\(3.000 - 3.500 = -0.500\)	\(\hat{\beta}_1 + \hat{\beta}_3\)

\(\beta_1\)	\(\beta_3\)	What happens to the effect of \(X_1\) as \(X_2\) increases	Sociological label
\(+\)	\(+\)	Positive effect grows stronger	Amplification / Cumulative advantage
\(+\)	\(-\)	Positive effect weakens; may reverse at \(X_2^* = -\beta_1/\beta_3\)	Buffering / Diminishing returns
\(-\)	\(+\)	Negative effect weakens; may reverse at \(X_2^* = -\beta_1/\beta_3\)	Mitigation / Protective factor
\(-\)	\(-\)	Negative effect grows stronger	Compounding / Cumulative disadvantage

Parameter	Estimate
\(\hat{\beta}_0\) (Intercept)	3.500
\(\hat{\beta}_1\) (Income)	0.080
\(\hat{\beta}_2\) (Age)	−0.010
\(\hat{\beta}_3\) (Income × Age)	0.002

Parameter	Estimate
\(\hat{\beta}_0\) (Intercept)	1.800
\(\hat{\beta}_1\) (Education)	0.120
\(\hat{\beta}_2\) (Immigrant)	0.400
\(\hat{\beta}_3\) (Education × Immigrant)	−0.050

Parameter	Estimate
\(\hat{\beta}_0\)	3.800
\(\hat{\beta}_1\) (Immigrant)	−0.150
\(\hat{\beta}_2\) (Uninsured)	−0.300
\(\hat{\beta}_3\) (Immigrant × Uninsured)	−0.350

Understanding Interaction Terms in Regression

Setup: Shared Packages

1. Continuous × Continuous Interaction

1.1 The Research Question

1.2 The Regression Model

1.3 Algebraic Interpretation

1.4 Step-by-Step Numeric Walkthrough

1.5 Reading the Direction: Four Sign Combinations of \(\beta_1\) and \(\beta_3\)

1.6 Simulating and Visualizing in R

1.7 Reading the Results

2. Categorical × Continuous Interaction

2.1 The Research Question

2.2 The Regression Model

2.3 Algebraic Interpretation

2.4 Step-by-Step Numeric Walkthrough

2.5 Simulating and Visualizing in R

2.6 Reading the Results

3. Categorical × Categorical Interaction

3.1 The Research Question

3.2 The Regression Model

3.3 Algebraic Interpretation

3.4 Step-by-Step Numeric Walkthrough

3.5 Simulating and Visualizing in R

3.6 Reading the Results

4. Centering and Scaling: A Practical Note

5. Testing and Reporting Interactions

5.1 Is the Interaction Statistically Significant?

5.2 Reporting Best Practices

6. Summary Comparison of Interaction Types

Directional Interpretation: Quick Reference

The Unifying Logic