Seminar on Interaction Effects with R

An example with easytable

Alfredo Hernandez Sanchez

Vilnius University

2026-04-30

When should we use interaction terms?

  • When the effect of a predictor variable \(x_1\) on the response variable \(\widehat{y}\) is not independent from another predictor \(x_2\)

  • For example, the interaction term tells us whether the relationship between \(x_{1}\) and \(\widehat{y}\) is different when \(x_{2} = 1\) compared with when \(x_{2} = 0\).

Multivariate OLS

\[ \widehat{y} = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \epsilon \]

Where

  • \(\widehat{y}\) is the dependent variable
  • \(\beta_{0}\) is the intercept
  • \(x_{1}\) is an independent variable
  • \(x_{2}\) is another independent variable
  • \(\epsilon\) is the error term

Regression with interaction term

\[ \widehat{y} = \beta_{0} + \beta_{1}x_{1} + \beta_{2}x_{2} + \beta_{3}(x_{1} \times x_{2}) + \epsilon \]

Where:

  • \(\beta_{3}\) is the coefficient for the interaction between \(x_{1}\) and \(x_{2}\)
  • the effect of \(x_{1}\) on \(\widehat{y}\) depends on the value of \(x_{2}\)

Interpreting the interaction terms

Assume \(x_{1}\) is continuous and \(x_{2}\) is binary.

When \(x_{2} = 0\):

\[ \widehat{y} = \beta_{0} + \beta_{1}x_{1} \]

When \(x_{2} = 1\):

\[ \widehat{y} = (\beta_{0} + \beta_{2}) + (\beta_{1} + \beta_{3})x_{1} \]

\(\beta_{2}\) is the change in the intercept when \(x_{2} = 1\).

\(\beta_{3}\) is the change in the slope of \(x_{1}\) when \(x_{2} = 1\).

Binary Interaction Results


term

Model 1

(Intercept)

1.26
(0.87)

aid_pct

0.1
(0.15)

democracy

0.23
(1.23)

aid_pct *
democracy

0.5 **
(0.21)

N

40

R sq.

0.5

Adj. R sq.

0.46

Significance: ***p < .01; **p < .05; *p < .1

Example of an interaction

Suppose we want to study whether foreign aid is associated with economic growth (RQ). But we suspect that aid does not work the same way everywhere.

  • H: Aid may be more useful when the state has stronger administrative capacity.

\[ \widehat{Growth} = 1.37 + 0.35 Aid + 0.52 Capacity + 0.20(Aid \times Capacity) \]

Where

  • \(Growth\) is annual GDP growth
  • \(Aid\) is foreign aid as a percentage of national income
  • \(Capacity\) is a state capacity index (median centered)
  • \(Aid \times Capacity\) is the interaction term

Interpretation

The effect of aid is not just \(0.35\).

Because aid is interacted with state capacity, the effect of aid is:

\[ \frac{\partial \widehat{Growth}}{\partial Aid} = 0.35 + 0.20 Capacity \]

For every one-point increase in state capacity, the association between aid and predicted GDP growth becomes 0.2 percentage points stronger.

Results


term

Model 1

(Intercept)

1.37 **
(0.52)

aid_pct

0.35 ***
(0.09)

state_capacity

0.52
(0.47)

aid_pct *
state_capacity

0.2 **
(0.08)

N

40

R sq.

0.64

Adj. R sq.

0.61

Significance: ***p < .01; **p < .05; *p < .1

Interaction Terms in R


#  OLS
m1 <- lm(gdp_growth ~ aid_pct + state_capacity, 
         data = df)

# Interaction 
m2 <- lm(gdp_growth ~ aid_pct * state_capacity, 
         data = df)


# Multiple Interactions
m3 <- lm(gdp_growth ~ 
           aid_pct * state_capacity + 
           aid_pct * democracy + 
           population_m, 
         data = df)

term

Model 1

Model 2

Model 3

(Intercept)

1.37 **
(0.56)

1.37 **
(0.52)

1.35 ***
(0.11)

aid_pct

0.35 ***
(0.1)

0.35 ***
(0.09)

0.1 ***
(0.02)

state_capacity

1.51 ***
(0.25)

0.52
(0.47)

0.53 ***
(0.06)

aid_pct *
state_capacity

0.2 **
(0.08)

0.2 ***
(0.01)

democracy

0.24 *
(0.13)

population_m

0
(0)

aid_pct *
democracy

0.5 ***
(0.02)

N

40

40

40

R sq.

0.58

0.64

0.99

Adj. R sq.

0.55

0.61

0.99

Significance: ***p < .01; **p < .05; *p < .1

Reporting regression results with easytable

Install easytable


# Install the package from R-Universe
install.packages("easytable", 
                 repos = c("https://alfredo-hs.r-universe.dev/easytable",
                           "https://cloud.r-project.org"))

# Load the library
library(easytable)

easytable() vs summary()

# Call summary() on object m1
summary(m1)

Call:
lm(formula = gdp_growth ~ aid_pct + state_capacity, data = df)

Residuals:
   Min     1Q Median     3Q    Max 
-3.515 -1.321  0.022  1.369  3.330 

Coefficients:
               Estimate Std. Error t value Pr(>|t|)    
(Intercept)     1.37244    0.55819   2.459  0.01874 *  
aid_pct         0.34556    0.09717   3.556  0.00105 ** 
state_capacity  1.51190    0.24582   6.150 3.93e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.738 on 37 degrees of freedom
Multiple R-squared:  0.577, Adjusted R-squared:  0.5542 
F-statistic: 25.24 on 2 and 37 DF,  p-value: 1.221e-07
# Call easytable() on object m1
easytable(m1)

term

Model 1

(Intercept)

1.37 **
(0.56)

aid_pct

0.35 ***
(0.1)

state_capacity

1.51 ***
(0.25)

N

40

R sq.

0.58

Adj. R sq.

0.55

Significance: ***p < .01; **p < .05; *p < .1

Comparing models with easytable()


# Activate easytable() on objects m1 y m2
easytable(m1, m2)

term

Model 1

Model 2

(Intercept)

1.37 **
(0.56)

1.37 **
(0.52)

aid_pct

0.35 ***
(0.1)

0.35 ***
(0.09)

state_capacity

1.51 ***
(0.25)

0.52
(0.47)

aid_pct *
state_capacity

0.2 **
(0.08)

N

40

40

R sq.

0.58

0.64

Adj. R sq.

0.55

0.61

Significance: ***p < .01; **p < .05; *p < .1

Other easytable() commands

# Highlight significant coefficients
easytable(m1, m2, m3,
          highlight = TRUE)

term

Model 1

Model 2

Model 3

(Intercept)

1.37 **
(0.56)

1.37 **
(0.52)

1.35 ***
(0.11)

aid_pct

0.35 ***
(0.1)

0.35 ***
(0.09)

0.1 ***
(0.02)

state_capacity

1.51 ***
(0.25)

0.52
(0.47)

0.53 ***
(0.06)

aid_pct *
state_capacity

0.2 **
(0.08)

0.2 ***
(0.01)

democracy

0.24 *
(0.13)

population_m

0
(0)

aid_pct *
democracy

0.5 ***
(0.02)

N

40

40

40

R sq.

0.58

0.64

0.99

Adj. R sq.

0.55

0.61

0.99

Significance: ***p < .01; **p < .05; *p < .1 

# Name models, set digits
easytable(m1, m2,
          digits = 1,
          model.names = c("A", "B"),
          highlight = TRUE)

term

A

B

(Intercept)

1.4 **
(0.6)

1.4 **
(0.5)

aid_pct

0.3 ***
(0.1)

0.3 ***
(0.1)

state_capacity

1.5 ***
(0.2)

0.5
(0.5)

aid_pct *
state_capacity

0.2 **
(0.1)

N

40

40

R sq.

0.58

0.64

Adj. R sq.

0.55

0.61

Significance: ***p < .01; **p < .05; *p < .1

Export and set control variables


# Define control variables
easytable(m1, m2, m3,
          control.var = c("population_m"),
          highlight = TRUE)

term

Model 1

Model 2

Model 3

(Intercept)

1.37 **
(0.56)

1.37 **
(0.52)

1.35 ***
(0.11)

aid_pct

0.35 ***
(0.1)

0.35 ***
(0.09)

0.1 ***
(0.02)

state_capacity

1.51 ***
(0.25)

0.52
(0.47)

0.53 ***
(0.06)

aid_pct *
state_capacity

0.2 **
(0.08)

0.2 ***
(0.01)

democracy

0.24 *
(0.13)

aid_pct *
democracy

0.5 ***
(0.02)

population_m

Y

N

40

40

40

R sq.

0.58

0.64

0.99

Adj. R sq.

0.55

0.61

0.99

Significance: ***p < .01; **p < .05; *p < .1 

# Export to word and csv
easytable(m1, m2, m3,
          control.var = c("cyl", "am"),
          model.names = c("A", "B", "C"),
          export.word = "table.docx",
          export.csv = "table.csv",
          highlight = TRUE)

Summary


library(easytable)
mtcars

# Define models
m1 <- lm(mpg ~ wt, data = mtcars)
m2 <- lm(mpg ~ wt + hp, data = mtcars)
m3 <- lm(mpg ~ wt + hp + cyl + am, data = mtcars)

# Table
easytable(m1, m2, m3,
          control.var = c("cyl", "am"),
          model.names = c("A", "B", "C"),
          export.word = "table.docx",
          export.csv = "table.csv",
          highlight = TRUE)

Activity

Work in pairs or groups of 3.

Your task is to use the gapminder data to test one interaction effect.

  • Choose an outcome variable, such as lifeExp
  • Choose one main explanatory variable, such as log_gdp
  • Choose a second variable that may change the relationship, such as continent, pop, or a dummy variable (binary) you create
  • Write one simple hypothesis
  • Write one short mechanism explaining why the interaction might exist
  • Estimate a regression model with an interaction term
  • Interpret the interaction coefficient in plain English

Thank you for your attention!

About the FIRSA Project

Disclaimer:

This project has received funding from the European Union Marie Skłodowska-Curie Postdoctoral Fellowships / ERA Fellowships action under grant agreement No. 101180601 under the title: Understanding FinTech Regulatory Sandbox Development in Europe (FIRSA).

Learn more at the project website.

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