1. What an interaction means (intuitive view)
An interaction effect means:
The effect of one variable on the outcome depends on the level of another variable.
So, the relationship between \(X_1\) and \(Y\) changes depending on the value of \(X_2\).
2. A simple regression without interaction
Say we model income as a function of gender and education:
\(\text{Income} = \beta_0 + \beta_1(\text{Gender}) + \beta_2(\text{Education}) + \epsilon\)
Here:
- \(\beta_1\): average difference between males and females (assuming Gender is dummy coded, say Male=1, Female=0)
- \(\beta_2\): effect of moving from one education level to another
- The two effects are independent — the gender gap is the same across all education levels.
3. Add an interaction term
Now we include a product term:
\(\text{Income} = \beta_0 + \beta_1(\text{Gender}) + \beta_2(\text{Education}) + \beta_3(\text{Gender} \times \text{Education}) + \epsilon\)
That last term — \(\beta_3(\text{Gender} \times \text{Education})\) — is the interaction.
Meaning:
It tells us whether the effect of education differs by gender.
- If \(\beta_3 = 0\): no interaction — both genders gain equally from higher education.
- If \(\beta_3 > 0\): the return to education (effect on income) is greater for males than females.
- If \(\beta_3 < 0\): the return to education is greater for females than males.
4. Example with numbers
Suppose you fit this model (Gender=1 for Male, 0 for Female; Education=years of schooling):
\(\text{Income} = 20{,}000 + 5{,}000(\text{Education}) + 3{,}000(\text{Gender}) + 1{,}000(\text{Gender} \times \text{Education})\)
Interpretation:
| Term | Meaning |
|---|---|
| 20,000 | Average income for a female with 0 years of education (baseline). |
| 5,000 | Each year of education increases income by $5,000 for females. |
| 3,000 | Males earn $3,000 more than females at 0 years of education. |
| 1,000 | For males, the effect of each additional year of education is $1,000 higher. |
So:
- For females: effect of education = 5,000 per year
- For males: effect of education = 5,000 + 1,000 = 6,000 per year
That difference (1,000) is the interaction effect.
5. In general
Interactions can occur between:
- Two categorical variables (e.g., gender × marital status)
- Categorical × continuous (e.g., gender × education years)
- Two continuous (e.g., age × income in predicting health)
Each type changes the interpretation slightly — but the core idea is always:
the slope (or effect) of one predictor depends on the value of another.