Centering a variable involves subtracting its mean from each observation, resulting in a new variable with a mean of zero. This transformation can reduce multicollinearity in multiple regression models, particularly when dealing with interaction terms or polynomial terms.
Consider a multiple regression model with an interaction term:
In this model, \(X_1\), \(X_2\), and \(X_1 \cdot X_2\) are often highly correlated, leading to multicollinearity.
Now, let's center \(X_1\) and \(X_2\) by subtracting their means:
The centered model becomes:
In this centered model, the correlation between \(X_1^c\), \(X_2^c\), and their interaction term \(X_1^c \cdot X_2^c\) is typically reduced, mitigating multicollinearity.
Let's consider a small dataset to illustrate this concept:
| Observation | X₁ | X₂ | Y |
|---|---|---|---|
| 1 | 1 | 4 | 10 |
| 2 | 2 | 5 | 15 |
| 3 | 3 | 6 | 22 |
| 4 | 4 | 7 | 31 |
| 5 | 5 | 8 | 42 |
Step 1: Calculate means of X₁ and X₂
\(\bar{X_1} = 3\), \(\bar{X_2} = 6\)
Step 2: Center X₁ and X₂
| Observation | X₁ᶜ = X₁ - 3 | X₂ᶜ = X₂ - 6 | Y |
|---|---|---|---|
| 1 | -2 | -2 | 10 |
| 2 | -1 | -1 | 15 |
| 3 | 0 | 0 | 22 |
| 4 | 1 | 1 | 31 |
| 5 | 2 | 2 | 42 |
Step 3: Calculate correlations
Original variables:
Centered variables:
In this example, we see that centering the variables has dramatically reduced the correlations between the main effects (X₁ᶜ and X₂ᶜ) and their interaction term (X₁ᶜ·X₂ᶜ). The correlation between X₁ᶜ and X₂ᶜ remains unchanged because centering doesn't affect the linear relationship between variables.
The reduction in correlations with the interaction term helps to mitigate multicollinearity, making the coefficient estimates more stable and easier to interpret. This is particularly useful when the main effects and their interactions are of interest in the analysis.