How to interpret whether centering improves linear regression model or not?

Question

I'm doing my first tests with linear regression and centering and on the linear regression model with dichotomic predictor that I have, I'm wondering how to interpret whether centering improves the results or makes them worse?

Without centering R's lm returns as the (Intercept) value the mean of those responses with predictor != 1. This value corresponds to what mean() gives, so it's correct.

However, when I center the predictor, I get an (Intercept) value that's about 11% larger than the earlier one. This does not correspond to what mean() gives. So is this (Intercept) better or worse than the earlier one? Does centering improve the results or make them worse?

Answer 1

Just to clarify all this as an answer

First, generate some data

y <- c(1:10)
x <- c(rep(0, 4), rep(1, 6))
dat <- data.frame(y, x)

now fit a linear model

summary(lm(y ~ x, data = dat))
Call:
lm(formula = y ~ x, data = dat)
Residuals:
   Min     1Q Median     3Q    Max 
 -2.50  -1.25   0.00   1.25   2.50 
Coefficients:
            Estimate Std. Error t value Pr(>|t|)   
(Intercept)   2.5000     0.8385   2.981  0.01756 * 
x             5.0000     1.0825   4.619  0.00171 **
Residual standard error: 1.677 on 8 degrees of freedom
Multiple R-squared:  0.7273,    Adjusted R-squared:  0.6932 
F-statistic: 21.33 on 1 and 8 DF,  p-value: 0.001713

Note that the intercept is indeed the mean of $y$ for $x==0$

Now centre $x$ and repeat

x2 <- x - mean(x)
x2
 [1] -0.6 -0.6 -0.6 -0.6  0.4  0.4  0.4  0.4  0.4  0.4
summary(lm(y ~ x2, data = dat))
Call:
lm(formula = y ~ x2, data = dat)
Residuals:
   Min     1Q Median     3Q    Max 
 -2.50  -1.25   0.00   1.25   2.50 
Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   5.5000     0.5303  10.371 6.46e-06 ***
x2            5.0000     1.0825   4.619  0.00171 ** 
Residual standard error: 1.677 on 8 degrees of freedom
Multiple R-squared:  0.7273,    Adjusted R-squared:  0.6932 
F-statistic: 21.33 on 1 and 8 DF,  p-value: 0.001713

Note that the coefficient is unchanged as the difference between the predicted means for the two categories remains unchanged but the intercept has been shifted to represent the hypothetical value predicted for $x2 == 0$.

Although centering can be an excellent idea it does not do much of interest with categorical predictors.