# Interaction term using centered variables hierarchical regression analysis? What variables should we center?

I'm running a hierarchical regression analysis and I have some little doubts:

1. Do we calculate the interaction term using the centered variables?

2. Do we have to center ALL the continuous variables we have in the dataset, except the dependent variable?

3. When we have to log some variables (because their s.d. is way higher than their mean), do we then center the variable that has just been logged or the initial one?

For example: Variable "Turnover" ---> Logged Turnover (because the s.d. is too high compared to the mean) ---> Centered_Turnover?

OR would it be directly Turnover --> Centered_Turnover (and we work with this one)

THANKS!!

You should center the terms involved in the interaction to reduce collinearity e.g.

set.seed(10204)
x1 <- rnorm(1000, 10, 1)
x2 <- rnorm(1000, 10, 1)
y <- x1 + rnorm(1000, 5, 5)  + x2*rnorm(1000) + x1*x2*rnorm(1000)

x1cent <- x1 - mean(x1)
x2cent <- x2 - mean(x2)
x1x2cent <- x1cent*x2cent

m1 <- lm(y ~ x1 + x2 + x1*x2)
m2 <- lm(y ~ x1cent + x2cent + x1cent*x2cent)

summary(m1)
summary(m2)

Output:

> summary(m1)

Call:
lm(formula = y ~ x1 + x2 + x1 * x2)

Residuals:
Min      1Q  Median      3Q     Max
-344.62  -66.29   -1.44   66.05  392.22

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  193.333    335.281   0.577    0.564
x1           -15.830     33.719  -0.469    0.639
x2           -14.065     33.567  -0.419    0.675
x1:x2          1.179      3.375   0.349    0.727

Residual standard error: 101.3 on 996 degrees of freedom
Multiple R-squared:  0.002363,  Adjusted R-squared:  -0.0006416
F-statistic: 0.7865 on 3 and 996 DF,  p-value: 0.5015

> summary(m2)

Call:
lm(formula = y ~ x1cent + x2cent + x1cent * x2cent)

Residuals:
Min      1Q  Median      3Q     Max
-344.62  -66.29   -1.44   66.05  392.22

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)     12.513      3.203   3.907 9.99e-05 ***
x1cent          -4.106      3.186  -1.289    0.198
x2cent          -2.291      3.198  -0.716    0.474
x1cent:x2cent    1.179      3.375   0.349    0.727
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 101.3 on 996 degrees of freedom
Multiple R-squared:  0.002363,  Adjusted R-squared:  -0.0006416
F-statistic: 0.7865 on 3 and 996 DF,  p-value: 0.5015

library(perturb)
colldiag(m1)
colldiag(m2)

Whether you center other variables is up to you; centering (as opposed to standardizing) a variable that is not involved in an interaction will change the meaning of the intercept, but not other things e.g.

x1 <- rnorm(1000, 10, 1)
x2 <- x1 - mean(x1)
y <- x1 + rnorm(1000, 5, 5)
m1 <- lm(y ~ x1)
m2 <- lm(y ~ x2)

summary(m1)
summary(m2)

Output:

> summary(m1)

Call:
lm(formula = y ~ x1)

Residuals:
Min       1Q   Median       3Q      Max
-16.5288  -3.3348   0.0946   3.4293  14.0678

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)   6.5412     1.6003   4.087 4.71e-05 ***
x1            0.8548     0.1591   5.373 9.63e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.082 on 998 degrees of freedom
Multiple R-squared:  0.02812,   Adjusted R-squared:  0.02714
F-statistic: 28.87 on 1 and 998 DF,  p-value: 9.629e-08

> summary(m2)

Call:
lm(formula = y ~ x2)

Residuals:
Min       1Q   Median       3Q      Max
-16.5288  -3.3348   0.0946   3.4293  14.0678

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  15.0965     0.1607  93.931  < 2e-16 ***
x2            0.8548     0.1591   5.373 9.63e-08 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.082 on 998 degrees of freedom
Multiple R-squared:  0.02812,   Adjusted R-squared:  0.02714
F-statistic: 28.87 on 1 and 998 DF,  p-value: 9.629e-08

But you should take logs of variables because it makes sense to do so or because the residuals from the model indicate that you should, not because they have a lot of variability. Regression does not make assumptions about the distribution of the variables, it makes assumptions about the distribution of the residuals.

• Thanks for your response, Peter! So, I assume then that first I would have to log the variables (all of the predictors?) and, after that, I would center only the independent variables required to calculate the interaction terms. One more question: Would you recommend centering or standardizing the variables? Again, thanks a lot!! – PhDstudent Jul 20 '13 at 22:32
• Yes, log before centering. Standardizing and centering do different things; neither is wrong. Some like standardizing, I usually prefer "raw" variables. – Peter Flom - Reinstate Monica Jul 20 '13 at 22:37
• I fail to see how defining the generating model as y <- x1 + rnorm(1000, 5, 5) + x2*rnorm(1000) + x1*x2*rnorm(1000) helps illustrating the answer. The mean of this is $x_1 +5$ and the variance is $1 + 25 + 1 + 1$, so there is no interaction term in the generating model. – Rufo Feb 25 at 18:52