# Calculating R-squared with same data but different regression model

Using the same data, two different regression models are produced.

e.g Regression 1: height= 5+ 3(father's height) + u

In regression two we have height= 0.02+ 5.1(father's height) + u

These models are different because in model 2, height was measured in feet.

Assuming these regressions used the same data, is the R-squared value identical between the two?

I was under the assumption that it would be the same.

• Won't change as long as your transformation is linear. Oct 14 '17 at 11:34

It appears that your question is "If I change the units, will R^2 change?" and the answer is "no". However, the intercept should not change. This is intuitive but can be demonstrated easily e.g. in R (everything after a # is comment):

set.seed(1234)  #Sets a seed
x1 <- rnorm(1000)
x2 <- x1*12  #Changes units
y <- 3*x1 + rnorm(1000,0,5)   #creates y
m1 <- lm(y~x1)  #fits a linear model
m2 <- lm(y~x2)

summary(m1)  #R^2 = 0.3083, y = 0.08 + 3.28x1
summary(m2) #R^2 = 0.3083, y = 0.08 + 0.27x1

• The OP says that "height" is measured in feet in the second model. I'd assume that this refers to both the IV (father's height) and the DV (height). In which case, yes, $R^2$ shouldn't change (the proportion of variance explained doesn't change), but the intercept should certainly do. Oct 14 '17 at 12:01
• @StephanKolassa If both IV and DV are measured in the same unit, the parameter 3 is dimensionless and should not change. Oct 14 '17 at 13:19
• @StephanKolassa Sure, but this observation shows that most definitely DV and IV are not measured in the same unit in both models, if I understand correctly. Oct 14 '17 at 13:31
• @FedericoPoloni: ah, I see your point. You are right. The OP's example seems to be broken. Thank goodness the question still makes sense. Oct 14 '17 at 13:35
• @StephanKolassa I agree -- the OP should specify more precisely what is measured in which unit, to make this a well-defined question. Oct 14 '17 at 13:39