Back-transformation

Question

I have a simple linear regression model, and I have done transformations on the response and the explanatory variable (see call in code below), and I obtained an r-squared of 0.415. When I back-transformed the response variable back to its original units, I get an r-squared of 0.29. My questions are two fold:

• First, why is there such a difference in the r-squared values when the same dataset is used;
• Second, if I report the root mean squared error for the second model (which I have read is the most appropriate action), can I still report the r-squared from the first model (i.e. r-squared of 0.42)?

Please see the two summary reports below. Thank you.

1. Call:
lm(formula = sqrt(Clay_Tot) ~ Dose, data = PedonsTx.11.LatLong)

Residuals:
    Min      1Q  Median      3Q     Max 
-3.9745 -0.6534 -0.1394  0.4876  4.5746 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) 1.203414   0.248964   4.834 2.42e-06 ***
Dose        0.075897   0.005848  12.977  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.161 on 235 degrees of freedom
Multiple R-squared:  0.4175,    Adjusted R-squared:  0.415 
F-statistic: 168.4 on 1 and 235 DF,  p-value: < 2.2e-16

2. Call:
lm(formula = Clay_Tot ~ Back.Dose.11, data = PedonsTx.11.LatLong)

Residuals:
    Min      1Q  Median      3Q     Max 
-22.316  -6.372  -2.337   2.649  60.211 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept)  7.8260736  1.4977945   5.225 3.84e-07 ***
Back.Dose.11 0.0070682  0.0007112   9.939  < 2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 11.75 on 235 degrees of freedom
Multiple R-squared:  0.2959,    Adjusted R-squared:  0.2929 
F-statistic: 98.78 on 1 and 235 DF,  p-value: < 2.2e-16

Answer 1

When you transform, you change many things:

1) the amount of variation of the response

2) the shape of the relationship between y and x

3) the spread about the relationship

So you cannot expect $R^2$ to be similar across transformations. A transformed response is not "the same data set" -- the transformation changes things!

You could fit a relationship and transform back and try to compare fits, but then the $R^2$ isn't really meaningful for the backtransformed response.

--

When you say "can I...?"; if you mean "does it make sense?" - that depends. What are you using the $R^2$ for a different model to convey? What are you saying about it?