Regression analysis Module 10: adjusted R2 practice problems
(The attached PDF file has better formatting.)
** Exercise 10.1: Adjusted R2
A statistician regresses the response variable Y on k explanatory variables X1, X2, …, Xk and one intercept.
The variance of the observed Y values is 10 and the estimated
ó2å (the error variance) is 2.
What is the total sum of squares (TSS)?
What is the residual sum of squares (RSS)?
What is the regression sum of squares?
What is the R2 of the regression?
What is the adjusted R2?
Part A:
The total sum of squares TSS is the observed variance × (N-1), where N = the observations.
Part B:
The residual sum of squares RSS is
ó2å × (N-k-1).
Note:
These relations are generally written in the reverse form: variance = TSS / (N-1);
ó2å = RSS / (N-k-1).
Part C:
The regression sum of squares RegSS is TSS – RSS.
Part D:
The R2 is ResSS / TSS.
Part E:
The adjusted R2 = [ RegSS / (N-k-1) ] / [ TSS / (N-1) ] = 1 –
ó2å / the variance of the response variable:
adjusted R2 = 1 – 2 / 10 = 80%.
The general formula is that the adjusted R2 = (var(Y) –
ó2å) / var(Y).
If we are given RSS and TSS (or RSS and RegSS, or RegSS and TSS), but not N (the number of observations) or k (the number of parameters), we can derive R2 but not the adjusted R2.
If we are given the variance of the Y values (the response variable) and the
ó2å, but not N (the number of observations) or k (the number of parameters), we can derive the adjusted R2 but not the simple R2.