I would like to know how the predicted R-Squared are calculate because I want to use it to determine whether I am overfitting a my regression model by including to many terms, based in Mr. Jim's blog http://statisticsbyjim.com/regression/interpret-adjusted-r-squared-predicted-r-squared-regression/
There is some way that it can be automatically calculated by SPSS or do we need to calculated it by hand?
NOTE: I am using SPSS, just in case if someone has a particular advice.
Thank you in advance!
When you use ordinary least squares (OLS) to estimate the parameters, the relationship between the error terms form a right triangle.
y=f(x) TRUE STATE OF NATURE
yi=f(xi)+error OBSERVED VALUES
y^i ESTIMATED VALUE USING OLS
y_ SAMPLE MEAN
By construction
Q = sum((yi-f(xi))^2)
the derivative set equat to zero forces orthognal vectors betwee E and R
T = yi - y_ TOTAL ERROR WITHOUT MODEL BETWEEN DATA AND MEAN
R = y^i - y_ THE ERROR BETWEEN THE THE ESTIMATED MODEL AND MEAN
E = yi-y^i THE ERROR BETWEEN THE DATA AND THE ESTIMATED MODEL
R is the explained error and E is the unexplained error and by the Pythagorean Theorem
T^2 = R^2 +E^2 MORE COMMONLY DENOTED AS
ST = SR +SE
Hence this is true for the sum of square errors
SST = SSR + SSE
Dividing by SST you have R^2 = SSR/SST the percent of the total sum of square errors explained by the model
Hope this helped.
RSQ PREDICTED = 1-(PRESS/SSTOTAL) , Where PRESS stands for Predicted Residual Sum of squares.
PRESS= SUM over [(ei/1-hii)^2], i =1,2,...n , hii are the diagonal elements of the Hat matrix H , H= A*[inv(A'*A)]*A', A is the full column rank Design matrix associated with the linear model.
ei's are the ordinary residuals at the experimental points 1,2,...n in the model experimental design.
Coefficient of determination (R Squared) tells us the relative variance in the outcome variable that is explained by the explanatory variables in the model. Therefore, it is calculated as:
R-square : Sum Square Regression (SSR)/Sum Square Total (SST)
Where, SST is equal SSR+SSE.
Within SPSS. Use the the Save option in the regression menu/syntax to save the deletion residuals. Use compute to square them and then sum those squares. That gives you PRESS. RSQ PREDICTED = 1-(PRESS/SSTOTAL) as noted above.
Thank you all, Rebecca D Wooten , Debopam Ghosh , Abolfazl Ghoodjani , Kebede Embaye Gezae , Thom S Baguley ,
Its very helpful, I am taking your suggestions into consideration.
Best regards!
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