MEASURES OF PREDICTIVE SUCCESS
Whatever method of validation is used, we need to have some measure of the success of the prediction procedure. One possibility is to use the sum of the losses in the calibration and the validation sample. Even this procedure contains an ambiguity that we need resolve. Are we more concerned with minimizing the expected loss, the average loss, or the maximum loss?
One measure of goodness of fit of the model is 
, where yi and 
denote the ith observed value and the corresponding value obtained from the model. The smaller this sum of squares, the better the fit.
If the observations are independent, then
The first sum on the right hand side of the equation is the total sum of squares (SST). Most statistics software use as a measure of fit R2 = 1 − SSE/SST. The closer the value of R2 is to 1, the better.
The automated entry of predictors into the regression equation using R2 runs the risk of overfitting, as R2 is guaranteed to increase with each predictor entering the model. To compensate, one may use the adjusted R2:
where n is the number of observations used ...
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