The Coefficient of Determination
The Quality of Prediction:
Beta weights help assess the magnitude of the relationship between an IV and a DV, but they do not provide information about how well the IV performs at predicting the DV. This is a substantial limitation because prediction is the heart of regression—it is the reason researchers use this technique. The coefficient of determination addresses the issue of the quality of prediction. It does this by comparing the actual, empirical values of y to the predicted values (ŷi ). A close match between these two sets of scores indicates that x does a
good job predicting y , whereas a poor correspondence signals that x is not a useful predictor. The coefficient of determination is given by
where = the correlation between the actual and predicted values of y.
The correlation between the y and ŷ and values is computed the same way that correlations between IVs and DVs are and so will not be shown here. In real life, SPSS generates this value for you. The correlation in this example is .74. This makes the coefficient of determination
.742 = .55
This means that 55% of the variance in y can be attributed to the influence of x. In the context of the present example, 55% of the variance in fine amounts is attributable to the number of charges. Again, this value looks familiar. This illustrates the close connection between correlation and regression at the bivariate level. Things get more complicated when additional IVs are added to the model.
