These residuals are referred to as \(HSGPA.SAT\), which means they are the residuals in \(HSGPA\) after having been predicted by \(SAT\). These errors of prediction are called "residuals" since they are what is left over in \(HSGPA\) after the predictions from \(SAT\) are subtracted, and represent the part of \(HSGPA\) that is independent of \(SAT\). In this example, the regression coefficient for \(HSGPA\) can be computed by first predicting \(HSGPA\) from \(SAT\) and saving the errors of prediction (the differences between \(HSGPA\) and \(HSGPA'\)). Interpretation of Regression CoefficientsĪ regression coefficient in multiple regression is the slope of the linear relationship between the criterion variable and the part of a predictor variable that is independent of all other predictor variables. Note that \(R\) will never be negative since if there are negative correlations between the predictor variables and the criterion, the regression weights will be negative so that the correlation between the predicted and actual scores will be positive. In this example, it is the correlation between \(UGPA'\) and \(UGPA\), which turns out to be \(0.79\). The multiple correlation (\(R\)) is equal to the correlation between the predicted scores and the actual scores. The values of \(b\) (\(b_1\) and \(b_2\)) are sometimes called "regression coefficients" and sometimes called "regression weights." These two terms are synonymous.
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