What is meant by regression coefficients in a multiple regression model?

What is meant by regression coefficients in a multiple regression model?

• A. A particular combination of coefficients maximizes the explained variance in the dependent variable
• B. They are the same as the ANOVA F-statistic
• C. They represent the cosines of angles between predictors
• D. They measure the residual variance

Answer: (A)

In multiple regression, regression coefficients are the estimated weights that, together, define the linear equation predicting the outcome from the predictors. In ordinary least squares, these coefficients are chosen to minimize the sum of squared differences between observed and predicted values. Because the total variation in the outcome is fixed, reducing the residual variance by fitting the model as well as possible is the same as increasing the variance explained by the model. So the coefficients represent the particular combination of predictor weights that yields the best linear explanation of the dependent variable, i.e., the maximum explained variance for that set of predictors. They are not the ANOVA F-statistic, not the cosines of angles between predictors, and not a measure of residual variance.

In multiple regression, regression coefficients are the estimated weights that, together, define the linear equation predicting the outcome from the predictors. In ordinary least squares, these coefficients are chosen to minimize the sum of squared differences between observed and predicted values. Because the total variation in the outcome is fixed, reducing the residual variance by fitting the model as well as possible is the same as increasing the variance explained by the model. So the coefficients represent the particular combination of predictor weights that yields the best linear explanation of the dependent variable, i.e., the maximum explained variance for that set of predictors. They are not the ANOVA F-statistic, not the cosines of angles between predictors, and not a measure of residual variance.