Multiple Regression Analysis Regression analysis with two or more independent variables. Leads to an improvement in accuracy of prediction models, higher R 2 s. Challenges to interpretation due to multicollinearity.

Linear model y = the predicted, or dependent variable x = the variable used to predict y, or the independent variable a = the intercept, or point where a “line” cuts the y axis when x = 0 b = the slope, or change in y for any 1 unit change in x

Terms for the Linear Model The use of term for y, dependent variable, is consistent with a belief that values of x (independent) determine values of y. Knowing values of x allows us to predict value of y. To develop a statistical model, the dependent variable, y, is “regressed on” values of x (the “regressor”).

Linear Model and Additivity x 1, x 2, x 3 are a set of independent regressors that additively influence our prediction of the dependent variable y. bs are coefficients, in the units of y, that convert a one unit change in x into a change in y.

Coefficients Unstandardized coefficients are in units of the dependent variable Standardized coefficients or Beta coefficients range from to +1.00

t-Values Reported with each coefficient Test if the coefficient is significantly different from zero Ratio of the coefficient to the standard error of the coefficient

Coefficient of Determination: R 2 Percentage of variance explained, ranges from near zero to 1.00 Squared value of a correlation coefficient in a simple, bivariate regression. Multiple R, is the equivalence of a correlation coefficient for a multiple regression

Adjusted R 2 R 2 increases with the addition of each regressor to the model This becomes very important with a small number of observations k = the number of regressors in the model n = the number of observations

Multicollinearity Violation of the independence assumption. Significance of coefficients from bivariate models reduced. Minimal improvement in prediction accuracy. Variation Inflation Factors approaching 10 It will be a problem in our models as we move from two to three census regressors.