To predict the value of the dependent variable of our interest (y) for the given, specific values of the independent variables (the x-variables) using a multiple regression model, we substitute the values of the x-variables into the multiple regression equation and solve for the value of y. The resulting value of y is a point estimate of y, a confidence interval of the average value of y, a prediction interval for a single value of y, or an exact forecast of y.
The regression coefficients of the independent variables (the x-variables) in a multiple regression model are referred to as partial regression coefficients.
In a multiple regression analysis, the numerical value of the adjusted R2 is always less than that of the coefficient of multiple determination (R2).
In a regression study, a multiple regression model with two explanatory variables is developed using a data set with 23 observations. In the ANOVA table for this model, the sum of squares total (SSyy) is 12500 and the sum of squares error (SSE) is 3000. The standard error of estimate (se) is 11.96.
In a regression study, a multiple regression model with two explanatory variables is developed using a data set with 23 observations. In the ANOVA table for this model, the sum of squares total (SSyy) is 12500 and the sum of squares error (SSE) is 3000. The coefficient of multiple determination (R2) is 0.76.
In a regression study, a multiple regression model with two explanatory variables is developed using a data set with 23 observations. In the ANOVA table for this model, the sum of squares total (SSyy) is 12500 and the sum of squares regression (SSR) is 9500. The number of degrees of freedom for the residual error is 20.
As n, the number of observations in the data set increases, the gap between R2 and adjusted R2 decreases.
The adjusted R2 accounts for the number of observations and the number of independent variables in the model.
The range for the coefficient of multiple determination is 0 to +1.
A measure of goodness of fit for a multiple regression model is the coefficient of multiple determination.
The coefficient of determination can take values of 0, 0.5, or 1.0, but not 1.5.
When additional independent variables are added to a simple linear regression, the coefficient of determination, R2, may increase or stay the same.
To test if an individual B coefficient in the population multiple regression model is significantly different from zero, a hypothesis test is conducted on the corresponding bi coefficient in the regression equation developed using sample data. This test is a t-test.
The test statistic used to test the overall significance of a multiple regression model, the null hypothesis that each one of the beta-coefficients of the x-variables in the model is equal to zero, is tested against the alternative hypothesis that at least one of the beta-coefficients of the x-variables in the model is ≠ zero, is the F value from the F-distribution tables.
In testing the overall significance of a multiple regression model, the null hypothesis that each one of the beta-coefficients of the x-variables in the model is equal to zero, is tested against the alternative hypothesis that at least one of the beta-coefficients of the x-variables in the model is ≠ zero.
A real estate analyst has developed a multiple regression line, y = 60 + 0.068 x1 – 2.5 x2, to predict y = the market price of a home (in $1,000s), using independent variables, x1 = the total number of square feet of living space, and x2 = the age of the house in years. The regression coefficient of x1 suggests this: The addition of 1 square foot area of living space results in a predicted increase of $0.068 in the price of the home if the age of the home were held constant.
A real estate analyst has developed a multiple regression line, y = 60 + 0.068 x1 – 2.5 x2, to predict y = the market price of a home (in $1,000s), using two independent variables, x1 = the total number of square feet of living space, and x2 = the age of the house in years. With this regression model, the predicted price of a 10-year old home with 2,500 square feet of living area is $205,000.00.
A real estate analyst has developed a multiple regression line, y = 60 + 0.068 x1 – 2.5 x2, to predict y = the market price of a home (in $1,000s), using independent variables, x1 = the total number of square feet of living space, and x2 = the age of the house in years. The regression coefficient of x2 suggests this: If the square feet area of living space is kept constant, a 1 year increase in the age of the homes will result in a predicted drop of $2500 in the price of the homes.