General Linear Model (GLM)

Regression Equations

  

The regression equation is an algebraic representation of the regression line and describes the relationship between the response and predictor variables. The regression equation takes the form of:

Response = constant + coefficient * predictor + ... + coefficient * predictor

or y = bo + b1X1 + b2X2 + ... + bkXk

Where:

·    Response (Y) is the value of the response.

·    Constant (bo) is the value of the response variable when the predictor variable(s) is zero. The constant is also called the intercept because it determines where the regression line intercepts (meets) the Y-axis.

·    Predictor(s) (X) is the value of the predictor variable(s). The predictor can be a polynomial term.

·    Coefficients (b1, b2, ... , bk) represent the estimated change in mean response for each unit change in the predictor value. In other words, it is the change in Y that occurs when X increases by one unit.

Minitab provides a single regression equation when the model contains only categorical factors or when there are more than 50 combinations of factor levels.

Example Output

Regression Equation

 

Salary = 2.7275 - 0.5775 Subject_1 + 0.0792 Subject_2 + 0.3858 Subject_3 + 0.1125 Subject_4

         - 0.3400 Degree_1 - 0.2600 Degree_2 + 0.6000 Degree_3 - 0.0100 Subject*Degree_1 1

         + 0.0600 Subject*Degree_1 2 - 0.0500 Subject*Degree_1 3 - 0.0167 Subject*Degree_2 1

         - 0.0267 Subject*Degree_2 2 + 0.0433 Subject*Degree_2 3 - 0.0233 Subject*Degree_3 1

         - 0.0033 Subject*Degree_3 2 + 0.0267 Subject*Degree_3 3 + 0.0500 Subject*Degree_4 1

         - 0.0300 Subject*Degree_4 2 - 0.0200 Subject*Degree_4 3

Interpretation

For the salary data, Minitab displays a single equation where the response variable is Salary and the predictors are the:

·    three levels of Subject

·    two levels of Degree

·    six combinations of Subject and Degree

You can interpret the slope value for each categorical factor as the change in salary based on group membership. For a professor with a Master's degree (degree = 2) in the Humanities (subject = 1 ), the variables Subject_1, Degree_2, and Subject*Degree_1 2 take on the value of 1, while the others are 0. Add the constant and the applicable terms to calculate the predicted value:

2.7275 - 0.5775 - 0.2600 + 0.0600 = 1.9500 or $1,950.00

Use Stat > ANOVA > General Linear Model > Predict to calculate the predictions and confidence intervals for predictor values that you specify.