Question

Question: There are six independent variables, x₁, x₂, x₃, x₄, x₅, and x₆, that might be useful in predicting a response y. A total of n = 50 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to be useful. The software fits all possible one-variable models of the form

where x_i is the ith independent variable, i = 1, 2, …, 6. The information in the table is provided from the computer printout.

$E\left(Y\right)={\beta }_0+{\beta }_1{x}_i$

a. Which independent variable is declared the best one variable predictor of y? Explain.

b. Would this variable be included in the model at this stage? Explain.

c. Describe the next phase that a stepwise procedure would execute.

Short answer

Answer

a. For x₁, the t-value is the highest (3.80), x₁ is the best one variable predictor of y.

b. Since x₁ is the best one-variable predictor of y, the variable will be included in the model at this stage.

c. The stepwise program now begins to search through the remaining (k – 1) independent variables for the best two-variable model of the form. Therefore, here the program begins searching remaining 5 independent variables for the best two-variable model.

Step-by-step solution

One-variable best predictor

T-test values for testing of single β parameter is$\frac{\widehat{{\beta }_i}}{s\widehat{{\beta }_i}}$

Here for x₁, x₂,x₃,x₄,x₅,and x₆t-values are 3.80, -90, 2.98, 1.21, -6.02, and 0.857 respectively. Since for x₁, the t-value is the highest, x₁ is the best one variable predictor of y.

Stepwise procedure

Since x₁ is the best one-variable predictor of y, the variable will be included in the model at this stage.

Stepwise procedure

The stepwise program now begins to search through the remaining (k – 1) independent variables for the best two-variable model of the form. Therefore, here the program begins searching remaining 5 independent variables for the best two-variable model.