Question: Entry-level job preferences. Benefits Quarterly published a study of entry-level job preferences. A number of independent variables were used to model the job preferences (measured on a 10-point scale) of 164 business school graduates. Suppose stepwise regression is used to build a model for job preference score (y) as a function of the following independent variables:

$$\begin{gathered} {\mathbf{\text{x}}}_{\mathbf{1}}\mathbf{=}\{1ifflextimeposition \\ 0ifnot \\ {\mathit{x}}_{\mathbf{2}}\mathbf{=}\{1iffdaycaresupportrequired \\ 0ifnot \\ {\mathit{x}}_{\mathbf{3}}\mathbf{=}\{1iffsupporttransfersupportrequired \\ 0ifnot \\ {\mathit{x}}_{\mathbf{4}}\mathbf{=}\mathit{N}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{s}\mathit{i}\mathit{c}\mathit{k}\mathbf{}\mathit{d}\mathit{a}\mathit{y}\mathit{s}\mathbf{}\mathit{a}\mathit{l}\mathit{l}\mathit{o}\mathit{w}\mathit{e}\mathit{d} \end{gathered}$$

$$\begin{gathered} {\mathbf{\text{x}}}_5\mathbf{=}\mathbf{\{}\mathbf{1}\mathbf{}\mathbf{iff}\mathbf{}\mathbf{1}\mathbf{}\mathit{a}\mathit{p}\mathit{p}\mathit{l}\mathit{i}\mathit{c}\mathit{a}\mathit{n}\mathit{t}\mathbf{}\mathit{m}\mathit{a}\mathit{r}\mathit{r}\mathit{i}\mathit{e}\mathit{d} \\ \mathbf{0}\mathbf{}\mathbf{if}\mathbf{}\mathbf{not} \\ {\mathit{x}}_6\mathbf{=}\mathit{N}\mathit{u}\mathit{m}\mathit{b}\mathit{e}\mathit{r}\mathbf{}\mathit{o}\mathit{f}\mathbf{}\mathit{c}\mathit{h}\mathit{i}\mathit{l}\mathit{d}\mathit{r}\mathit{e}\mathit{n}\mathbf{}\mathit{a}\mathit{p}\mathit{p}\mathit{l}\mathit{i}\mathit{c}\mathit{a}\mathit{n}\mathit{t} \\ {\mathit{x}}_{\mathbf{6}}\mathbf{=}\mathbf{\{}\mathbf{1}\mathbf{}\mathbf{iff}\mathbf{}\mathbf{male}\mathbf{}\mathbf{applicant} \\ \mathbf{0}\mathbf{}\mathbf{if}\mathbf{}\mathbf{female}\mathbf{}\mathbf{applicant} \end{gathered}$$

a. How many models are fit to the data in step 1? Give the general form of these models.

b. How many models are fit to the data in step 2? Give the general form of these models.

c. How many models are fit to the data in step 3? Give the general form of these models.

d. Explain how the procedure determines when to stop adding independent variables to the model.

e. Describe two major drawbacks to using the final stepwise model as the best model for job preference score y.

There are total seven independent variables out of which five are qualitative (binary) while two are quantitative variables.

In step 1 of the stepwise regression, linear model in one independent variable is modelled for all the k no of variables in the question.

So, in this situation since there are 7 variables, 7 linear models in one variable is fitted to the data for 7 independent variables.

The general model for step 1 is for$E\left(y\right)={\beta }_0+{\beta }_1{x}_i$.

In step 2 of the stepwise regression, linear model in two independent variables is modelled for selected independent variable in step 1 and all the remaining (k=1) no of variables in the question.

Hence, in this situation since there are 7 independent variables, combination formula is used to choose i items from a total of n items) linear models in two variables is fitted to the data for 7 independent variables. The general model for step 1 is for $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_i$.

In step 3 of the stepwise regression, linear model in three independent variables is modelled for selected independent variables in step 2 and all the remaining (k-2) no of variables in the question.

Therefore, in this situation since there are 7 independent variables, (combination formula is used to choose x items from a total of n items) linear models in three variables is fitted to the data for 7 independent variables. The general model for step 1 is for$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_i$

The stepwise regression keeps on adding independent variables till no further independent variable can be added that gives significant t-values. In the question since there are 7 independent variables, the step-wise regression will be run till step 7 and t-test will be conducted to check the significance of each added variable.

The final model reached with step wise regression doesn’t account for interaction or higher order terms which might be more fitted for the data. Also since for every added variable, t-tests are conducted which might lead to the high probability of making type I or type II error.