Question

Question: Bus Rapid Transit study. Bus Rapid Transit (BRT) is a rapidly growing trend in the provision of public transportation in America. The Center for Urban Transportation Research (CUTR) at the University of South Florida conducted a survey of BRT customers in Miami (Transportation Research Board Annual Meeting, January 2003). Data on the following variables (all measured on a 5-point scale, where 1 = very unsatisfied and 5 = very satisfied) were collected for a sample of over 500 bus riders: overall satisfaction with BRT (y), safety on bus (x₁), seat availability (x₂), dependability (x₃), travel time (x₄), cost (x₅), information/maps (x₆), convenience of routes (x₇), traffic signals (x₈), safety at bus stops (x₉), hours of service (x₁₀), and frequency of service (x₁₁). CUTR analysts used stepwise regression to model overall satisfaction (y).

a. How many models are fit at step 1 of the stepwise regression?

b. How many models are fit at step 2 of the stepwise regression?

c. How many models are fit at step 11 of the stepwise regression?

d. The stepwise regression selected the following eight variables to include in the model (in order of selection): x₁₁, x₄, x₂, x₇, x₁₀, x₁, x₉, and x₃. Write the equation for E(y) that results from stepwise regression.

e. The model, part d, resulted in R² = 0.677. Interpret this value.

f. Explain why the CUTR analysts should be cautious in concluding that the best model for E(y) has been found.

Short answer

Answer

a. 11 1-variable models are fitted.

b. 10 2-variable models are fitted.

c. 1 11-variable model is fitted.

d. Through the stepwise regression, 8 variables are selected. The equation for E(y) can be written as $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_5+{\beta }_7{x}_7+{\beta }_8{x}_8$.

e. The value of R² is 0.677 indicating that approximately 67% of the variation in the regression is explained by the model. 67% is a high value indicating that the model is a good fit for the data.

f. Precautions while using stepwise model - First, an extremely large number of t-tests have been conducted, leading to a high probability of making one or more Type I or Type II errors. Second, the stepwise model does not include any higher-order or interaction terms.

Step-by-step solution

1-variable model

Since there are 11 independent variables, k no of models are 1-variable models are fitted in step 1 of stepwise regression.

So, 11 1-variable models are fitted.

2-variable model

Since there are 11 independent variables, (k-1) no of models are 2-variable models are fitted in step 2 of stepwise regression.

Thus, 10 2-variable models are fitted.

11-variable model

Since there are 11 independent variables, (k-10) no of models are 3-variable models are fitted in step 3 of stepwise regression.

Hence, 1 11-variable model is fitted.

Equation of the stepwise model

Through the stepwise regression, 8 variables are selected. The equation for E(y) can be written as $E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_5+{\beta }_7{x}_7+{\beta }_8{x}_8$.

Interpretation of R2

The value of R² is 0.677 indicating that approximately 67% of the variation in the regression is explained by the model. Higher the value of R², the better fit the model is to the data. 67% is a high value indicating that the model is a good fit for the data.

Precautions while using stepwise model  

Precautions while using stepwise model -

First, an extremely large number of t-tests have been conducted, leading to a high probability of making one or more Type I or Type II errors. Second, the stepwise model does not include any higher-order or interaction terms.