Exam 15: Multiple Regression Model Building

TABLE 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Let Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination () of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Adjusted Model Variables R. Square R. Square Std. Error 1 1 3.05 3 0.6024 0.593 10.5787 2 12 3.66 3 0.6145 0.597 10.535 3 123 4.00 4 0.6288 0.6029 10.4570 4 13 2.00 3 0.6288 0.6119 10.3375 5 2 67.35 3 0.0474 0.026 16.375 6 23 64.30 3 0.0910 0.0497 16.1768 7 3 62.33 3 0.0907 0.0705 15.9984 Following is the residual plot for % Attendance: Following is the output of several multiple regression models: Model (I): Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 2.88-09 -957.340 -549.5050 \% Attendance 8.5014 1.0771 7.8929 6.73-1 6.3292 10.6735 Salary 6.85-07 0.0006 0.0011 0.9991 -0.0011 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 Model (II): Coefficients Standard Error t Stat P-value Intercept -753.4086 99.1451 -7.5991 0.0000 \% Attendance 8.5014 1.0645 7.9862 0.0000 Spending 0.0060 0.0034 1.767 0.0840 Model (III): df SS MS F Significance F Regression 2 8162.9429 4081.4714 39.8709 0.0000 Residual 44 4504.1635 102.3674 Total 46 12667.1064 Coefficients Standard Error t Stat P-value Intercept 6672.8367 3267.7349 2.0420 0.0472 \% Attendance -150.5694 69.9519 -2.1525 0.0369 \% Attendance Squared 0.8532 0.3743 2.2792 0.0276 -Referring to Table 15-4, what is the p-value of the test statistic to determine whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance?

Using the best-subsets approach to model building, models are being considered when their

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y) and the independent variables are the age of the worker (X₁), the number of years of education received (X₂), the number of years at the previous job (X₃), a dummy variable for marital status (X₄: 1 = married, 0 = otherwise), a dummy variable for head of household (X₅: 1 = yes, 0 = no) and a dummy variable for management position (X₆: 1 = yes, 0 = no). The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.2628, 0.1240, 0.2404, 0.3510, 0.3342 and 0.0993. The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Table 15-6, the model that includes X?, X? and X? should be selected using the adjusted r² statistic.

Which of the following will not change a nonlinear model into a linear model?

TABLE 15-5 What are the factors that determine the acceleration time (in sec.) from 0 to 60 miles per hour of a car? Data on the following variables for 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 5 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, what is the value of the variance inflationary factor of MPG?

The parameter estimates are biased when collinearity is present in a multiple regression equation.

TABLE 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Let Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination () of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Adjusted Model Variables R. Square R. Square Std. Error 1 1 3.05 3 0.6024 0.593 10.5787 2 12 3.66 3 0.6145 0.597 10.535 3 123 4.00 4 0.6288 0.6029 10.4570 4 13 2.00 3 0.6288 0.6119 10.3375 5 2 67.35 3 0.0474 0.026 16.375 6 23 64.30 3 0.0910 0.0497 16.1768 7 3 62.33 3 0.0907 0.0705 15.9984 Following is the residual plot for % Attendance: Following is the output of several multiple regression models: Model (I): Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 2.88-09 -957.340 -549.5050 \% Attendance 8.5014 1.0771 7.8929 6.73-1 6.3292 10.6735 Salary 6.85-07 0.0006 0.0011 0.9991 -0.0011 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 Model (II): Coefficients Standard Error t Stat P-value Intercept -753.4086 99.1451 -7.5991 0.0000 \% Attendance 8.5014 1.0645 7.9862 0.0000 Spending 0.0060 0.0034 1.767 0.0840 Model (III): df SS MS F Significance F Regression 2 8162.9429 4081.4714 39.8709 0.0000 Residual 44 4504.1635 102.3674 Total 46 12667.1064 Coefficients Standard Error t Stat P-value Intercept 6672.8367 3267.7349 2.0420 0.0472 \% Attendance -150.5694 69.9519 -2.1525 0.0369 \% Attendance Squared 0.8532 0.3743 2.2792 0.0276 -Referring to Table 15-4, the residual plot suggests that a nonlinear model on % attendance may be a better model.

In multiple regression, the ________ procedure permits variables to enter and leave the model at different stages of its development.

TABLE 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Let Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination () of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Adjusted Model Variables R. Square R. Square Std. Error 1 1 3.05 3 0.6024 0.593 10.5787 2 12 3.66 3 0.6145 0.597 10.535 3 123 4.00 4 0.6288 0.6029 10.4570 4 13 2.00 3 0.6288 0.6119 10.3375 5 2 67.35 3 0.0474 0.026 16.375 6 23 64.30 3 0.0910 0.0497 16.1768 7 3 62.33 3 0.0907 0.0705 15.9984 Following is the residual plot for % Attendance: Following is the output of several multiple regression models: Model (I): Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 2.88-09 -957.340 -549.5050 \% Attendance 8.5014 1.0771 7.8929 6.73-1 6.3292 10.6735 Salary 6.85-07 0.0006 0.0011 0.9991 -0.0011 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 Model (II): Coefficients Standard Error t Stat P-value Intercept -753.4086 99.1451 -7.5991 0.0000 \% Attendance 8.5014 1.0645 7.9862 0.0000 Spending 0.0060 0.0034 1.767 0.0840 Model (III): df SS MS F Significance F Regression 2 8162.9429 4081.4714 39.8709 0.0000 Residual 44 4504.1635 102.3674 Total 46 12667.1064 Coefficients Standard Error t Stat P-value Intercept 6672.8367 3267.7349 2.0420 0.0472 \% Attendance -150.5694 69.9519 -2.1525 0.0369 \% Attendance Squared 0.8532 0.3743 2.2792 0.0276 -Referring to Table 15-4, which of the following models should be taken into consideration using the Mallows' Cp statistic?

The Cp statistic is used

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y) and the independent variables are the age of the worker (X₁), the number of years of education received (X₂), the number of years at the previous job (X₃), a dummy variable for marital status (X₄: 1 = married, 0 = otherwise), a dummy variable for head of household (X₅: 1 = yes, 0 = no) and a dummy variable for management position (X₆: 1 = yes, 0 = no). The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.2628, 0.1240, 0.2404, 0.3510, 0.3342 and 0.0993. The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Table 15-6, the variable X? should be dropped to remove collinearity.

TABLE 15-4 The superintendent of a school district wanted to predict the percentage of students passing a sixth-grade proficiency test. She obtained the data on percentage of students passing the proficiency test (% Passing), daily mean of the percentage of students attending class (% Attendance), mean teacher salary in dollars (Salaries), and instructional spending per pupil in dollars (Spending) of 47 schools in the state. Let Y = % Passing as the dependent variable, X₁ = % Attendance, X₂ = Salaries and X₃ = Spending. The coefficient of multiple determination () of each of the 3 predictors with all the other remaining predictors are, respectively, 0.0338, 0.4669, and 0.4743. The output from the best-subset regressions is given below: Adjusted Model Variables R. Square R. Square Std. Error 1 1 3.05 3 0.6024 0.593 10.5787 2 12 3.66 3 0.6145 0.597 10.535 3 123 4.00 4 0.6288 0.6029 10.4570 4 13 2.00 3 0.6288 0.6119 10.3375 5 2 67.35 3 0.0474 0.026 16.375 6 23 64.30 3 0.0910 0.0497 16.1768 7 3 62.33 3 0.0907 0.0705 15.9984 Following is the residual plot for % Attendance: Following is the output of several multiple regression models: Model (I): Coefficients Standard Error t Stat P-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 2.88-09 -957.340 -549.5050 \% Attendance 8.5014 1.0771 7.8929 6.73-1 6.3292 10.6735 Salary 6.85-07 0.0006 0.0011 0.9991 -0.0011 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 Model (II): Coefficients Standard Error t Stat P-value Intercept -753.4086 99.1451 -7.5991 0.0000 \% Attendance 8.5014 1.0645 7.9862 0.0000 Spending 0.0060 0.0034 1.767 0.0840 Model (III): df SS MS F Significance F Regression 2 8162.9429 4081.4714 39.8709 0.0000 Residual 44 4504.1635 102.3674 Total 46 12667.1064 Coefficients Standard Error t Stat P-value Intercept 6672.8367 3267.7349 2.0420 0.0472 \% Attendance -150.5694 69.9519 -2.1525 0.0369 \% Attendance Squared 0.8532 0.3743 2.2792 0.0276 -Referring to Table 15-4, the null hypothesis should be rejected when testing whether the quadratic effect of daily average of the percentage of students attending class on percentage of students passing the proficiency test is significant at a 5% level of significance.

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y) and the independent variables are the age of the worker (X₁), the number of years of education received (X₂), the number of years at the previous job (X₃), a dummy variable for marital status (X₄: 1 = married, 0 = otherwise), a dummy variable for head of household (X₅: 1 = yes, 0 = no) and a dummy variable for management position (X₆: 1 = yes, 0 = no). The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.2628, 0.1240, 0.2404, 0.3510, 0.3342 and 0.0993. The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Table 15-6, what is the value of the variance inflationary factor of Head?

A regression diagnostic tool used to study the possible effects of collinearity is

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y) and the independent variables are the age of the worker (X₁), the number of years of education received (X₂), the number of years at the previous job (X₃), a dummy variable for marital status (X₄: 1 = married, 0 = otherwise), a dummy variable for head of household (X₅: 1 = yes, 0 = no) and a dummy variable for management position (X₆: 1 = yes, 0 = no). The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.2628, 0.1240, 0.2404, 0.3510, 0.3342 and 0.0993. The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Table 15-6, the variable X? should be dropped to remove collinearity.

The Variance Inflationary Factor (VIF)measures the

Collinearity is present if the dependent variable is linearly related to one of the explanatory variables.

A high value of R² significantly above 0 in multiple regression accompanied by insignificant t-values on all parameter estimates very often indicates a high correlation between independent variables in the model.

TABLE 15-6 Given below are results from the regression analysis on 40 observations where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Y) and the independent variables are the age of the worker (X₁), the number of years of education received (X₂), the number of years at the previous job (X₃), a dummy variable for marital status (X₄: 1 = married, 0 = otherwise), a dummy variable for head of household (X₅: 1 = yes, 0 = no) and a dummy variable for management position (X₆: 1 = yes, 0 = no). The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.2628, 0.1240, 0.2404, 0.3510, 0.3342 and 0.0993. The partial results from best-subset regression are given below: Model R Square Adj. R Square Std. Error X1X5X6 0.4568 0.4116 18.3534 X1X2X5X6 0.4697 0.4091 18.3919 X1X3X5X6 0.4691 0.4084 18.4023 X1X2X3X5X6 0.4877 0.4123 18.3416 X1X2X3X4X5X6 0.4949 0.4030 18.4861 -Referring to Table 15-6, the variable X? should be dropped to remove collinearity.

TABLE 15-5 What are the factors that determine the acceleration time (in sec.) from 0 to 60 miles per hour of a car? Data on the following variables for 171 different vehicle models were collected: Accel Time: Acceleration time in sec. Cargo Vol: Cargo volume in cu. ft. HP: Horsepower MPG: Miles per gallon SUV: 1 if the vehicle model is an SUV with Coupe as the base when SUV and Sedan are both 0 Sedan: 1 if the vehicle model is a sedan with Coupe as the base when SUV and Sedan are both 0 The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 5 variables Xⱼ as the dependent variable and all other X variables as independent variables are, respectively, 0.7461, 0.5676, 0.6764, 0.8582, 0.6632. -Referring to Table 15-5, there is reason to suspect collinearity between some pairs of predictors based on the values of the variance inflationary factor.