Exam 15: Multiple Regression Model Building

TABLE 15-9 Many factors determine the attendance at Major League Baseball games. These factors can include when the game is played, the weather, the opponent, whether or not the team is having a good season, and whether or not a marketing promotion is held. Data from 80 games of the Kansas City Royals for the following variables are collected. ATTENDANCE = Paid attendance for the game TEMP = High temperature for the day WIN% = Team's winning percentage at the time of the game OPWIN% = Opponent team's winning percentage at the time of the game WEEKEND - 1 if game played on Friday, Saturday or Sunday; 0 otherwise PROMOTION - 1 = if a promotion was held; 0 = if no promotion was held The regression results using attendance as the dependent variable and the remaining five variables as the independent variables are presented below. Regression Statistics Multiple R 0.5487 R Square 0.3011 Adjusted R Square 0.2538 Standard Error 6442.4456 Observations 80 ANOVA SS MS F Significance Regression 5 1322911703.0671 264582340.6134 6.3747 0.0001 Residual 74 3071377751.1204 41505104.7449 Total 79 4394289454.1875 Coefficients Standard Error t Stat p-value Intercept -3862.4808 6180.9452 -0.6249 0.5340 Temp 51.7031 62.9439 0.8214 0.4140 Win\% 21.1085 16.2338 1.3003 0.1975 OpWin\% 11.3453 6.4617 1.7558 0.0833 Weekend 367.5377 2786.2639 0.1319 0.8954 Promotion 6927.8820 2784.3442 2.4882 0.0151 The coefficient of multiple determination ( R ² j) of each of the 5 predictors with all the other remaining predictors are, respectively, 0.2675, 0.3101, 0.1038, 0.7325, and 0.7308. -Referring to Table 15-9, there is enough evidence to conclude that TEMP makes a significant contribution to the regression model in the presence of the other independent variables at a 5% level of significance.

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

The Variance Inflationary Factor (VIF) measures the correlation of the X variables with the Y variable.

TABLE 15-9 Many factors determine the attendance at Major League Baseball games. These factors can include when the game is played, the weather, the opponent, whether or not the team is having a good season, and whether or not a marketing promotion is held. Data from 80 games of the Kansas City Royals for the following variables are collected. ATTENDANCE = Paid attendance for the game TEMP = High temperature for the day WIN% = Team's winning percentage at the time of the game OPWIN% = Opponent team's winning percentage at the time of the game WEEKEND - 1 if game played on Friday, Saturday or Sunday; 0 otherwise PROMOTION - 1 = if a promotion was held; 0 = if no promotion was held The regression results using attendance as the dependent variable and the remaining five variables as the independent variables are presented below. Regression Statistics Multiple R 0.5487 R Square 0.3011 Adjusted R Square 0.2538 Standard Error 6442.4456 Observations 80 ANOVA SS MS F Significance Regression 5 1322911703.0671 264582340.6134 6.3747 0.0001 Residual 74 3071377751.1204 41505104.7449 Total 79 4394289454.1875 Coefficients Standard Error t Stat p-value Intercept -3862.4808 6180.9452 -0.6249 0.5340 Temp 51.7031 62.9439 0.8214 0.4140 Win\% 21.1085 16.2338 1.3003 0.1975 OpWin\% 11.3453 6.4617 1.7558 0.0833 Weekend 367.5377 2786.2639 0.1319 0.8954 Promotion 6927.8820 2784.3442 2.4882 0.0151 The coefficient of multiple determination ( R ² _j ) of each of the 5 predictors with all the other remaining predictors are, respectively, 0.2675, 0.3101, 0.1038, 0.7325, and 0.7308 -Referring to Table 15-9, which of the following assumptions is most likely violated based on the residual plot for OPWIN%?

TABLE 15- 8 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 average of the percentage of students attending class (% Attendance), average 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 (R ² _j) 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 X1 3.05 2 0.6024 0.5936 10.5787 2 X1X2 3.66 3 0.6145 0.5970 10.5350 3 X1X2X3 4.00 4 0.6288 0.6029 10.4570 4 X1X3 2.00 3 0.6288 0.6119 10.3375 5 X2 67.35 2 0.0474 0.0262 16.3755 6 X2X3 64.30 3 0.0910 0.0497 16.1768 7 X3 62.33 2 0.0907 0.0705 15.9984 Following is the residual plot for % Attendance: Following is the output of several multiple regression models: $\text {Model (I):}$ Coefficients Std Error Stat p-value Lower 95\% Upper 95\% Intercept -753.4225 101.1149 -7.4511 2.88-09 -957.3401 -549.5050 \% Attend 8.5014 1.0771 7.8929 6.73-10 6.3292 10.6735 Salary 6.85-07 0.0006 0.0011 0.9991 -0.0013 0.0013 Spending 0.0060 0.0046 1.2879 0.2047 -0.0034 0.0153 $\text {Model (II):}$ Coefficients Standard Error t Stat p -value Intercept -753.4086 99.1451 -7.5991 1.5291-09 \% Attendance 8.5014 1.0645 7.9862 4.223-10 Spending 0.0060 0.0034 1.7676 0.0840 $\text {Model (III):}$ d f SS MS F Significance F Regression 2 8162.9429 4081.4714 39.8708 1.3201-10 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 15-8, what are, respectively, the values of the variance inflationary factor of the 3 predictors?

Only when all three of the hat matrix elements hi, the Studentized deleted residuals ti and the Cook's distance statistic D_i reveal consistent result should an observation be removed from the regression analysis.

TABLE 15-7 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." SUMMARY OUTPUT Regression Statistics Multiple R 0.747 RSquare 0.558 Adjusted R Square 0.478 Standard Error 863.1 Observations 14 ANOVA df SS MS F Significance F Regression 2 10344797 5172399 6.94 0.0110 Residual 11 8193929 744903 Total 13 18538726 Coeff Std Error t Stut p -value Intercept 1283.0 352.0 3.65 0.0040 CenDose 25.228 8.631 2.92 0.0140 CenDoseSq 0.8604 0.3722 2.31 0.0410 -Referring to Table 15-7, the prediction of time to relief for a person receiving a dose of the drug 10 units above the average dose , is____ .

The_____ (larger/smaller) the value of the Variance Inflationary Factor, the higher is the collinearity of the X variables.

One of the consequences of collinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

TABLE 15-7 A chemist employed by a pharmaceutical firm has developed a muscle relaxant. She took a sample of 14 people suffering from extreme muscle constriction. She gave each a vial containing a dose (X) of the drug and recorded the time to relief (Y) measured in seconds for each. She fit a "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." SUMMARY OUTPUT Regression Statistics Multiple R 0.747 RSquare 0.558 Adjusted R Square 0.478 Standard Error 863.1 Observations 14 ANOVA df SS MS F Significance F Regression 2 10344797 5172399 6.94 0.0110 Residual 11 8193929 744903 Total 13 18538726 Coeff Std Error t Stut p -value Intercept 1283.0 352.0 3.65 0.0040 CenDose 25.228 8.631 2.92 0.0140 CenDoseSq 0.8604 0.3722 2.31 0.0410 -Referring to Table 15-7, suppose the chemist decides to use an F test to determine if there is a significant curvilinear relationship between time and dose. If she chooses to use a level of significance of 0.05, she would decide that there is a significant curvilinear relationship.

Which of the following is not used to determine observations that have influential effect on the fitted model?

TABLE 15-9 Many factors determine the attendance at Major League Baseball games. These factors can include when the game is played, the weather, the opponent, whether or not the team is having a good season, and whether or not a marketing promotion is held. Data from 80 games of the Kansas City Royals for the following variables are collected. ATTENDANCE = Paid attendance for the game TEMP = High temperature for the day WIN% = Team's winning percentage at the time of the game OPWIN% = Opponent team's winning percentage at the time of the game WEEKEND - 1 if game played on Friday, Saturday or Sunday; 0 otherwise PROMOTION - 1 = if a promotion was held; 0 = if no promotion was held The regression results using attendance as the dependent variable and the remaining five variables as the independent variables are presented below. Regression Statistics Multiple R 0.5487 R Square 0.3011 Adjusted R Square 0.2538 Standard Error 6442.4456 Observations 80 ANOVA SS MS F Significance Regression 5 1322911703.0671 264582340.6134 6.3747 0.0001 Residual 74 3071377751.1204 41505104.7449 Total 79 4394289454.1875 Coefficients Standard Error t Stat p-value Intercept -3862.4808 6180.9452 -0.6249 0.5340 Temp 51.7031 62.9439 0.8214 0.4140 Win\% 21.1085 16.2338 1.3003 0.1975 OpWin\% 11.3453 6.4617 1.7558 0.0833 Weekend 367.5377 2786.2639 0.1319 0.8954 Promotion 6927.8820 2784.3442 2.4882 0.0151 -Referring to Table 15-9, what is the correct interpretation for the estimated coefficient for TEMP?

The C_p statistic is used

As a project for his business statistics class, a student examined the factors that determined parking meter rates throughout the campus area. Data were collected for the price per hour of parking, blocks to the quadrangle, and one of the three jurisdictions: on campus, in downtown and off campus, or outside of downtown and off campus. The population regression model hypothesized is $Y_{i}=\alpha+\beta_{1} X_{1 i}+\beta_{2} X_{2 i}+\beta_{3} X_{3 i}+\varepsilon$ where Y is the meter price X₁ is the number of blocks to the quad X₂ is a dummy variable that takes the value 1 if the meter is located in downtown and off campus and the value 0 otherwise X₃ is a dummy variable that takes the value 1 if the meter is located outside of downtown and off campus, and the value 0 otherwise Suppose that whether the meter is located on campus is an important explanatory factor. Why should the variable that depicts this attribute not be included in the model?

TABLE 15-9 Many factors determine the attendance at Major League Baseball games. These factors can include when the game is played, the weather, the opponent, whether or not the team is having a good season, and whether or not a marketing promotion is held. Data from 80 games of the Kansas City Royals for the following variables are collected. ATTENDANCE = Paid attendance for the game TEMP = High temperature for the day WIN% = Team's winning percentage at the time of the game OPWIN% = Opponent team's winning percentage at the time of the game WEEKEND - 1 if game played on Friday, Saturday or Sunday; 0 otherwise PROMOTION - 1 = if a promotion was held; 0 = if no promotion was held The regression results using attendance as the dependent variable and the remaining five variables as the independent variables are presented below. Regression Statistics Multiple R 0.5487 R Square 0.3011 Adjusted R Square 0.2538 Standard Error 6442.4456 Observations 80 ANOVA SS MS F Significance Regression 5 1322911703.0671 264582340.6134 6.3747 0.0001 Residual 74 3071377751.1204 41505104.7449 Total 79 4394289454.1875 Coefficients Standard Error t Stat p-value Intercept -3862.4808 6180.9452 -0.6249 0.5340 Temp 51.7031 62.9439 0.8214 0.4140 Win\% 21.1085 16.2338 1.3003 0.1975 OpWin\% 11.3453 6.4617 1.7558 0.0833 Weekend 367.5377 2786.2639 0.1319 0.8954 Promotion 6927.8820 2784.3442 2.4882 0.0151 The coefficient of multiple determination ( R ² _j) of each of the 5 predictors with all the other remaining predictors are, respectively, 0.2675, 0.3101, 0.1038, 0.7325, and 0.7308. $j$ -Referring to Table 15-9, what is the correct interpretation for the estimated coefficient for PROMOTION?

The Variance Inflationary Factor (VIF) measures the

Using the Cook's distance statistic Di to determine influential points in a multiple regression model with k independent variable and n observations and letting Fv1,v 2 denote the critical value of an F distribution with v1 and v2 degrees of freedom at a 0.50 level of significance, Xi is an influential point if

Two simple regression models were used to predict a single dependent variable. Both models were highly significant, but when the two independent variables were placed in the same multiple regression model for the dependent variable, R2 did not increase substantially and the parameter estimates for the model were not significantly different from 0. This is probably an example of collinearity.

Collinearity is present when there is a high degree of correlation between the dependent variable and any of the independent variables.

TABLE 15-9 Many factors determine the attendance at Major League Baseball games. These factors can include when the game is played, the weather, the opponent, whether or not the team is having a good season, and whether or not a marketing promotion is held. Data from 80 games of the Kansas City Royals for the following variables are collected. ATTENDANCE = Paid attendance for the game TEMP = High temperature for the day WIN% = Team's winning percentage at the time of the game OPWIN% = Opponent team's winning percentage at the time of the game WEEKEND - 1 if game played on Friday, Saturday or Sunday; 0 otherwise PROMOTION - 1 = if a promotion was held; 0 = if no promotion was held The regression results using attendance as the dependent variable and the remaining five variables as the independent variables are presented below. Regression Statistics Multiple R 0.5487 R Square 0.3011 Adjusted R Square 0.2538 Standard Error 6442.4456 Observations 80 ANOVA SS MS F Significance Regression 5 1322911703.0671 264582340.6134 6.3747 0.0001 Residual 74 3071377751.1204 41505104.7449 Total 79 4394289454.1875 Coefficients Standard Error t Stat p-value Intercept -3862.4808 6180.9452 -0.6249 0.5340 Temp 51.7031 62.9439 0.8214 0.4140 Win\% 21.1085 16.2338 1.3003 0.1975 OpWin\% 11.3453 6.4617 1.7558 0.0833 Weekend 367.5377 2786.2639 0.1319 0.8954 Promotion 6927.8820 2784.3442 2.4882 0.0151 The coefficient of multiple determination ( R ² _j ) of each of the 5 predictors with all the other remaining predictors are, respectively, 0.2675, 0.3101, 0.1038, 0.7325, and 0.7308. -Referring to Table 15-9, what are, respectively, the values of the variance inflationary factor of the 5 predictors?