Exam 15: Multiple Regression

One of the consequences of collinearity in multiple regression is biased estimates on the slope coefficients.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, the model that includes only $X _ { 1 } \text { and } X _ { 2 }$ should be among the appropriate models using the Mallow's Cp statistic.

Using the best-subsets approach to model building, models are being considered when their a) $C _ { p } > k$ b) $C _ { p } \leq k$ c) $C _ { p } > ( k + 1 )$ d) $C _ { p } \leq ( k + 1 )$

SCENARIO 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. -Referring to Scenario 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.

So that we can fit curves as well as lines by regression, we often use mathematical manipulations for converting one variable into a different form. These manipulations are called dummy variables.

SCENARIO 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. -Referring to Scenario 15-4, there is reason to suspect collinearity between some pairs of predictors.

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

SCENARIO 15-3 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 curvilinear model to this data. The results obtained by Microsoft Excel follow -Referring to Scenario 15-3, suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a curvilinear model that includes a linear term. If she used a level of significance of 0.01, she would decide that the linear model is sufficient.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, the value of the t test statistic for testing whether the quadratic term for the number of milking cows is statistically significant after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers is _____.

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

SCENARIO 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. -Referring to Scenario 15-4, the better model using a 5% level of significance derived from the "best" model above is a) $X _ { 1 }$ b) $X _ { 3 }$ c) $X _ { 1 } , X _ { 3 }$ d) $X _ { 1 } , X _ { 2 } , X _ { 3 }$

SCENARIO 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 $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ 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 Scenario 15-6, what is the value of the variance inflationary factor of Age?

In data mining where huge data sets are being explored to discover relationships among a large number of variables, the best-subsets approach is more practical than the stepwise regression approach.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, the p value of the t test statistic for testing whether the quadratic term for the number of laborers is statistically significant after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers is _____.

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, what is your decision on testing whether the quadratic term for the number of milking cows is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers?

The Cp statistic is used

SCENARIO 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 $X _ { j }$ 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 Scenario 15-5, what is the value of the variance inflationary factor of MPG?

The logarithm transformation can be used

SCENARIO 15-7-A You are the CEO of a dairy company. You are planning to expand milk production by purchasing additional cows, lands and hiring more workers. From the existing 50 farms owned by the company, you have collected data on total milk production (in liters), the number of milking cows, land size (in acres) and the number of laborers. The data are shown below and also available in the Excel file Scenario15-7-DataA.XLSX. You believe that the number of milking cows $\left( X _ { 1 } \right)$ , land size $\left( X _ { 2 } \right)$ and the number of laborers $\left( X _ { 3 } \right)$ are the best predictors for total milk production on any given farm. -Referring to Scenario 15-7-A, there is sufficient evidence to conclude that the quadratic term for the number of milking cows is statistically significant at the 10% level of significance after you have performed a multiple regression that includes the quadratic terms for the number of milking cows, land size and the number of laborers.

SCENARIO 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 $\left( X _ { 1 } \right)$ , the number of years of education received $\left( X _ { 2 } \right)$ , the number of years at the previous job $\left( X _ { 3 } \right)$ , a dummy variable for marital status ( $X _ { 4 } : 1 =$ married, $0 =$ otherwise), a dummy variable for head of household $\left( X _ { 5 } : 1 = \right.$ yes, $0 =$ no) and a dummy variable for management position $\left( X _ { 6 } : 1 = \right.$ yes, $0 =$ no $)$ . The coefficient of multiple determination $\left( R _ { j } ^ { 2 } \right)$ for the regression model using each of the 6 variables $X _ { j }$ 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 Scenario 15-6, the variable X3 should be dropped to remove collinearity?