Exam 16: Multiple Regression Model Building

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

Instruction 16-7 What are the factors that determine the acceleration time (in sec. )from 0 to 60 kilometres 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.cm. EP: Engine power KPL: Kilometres per litre 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 (R²_j)for the regression model using each of the five 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 Instruction 16-7,what is the value of the variance inflationary factor of EP?

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the Mallow's C_p statistic for the model that includes X₁,X₂,X₃,X₅ and X₆?

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the Mallow's C_p statistic for the model that includes all the six independent variables?

For a model with 5 independent variables and data set with 50 observations,the numerator degrees of freedom for the Cook's Distance Statistic would be 4.

The logarithm transformation can be used

Instruction 16-5 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 quadratic model to this data.The results obtained by Microsoft Excel follow. SUMMARY output Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 363.1 Observations 14 ANOVA df SS MS F Signưf F Regression 2 10344797 5172399 6.94 0.0110 Residual 11 8193929 744903 Total 13 18538726 Coeff StdErior Stat P -value Intercept 1283.0 352.0 3.65 0.0040 CenDose 25.228 3.631 2.92 0.0140 CenDoseSq 0.8604 0.3722 2.31 0.0410 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 16-5,the prediction of time to relief for a person receiving a dose of the drug 10 units above the average dose ,is ________.

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,the model that includes all six independent variables should be selected using the Mallow's C_p statistic.

Instruction 16-3 In Hawaii,condemnation proceedings are under way to enable private citizens to own the property upon which their homes are built.Until recently,only estates were permitted to own land,and homeowners leased the land from the estate.In order to comply with the new law,a large Hawaiian estate wants to use regression analysis to estimate the fair market value of the land.The following model was fit to data collected for n = 20 properties,10 of which are located near a cove.Model 1: Y = β 0 + β 1X1 + β 2X2 + β 3X1X2 + β 4+ β 5X2 + ε where Y = Sale price of property in thousands of dollars X1 = Size of property in thousands of square metres X2 = 1 if property located near cove,0 if not Using the data collected for the 20 properties,the following partial output obtained from Microsoft Excel is shown: SUMMARY OUTPUT Regression Statistics Multiple R 0.985 R Square 0.970 Std. Error 9.5 Observations 20 ANOVA df SS MS F Signif F Regression 5 28324 5664 62.2 0.0001 Residual 14 1279 91 Total 19 29603 Coeff StdError t stat P-Value Intercept -32.1 35.7 -0.90 0.3834 Size 12.2 5.9 2.05 0.0594 Cove -104.3 53.5 -1.95 0.0715 Size Cove 17.0 8.5 1.99 0.0661 SizeSq -0.3 0.2 -1.28 0.2204 SizeSq -0.3 0.3 -1.13 0.2749 Note: Std.Error = Standard Error -Referring to Instruction 16-3,given a quadratic relationship between sale price (Y)and property size (X1),what null hypothesis would you test to determine whether the curves differ from cove and non-cove properties?

Instruction 16-5 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 quadratic model to this data.The results obtained by Microsoft Excel follow. SUMMARY output Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 363.1 Observations 14 ANOVA df SS MS F Signưf F Regression 2 10344797 5172399 6.94 0.0110 Residual 11 8193929 744903 Total 13 18538726 Coeff StdErior Stat P -value Intercept 1283.0 352.0 3.65 0.0040 CenDose 25.228 3.631 2.92 0.0140 CenDoseSq 0.8604 0.3722 2.31 0.0410 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 16-5,suppose the chemist decides to use an F test to determine if there is a significant quadratic relationship between time and dose.If she chooses to use a level of significance of 0.01 she would decide that there is a significant quadratic relationship.

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.

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the variance inflationary factor of Married?

Using the Studentised residuals t_i to determine influential points in a multiple regression model with k independent variable and n observations and letting t_n-k-2 denote the upper critical value of a two-tail t test with a 0.10 level of significance,X_i is an influential point if

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,the variable X₃ should be dropped to remove collinearity.

Instruction 16-2 A certain type of rare gem serves as a status symbol for many of its owners.In theory,for low prices,the demand decreases as the price of the gem increases.However,experts hypothesise that when the gem is valued at very high prices,the demand increases with price due to the status owners believe they gain in obtaining the gem.Thus,the model proposed to best explain the demand for the gem by its price is the quadratic model: Y = β₀ + β₁X + β₂X² + ^ε where Y = demand (in thousands)and X = retail price per carat. This model was fit to data collected for a sample of 12 rare gems of this type.A portion of the computer analysis obtained from Microsoft Excel is shown below: SUMMARY output Regression Statistics Multiple R 0.994 R Square 0.988 Std. Error 12.42 Observations 12 ANOVA dff SS MS F Siguif F Regression 2 115145 57573 373 0.0001 Residual 9 1388 154 Total 11 116533 Coeff StdError t Stat P-Value Intercept 286.42 9.66 29.64 0.0001 Price -0.31 0.06 -5.14 0.0006 Price Sq p.000067 p.00007 p.95 p.3647 Note: Std.Error = Standard Error -Referring to Instruction 16-2,what is the p-value associated with the test statistic for testing whether the quadratic term is necessary in fitting the response curve relating the demand (Y)and the price (X)?

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the variance inflationary factor of Edu?

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

Instruction 16-6 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 (R²_j)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 X1X5Х6 0.4568 0.4116 18.3534 \times1\times2\times5\times6 0.4697 0.4091 18.3919 \times1\times3\times5\times6 0.4691 0.4084 18.4023 \times1\times2\times3\times5\times6 0.4877 0.4123 18.3416 \times1\times2\times3\times4\times5\times6 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the variance inflationary factor of Job Y_r?

Instruction 16-5 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 quadratic model to this data.The results obtained by Microsoft Excel follow. SUMMARY output Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 363.1 Observations 14 ANOVA df SS MS F Signưf F Regression 2 10344797 5172399 6.94 0.0110 Residual 11 8193929 744903 Total 13 18538726 Coeff StdErior Stat P -value Intercept 1283.0 352.0 3.65 0.0040 CenDose 25.228 3.631 2.92 0.0140 CenDoseSq 0.8604 0.3722 2.31 0.0410 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 16-5,suppose the chemist decides to use a t test to determine if the linear term is significant.Using a level of significance of 0.05,she would decide that the quadratic model should include a linear term.

Instruction 16-5 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 quadratic model to this data.The results obtained by Microsoft Excel follow. SUMMARY output Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 363.1 Observations 14 ANOVA df SS MS F Signưf F Regression 2 10344797 5172399 6.94 0.0110 Residual 11 8193929 744903 Total 13 18538726 Coeff StdErior Stat P -value Intercept 1283.0 352.0 3.65 0.0040 CenDose 25.228 3.631 2.92 0.0140 CenDoseSq 0.8604 0.3722 2.31 0.0410 Note: Adj.R Square = Adjusted R Square;Std.Error = Standard Error -Referring to Instruction 16-5,suppose the chemist decides to use an F test to determine if there is a significant quadratic 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.