Exam 16: Multiple Regression Model Building

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.

Instruction 16-2 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. OUTPU? Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 863.1 Observations 14 ANOVA df Ss MS F Signif F Regression 2 10,344,797 5,172,399 6.94 0.0110 Residual 11 8,193,929 744,903 Total 13 18,538,726 Coeff Std. Error t stat 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 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 there is a significant difference between a linear model and a quadratic curvilinear model that includes a linear term.The p-value of the test statistic for the contribution of the quadratic curvilinear term is _______.

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 Cargo Vol?

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

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

The logarithm transformation can be used

Instruction 16-2 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. OUTPU? Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 863.1 Observations 14 ANOVA df Ss MS F Signif F Regression 2 10,344,797 5,172,399 6.94 0.0110 Residual 11 8,193,929 744,903 Total 13 18,538,726 Coeff Std. Error t stat 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 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 16-2,suppose the chemist decides to use an F test to determine if there is a significant quadratic curvilinear relationship between time and dose.The value of the test statistic is _______.

Using the hat matrix elements h_i to determine influential points in a multiple regression model with k independent variable and n observations,X_i is an influential point if

Which of the following is used to determine observations that have influential effect on the fitted model?

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 0.4568 0.4116 18.3534 0.4697 0.4091 18.3919 0.4691 0.4084 18.4023 0.4877 0.4123 18.3416 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?

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 0.4568 0.4116 18.3534 0.4697 0.4091 18.3919 0.4691 0.4084 18.4023 0.4877 0.4123 18.3416 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the variance inflationary factor of Job Yr?

An independent variable X_j is considered highly correlated with the other independent variables if

In stepwise regression,an independent variable is not allowed to be removed from the model once it has entered into the model.

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 0.4568 0.4116 18.3534 0.4697 0.4091 18.3919 0.4691 0.4084 18.4023 0.4877 0.4123 18.3416 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,the model that includes X₁,X₂,X₃,X₅ and X₆ should be selected using the adjusted r² statistic.

Instruction 16-5 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 = β₀ + β₁X₁ + β₂X₂ + β₃X₁X₂ + β₄+ β₅X₂ + 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: Note: Std. Error = Standard Error -Referring to Instruction 16-5,is the overall model statistically adequate at a 0.05 level of significance for predicting sale price (Y)?

Instruction 16-2 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. OUTPU? Regression Statistics Multiple R 0.747 R Square 0.558 Adj. R Square 0.478 Std. Error 863.1 Observations 14 ANOVA df Ss MS F Signif F Regression 2 10,344,797 5,172,399 6.94 0.0110 Residual 11 8,193,929 744,903 Total 13 18,538,726 Coeff Std. Error t stat 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 Note: Adj. R Square = Adjusted R Square; Std. Error = Standard Error -Referring to Instruction 16-2,suppose the chemist decides to use a t test to determine if there is a significant difference between a linear model and a quadratic model that includes a linear term.If she used a level of significance of 0.05,she would decide that the linear model is sufficient.

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.

As a project for his business statistics class,a student examined the factors that determined parking metre 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 hypothesised is Y_i = $\alpha$ + $\beta$ ₁x_1i + $\beta$ ₂x_2i + $\beta$ ₃x_3i + $\beta$ _i Where Y is the metre price X₁ is the number of blocks to the quad X₂ is a dummy variable that takes the value 1 if the metre is located in downtown And off-campus and the value 0 otherwise X₃ is a dummy variable that takes the value 1 if the metre is located outside of Downtown and off-campus,and the value 0 otherwise Suppose that whether the metre is located on campus is an important explanatory factor.Why should the variable that depicts this attribute not be included in the model?

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 0.4568 0.4116 18.3534 0.4697 0.4091 18.3919 0.4691 0.4084 18.4023 0.4877 0.4123 18.3416 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,the variable X₃ should be dropped to remove collinearity.

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 0.4568 0.4116 18.3534 0.4697 0.4091 18.3919 0.4691 0.4084 18.4023 0.4877 0.4123 18.3416 0.4949 0.4030 18.4861 -Referring to Instruction 16-6,what is the value of the variance inflationary factor of Head of Household?