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Exam 16: Multiple Regression Model Building

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Question 41

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

A) stepwise regression
B) forward selection
C) backward elimination
D) residual analysis

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Question 42

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Cook's Distance Statistic can be used to analyze the influence of individual data points.

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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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,there is reason to suspect collinearity between some pairs of predictors based on the values of the variance inflationary factor.

One use of VIF in multiple regression is deciding which variable to include in a 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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,the variable X₅ should be dropped to remove collinearity.

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. $\begin{array}{l|l|l|l|l|l|}\text { SUMMARY } & output& \\\hline \text { Regression } & \text { Statistics } & \\\hline \text { Multiple R } & & 0.747 \\\hline \text { R Square } & & 0.558 \\\hline \text { Adj. R Square } & & 0.478 \\\hline \text { Std. Error } & 363.1 \\\hline \text { Observations } & 14 \\\hline\\\hline \text { ANOVA } & & & & & \\\hline & \text { df } & \text { SS } & \text { MS } & F & \text { Signưf } F \\\hline \text { Regression } & 2 & 10344797 & 5172399 & 6.94 & 0.0110 \\\hline \text { Residual } & 11 & 8193929 & 744903 & & \\\hline \text { Total } & 13 & 18538726 & & & \\\hline\\\hline & \text { Coeff } & \text { StdErior } & \text { Stat } & P \text {-value } \\\hline \text { Intercept } & 1283.0 & 352.0 & 3.65 & 0.0040 \\\hline \text { CenDose } & 25.228 & 3.631 & 2.92 & 0.0140 \\\hline \text { CenDoseSq } & 0.8604 & 0.3722 & 2.31 & 0.0410\\\hline\end{array}$ 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 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.

Calculating Cook's Distance Statistic requires the use of matrix algebra.

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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,what is the value of the variance inflationary factor of Manager?

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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,the model that includes X₁,X₅ and X₆ should be selected using the adjusted r² statistic.

Which of the following is NOT 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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,the model that includes X₁,X₂,X₅ and X₆ should be among the appropriate models using the Mallow's C_p statistic.

The C_p statistic is used

One difference between simple regression and multiple regression is that,to avoid pitfalls in multiple regression,you must evaluate interaction terms.

Using the C_p statistic in model building,all models with C_p ≤ (k + 1)are equally good.

Applying a transformation to a data set,original values of Y of 1.6 and 4.2 become transformed values of 11.5 and 33.9.What transformation was used?

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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,what is the value of the Mallow's C_p statistic for the model that includes X₁,X₃,X₅ and X₆?

A microeconomist wants to determine how corporate sales are influenced by capital and wage spending by companies.She proceeds to randomly select 26 large corporations and record information in millions of dollars.A statistical analyst discovers that capital spending by corporations has a significant inverse relationship with wage spending.What should the microeconomist who developed this multiple regression model be particularly concerned with?

Collinearity is present if the dependent variable is linearly related to one of the explanatory 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: $\begin{array} { | l | l | l | l | } \hline \text { Model } & \text { R Square } & \text { Adj. R Square } & \text { Std. Error } \\\hline \text { X1X5Х6 } & 0.4568 & 0.4116 & 18.3534 \\\hline \times 1 \times 2 \times 5 \times 6 & 0.4697 & 0.4091 & 18.3919 \\\hline \times 1 \times 3 \times 5 \times 6 & 0.4691 & 0.4084 & 18.4023 \\\hline \times 1 \times 2 \times 3 \times 5 \times 6 & 0.4877 & 0.4123 & 18.3416 \\\hline \times 1 \times 2 \times 3 \times 4 \times 5 \times 6 & 0.4949 & 0.4030 & 18.4861 \\\hline\end{array}$ -Referring to Instruction 16-6,the model that includes X₁,X₃,X₅ and X₆ should be among the appropriate models using the Mallow's C_p statistic.

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