Exam 18: Model Building

The model $\tilde { y } = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 }$ is used whenever the statistician believes that,on average,y is linearly related to x₁ and x₂ and the predictor variables do not interact.

The model $\gamma _ { i } = \beta _ { 0 } + \beta _ { 1 } x _ { i } + \beta _ { 2 } x _ { i } ^ { 2 } + \ldots \ldots \ldots + \beta _ { y } x _ { i } ^ { y } + \varepsilon _ { i }$ is referred to as a polynomial model with one predictor variable.

Hockey Teams An avid hockey fan was in the process of examining the factors that determine the success or failure of hockey teams.He noticed that teams with many rookies and teams with many veterans seem to do quite poorly.To further analyze his beliefs he took a random sample of 20 teams and proposed a second-order model with one independent variable,average years of professional experience.The selected model is y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ ,where y = winning team's percentage,and x = average years of professional experience.The computer output is shown below. THE REGRESSION EQUATION IS y = 32.6 + 5.96x - .48x² Predictar Coef StDev Constant 32.6 19.3 1.689 5.96 2.41 2.473 -.48 .22 -2.182 S = 16.1 $\quad$ R - S q = 43.9 % ANALYSIS OF VARIANCE Source of Variation Regressian 2 3452 1726 6.663 Error 17 4404 259.059 Total 19 7856 -{Hockey Teams Narrative} Predict the winning percentage for a hockey team with an average of 6 years of professional experience.

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -In stepwise regression procedure,the independent variable with the largest F-statistic,or equally with the smallest p-value,is chosen as the first entering variable.The standard,also called the F-to-enter,is usually set at F equals:

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -If two indicator variables are used in a logistic regression model,then the nominal variable they represent has only two categories.

Suppose that the sample regression equation of a second-order model is given by $\tilde { y } = 2.50 + 0.15 x + 0.45 x ^ { 2 }$ .Then,the value 4.60 is the:

Incomes of Physicians An economist is analyzing the incomes of physicians (general practitioners,surgeons,and psychiatrists).He realizes that an important factor is the number of years of experience.However,he wants to know if there are differences among the three professional groups.He takes a random sample of 125 physicians and estimates the multiple regression model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = annual income (in $1,000),x₁ = years of experience,x₂ = 1 if physician and 0 if not,and x₃ = 1 if surgeons and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 71.65 + 2.07x₁ + 10.16x₂ - 7.44x₃ Predictar Codef StDev T Constant 71.65 18.56 3.860 2.07 .81 2.556 10.16 3.16 3.215 -7.44 2.85 -2.611 $S = 42.6 \quad R - S q = 30.9 \%$ ANALYSIS OF VARIANCE -{Incomes of Physicians Narrative} Estimate the annual income for a psychiatrist with 15 years of experience.

{Computer Training Narrative} Find the coefficient of determination of this simple linear model.What does this statistic tell you about the model?

We can incorporate any nominal variable into regression analysis by creating one or more dummy variables,also known as:

Motorcycle Fatalities A traffic consultant has analyzed the factors that affect the number of motorcycle fatalities.She has come to the conclusion that two important variables are the number of motorcycle and the number of cars.She proposed the model $\tilde { y } = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ (the second-order model with interaction),where y = number of annual fatalities per county,x₁ = number of motorcycles registered in the county (in 10,000),and x₂ = number of cars registered in the county (in 1000).The computer output (based on a random sample of 35 counties)is shown below: THE REGRESSION EQUATION IS $y = 69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ Predictor Coef StDev T Constant 69.7 41.3 1.688 11.3 5.1 2.216 7.61 2.55 2.984 -1.15 .64 -1.797 -.51 .20 -2.55 -.13 .10 -1.30 $S= 15.2 \quad \mathrm { R } - \mathrm { Sq } = 47.2 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 5 5959 1191.800 5.181 Error 29 6671 230.034 Total 34 12630 -{Motorcycle Fatalities Narrative} What is the multiple coefficient of determination? What does this statistic tell you about the model?

In order to represent 3 categories of a nominal variable,we need to create _______________ indicator variables.

In general,to represent a nominal independent variable that has c possible categories,we would create (c -1)dummy variables.

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -In a stepwise regression procedure,if two independent variables are highly correlated,then:

Senior Medical Students A professor of Anatomy wanted to develop a multiple regression model to predict the students' grades in her fourth-year medical course.She decides that the two most important factors are the student's grade point average in the first three years and the student's major.She proposes the model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = Fourth-year medical course final score (out of 100),x₁ = G.P.A.in first three years (range from 0 to 12),x₂ = 1 if student's major is medicine and 0 if not,and x₃ = 1 if student's major is biology and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 9.14 + 6.73x₁ + 10.42x₂ + 5.16x₃ Predictor Coef StDev T Constant 9.14 7.10 1.287 6.73 1.91 3.524 10.42 4.16 2.505 5.16 3.93 1.313 $S = 15.0 \quad R - S q = 44.2 \%$ ANALYSIS OF VARIANCE Source of Variation df SS MS F Regression 3 17098 5699.333 25.386 Error 96 21553 224.510 Total 99 38651 -In general,on what basis are independent variables selected for entry into the equation during stepwise regression?

Incomes of Physicians An economist is analyzing the incomes of physicians (general practitioners,surgeons,and psychiatrists).He realizes that an important factor is the number of years of experience.However,he wants to know if there are differences among the three professional groups.He takes a random sample of 125 physicians and estimates the multiple regression model y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₃ + $\varepsilon$ ,where y = annual income (in $1,000),x₁ = years of experience,x₂ = 1 if physician and 0 if not,and x₃ = 1 if surgeons and 0 if not.The computer output is shown below. THE REGRESSION EQUATION IS y = 71.65 + 2.07x₁ + 10.16x₂ - 7.44x₃ Predictar Codef StDev T Constant 71.65 18.56 3.860 2.07 .81 2.556 10.16 3.16 3.215 -7.44 2.85 -2.611 $S = 42.6 \quad R - S q = 30.9 \%$ ANALYSIS OF VARIANCE -{Incomes of Physicians Narrative} Is there enough evidence at the 10% significance level to conclude that surgeons earn less on average than psychiatrists?

Motorcycle Fatalities A traffic consultant has analyzed the factors that affect the number of motorcycle fatalities.She has come to the conclusion that two important variables are the number of motorcycle and the number of cars.She proposed the model $\tilde { y } = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 2 } + \beta _ { 3 } x _ { 1 } ^ { 2 } + \beta _ { 4 } x _ { 2 } ^ { 2 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \varepsilon$ (the second-order model with interaction),where y = number of annual fatalities per county,x₁ = number of motorcycles registered in the county (in 10,000),and x₂ = number of cars registered in the county (in 1000).The computer output (based on a random sample of 35 counties)is shown below: THE REGRESSION EQUATION IS $y = 69.7 + 11.3 x _ { 1 } + 7.61 x _ { 2 } - 1.15 x _ { 1 } ^ { 2 } - 0.51 x _ { 2 } ^ { 2 } - 0.13 x _ { 1 } x _ { 2 }$ Predictor Coef StDev T Constant 69.7 41.3 1.688 11.3 5.1 2.216 7.61 2.55 2.984 -1.15 .64 -1.797 -.51 .20 -2.55 -.13 .10 -1.30 $S= 15.2 \quad \mathrm { R } - \mathrm { Sq } = 47.2 \%$ ANALYSIS OF VARIANCE Source of Variation Repressian 5 5959 1191.800 5.181 Error 29 6671 230.034 Total 34 12630 -{Motorcycle Fatalities Narrative} What does the coefficient of $x _ { 1 } ^ { 2 }$ tell you about the model?

Silver Prices An economist is in the process of developing a model to predict the price of silver.She believes that the two most important variables are the price of a barrel of oil (x₁)and the interest rate (x₂).She proposes the first-order model with interaction: y = $\beta$ ₀ + $\beta$ ₁x₁ + $\beta$ ₂x₂ + $\beta$ ₃x₁x₃ + $\varepsilon$ .A random sample of 20 daily observations was taken.The computer output is shown below. THE REGRESSION EQUATION IS y = 115.6 + 22.3x₁ + 14.7x₂ - 1.36x₁x₂ Predictar Coef StDev T Constant 115.6 78.1 1.480 22.3 7.1 3.141 14.7 6.3 2.333 -1.36 .52 -2.615 $S= 20.9 \quad R - S q = 55.4 \%$ ANALYSIS OF VARIANCE Source of Variation Regressian 3 8661 2887.0 6.626 Errorr 16 6971 435.7 Total 19 15632 -{Silver Prices Narrative} Interpret the coefficient b₂.

{Computer Training Narrative} Use the quadratic model to predict the value of y when x = 45.

Hockey Teams An avid hockey fan was in the process of examining the factors that determine the success or failure of hockey teams.He noticed that teams with many rookies and teams with many veterans seem to do quite poorly.To further analyze his beliefs he took a random sample of 20 teams and proposed a second-order model with one independent variable,average years of professional experience.The selected model is y = $\beta$ ₀ + $\beta$ ₁x + $\beta$ ₂x² + $\varepsilon$ ,where y = winning team's percentage,and x = average years of professional experience.The computer output is shown below. THE REGRESSION EQUATION IS y = 32.6 + 5.96x - .48x² Predictar Coef StDev Constant 32.6 19.3 1.689 5.96 2.41 2.473 -.48 .22 -2.182 S = 16.1 $\quad$ R - S q = 43.9 % ANALYSIS OF VARIANCE Source of Variation Regressian 2 3452 1726 6.663 Error 17 4404 259.059 Total 19 7856 -{Hockey Teams Narrative} Do these results allow us to conclude at the 5% significance level that the model is useful in predicting the team's winning percentage?

Dummy variables are variables that can take on only two values (namely,0 or 1)and that are used to indicate the absence or presence of a particular nominal characteristic.