Exam 15: Multiple Regression Model Building

The Variance Inflationary Factor (VIF) measures the correlation of the X variables with the Y variable.

TABLE 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 (R) for the regression model using each of the 5 variables Xⱼ 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 Table 15-5, what is the value of the variance inflationary factor of SUV?

The ________ (larger/smaller) the value of the Variance Inflationary Factor, the higher is the collinearity of the X variables.

TABLE 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 (R) for the regression model using each of the 5 variables Xⱼ 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 Table 15-5, what is the value of the variance inflationary factor of MPG?

The logarithm transformation can be used

Using the best-subsets approach to model building, models are being considered when their

Collinearity is present if the dependent variable is linearly related to one of the explanatory variables.

TABLE 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 "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." -Referring to Table 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. The p-value of the test statistic for the contribution of the curvilinear term is ________.

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

TABLE 15-1 A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it decreases as the price of the gem increases. However, experts hypothesize 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 + β₁X² + 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: -Referring to Table 15-1, what is the p-value associated with the test statistic for testing whether there is an upward curvature in the response curve relating the demand (Y) and the price (X)?

TABLE 15-1 A certain type of rare gem serves as a status symbol for many of its owners. In theory, for low prices, the demand increases and it decreases as the price of the gem increases. However, experts hypothesize 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 + β₁X² + 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: -Referring to Table 15-1, a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).

TABLE 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 (R) for the regression model using each of the 5 variables Xⱼ 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 Table 15-5, there is reason to suspect collinearity between some pairs of predictors based on the values of the variance inflationary factor.

A high value of R² significantly above 0 in multiple regression accompanied by insignificant t-values on all parameter estimates very often indicates a high correlation between independent variables in the model.

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

TABLE 15-2 -Referring to Table 15-2, is the overall model statistically adequate at a 0.05 level of significance for predicting sale price (Y)?

Collinearity is present when there is a high degree of correlation between independent variables.

The Cp statistic is used

The logarithm transformation can be used

TABLE 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 "centered" curvilinear model to this data. The results obtained by Microsoft Excel follow, where the dose (X) given has been "centered." -Referring to Table 15-3, 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 linear term is significant.

A real estate builder wishes to determine how house size (House) is influenced by family income (Income), family size (Size), and education of the head of household (School). House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is in years. The builder randomly selected 50 families and constructed the multiple regression model. The business literature involving human capital shows that education influences an individual's annual income. Combined, these may influence family size. With this in mind, what should the real estate builder be particularly concerned with when analyzing the multiple regression model?