Exam 16: Multiple Regression Model Building

One of the consequences of collinearity in multiple regression is inflated standard errors in some or all of the estimated slope coefficients.

One transformation that may help overcome violations to the assumption of equal variance is the square root transformation.

Instruction 16-3 To explain personal consumption (CONS) measured in dollars, data is collected for INC: personal income in dollars CRDTLIM: \1 plus the credit limit in dollars available to the individual APR: average annualised percentage interest rate for borrowing for the individual per person advertising expenditure in dollars by manufacturers in the city where the ADVT: GENDER: gender of the individual; 1 if female, 0 if male A regression analysis was performed with CONS as the dependent variable and log(CRDTLIM), log(APR), log(ADVT) and GENDER as the independent variables. The estimated model was = 2.28 ? 0.29 log(CRDTLIM) + 5.77 log(APR) + 2.35 log(ADVT) + 0.39 GENDER -Referring to Instruction 16-3,and noting that ADVT has been transformed using the log transformation,what is the correct interpretation for the estimated coefficient for ADVT?

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

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 X₁ should be dropped to remove collinearity.

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: $Y = B _ { 0 } + B _ { 1 } X + B_ { 2 } X ^ { 2 } + \varepsilon$ where Y = demand (in thousands) and X = retail price per carat. SUMMARY OUTPUT Regression Statistics Multiple R 0.994 R Square 0.988 Std. Error 12.42 Observations 12 ANOVA df SS MS F Signif F Regression 2 115,145 57,573 373 0.0001 Residual 9 1,388 154 Total 11 116,533 Coeff Std. Error t Stat P -Value Intercept 286.42 9.66 29.64 0.0001 Price -0.31 0.06 -5.14 0.0006 Price Sq 0.000067 0.00007 0.95 0.3647 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 Instruction 16-1,a more parsimonious simple linear model is likely to be statistically superior to the fitted curvilinear for predicting sale price (Y).

The goals of model building are to find a good model with the fewest independent variables that is easier to interpret and has lower probability of collinearity.

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?

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

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

Instruction 16-3 To explain personal consumption (CONS) measured in dollars, data is collected for INC: personal income in dollars CRDTLIM: \1 plus the credit limit in dollars available to the individual APR: average annualised percentage interest rate for borrowing for the individual per person advertising expenditure in dollars by manufacturers in the city where the ADVT: GENDER: gender of the individual; 1 if female, 0 if male A regression analysis was performed with CONS as the dependent variable and log(CRDTLIM), log(APR), log(ADVT) and GENDER as the independent variables. The estimated model was = 2.28 ? 0.29 log(CRDTLIM) + 5.77 log(APR) + 2.35 log(ADVT) + 0.39 GENDER -Referring to Instruction 16-3,and noting that ADVT has been transformed using the log transformation,what is the correct interpretation for the estimated coefficient for APR?

Two simple regression models were used to predict a single dependent variable.Both models were highly significant,but when two independent variables were placed in the same multiple regression model for the dependent variable,R² did not increase substantially and the parameter estimates for the model were not significantly different from 0.This is probably an example of collinearity.

To avoid pitfalls in multiple regression,it is important to evaluate a residual plot for at least one independent variable.

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 relationship between time and dose.The p-value of the test is _______.

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 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.

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?

The C_p statistic is used

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,given a quadratic relationship between sale price (Y)and property size (X1),what test should be used to test whether the curves differ from cove and non-cove properties?

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,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?

Collinearity will result in excessively low standard errors of the parameter estimates reported in the regression output.