1 Chapter 20 Model Building

Introduction Regression analysis is one of the most commonly used techniques in statistics. It is considered powerful for several reasons: –It can cover a variety of mathematical models linear relationships. non - linear relationships. nominal independent variables. –It provides efficient methods for model building

3 Selecting a Model Several models have been introduced. How do we select the right model? Selecting a model: –Use your knowledge of the problem (variables involved and the nature of the relationship between them) to select a model. –Test the model using statistical techniques.

Nominal Independent Variables In many real-life situations one or more independent variables are nominal. Including nominal variables in a regression analysis model is done via indicator variables. An indicator variable (I) can assume one out of two values, “zero” or “one”. 1 if a first condition out of two is met 0 if a second condition out of two is met I= 1 if data were collected before if data were collected after if the temperature was below 50 o 0 if the temperature was 50 o or more 1 if a degree earned is in Finance 0 if a degree earned is not in Finance

5 Nominal Independent Variables; Example: Auction Car Price (II) Example revised (Xm18-02a)Xm18-02a –Recall: A car dealer wants to predict the auction price of a car. –The dealer believes now that odometer reading and the car color are variables that affect a car’s price. –Three color categories are considered: White Silver Other colors Note: Color is a nominal variable.

6 Example revised (Xm18-02b)Xm18-02b I 1 = 1 if the color is white 0 if the color is not white I 2 = 1 if the color is silver 0 if the color is not silver The category “Other colors” is defined by: I 1 = 0; I 2 = 0 Nominal Independent Variables; Example: Auction Car Price (II)

7 Note: To represent the situation of three possible colors we need only two indicator variables. Conclusion: To represent a nominal variable with m possible categories, we must create m-1 indicator variables. How Many Indicator Variables?

8 Solution –the proposed model is y =  0 +  1 (Odometer) +  2 I 1 +  3 I 2 +  –The data White car Other color Silver color Nominal Independent Variables; Example: Auction Car Price

9 Odometer Price Price = (Odometer) (0) (1) Price = (Odometer) (1) (0) Price = (Odometer) (0) + 148(0) (Odometer) (Odometer) (Odometer) The equation for an “other color” car. The equation for a white color car. The equation for a silver color car. From Excel (Xm18-02b) we get the regression equationXm18-02b PRICE = (Odometer)+90.48(I-1) (I-2) Example: Auction Car Price The Regression Equation

10 From Excel we get the regression equation PRICE = (Odometer)+90.48(I-1) (I-2) A white car sells, on the average, for $90.48 more than a car of the “Other color” category A silver color car sells, on the average, for $ more than a car of the “Other color” category. For one additional mile the auction price decreases by 5.55 cents. Example: Auction Car Price The Regression Equation

11 There is insufficient evidence to infer that a white color car and a car of “other color” sell for a different auction price. There is sufficient evidence to infer that a silver color car sells for a larger price than a car of the “other color” category. Xm18-02b Example: Auction Car Price The Regression Equation

12 Recall: The Dean wanted to evaluate applications for the MBA program by predicting future performance of the applicants. The following three predictors were suggested: –Undergraduate GPA –GMAT score –Years of work experience It is now believed that the type of undergraduate degree should be included in the model. Nominal Independent Variables; Example: MBA Program Admission (MBA II) MBA IIMBA II Note: The undergraduate degree is nominal data.

13 Nominal Independent Variables; Example: MBA Program Admission (II) I 1 = 1 if B.A. 0 otherwise I 2 = 1 if B.B.A 0 otherwise The category “Other group” is defined by: I 1 = 0; I 2 = 0; I 3 = 0 I 3 = 1 if B.Sc. or B.Eng. 0 otherwise

14 Nominal Independent Variables; Example: MBA Program Admission (II) MBA-II

Applications in Human Resources Management: Pay-Equity Pay-equity can be handled in two different forms: –Equal pay for equal work –Equal pay for work of equal value. Regression analysis is extensively employed in cases of equal pay for equal work.

16 Human Resources Management: Pay-Equity Example 20.3 (Xm20-03)Xm20-03 –Is there sex discrimination against female managers in a large firm? –A random sample of 100 managers was selected and data were collected as follows: Annual salary Years of education Years of experience Gender

17 Solution –Construct the following multiple regression model: y =  0 +  1 Education +  2 Experience +  3 Gender +  –Note the nature of the variables: Education – Interval Experience – Interval Gender – Nominal (Gender = 1 if male; =0 otherwise). Human Resources Management: Pay-Equity

18 Solution – Continued (Xm20-03)Xm20-03 Human Resources Management: Pay-Equity Analysis and Interpretation The model fits the data quite well. The model is very useful. Experience is a variable strongly related to salary. There is no evidence of sex discrimination.