# The Regression Equation  A predicted value on the DV in the bi-variate case is found with the following formula: Ŷ = a + B (X1)

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The Regression Equation  A predicted value on the DV in the bi-variate case is found with the following formula: Ŷ = a + B (X1)

For Multiple Regression Ŷ = a + B1(X1) + B2 (X2) + B3 (X3)

Example:  Income (Y) regressed on R’s education (IV1) and Father’s education (IV2).  Constant (a) = 15,000  B1 (education) = 125  B2 (fathers educ) =25 Predict a value for Y when respondents educ = 14 years & fathers educ = 12 years.

Answer Ŷ= a + B1(14) + B2 (12) B1 = Respondent’s Educ B2 = Father’s Educ Ŷ= 15,000 + 125 (14) + 25 (12) Ŷ = 15,000 + 1,750 + 300 Ŷ = 17,050

Dummy Variables  Remember that multiple regression is used when: variables are interval/ratio variables are interval/ratio  Dummy variables allows us to use nominal data.  Most often we are comparing groups of individuals (i.e., men & women; Blacks & Whites; Republicans & Democrats)

A dummy variable is:  A variable coded 1 to indicate the presence of an attribute and coded 0 to indicate its absence.  Dummy variables are used with nominal data like gender, religion and race.  Dummy Variables allow us to assess how the relationships theorized in the multiple regression model hold for different groups (e.g., men and women).

Coding dummy variables  For our purposes we will have only 2 categories Categories will be coded 0 & 1 Categories will be coded 0 & 1 Example (females =0; males =1) Example (females =0; males =1) The category coded 0 is considered the “left out” category (group). The category coded 0 is considered the “left out” category (group). The category coded 1 is the comparison group The category coded 1 is the comparison group In other words: You are comparing the group coded 1 with the group coded 0. You are comparing the group coded 1 with the group coded 0.

EXAMPLE Sex  If you code females = 0 & males =1,  you are comparing men to women.  If you coded males = 0 & females=1,  you are comparing women to men.

Example  We regress income on education and family background (father’s education).  But, how does gender influence this relationship?  Regression with dummy variables answers this question.

Income regressed on R’s educ., fathers educ., and gender (coded 0=females, 1=males): Ŷ = a + B1(X1) + B2 (X2) + B3 (X3) a= 15,000 B1(educ)= 110 B2 (faeduc)= 15 B3 (gender)= 150 Let’s interpret B3: 2 possible values for gender (0 & 1) so 150 (0) = 0 (females) 150 (1) = 150 (males) So males earn \$150 more than women. So males earn \$150 more than women.

 Let’s plug values into the equation: Predict income for a male respondent with14 years of education and a father who has 12 years of education. Ŷ = 15,000 + 110 (14) + 15 (12) + 150 (1) = 15,000 + 1,540 + 180 + 150 = 16,870 What if the respondent were a female? Ŷ = 15,000 +110 (14) + 15 (12) + 150 (0) = 15,000 + 1,540 + 180 + 0 = 16,720

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