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The Regression Equation

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).

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The Regression Equation

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

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

  3. 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.

  4. 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

  5. Dummy Variables • Remember that multiple regression is used when: • 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)

  6. 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).

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

  8. 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.

  9. 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.

  10. 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.

  11. 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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