Logistic Regression. BUS 735: Business Decision Making and Research
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2 Goals of this section 2/ 8 Specific goals: Learn how to conduct regression analysis with a dummy independent variable. Learning objectives: LO2: Be able to construct and use multiple regression models (including some limited dependent variable models) to construct and test hypotheses considering complex relationships among multiple variables. LO6: Be able to use standard computer packages such as SPSS and Excel to conduct the quantitative analyses described in the learning objectives above. LO7: Have a sound familiarity of various statistical and quantitative methods in order to be able to approach a business decision problem and be able to select appropriate methods to answer the question.
3 Goals of this section 2/ 8 Specific goals: Learn how to conduct regression analysis with a dummy independent variable. Learning objectives: LO2: Be able to construct and use multiple regression models (including some limited dependent variable models) to construct and test hypotheses considering complex relationships among multiple variables. LO6: Be able to use standard computer packages such as SPSS and Excel to conduct the quantitative analyses described in the learning objectives above. LO7: Have a sound familiarity of various statistical and quantitative methods in order to be able to approach a business decision problem and be able to select appropriate methods to answer the question.
4 3/ 8 : method for estimating a regression with a dummy dependent variable. Will a potential customer purchase a product? (YES=1, NO=0). Might use explanatory variables: age, gender, income, etc. Will a potential employee be retained after one year? (YES=1, NO=0). Might use explanatory variables: age, gender, years experience, past income, education dummy (4-year = 1, otherwise = 0). Why not run a regression? Which assumption is violated?
5 3/ 8 : method for estimating a regression with a dummy dependent variable. Will a potential customer purchase a product? (YES=1, NO=0). Might use explanatory variables: age, gender, income, etc. Will a potential employee be retained after one year? (YES=1, NO=0). Might use explanatory variables: age, gender, years experience, past income, education dummy (4-year = 1, otherwise = 0). Why not run a regression? Which assumption is violated?
6 3/ 8 : method for estimating a regression with a dummy dependent variable. Will a potential customer purchase a product? (YES=1, NO=0). Might use explanatory variables: age, gender, income, etc. Will a potential employee be retained after one year? (YES=1, NO=0). Might use explanatory variables: age, gender, years experience, past income, education dummy (4-year = 1, otherwise = 0). Why not run a regression? Which assumption is violated?
7 3/ 8 : method for estimating a regression with a dummy dependent variable. Will a potential customer purchase a product? (YES=1, NO=0). Might use explanatory variables: age, gender, income, etc. Will a potential employee be retained after one year? (YES=1, NO=0). Might use explanatory variables: age, gender, years experience, past income, education dummy (4-year = 1, otherwise = 0). Why not run a regression? Which assumption is violated?
8 3/ 8 : method for estimating a regression with a dummy dependent variable. Will a potential customer purchase a product? (YES=1, NO=0). Might use explanatory variables: age, gender, income, etc. Will a potential employee be retained after one year? (YES=1, NO=0). Might use explanatory variables: age, gender, years experience, past income, education dummy (4-year = 1, otherwise = 0). Why not run a regression? Which assumption is violated?
9 3/ 8 : method for estimating a regression with a dummy dependent variable. Will a potential customer purchase a product? (YES=1, NO=0). Might use explanatory variables: age, gender, income, etc. Will a potential employee be retained after one year? (YES=1, NO=0). Might use explanatory variables: age, gender, years experience, past income, education dummy (4-year = 1, otherwise = 0). Why not run a regression? Which assumption is violated? Dependent variable is not interval. As a result, residual will not be normally distributed.
10 Structural Form 4/ 8 Normal regression: y i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Logistic regression: log (Odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Odds = P(y i = 1) 1 P(y i = 1) = P(y i = 1) P(y i = 0) Remember, y i 0, 1 indicates YES=1 event did occur, or NO=0 event did not occur.
11 Structural Form 4/ 8 Normal regression: y i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Logistic regression: log (Odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Odds = P(y i = 1) 1 P(y i = 1) = P(y i = 1) P(y i = 0) Remember, y i 0, 1 indicates YES=1 event did occur, or NO=0 event did not occur.
12 Structural Form 4/ 8 Normal regression: y i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Logistic regression: log (Odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Odds = P(y i = 1) 1 P(y i = 1) = P(y i = 1) P(y i = 0) Remember, y i 0, 1 indicates YES=1 event did occur, or NO=0 event did not occur.
13 Structural Form 4/ 8 Normal regression: y i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Logistic regression: log (Odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Odds = P(y i = 1) 1 P(y i = 1) = P(y i = 1) P(y i = 0) Remember, y i 0, 1 indicates YES=1 event did occur, or NO=0 event did not occur.
14 Structural Form 4/ 8 Normal regression: y i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Logistic regression: log (Odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Odds = P(y i = 1) 1 P(y i = 1) = P(y i = 1) P(y i = 0) Remember, y i 0, 1 indicates YES=1 event did occur, or NO=0 event did not occur.
15 Structural Form 4/ 8 Normal regression: y i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Logistic regression: log (Odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i + e i Odds = P(y i = 1) 1 P(y i = 1) = P(y i = 1) P(y i = 0) Remember, y i 0, 1 indicates YES=1 event did occur, or NO=0 event did not occur.
16 Predicted Value 5/ 8 Predicted value from a regular regression: ŷ i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i For a logistic regression, you can get predicted logist (not too interesting yet): ˆl i = ln(odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i To uncover the predicted probability of the event occuring: P(y 1 i = 1) = 1 + e ( ˆl i )
17 Predicted Value 5/ 8 Predicted value from a regular regression: ŷ i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i For a logistic regression, you can get predicted logist (not too interesting yet): ˆl i = ln(odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i To uncover the predicted probability of the event occuring: P(y 1 i = 1) = 1 + e ( ˆl i )
18 Predicted Value 5/ 8 Predicted value from a regular regression: ŷ i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i For a logistic regression, you can get predicted logist (not too interesting yet): ˆl i = ln(odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i To uncover the predicted probability of the event occuring: P(y 1 i = 1) = 1 + e ( ˆl i )
19 Predicted Value 5/ 8 Predicted value from a regular regression: ŷ i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i For a logistic regression, you can get predicted logist (not too interesting yet): ˆl i = ln(odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i To uncover the predicted probability of the event occuring: P(y 1 i = 1) = 1 + e ( ˆl i )
20 Predicted Value 5/ 8 Predicted value from a regular regression: ŷ i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i For a logistic regression, you can get predicted logist (not too interesting yet): ˆl i = ln(odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i To uncover the predicted probability of the event occuring: P(y 1 i = 1) = 1 + e ( ˆl i )
21 Predicted Value 5/ 8 Predicted value from a regular regression: ŷ i = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i For a logistic regression, you can get predicted logist (not too interesting yet): ˆl i = ln(odds) = b 0 + b 1 X 1,i + b 2 X 2,i b k 1 X k 1,i To uncover the predicted probability of the event occuring: P(y 1 i = 1) = 1 + e ( ˆl i )
22 Marginal Effects 6/ 8 Marginal effect for regression: measure of how much y changes when x increases by 1. Example: How much does public expenditure per capita increase (or decrease) when economic ability increases by one unit? Marginal effect for logit: measure of how much P(y i = 1) changes when x increases by 1. How much more (or less) likely will an interview candicate be working here after one year if she/he has a four year college education?
23 Marginal Effects 6/ 8 Marginal effect for regression: measure of how much y changes when x increases by 1. Example: How much does public expenditure per capita increase (or decrease) when economic ability increases by one unit? Marginal effect for logit: measure of how much P(y i = 1) changes when x increases by 1. How much more (or less) likely will an interview candicate be working here after one year if she/he has a four year college education?
24 Marginal Effects 6/ 8 Marginal effect for regression: measure of how much y changes when x increases by 1. Example: How much does public expenditure per capita increase (or decrease) when economic ability increases by one unit? Marginal effect for logit: measure of how much P(y i = 1) changes when x increases by 1. How much more (or less) likely will an interview candicate be working here after one year if she/he has a four year college education?
25 Marginal Effects 6/ 8 Marginal effect for regression: measure of how much y changes when x increases by 1. Example: How much does public expenditure per capita increase (or decrease) when economic ability increases by one unit? Marginal effect for logit: measure of how much P(y i = 1) changes when x increases by 1. How much more (or less) likely will an interview candicate be working here after one year if she/he has a four year college education?
26 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
27 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
28 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
29 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
30 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
31 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
32 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
33 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
34 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
35 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
36 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
37 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
38 Coefficients versus Marginal Effects 7/ 8 For a regular regression, for coefficient b 2 : The magnitude tells how much y increases when increasing x 2 by 1. Coefficient = Marginal Effect. For a logistic regression, the coefficient b 2 : The magnitude is pretty meaningless. Need to do more to figure out marginal effect. Because SPSS is stupid, it cannot figure out marginal effects. For a specific individual (specific values for X s), compute predicted probabilities by hand for: The individual s actual set of values for X s. Same set of X s, except increase one of them by 1. Take the difference = Marginal effect.
39 Example 8/ 8 Well Being Dataset (Described on pages of SAB). 182 college students in Washington, D.C. area. Variables of interest: Weight (4 categories: UNDERWEIGHT=1, NORMAL=2, OVERWEIGHT=3, OBESE=4) Gender (2 categories: MALE=0, FEMALE=1) Age Race (AFRICAN AMERICAN=1, WHITE=2, OTHER=3) Marital Status (SINGLE=1, MARRIED=2, OTHER=3) Financial Status (OVEREXTENDED=1, MAKING ENDS MEET=2, COMFORTABLE=3) Physical Health (5 point Likert scale) Depression Scale (0-60, values > 15 indicate depression) Find preliminary evidence whether each variable is related to weight. Analysis.
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