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Limited Dependent Variables: Binary Models Erik Nesson Ball State University MBSW

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Outline 1.Overview of LDVs 2.Binary Outcome Models a.Linear Probability Model b.Logit and Probit 3.Interpretation of Coefficients a.Odds ratios vs. marginal effects b.Implementation in Stata 2

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Binary Outcome Models 3

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Example: Secondhand Smoke Half of adult non-smokers are exposed to environmental tobacco smoke (ETS) Main point of public policies is to reduce ETS exposure in specific areas Ex: smokefree air laws are meant to reduce ETS exposure at work But most research about tobacco control policies focuses on reducing smoking Main question: Do tobacco control policies reduce ETS exposure in the workplace? 4

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How to measure ETS exposure? NHANES Dataset Repeated cross-section dataset covering and Roughly 10k individuals interviewed every year All individuals complete extensive survey AND receive physical Contains extensive demographic and health history information Sample Non-smoking, employed individuals age 18 to 65 8,554 individuals 5

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Dependent and Independent Variables 1.Indicators of ETS Exposure a.Q: “… how many hours per day can you smell the smoke from other people’s cigarettes, cigars, and/or pipes?” b.Serum cotinine levels Cotinine is major metabolite of nicotine with 8-16 hr half life Very low levels can be detected 2.Tobacco Control Policies a.Cigarette Taxes measured in real $2009 b.The percent of each individual’s state living under a workplace SFA law (from 0 to 100) 6

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Summary Statistics All Workers (N=8554) White Collar Workers (N=4825) Blue Collar Workers (N=3729) MeanStd. Dev.MeanStd. Dev.MeanStd. Dev.T-Test Smell Smoke at Work Observable Cotinine Level Cigarette Excise Tax Female Age Black Hispanic Married Income to Poverty Ratio B.A. Degree Some College H.S. Degree Less than H.S Family Size Rooms in Home

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Self-Reported ETS Exposure at Work by Job Category 8

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Tabulation of Observable Cotinine Levels by Job Category 9

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Basic Model 10

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Linear Probability Model 11

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Binary Outcome Models 12

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Linear Probability Results Table shows marginal effects with standard errors in parentheses Self-Reported ExposureObservable Cotinine Levels All Workers White Collar Workers Blue Collar WorkersAll Workers White Collar Workers Blue Collar Workers Cigarette Excise Tax * (0.03) (0.07)(0.03)(0.04)(0.05) Work SFA Law-0.001* ***-0.002***-0.003*** (0.00) 13

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Logit or Probit 14

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Interpretation of Coefficients 15

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Interpretation of Coefficients 16

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Calculating Marginal Effects For both continuous and discrete variables, other coefficients and variable values enter into marginal effects calculation Three common approaches: 1.Marginal effect at the mean: Use mean values of other variables 2.Average marginal effect: Calculate marginal effect for each observation 3.Marginal effect at some other value of coefficients 17

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Calculating Marginal Effects 18

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Calculating Marginal Effects 19

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Implementation in Stata Code for estimating marginal effect at the mean in Stata: logit y x margins, dydx(varlist) atmeans Code for estimating average marginal effect in Stata: logit y x margins, dydx(varlist) Usually Stata is smart enough to determine which independent variables are binary 20

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Odds Ratios 21

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Odds Ratios and Logit 22

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Odds Ratios and Logit 23

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Odds Ratios and Logit What does an odds ratio of 2 mean? Odds of y=1 are 2x greater when x=1 than when x=0 Could be that Odds that y=1|x=1 = 4 and odds that y=1|x=0 = 2 Odds that y=1|x=1 = 3 and odds that y=1|x=0 = 1.5 Odds that y=1|x=1 = 2 and odds that y=1|x=0 = 1 So odds ratios are not equal to marginal effects Do not tell us about differences in probability 24

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Odds Ratios and Marginal Effects Odds Ratio Marginal Effect

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Logit Results Table shows odds ratios, standard errors in parentheses, and marginal effects in brackets Self-Reported ExposureObservable Cotinine Levels All Workers White Collar Workers Blue Collar WorkersAll Workers White Collar Workers Blue Collar Workers Cigarette Excise * Tax(1.17)(0.79)(3.49)(0.02)(0.05)(0.02) [0.030][0.039][0.027][0.001][0.000][0.003] Work SFA Law0.993*** ***0.990***0.987***0.997 (-0.32)(-0.86)(-0.38)(0.00) [-0.001] [-0.003][-0.002] [-0.000] 26

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Other Issues Standard errors are complicated Be wary of canned programs (like Stata!) which allow calculation of robust variance/covariance matrices. Interaction terms are also complicated Odds ratios can be difficult to interpret Marginal effects are better! 27

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