Discrete Regression, Logit, Obesity

1. Predicting Obesity Rates in the US
D3M

2. Discrete Choice Models
 Regression models we were studying so far had a continuous
dependent variable (e.g. sales of a product, House prices etc.)
o Predictor variables could be continuous or discrete (dummy variables)
 Often the phenomenon of interest (i.e. our dependent
variable) is discrete
o Vote or not
o Customer acquisition/defection
o Buy/no-buy
o Click on a banner Ad
o Survive/Don’t survive
With discrete outcomes, we are predicting probabilities (of say
customer defection).
Properties of probabilities?

3. 3
• With binary or categorical dependent variables
standard regression analysis is not appropriate
• Example
• binary dependent variable y coded to be zero for non-purchases and
one for purchases
• X is a continuous metric say price
• Problems
 The error terms are heteroskedastic (variance of the dependent
variable is different with different values of the independent variables
 Does not meet the assumptions of standard ols regression
 Prediction often below zero and values above one
Why Regression does not work
with
0 for non purchases
1 for purchases
y x
y
    

 


4. 4
Discrete choice models
 Generalize the regression model for the situations
where y is a non-metric variable
o a binary (0-1) variable or
o an ordinal variable (like a questionnaire item assuming the
values completely disagree, disagree, neither, agree,
completely agree) or
o a categorical variable (for example a nominal variable
recording the preferred Brand).
 The right-hand side variables can be discrete or continuous
 Similar to linear regression but interpretations are different
i iy x    

5. 5

6. Logit: We want our predictions to be a probability
Solution: instead of estimating
we estimate the model
which, after rearranging, equals
nn xcxcxcc
yp
yp



...
)1(1
)1(
ln 22110
nn
nn
xcxcc
xcxcc
e
e
yp 


 ...
...
110
110
1
)1(

7. Case Study: Predicting Obesity Rates

8. 8
Obesity Trends* Among U.S. Adults
BRFSS, 1990
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14%

9. 9
Obesity Trends* Among U.S. Adults
BRFSS, 1991
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19%

10. 10
Obesity Trends* Among U.S. Adults
BRFSS, 1992
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19%

11. 11
Obesity Trends* Among U.S. Adults
BRFSS, 1993
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19%

12. 12
Obesity Trends* Among U.S. Adults
BRFSS, 1994
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19%

13. 13
Obesity Trends* Among U.S. Adults
BRFSS, 1995
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19%

14. 14
Obesity Trends* Among U.S. Adults
BRFSS, 1996
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19%

15. 15
Obesity Trends* Among U.S. Adults
BRFSS, 1997
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% ≥20%

16. 16
Obesity Trends* Among U.S. Adults
BRFSS, 1998
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% ≥20%

17. 17
Obesity Trends* Among U.S. Adults
BRFSS, 1999
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% ≥20%

18. 18
Obesity Trends* Among U.S. Adults
BRFSS, 2000
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% ≥20%

19. 19
Obesity Trends* Among U.S. Adults
BRFSS, 2001
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% 20%–24% ≥25%

20. (*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
Obesity Trends* Among U.S. Adults
BRFSS, 2002
No Data <10% 10%–14% 15%–19% 20%–24% ≥25%
20

21. 21
Obesity Trends* Among U.S. Adults
BRFSS, 2003
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% 20%–24% ≥25%

22. Obesity Trends* Among U.S. Adults
BRFSS, 2004
22
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14% 15%–19% 20%–24% ≥25%

23. Obesity Trends* Among U.S. Adults
BRFSS, 2005
23
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14 15%–19% 20%–24% 25%–29% ≥30%

24. Obesity Trends* Among U.S. Adults
BRFSS, 2006
24
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14 15%–19% 20%–24% 25%–29% ≥30%

25. Obesity Trends* Among U.S. Adults
BRFSS, 2007
25
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14 15%–19% 20%–24% 25%–29% ≥30%

26. Obesity Trends* Among U.S. Adults
BRFSS, 2008
26
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14 15%–19% 20%–24% 25%–29% ≥30%

27. Obesity Trends* Among U.S. Adults
BRFSS, 2009
27
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14 15%–19% 20%–24% 25%–29% ≥30%

28. Obesity Trends* Among U.S. Adults
BRFSS, 2010
28
(*BMI ≥30, or ~ 30 lbs. overweight for 5’ 4” person)
No Data <10% 10%–14 15%–19% 20%–24% 25%–29% ≥30%

29. Data Source
BRFSS Survey by CDC

30. BRFSS Data at CDC
• Behavioral Risk Factor Surveillance System
(BRFSS) at CDC:
 World’s largest survey
• Monthly telephone interviews 18 years old of age or
older living in households
• 50 states, the District of Columbia, Puerto Rico,
Guam and the Virgin Islands
 Pooled data for 2001-2010.
 Approximately 3 million observations.
 We are using a random sample from 2006-2010

31. Data

32. Always start by summary statistics

33. OLS Coefficients

34. OLS

35. Logit Coefficients

36. Logit

37. Marginal Effects

38. Interpretation
 For continuous variables: Change in probability of being
Obese for a 1 unit change in the variable. In our case, only
AGE is continuous. Increasing AGE by 1 year, lowers
probability of being obese by 0.1%. Small effect but see later.
 For dummy variables, change in probability compared to the
reference category. Person with “No High School” has 7%
higher likelihood of being obese compared to a person with
college degree (holding everything else fixed)

39. Capturing non-linear Age Effects
OLS Regression

40. Capturing non-linear Age Effects
Logit

41. Marginal Effects
What do we conclude about Age now?

42. Age Effect is non-linear