This document discusses predicting obesity rates in the US using discrete choice models. It first provides background on discrete choice models and why standard regression is not appropriate for binary dependent variables. It then analyzes Behavioral Risk Factor Surveillance System (BRFSS) survey data from 2006-2010 to predict obesity. Logistic regression and marginal effects are used to model the probability of being obese based on demographic and socioeconomic variables. The results show age has a non-linear effect, initially decreasing obesity probability but then increasing it at older ages.

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