1. Lecture 10: cohort analysis (part 1): Intro to regression and dummy variables
Jeffrey E. Korte, PhD
BMTRY 747: Foundations of Epidemiology II
Department of Public Health Sciences
Medical University of South Carolina
Spring 2015

2. Measures of association: linked to study design
e.g. if you have a dichotomous outcome:
Linear regression – not okay
Assumes continuous normal outcome
Logistic regression – okay
Assumes follow-up is the same for all subjects
Predictor variables can be continuous, dichotomous, ordinal, nominal
Produces odds ratio for each variable
Regression coefficient is natural log of adjusted OR

3. Measures of association: linked to study design
If you have a dichotomous outcome (cont):
Proportional hazards regression – usually better
Also called “Cox models”, “survival analysis”
Takes into account different follow-up possible for each subject; different risk sets for each case
Predictor variables can be continuous, dichotomous, ordinal, nominal
Produces hazard ratio for each variable
Assumes that the hazard ratio is constant over time

4. Measures of association: linked to study design
If you have a dichotomous outcome (cont):
Log binomial or negative binomial
Can directly estimate relative risk
Preferable to logistic regression when outcome is common
Good for causal models in cohort studies where you are not modeling time-to-event

5. Measures of association
Odds ratio
Can be obtained from cross-sectional, case-control, cohort study, randomized controlled trial
Rate ratio, risk ratio, hazard ratio
Poisson regression, log binomial, survival analysis
Can be obtained from cohort study, randomized controlled trial
Cross-sectional study can give you risk ratio, prevalence ratio, odds ratio

6. Measures of association
Analysis strategy should progress in stages
Univariate analysis
Bivariate analysis
Modeling
Analysis methods are linked to:
Study design
Question of interest
Structure of data and variable coding
Appropriate measure of association

7. Linear regression Assume continuous normal outcome (e.g. hypertension)
Model the actual outcome (no link function)
Coefficient is the weight for each variable
Model predicts value of Y among unexposed (i.e. all X=0), and change in Y associated with a one-unit increase for each variable, adjusted for other variables
Can obtain predicted risk for an individual

8. Logistic regression Assume dichotomous outcome (e.g. preterm birth)
Model the “logit” link function (log odds)
Coefficient is the weight for each variable
Intercept is log odds of outcome among unexposed
Other betas are the increased “log odds” associated with a one-unit increase for each variable, adjusted for the other variables

9. Logistic regression
This works out so that each beta (except for the intercept) is the adjusted log odds ratio for a one-unit increase in the exposure variable of interest
To obtain adjusted odds ratio, exponentiate the beta

10. Why that works
Odds:

11. Why that works
Odds ratio: estimate odds of disease in exposed (X1=1) versus unexposed (X1=0)

12. Why that works
Exponential mathematics: e(x+y) = ex * ey

13. Why that works
Adjusted OR for X1 (hold X2 steady)

14. Why that works If X1 is (1=yes, 0=no), then: β1X1e= β1 β1X1u=0
Note that e0=1

15. Why that works
So therefore, in logistic regression:

16. Why that works
Note: exponentiated beta is simply the adjusted OR for a one-unit increase in the variable, no matter how it is coded
Dichotomous 1=yes, 0=no
Dichotomous 0=yes, 1=no
Dichotomous 85=yes, 0=no
Continuous ranging from 85 to 450
Ordinal (1, 2, 3, 4, 5)

17. Why that works
One-unit increase for continuous variable:

18. Implication
Odds ratio is assumed to be the same for a one-unit increase anywhere along the scale of the exposure variable
e.g. 1 versus 0, 2 versus 1, 3 versus 2, etc.
i.e. the relationship is linear in the logit
This is why it is called a “multiplicative model”
Each unit increase in exposure is assumed to multiply risk, rather than add to the risk, by a defined amount

19. Linearity in the logit Dichotomous variable: okay
Two points always form a perfect line
Continuous or ordinal variables: maybe
Need to test the assumption by constructing “dummy variables”
Categorize the variable with regular intervals
Define reference category
All other categories will be compared to it
Assess whether dose-response curve is smooth

20. Dummy variables: example
Outcome variable: Type II diabetes
Continuous exposure variable: age
Fit as continuous variable: logistic regression model will assume the OR is the same for each one-unit increase in age
e.g. OR=1.14 (95% CI: – 1.19)
Does this fit the data well? Find out by testing dummy variables for age groups

21. Dummy variables: example
Age range:
Choose categories with regular intervals to test linearity in logit
Reference category should be chosen as described earlier
Robust sample size
May be convenient to have it be the category with expected lowest (or highest) risk

22. Dummy variables: example
Make 7 categories:
18-29 (reference)
30-39
40-49
50-59
60-69
70-79
80-85

23. Dummy variables: example
6 dummy variables for 7 categories: each dichotomous (1=yes, 0=no)
18-29 (reference)
30-39: dummy variable 1 (AGE2)
40-49: dummy variable 2 (AGE3)
50-59: dummy variable 3 (AGE4)
60-69: dummy variable 4 (AGE5)
70-79: dummy variable 5 (AGE6)
80-85: dummy variable 6 (AGE7)

24. Dummy variables: example
AGE1
AGE2
AGE3
AGE4
AGE5
AGE6
18-29
30-39 1
40-49
50-59
60-69
70-79
80-85

25. Dummy variables: example
This coding strategy results in independent comparisons of risk between the reference group and each other category
Each dummy variable corresponds to the comparison between that category (dummy variable = 1) and the reference category

26. Dummy variables: example
Possible results (logistic regression):
18-29 (reference)
30-39: AGE2: OR=1.5 (0.7 – 2.2)
40-49: AGE3: OR=2.0 (0.9 – 2.9)
50-59: AGE4: OR=2.5 (1.3 – 3.1)
60-69: AGE5: OR=3.0 (1.8 – 4.1)
70-79: AGE6: OR=3.5 (2.0 – 4.9)
80-85: AGE7: OR=4.0 (2.4 – 6.8)

27. Dummy variables: example
Possible results (logistic regression):
18-29 (reference)
30-39: AGE2: OR=1.5 (0.7 – 2.2)
40-49: AGE3: OR=2.0 (0.9 – 2.9)
50-59: AGE4: OR=2.5 (1.3 – 3.1)
60-69: AGE5: OR=2.6 (1.2 – 3.7)
70-79: AGE6: OR=2.4 (1.1 – 3.3)
80-85: AGE7: OR=2.5 (1.1 – 3.4)

28. Dummy variables: example
Possible results (logistic regression):
18-29 (reference)
30-39: AGE2: OR=0.92 (0.64 – 1.4)
40-49: AGE3: OR=1.2 (0.76 – 1.5)
50-59: AGE4: OR=1.6 (1.0 – 2.4)
60-69: AGE5: OR=2.6 (1.2 – 3.7)
70-79: AGE6: OR=4.8 (2.1 – 7.1)
80-85: AGE7: OR=7.5 (2.9 – 13.4)

29. Dummy variables To confirm linearity in the logit:
Beta coefficients for successive categories should progress in a linear scale (e.g. each category has a beta approximately 0.4 units higher than the previous beta)
Odds ratios for successive categories should progress in a multiplicative scale (e.g. each category has an odds ratio approximately 1.8 times higher than the previous odds ratio)

30. Dummy variables
If the exposure-disease relationship is not linear in the logit, may be advisable to create new dummy variables to use in modeling
Choose categories (including the designation of reference category) in some meaningful way based on theoretical considerations, sample size
This is in contrast to choosing standard-width categories to assess log-linearity

31. Dummy variables Final notes:
Any simple dichotomous variable is an example of a dummy variable
Also known as “nominal” variables
Three types of variables: continuous, ordinal, nominal