1. Lecture 17: Regression for Case-control Studies BMTRY 701 Biostatistical Methods II

2. Old business: Comparing AUCs  Good reference: Hanley and McNeill “Comparing AUCs for ROC curves based on the same data” See class website for pdf.

3. Additional Reading in Logistic REgression  Hosmer and Lemeshow, Applied Logistic Regression  http://en.wikipedia.org/wiki/Logistic_regression http://en.wikipedia.org/wiki/Logistic_regression  http://luna.cas.usf.edu/~mbrannic/files/regressio n/Logistic.html http://luna.cas.usf.edu/~mbrannic/files/regressio n/Logistic.html  http://www.statgun.com/tutorials/logistic- regression.html http://www.statgun.com/tutorials/logistic- regression.html  http://www.bus.utk.edu/stat/Stat579/Logistic%20 Regression.pdf http://www.bus.utk.edu/stat/Stat579/Logistic%20 Regression.pdf  Etc: Google “logistic regression”

4. Case Control Studies in Logistic Regression  http://www.oxfordjournals.org/our_journals/tropej /online/ma_chap11.pdf http://www.oxfordjournals.org/our_journals/tropej /online/ma_chap11.pdf  How is a case-control study performed?  What is the outcome and what is the predictor in the regression setting?

5. Recall the simple 2x2 example  Odds ratio for 2x2 table can be used in case- control studies  Similarly, the logistic regression model can be used treating ‘case’ status as the outcome.  It has been shown that the results do not depend on the sampling (i.e., cohort vs. case-control study).

6. Example: Case control study of HPV and Oropharyngeal Cancer  Gillison et al. (http://content.nejm.org/cgi/content/full/356/19/1 944)http://content.nejm.org/cgi/content/full/356/19/1 944  100 cases and 200 controls with oropharyngeal cancer  How was the sampling done?

7. Data on Case vs. HPV > table(data$hpv16ser, data$control) 0 1 0 186 43 1 14 57 > epitab(data$hpv16ser, data$control) $tab Outcome Predictor 0 p0 1 p1 oddsratio lower upper p.value 0 186 0.93 43 0.43 1.00000 NA NA NA 1 14 0.07 57 0.57 17.61130 8.99258 34.49041 4.461359e-21

8. Multiple Logistic Regression  This is not ‘randomized’ study  there are lots of other predictors that may be associated with the cancer  Examples: smoking alcohol age gender

9. Fit the model:  Write down the model assume main effects of tobacco, alcohol and their interaction  What is the likelihood function?  What are the MLEs?

10. How do we interpret the results?  Is there an effect of tobacco?  Is there an effect of alcohol?  Is there an interaction?

11. Interpreting the interaction  What is the OR for smoker/non-drinker versus a non-smoker/non-drinker?  What is the OR for a smoker/drinker versus a non-smoker/drinker?

12. How can we assess if the effect of smoking differs by HPV status? 

13. How likely is it that someone who smokes and drinks will get oropharyngeal cancer?  How can we estimate the chance?

14. Matched case control studies  References: Hosmer and Lemeshow, Applied Logistic Regression http://staff.pubhealth.ku.dk/~bxc/SPE.2002/Slides/mc c.pdfhttp://staff.pubhealth.ku.dk/~bxc/SPE.2002/Slides/mc c.pdf http://staff.pubhealth.ku.dk/~bxc/Talks/Nested- Matched-CC.pdfhttp://staff.pubhealth.ku.dk/~bxc/Talks/Nested- Matched-CC.pdf http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap49/s ect35.htmhttp://www.tau.ac.il/cc/pages/docs/sas8/stat/chap49/s ect35.htm http://www.ats.ucla.edu/stat/sas/library/logistic.pdf (beginning page 5)http://www.ats.ucla.edu/stat/sas/library/logistic.pdf

15. Matched design  Matching on important factors is common  OP cancer: age gender  Why? forces the distribution to be the same on those variables removes any effects of those variables on the outcome eliminates confounding

16. 1-to-M matching  For each ‘case’, there is a matched ‘control  Process usually dictates that the case is enrolled, then a control is identified  For particularly rare diseases or when large N is required, often use more than one control per case

17. Logistic regression for matched case control studies  Recall independence  But, if cases and controls are matched, are they still independent?

18. Solution: treat each matched set as a stratum  one-to-one matching: 1 case and 1 control per stratum  one-to-M matching: 1 case and M controls per stratum  Logistic model per stratum: within stratum, independence holds.  We assume that the OR for x and y is constant across strata

19. How many parameters is that?  Assume sample size is 2n and we have 1-to-1 matching:  n strata + p covariates = n+p parameters  This is problematic: as n gets large, so does the number of parameters too many parameters to estimate and a problem of precision  but, do we really care about the strata-specific intercepts?  “NUISANCE PARAMETERS”

20. Conditional logistic regression  To avoid estimation of the intercepts, we can condition on the study design.  Huh?  Think about each stratum: how many cases and controls? what is the probability that the case is the case and the control is the control? what is the probability that the control is the case and the case the control?  For each stratum, the likelihood contribution is based on this conditional probability

21. Conditioning  For 1 to 1 matching: with two individuals in stratum k where y indicates case status (1 = case, 0 = control)  Write as a likelihood contribution for stratum k:

22. Likelihood function for CLR Substitute in our logistic representation of p and simplify:

23. Likelihood function for CLR  Now, take the product over all the strata for the full likelihood  This is the likelihood for the matched case-control design  Notice: there are no strata-specific parameters cases are defined by subscript ‘1’ and controls by subscript ‘2’  Theory for 1-to-M follows similarly (but not shown here)

24. Interpretation of β  Same as in ‘standard’ logistic regression  β represents the log odds ratio comparing the risk of disease by a one unit difference in x

25. When to use matched vs. unmatched?  Some papers use both for a matched design  Tradeoffs: bias precision  Sometimes matched design to ensure balance, but then unmatched analysis  They WILL give you different answers  Gillison paper

26. Another approach to matched data  use random effects models  CLR is elegant and simple  can identify the estimates using a ‘transformation’ of logistic regression results  But, with new age of computing, we have other approaches  Random effects models: allow strata specific intercepts not problematic estimation process additional assumptions: intercepts follow normal distribution Will NOT give identical results

27. . xi: clogit control hpv16ser, group(strata) or Iteration 0: log likelihood = -72.072957 Iteration 1: log likelihood = -71.803221 Iteration 2: log likelihood = -71.798737 Iteration 3: log likelihood = -71.798736 Conditional (fixed-effects) logistic regression Number of obs = 300 LR chi2(1) = 76.12 Prob > chi2 = 0.0000 Log likelihood = -71.798736 Pseudo R2 = 0.3465 ------------------------------------------------------------------------------ control | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- hpv16ser | 13.16616 4.988492 6.80 0.000 6.26541 27.66742 ------------------------------------------------------------------------------

28. . xi: logistic control hpv16ser Logistic regression Number of obs = 300 LR chi2(1) = 90.21 Prob > chi2 = 0.0000 Log likelihood = -145.8514 Pseudo R2 = 0.2362 ------------------------------------------------------------------------------ control | Odds Ratio Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- hpv16ser | 17.6113 6.039532 8.36 0.000 8.992582 34.4904 ------------------------------------------------------------------------------

29. . xi: gllamm control hpv16ser, i(strata) family(binomial) number of level 1 units = 300 number of level 2 units = 100 Condition Number = 2.4968508 gllamm model log likelihood = -145.8514 ------------------------------------------------------------------------------ control | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- hpv16ser | 2.868541.3429353 8.36 0.000 2.1964 3.540681 _cons | -1.464547.1692104 -8.66 0.000 -1.796193 -1.1329 ------------------------------------------------------------------------------ Variances and covariances of random effects ------------------------------------------------------------------------------ ***level 2 (strata) var(1): 4.210e-21 (2.231e-11) ------------------------------------------------------------------------------ OR = 17.63