Lecture 17:Regression for Case-control Studies

BMTRY 701

Biostatistical Methods II

Old business: Comparing AUCs

Good reference: Hanley and McNeill

“Comparing AUCs for ROC curves based on the same data”

See class website for pdf.

Additional Reading in Logistic REgression

Hosmer and Lemeshow, Applied Logistic Regression

http://en.wikipedia.org/wiki/Logistic_regression http://luna.cas.usf.edu/~mbrannic/files/

regression/Logistic.html http://www.statgun.com/tutorials/logistic-

regression.html http://www.bus.utk.edu/stat/Stat579/Logistic

%20Regression.pdf Etc: Google “logistic regression”

Case Control Studies in Logistic Regression

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?

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).

Example: Case control study of HPV and Oropharyngeal Cancer

Gillison et al. (http://content.nejm.org/cgi/content/full/356/19/1944)

100 cases and 200 controls with oropharyngeal cancer

How was the sampling done?

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

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

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?

How do we interpret the results?

Is there an effect of tobacco?

Is there an effect of alcohol?

Is there an interaction?

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?

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

How likely is it that someone who smokes and drinks will get oropharyngeal cancer?

How can we estimate the chance?

Matched case control studies

References:• Hosmer and Lemeshow, Applied Logistic Regression• http://staff.pubhealth.ku.dk/~bxc/SPE.2002/Slides/

mcc.pdf• http://staff.pubhealth.ku.dk/~bxc/Talks/Nested-

Matched-CC.pdf• http://www.tau.ac.il/cc/pages/docs/sas8/stat/chap49/

sect35.htm• http://www.ats.ucla.edu/stat/sas/library/logistic.pdf

(beginning page 5)

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

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

Logistic regression for matched case control studies

Recall independence

But, if cases and controls are matched, are they still independent?

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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

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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”

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

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:

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)0,1()0,1(

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Likelihood function for CLR

Substitute in our logistic representation of p and simplify:

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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)

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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

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

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

. 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 = 0.0000Log 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------------------------------------------------------------------------------

. xi: logistic control hpv16ser

Logistic regression Number of obs = 300 LR chi2(1) = 90.21 Prob > chi2 = 0.0000Log 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------------------------------------------------------------------------------

. xi: gllamm control hpv16ser, i(strata) family(binomial)

number of level 1 units = 300number 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