1. Lecture 18 Matched Case Control Studies
BMTRY 701
Biostatistical Methods II

2. Matched case control studies
References:
Hosmer and Lemeshow, Applied Logistic Regression
(beginning page 5)

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

4. 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

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

6. 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

7. 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”

8. 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

9. 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:

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

11. 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)

12. 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

13. 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

14. 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

15. . xi: clogit control hpv16ser, group(strata) or
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Conditional (fixed-effects) logistic regression Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
control | Odds Ratio Std. Err z P>|z| [95% Conf. Interval]
hpv16ser |

16. . xi: logistic control hpv16ser
Logistic regression Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
control | Odds Ratio Std. Err z P>|z| [95% Conf. Interval]
hpv16ser |

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