1. Lecture 13: Case-control studies: introduction to matching
Jeffrey E. Korte, PhD
BMTRY 747: Foundations of Epidemiology II
Department of Public Health Sciences
Medical University of South Carolina
Spring 2015

2. Matching: overview Control confounding in the design stage
Residual confounding may occur with matched variables, or with other covariates
Matched analysis: pairs of subjects with the same exposure status (in a case-control study) are non-informative
Only pay attention to discordant pairs

3. Matching (definition)
Matching refers to the selection of a reference series (unexposed subjects in a cohort study or controls in a case-control study)– that is identical, or nearly so, to the index series with respect to the distribution of one or more potentially confounding factors
- Rothman and Greenland, Modern Epidemiology, 1998

4. Matching Matching is most commonly done in case-control studies
Cases (usually those with the disease of interest) are matched with controls (those without the disease) based on the value of a certain variable.
Can be employed in other studies
Exposed members of a cohort are matched with unexposed members and followed longitudinally

5. Types of Matching Individual Matching Frequency Matching
One or more reference subjects with matching factor values equal to those of the index subject
Frequency Matching
Selection of an entire stratum of reference subjects with matching-factor values equal to that of a stratum of index subjects

6. Advantages of Matching
Controls factors that are unknown or unmeasureable
Matching siblings can control for many genetic and enviromental factors
Improves precision (study validity)
Convenience
Narrows down a large number of controls
Can reduce sampling variability
Less heterogeneity across paired subjects
Can increase statistical power
Can improve efficiency by reducing sample size needed for a study

7. Disadvantages of Matching
Lose the ability to assess the matched variable as a risk factor
Additional time and expense
Decision is irreversible
Requires special analytic techniques
If the matching variable is not an independent risk factor for the outcome, matching is wasteful.
If the matching variable is not an independent risk factor for the outcome, but is associated with the risk factor, matching is wasteful and inefficient.
Potentially alters the distribution of the risk factor

8. Elements of Matching
Unit of analysis is the pair of components with something in common.
Twins
Siblings
Neighborhood Controls
Two components are selected to have different exposures (RCT or cohort study) or different outcomes (Case-Control)

9. Elements of Matching
Matched pairs can be used in different study designs:
Cross-sectional
Cohort
Case-Control
Randomized Clinical Trial

10. Elements of Matching Same Subject Matching (within subject)
Data pairs could be measured on the same subject
Example:
Compare skin-grafting techniques by applying each treatment to the same person
Compare intra-ocular pressure in both eyes of one subject, after giving each eye a different treatment.
Notice the data structure may bear a superficial resemblance to multi-level analysis…both analytic strategies take “groups” into account

11. Overmatching
The factor to be matched is (partially or wholly) on the causal pathway between the risk factor and the outcome.
Example
Study Goal: Assess the association between alcohol consumption and heart disease (case-control study)
Matching Variable: sleep apnea (want to control for conditions that may influence oxygenation)
If sleep apnea is a condition that occurs as a result of alcohol consumption and is a risk factor for heart disease, then matching
Forces cases and controls to have a more similar distribution of alcohol consumption
Accordingly, attenuates the odds ratio
The true association between alcohol consumption and heart disease cannot be assessed.

12. Example 1: Number of Sick Days
Twin study (children) – 10 pairs of twins
One twin is immunized
The other twin is not
Subjects are followed through school year
This is a matched study – each pair of twins is a matched set for the clinical trial

13. Example 1: Number of Sick Days
Pair
Control
Treated
Control-Treated 1 4 2 8 6 3 5 7 -2 9 10

14. Example 1: Number of Sick Days
Mean of column 3 = 1.5 days
Standard deviation =
= 1.78 days

15. Example 1: Number of Sick Days
Standard error =
= = 0.56 days
95% confidence interval: 1.5 ± 1.96*(0.56)
1.5 (0.40, 2.60) more days in control group
t-test: p=0.026

16. Example 1: Unmatched analysis?
Control mean = 4.6 days
Treated mean = 3.1 days
Difference between means:
Estimated benefit = 1.5 days
95% CI (-0.47, 3.47)
Two-sample t-test: p=0.145
Control
Treated 4 8 6 5 3 2 7

17. Example 1: Number of Sick Days
Conclusion: a study based on matched pairs can be more powerful than an unmatched study
Tighter confidence limits; easier to show statistical significance
In this case, the unmatched analysis did not result in bias (we still estimated 1.5 fewer sick days in the treated group)

18. Statistical Methods for Analyzing Matched Data
Unmatched study: one entry for each subject
Exposure E _
Outcome D a b c d

19. Statistical Methods for Analyzing Matched Data
Matched cohort study: one entry for each pair
Exposed D _
Unexposed a b c d

20. Statistical Methods for Analyzing Matched Data
Matched case-control study: one entry for each pair
Control E _
Case a b c d

21. Statistical Methods for Analyzing Matched Data
Matched case-control study:
Matched pair both exposed: add 1 to a
Case exposed; control unexposed: add 1 to b
Control exposed; case unexposed: add 1 to c
Matched pair both unexposed: add 1 to d
Only discordant pairs (in cells “b” and “c”) give useful information

22. Statistical Methods for Analyzing Matched Data
Discordant pairs  estimate odds ratio!
p1 = probability of exposure in cases
p0 = probability of exposure in controls
Therefore:
probability of cell “b” = p1(1-p0)
probability of cell “c” = p0(1-p1)

23. Statistical Methods for Analyzing Matched Data
Formula for standard error of odds ratio:
95% confidence limits:
lnOR ± 1.96(SE(lnOR))
(must then exponentiate confidence limits of lnOR to obtain 95% CI for odds ratio)

24. Statistical Methods for Analyzing Matched Data
McNemar’s test
Chi-squared statistic for matched data
with 1 degree of freedom

25. Example 2: Matched Case-Control Study
Research question: Is there an association between the amount of time a mother spends on her feet during a pregnancy and the likelihood of preterm birth?
Study Sample: 223 matched case-control pairs of women who had given birth at a local hospital,
Disease
1= preterm birth (<37 weeks gestation)
0= no preterm birth
Exposure
1= mother’s work required standing
0= mother’s work did not require standing

26. Example 2 (continued) Matching
Each case (disease=1) was matched with a control (disease=0) on the basis of Maternal Age (<3 years) and Parity (1, 0)
4 Possible Exposure Combinations of Matched Pairs
Both Case and Control are EXPOSED
Case EXPOSED and Control NON-EXPOSED
Case NON-EXPOSED and Control EXPOSED
Both Case and Control NON-EXPOSED

27. A Look at the Raw Data ID/ Matched Pair Preterm Birth (case/control)
Age
Parity
Work Standing
(exposure) 1 22 23 2 28 27 3 19 4 32

28. Example 2 (continued) Control (Standing) Control (Not Standing)
Case (Standing)
147 31
Case 14
Note: Relevant information is confined to discordant pairs. OR cannot be estimated in studies in which all matched pairs have the same level of exposure

29. Example 2 (continued)
Odds ratio for matched case-control study: ratio of the number of positive to negative discordant pairs.
31 pairs (in which exposed member experienced the outcome and the non-exposed member did not)
14 pairs (in which these outcomes were reversed)
Odds Ratio= 31/14= 2.21

30. Example 2 (continued) Approximate 95% confidence interval:
Two standard deviations on each side of the estimated log odds ratio
Exponentiate the result (take the anti-logarithm)
Confidence interval ranges from 1.16, 4.22
Conclusion: standing is associated with pre-term birth.
Note: McNemar’s Chi-square statistic will provide the p-value testing the null hypothesis of no association between exposure and disease

31. Example 2: What happens if you ignore the matching?
Four possible combinations of Matched Pairs
Unexposed Control, Unexposed Case
Unexposed Control, Exposed Case
Exposed Control, Unexposed Case
Exposed Control, Exposed Case
Unexp
Exp
Control
1 + 2
3 + 4
Case
1 + 3
2 + 4

32. Unmatched Data from the same study
Standing
Not Standing
Preterm
178 45
Term
161 62
OR= (178 x 62)/ (45 x 161) = 1.52
95% CI= 0.98, 2.36

33. When Matching is ignored?
A noticeable difference between the matched and unmatched analyses
Matched: OR = 2.21 (1.16, 4.22)
Unmatched: OR = 1.52 (0.98, 2.36)
Unmatched analysis ignores any correlation in exposure status between the case and control in the matched pair.
If this correlation is substantial, then the unmatched analysis gives a biased result.

34. Should a matched analysis always be used for matched data?
If there is no evidence of a correlation within pairs, should you still proceed with a matched analysis?
NOT NECESSARILY
Matched analyses can give an unstable result if the sample size is too small

35. Example 3: Matched Cohort Study
Research question: Is there an association between vasectomy and myocardial infarction?
Study Sample: 4830 exposed-unexposed pairs of men
Matching Variables: Age (5-year band), current smoking status (yes/no)
Disease Outcome
1= MI
0= No MI
Exposure
1= Vasectomy
0= No vasectomy

36. Example 3: Matched Cohort Study Analysis
The previous odds ratio computational methodology applies to pair-matched cross-sectional or cohort studies with binary outcomes.
Matching
Each matched pair contains one exposed and one un-exposed individual
4 Possible Exposure Combinations of Matched Pairs
Unexposed has No Disease / Exposed has No Disease
Unexposed has No Disease / Exposed has Disease
Unexposed has Disease / Exposed has No Disease
Unexposed has Disease / Exposed has Disease

37. Example 3 (continued) No Vasectomy MI No MI Vasectomy 20 16 4,79420 16
4,794
Note: Relevant information is confined to discordant pairs. OR cannot be estimated in studies in which all matched pairs have the same disease outcome

38. Example 3: OR computation
The odds ratio of a matched cohort study may be estimated by taking the ratio of the number of positive to negative discordant pairs.
20 pairs (in which exposed member experienced the outcome and the non-exposed member did not)
16 pairs (in which these outcomes were reversed)
Odds Ratio= 20/16= 1.25 (0.65, 2.41)
No association between having a vasectomy and suffering an MI.

39. Matched analysis in modeling
Use “conditional logistic regression”
Produce matched OR and confidence interval
Control for confounders
Ignores pairs that are concordant on all variables
Logistic regression in unmatched data is “unconditional logistic regression”
Either one of these can be done using PROC LOGISTIC

40. Conditional logistic regression in SAS
proc logistic
data=one;
strata ID;
model outcome=expose cov1 cov2;
run;

41. Summary: reasons to match
Control for confounding in design phase
Nuisance variables
Not important predictors you hope to assess
Improve study efficiency
Useful if sample size is limited
May clarify or simplify decision-making about control recruitment

42. Summary: problems with matching
Risk of over-matching, or unnecessary matching
May add a layer of complexity and difficulty to the study implementation
May be difficult or impossible to find a match for some individuals
May therefore add expense
Once matching is done, cannot be undone
Must (usually) use matched analysis