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# Matched studies

Matching (or pairing) is commonly used in study designs as a means of controlling for variability. It is especially common in case-control designs where each case is matched to one (or more) controls. Matching is most frequently done for age, but may be done for any potentially confounding factor.

Significance testing may be carried out using either McNemar's test for significance of change or the binomial test - topics covered in the More Information page on Binomial and related tests. However, one may also wish to estimate an odds ratio as an effect measure with its associated confidence interval.

 Correct tabulation for matched case control study Case exposed to risk factor Control exposed to risk factor Yes No Yes a b No c d
The data from a matched study should be displayed so as to retain the information on the pairings. Pairs in which case and control have the same exposure to the risk factor are termed concordant pairs (a and d) whilst pairs in which case and control have different exposure are termed discordant pairs (b and c).

### Estimation of odds ratio

The odds ratio is estimated simply as the ratio of the number of discordant pairs. Specifically it is the number of pairs in which cases are exposed to the risk factor and controls are not divided by the number of pairs in which controls are exposed to the risk factor and cases are not:

#### Algebraically speaking -

 OR  = b c
Where:
• b and c are the number of discordant pairs as shown above.

### Estimation of confidence interval

If the number of discordant pairs is large, a normal approximation confidence interval may be used. It is obtained in two steps:

1. estimate the upper and lower 95% confidence limits to the proportion (p) that the number of pairs in which cases are exposed to the risk factor make up of the total number of discordant pairs - namely b/(b+c) - using a simple normal approximation interval for a proportion - namely p ± 1.96 √ (pq/n).
2. These upper and lower limits are then converted to limits for the odds ratio using the relationship:
OR = p / (1-p)

If the number of discordant pairs is not large, an 'exact' interval should be estimated based on the F distribution:

ORL = c/[b + 1)F L       ORU = [c + 1)FU / b
where
• FL and FU are the upper 2½% points of F with (2(b+1), 2c) and (2(c+1), 2b) degrees of freedom respectively. Except where otherwise specified, all text and images on this page are copyright InfluentialPoints, all rights reserved. Images not copyright InfluentialPoints credit their source on web-pages attached via hypertext links from those images.