## Table format for an individually matched study

The results of matched case control study performed during a Echovirus outbreak in Germany in 2001 [1]  are outlined in the table. The study hypothesis was that swimming in a specific pond (pond A) was suspected of increasing the risk of aseptic meningitis due to an Echovirus . In the study, each case was matched to a single control (individual matching), therefore constituting a matched pair.

Table. Cases of Echovirus meningitis and controls according to swimming in pond A, Hesse, Germany, 2001, according to a matched table showing pairs.

 Controls Exposed to pond A Unexposed Total Cases Exposed to pond A 194 46 240 Unexposed 6 29 35 Total 200 75 275

The figures inside the four cells of the two-by-two table no longer count individuals but pairs. We have therefore 275 pairs in the study including 275 cases and 275 controls. A pair in which both case and control have the same exposure is called a concordant pair. Alternatively if exposure differs between a case and its matched control they constitute a discordant pair.

In the table pairs are distributed as follows:

• 194 concordant pairs in which both the case and the control are exposed
• 29 concordant pairs in which both the case and the control are unexposed
• 46 discordant pairs in which the case is exposed and the control is nots
• 6 discordant pair in which the case is unexposed and the control is exposed

As a general rule, in order to distinguish an unmatched two-by-two table from a matched table showing pairs, the letters (e, f, g, and h) are used rather than (a, b, c, d) to identify the discordant and concordant pairs.

 Cases Controls Total Exposed to pond A e f e+f Unexposed g h g+h Total e+g f+h e+f+g+h

It is important to draw such a table showing the pairs in order to get familiar with our data. It is also important to draw such a table if we need to do an analysis by hand.

From the table showing pairs we can reconstruct the table we would have had if doing an unmatched analysis of the same data. We can see that the marginal totals of the matched table correspond to each of the 4 inner cells of the unmatched table. However only the matched table is appropriate when analysing a matched study.

Table. Cases of Echovirus meningitis and controls according to swimming in pond A, Hesse, Germany, 2001, according to an unmatched table

 Cases Controls Total Exposed to pond A 240 200 240 Unexposed 35 75 110 Total 275 275 550

## Analysis of an individually matched case control study

In matched case control study odds ratio are calculated, just as in any case control study. Matched odds ratio are calculated by using pairs instead of individuals. We have seen from the table that a pair is equivalent to a stratum and that a matched analysis is a stratified analysis. We will therefore conduct a stratified analysis in which there will be as many strata as pairs.

The following tables illustrates the four types of strata one can have in a matched pair analysis. Taking the example the table we would then have:

• 194 strata with a concordant pair of type e,
• 46 strata with a discordant pair of type f,
• 6 strata with a discordant pair of type g
• 29 strata with a concordant pair of type h.

Since we are doing a stratified analysis we will use the Mantel Haenszel method to calculate an odds ratio.

Formula:

Tables with situation e, f, g, h

Situation e (case and control are both exposed, concordant pairs)

 Case Control Total Exposed 1 1 2 Unexposed 0 0 0 Total 1 1 2

In situation e, the calculation using the formula yields to 0:

ad =  1x0       bc =1X0        T =2            ad/T   = 0/2       bc/T = 0/2

Situation f (case is exposed and control is unexposed, discordant pairs)

 Case Control Total Exposed 1 0 1 Unexposed 0 1 1 Total 1 1 2

In situation f, the calculation using the formula yields to 1/2:

ad =  1x1       bc =0X0        T =2            ad/T   = 1/2       bc/T = 0/2

Situation g (case is unexposed and control is exposed, discordant pairs)

 Case Control Total Exposed 0 1 1 Unexposed 1 0 1 Total 1 1 2

In situation g, the calculation using the formula yields to 1/2:

ad =  0x0       bc =1X1        T =2            ad/T   = 0/2       bc/T = 1/2

Situation h (case is unexposed and control is unexposed, concordant pairs)

 Case Control Total Exposed 0 0 0 Unexposed 1 1 2 Total 1 1 2

In situation h, the calculation using the formula yields to 0:

ad =  0x1       bc =0X1        T =2            ad/T   = 0/2       bc/T = 0/2

From the above tables and from the Mantel Haenszel formula for the odds ratio, we understand that concordant pairs (e and h) contribute neither to the numerator nor to the denominator of the ORMH.

Each discordant pair of type f contributes for ½ to the numerator and each discordant pair of type g contributes for ½ to the denominator of the ORMH. The ORMH calculated from the example on the Echovirus matched case control study  is therefore:

½ x 46
which is equal to  =  ----------------- = 46/6 = 7,7
½ x 6

In other words the ORMH is the ratio of the number of discordant pairs in which the case is exposed (f) over the number of discordant pairs in which the case is not exposed (g).

ORMH  =   f / g

When more than one control per case are selected, the same principle applies. Let's supposed we selected two controls per case. For the stratified analysis each stratum includes therefore 3 individuals (the case and its two controls). This leads to the 6 following possibilities for that type of strata.

From a triplet (1 case and 2 controls) we in fact constitute two pairs, the case with the first control and the same case with the second control. They are discordant or not with respect to exposure. The ORMH will here also be the ratio of the sum of the discordant pairs in which the case is exposed over the sum of the discordant pairs in which the case is not exposed. Concordant pairs (in which the case and a control are either both exposed or both unexposed) do not contribute to the numerator nor to the denominator of the ORMH.

Statistical sofwares help us in the calculation of the matched OR. However what we need to do as epidemiologists is to "tell" the software that we are "interested" in a matched OR, and not in an unmatched OR.

## Consequences of breaking the match (conducting an unmatched instead of a matched analysis)

One of the question frequently asked regarding matching is: "Why do we need to do a matched analysis since the groups we have created (cases and matched controls) are already equal with respect to the distribution of the confounding factor?

This is because, by matching, we have superimposed on the original confounding a selection bias that acts as an additional confounding. It is of course a very special type of selection bias, because the investigator is fully aware that this is occurring. We therefore need to control for that additional confounding by performing a stratified analysis. Another way to look at this bias is to consider that since the controls are not randomly selected (but selected according to the matching criteria), they may no longer be representative of the population giving rise to cases. We therefore need to control for that selection bias. Failing to do so (i.e. doing an unmatched analysis) would usually bring the OR towards one.

The matched  Mantel Haenszel odds ratio is controlling for the confounding effect we have introduced (selection bias) with the matched design. This is the reason why a matched analysis is required when matching. If the matched OR differs from the unmatched OR, this means we had introduced confounding when matching. We need to use the matched OR since it controls for the confounding we have introduced. If the matched odds ratio is equal to the unmatched odds ratio, this means that matching did not introduce confounding. It does not mean that matching was unnecessary. It does not mean that the confounding factor on which we have matched is not a confounder.

From matched table we have seen that the ORMH was equal to 7.7 (46/6). If we break the matched pairs and do an unmatched analysis with the unmatched table is the unmatched OR is equal to:

240 x 75
OR =  —————  =  2.57
200 x 35

The unmatched OR (2.57) is different from the matched OR (7.7) strongly suggesting that confounding was introduced by matched and that the ORMH should be used to describe the results.

### How to analysis sets with unequal numbers of controls per case

A matched case control study may have one to several controls per case, in the same study. This means that in a particular study some cases might be matched to one control, while other might be matched to two or more controls. This may happen easily, because controls might be difficult to enroll.  In the straitified analysis each stratum includes one case and  one to several controls. If each stratum does not include the same number of controls per case we could end up with strata including 2 individuals (1 case and one control), and strata with 3 individuals (1 case and two controls), etc. The computation of the stratum specific ad/T or bc/T would then change accordingly (T being equal respectively to 2 or 3, etc.). This could be a tedious process. Fortunately some statistical packages permit the analysis of unequal number of controls per case.