## Swots Corner: What is an odds ratio?

Bandolier readers will know that we favour the number-needed-to-treat (NNT) as a way of describing the benefits (or harms) of treatments, both in individual trials and in systematic reviews. Few papers report results using this easily interpretable measure, so Bandolier has had to get its head around how to do the calculations.

NNT calculations, though, come second to working out whether an effect of treatment in one group of patients is different from that found in the control groups. Many studies, and particularly systematic reviews, report their results as odds ratios, or as a reduction in odds ratios, and some trials do the same. Odds ratios are also commonly used in epidemiological studies to describe the likely harm an exposure might cause.

Bandolier therefore turned to Jon Deeks, of the Centre for Statistics in Medicine, to answer the question - what the heck is an odds ratio?

### Calculating the odds

The odds of an event are calculated as the number of events divided by the number of non-events. For example, on average 51 boys are born in every 100 births, so the odds of any randomly chosen delivery being that of a boy is:

number of boys 51 / number of girls 49, or about 1.04

Equivalently we could have calculated the same answer as the ratio of the baby being a boy (0.51) and it not being a boy (0.49). If the odds of an event are greater than one the event is more likely to happen than not (the odds of an event that is certain to happen are infinite); if the odds are less than one the chances are that the event won't happen (the odds of an impossible event are zero).

### An odds ratio is.....

An odds ratio is calculated by dividing the odds in the treated or exposed group by the odds in the control group.

Epidemiological studies generally try to identify factors that cause harm - those with odds ratios greater than one. For example, Bandolier 20 looked at case-control studies investigating the potential harm of giving high doses of calcium channel blockers for hypertension.

Clinical trials typically look for treatments which reduce event rates, and which have odds ratios of less than one. In these cases a percentage reduction in the odds ratios is often quoted instead of the odds ratio. For example, the ISIS-4 trial reported a 7% reduction in the odds of mortality with captopril, rather than reporting an odds ratio of 0.93.

### Relative risks

Few people have a natural ability to interpret event rates which are reported in terms of odds ratios (which may be why bookmakers always use them). Understanding risks, and relative risks, seems to be something easier to grasp.

The risk (or probability) of having a boy is simply 51/100, or 0.51. If for some reason we were told that the risk had doubled (relative risk = 2) or halved (relative risk = 0.5) we feel we have a clear perception as to what this would mean: the event would be twice as likely, or half as likely to occur.

### Risks and odds

In many situations in medicine we can get a long way in interpreting odds ratios by pretending that they are relative risks. When events are rare, risks and odds are very similar. For example, in the ISIS-4 study 2,231 of 29,022 patients in the control group died within 35 days: a risk of 0.077 [2,231/29,022] or an odds of 0.083 [2,231/(29,022 - 2,231)]. This is an absolute difference of 6 in 1000, or a relative error of about 7%.This close approximation holds when we talk about odds ratios and relative risks, providing the events are rare.

### Assessing harm

The first figure shows the relationship between ORs and RRs for studies which are assessing harm. Each line on the graph relates to a different baseline prevalence, or event rate in the control group. We can use this graph to get a grasp of how misleading it could be to interpret ORs as if they were RRs. It is clear that when the prevalence of the event is low, say 1%, the RR is a good approximation of the OR. For example, when the OR is 10, the RR is 9, an error of 10%. click here for Figure 1 which will be slow to download (30+k)

That sort of error is unlikely to be seriously misleading, especially when you remember the likely width of the confidence intervals which go along with the estimates. However, as both the prevalence and OR increase, the error in the approximation quickly becomes unacceptable: if the baseline prevalence is 10% an OR of 4 is equivalent to a RR of 3, a discrepancy of 25%.

### Assessing benefit

The second figure indicates the relationship between OR and RR for studies which are assessing benefit. Again, when event rates are very low the approximation is close, but breaks down as event rates increase. For example, if the event rate is 50% and there is a 20% reduction in the odds, the relative risk adjustment will be little over 10%.

However, in the case of ISIS-4 the 7% reduction in the odds of mortality with captopril corresponds to a 7.7% reduction in risk. So providing that the events are rare and the treatment not too successful, the approximation can be used in these circumstances as well.

### Why use an OR rather than RR?

So if odds ratios are difficult to interpret, why do we not always use relative risks instead? Many academics agree and have argued that there is no place for describing treatment effects in clinical trials using odds ratios. However, they continue to be used, especially in systematic reviews.

There are several reasons for this, most of which relate to the superior mathematical properties of odds ratios. Odds ratios can always take values between zero and infinity, which is not the case for relative risk. A keen reader may have already spotted a problem in the sex ratio example cited above: if the baseline risk of having a boy is 0.51 it is not possible to double it!

The range that relative risk can take therefore depends on the baseline event rate. This could obviously cause problems if we were performing a meta-analysis of relative risks in trials with greatly different event rates. Odds ratios also possess a symmetrical property: if you reverse the outcomes in the analysis and look at good outcomes rather than bad, the relationships will have reciprocal odds ratios. This again is not true for relative risks.

Odds ratios are always used in case-control studies where disease prevalence is not known: the apparent prevalence there depends solely on the ratio of sampling cases to controls, which is totally artificial. To use an effect measure which is altered by prevalence in these circumstances would obviously be wrong, so odds ratios are the ideal choice. This in fact provides the historical link with their use in meta-analyses: the statistical methods that are routinely used are based on methods first published in the 1950s for the analysis of stratified case-control studies. Meta-analytical methods are now available which combine relative risks and absolute risk reductions, and will soon be provided in the Cochrane Database of Systematic Reviews, but more caution is required in their application, especially when there are large variations in baseline event rates.

A fourth point of convenience occurs if we need to make adjustments for confounding factors using multiple regression. When we are measuring event rates the correct approach is to use logistic regression models which work in terms of odds, and report effects as odds ratios.

All of which makes odds ratios likely to be with us for some time - so we need to understand how to use them. Of course it's important to consider the statistical significance of an effect as well as its size: as with relative risks we can most easily spot statistically significant odds ratios by noting whether their 95% confidence intervals do not include 1, that's analogous to there being less than a 1 in 20 chance (or a probability of less than 0.05, or gambling odds of better than 19 to 1) that the reported effect is solely due to chance.

Jon Deeks

Centre for Statistics in Medicine Oxford

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