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## Swots corner: what is an odds ratio?

**medicine.ox.ac.uk**
**Swots Corner: What is an odds ratio?**
**Bandolier** readers wil 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 onegroup of patients is different from that found in the control groups. Many studies, and particularlysystematic reviews, report their results as odds ratios, or as a reduction in odds ratios, and sometrials do the same. Odds ratios are also commonly used in epidemiological studies to describe thelikely harm an exposure might cause.

Bandolier therefore turned to Jon Deeks, of the Centre for Statistics in Medicine, to answer thequestion - 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 randomlychosen 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 likelyto happen than not (the odds of an event that is certain to happen are infinite); if the odds areless than one the chances are that the event won't happen (the odds of an impossible event arezero).

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

Epidemiological studies general y try to identify factors that cause harm - those with odds ratiosgreater than one. For example, looked at case-control studies investigating thepotential harm of giving high doses of calcium channel blockers for hypertension.

Clinical trials typical y look for treatments which reduce event rates, and which have odds ratios ofless than one. In these cases a percentage reduction in the odds ratios is often quoted instead of
less than one. In these cases a percentage reduction in the odds ratios is often quoted instead ofthe odds ratio. For example, the ISIS-4 trial reported a 7% reduction in the odds of mortality withcaptopril, 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 oddsratios (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 toldthat the risk had doubled (relative risk = 2) or halved (relative risk = 0.5) we feel we have a clearperception 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 thatthey are relative risks. When events are rare, risks and odds are very similar. For example, in theISIS-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 in1000, or a relative error of about 7%.This close approximation holds when we talk about oddsratios and relative risks, providing the events are rare.

**Assessing harm**
The first figure shows the relationship between ORs and RRs for studies which are assessingharm. Each line on the graph relates to a different baseline prevalence, or event rate in the controlgroup. We can use this graph to get a grasp of how misleading it could be to interpret ORs as ifthey were RRs. It is clear that when the prevalence of the event is low, say 1%, the RR is a goodapproximation of the OR. For example, when the OR is 10, the RR is 9, an error of 10%. click herefor which wil be slow to download (30+k)
That sort of error is unlikely to be seriously misleading, especial y when you remember the likelywidth of the confidence intervals which go along with the estimates. However, as both theprevalence and OR increase, the error in the approximation quickly becomes unacceptable: if thebaseline 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 assessingbenefit. Again, when event rates are very low the approximation is close, but breaks down asevent rates increase. For example, if the event rate is 50% and there is a 20% reduction in theodds, the relative risk adjustment wil be little over 10%.

click here for which wil be slow to download (90k)
However, in the case of ISIS-4 the 7% reduction in the odds of mortality with captopril correspondsto a 7.7% reduction in risk. So providing that the events are rare and the treatment not toosuccessful, the approximation can be used in these circumstances as wel .

**Why use an OR rather than RR?**
So if odds ratios are difficult to interpret, why do we not always use relative risks instead? Manyacademics agree and have argued that there is no place for describing treatment effects in clinicaltrials using odds ratios. However, they continue to be used, especial y in systematic reviews.

There are several reasons for this, most of which relate to the superior mathematical properties ofodds ratios. Odds ratios can always take values between zero and infinity, which is not the casefor relative risk. A keen reader may have already spotted a problem in the sex ratio example citedabove: 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 couldobviously cause problems if we were performing a meta-analysis of relative risks in trials withgreatly different event rates. Odds ratios also possess a symmetrical property: if you reverse theoutcomes in the analysis and look at good outcomes rather than bad, the relationships wil havereciprocal 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: theapparent prevalence there depends solely on the ratio of sampling cases to controls, which istotal y artificial. To use an effect measure which is altered by prevalence in these circumstanceswould obviously be wrong, so odds ratios are the ideal choice. This in fact provides the historicallink with their use in meta-analyses: the statistical methods that are routinely used are based onmethods 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, andwil soon be provided in the Cochrane Database of Systematic Reviews, but more caution isrequired in their application, especial y when there are large variations in baseline event rates.

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

Al of which makes odds ratios likely to be with us for some time - so we need to understand howto use them. Of course it's important to consider the statistical significance of an effect as wel asits size: as with relative risks we can most easily spot statistical y significant odds ratios by notingwhether their 95% confidence intervals do not include 1, that's analogous to there being less thana 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.

Source: http://ctu.edu.vn/~dvxe/econometrics/What_is_an_odds_ratio.pdf

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