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Information in pmcdce Our system can be adapted for use by any spedaky. Unlike other systerm, no custom built equipment or software is required and learning to use it is easy. Although the internet has been used to transmit medi- cal image~~ this is the first report of ruing the world wide web in an em~cy that we are awa~ of. Image quality i~ paramount to the ~u:ce~ of~uch a syste~ Pre~om report~ of similar quality imag~ indicated ~ interpretation of trammitted images ~isfactory.~ However, we recommend that any department adopting this approach to patient care should audit its use, as well as emuring compliance with the Data Protection Act and its prindples. Conu'ibe~r~: PB ~ up the computer ,Diem and ~ n~- • e pmj~ DSJ ~d PH ~ ~e s~ d~ pa~ ~ ~jo~fly by DSJ, ~. P~ ~d P~ DSJ h for ~e pa~. Fun~ None• Co~ of ~t~ Non~ Fig 2 Lateral radiograph of palJent's fight anlde (left) and as viewed in a web browser (right) When can odds rados mislead? Huw Talfryn Oaldey Davies, lain Kinloch Crombie, Manouche Tavakoli Odds ratios are a common measure of the size of an effect and may be reported in ca~e<ontrol studies, cohort studies, or clinical trial~. Increasingly, they are also used to report the findings from systematic reviews and meta-anaiyse~ Odds ratios are hard'to comprehend directly and are usually interpreted as being equivalent to the relative risk. Unfortunately, there is a recognised problem that odds ratios do not approximate well to the relative risk when the initial risk (that i~ the prevalence of the outcome of interest) ~e' ~ Thus there is a danger that if odds ratios are ted as though they were relative risks then they may mislead. The advice given in many texts is unusually coy on the matter. For example: ~fhe odds ratio is approximately the same as the relative risk if the outcome of Lrtte~;t is t-a_re. For cotTtmort events, however, they can be quite different." How close is "approximately the same," how uncommon does an event have to be to qualify, ~s ~are" and how different is "quite different"? This short note quantifies the discrepancy, between odds ratios and relative risks in different circumstances. and assesses whether such a discrepancy may seriously mislead if an odds ratio is used as an estimate of the relative risk. Odds and risk There is a problem with odds: unlike risks, they are dif- ficult to understand. The risk of=m event happening is Snmmary points If the odds ratio i~ interpreted as a relative risk it will always over, tare any effect size: the odds ratio is smaller t/ran the relative risk for odds ratios of less tlmn one, and bigger than the r~4arive risk for odds ratio, of greater than one The extent of oversmmment increas~ as both the initial risk increases and the odcla ratio departs f~om unity However, serious ~vergence between the odds ratio and the re_!arlve risk occm~ only with large effects on groups at high ix.ida/risk. Therefore qualitative judgments based on interpreting odds ratioa as though they were rdative risks are unlikely to be seriously in error • i In studies which show reductions in risk (odds I ratios o~l~ss than one), the odds ratio will never I underestimate the reladve risk by a greamr ~ percentage than the level of initial risk In studies which show ina-ease$ in risk (odds ratios of greater than one), the odds ratio will be no more than twice the relative risk so long as the odds ratio times the initial risk is tess tha~ 100% Department of ,Management. University of $~ ?,ndrev~ St Andrews KY16 9AL Huw Talf~'n Oaldey Davies. la~urer m htalth care Manouch¢ lta'w'~ in htalth and ind,,~wi,,~ tco~omic~ Department of Epidemio|ogy and Pubic Health. Uni~tsi~" of Dundee. Ninew¢lIs Hospital mad Dundee DD 1 9SY lain KLnioch Crorabia Dr Davie~ 8M] 1998',316,-989-91 ,SM~ VOLL'.ME :I Itl '2~q .~L-~RCH 199~ 9~9

Information in practice [ Table I Comparing risks and odds Riz~ Oddz 0.05 or 5% 0.053 0.1 or 10% 0.11 0.2 or 20% 0.25 0.3 or 30% 0.43 0.4 or 40% 0.67 0.5 or 50% 1 0.6 or C~]% 1.5 0.7 or 70% 2.3 0.8 or 80% 4 0.9 or 90% 9 0.95 or 95% 19 simply the number of" those who experience the event divided by the total number of people at risk of'having that event. It is usually expressed as a proportion or as a percentage. In either case the meaning is usually clear. In contrast, the odds of an event is the number of those who experience the event divided by the number of those who do not. It is expressed as a number from zero (event will never happen) to infird~, (event is cer- tain to happen). Odds are fairly easy, to visualise when they are greater than one. but are less easily grasped when the ~-alue is less than one. Thus odds of six (that is, six to one) mean that six people will experience the e~nt for ever), one that does not (a risk of six out of seven or 86q~0. An odds of 0.2° however seems less intuitive: 0-o people will experience the event for every one that does not.This translates to one event for ev~," five non-events (a risk of one in six or 17%). A ~econd problem with odds is that, although they are r,-h-,d to risk, the relation is not straightforward. The table shows the odds for various risks. For risks of less than about 20% the odds are not greatly dissimilar to the risk, but as the risk climbs above 50% the odds start to look very different. Relative risks and odds ratios The relative risk ot'one group compared with another is simply the rado of the risks in the two groups. Thus the relative risk tells us how much risk is increased or decreased fi-om an initial leveL Again it is readily understood: a relative risk of 0.5 shows that the initial risk has been halved; a relative risk of 3 shows that the initial risk has been increased threefold. The odds ratio is calculated in a similar way: it is simply the rado of the odds in the two groups ofinter- es~ We know that if the odds ratio is less than one then the odds (and therefore the risk too) has decreased, and if the odds ratio is greater than one then they have increased. But by how much? How do we interpret an odds ratio of', say, 0.5 or an odds ratio of 3? A lack of familiarity with odds means that many people have no intuitive fed for the size 'of the difference when expressed in this waF When the risks (or odds) in the two groups being compared are both small (say less than 20%) then the odds will approximate to the risks and the odds ratio will approximate to the relative risk. Then interpretation is easv. But as the risk in either group rises above 20°/, the gap between the odds ratio and the relative risk ~dll widen. A recent article in Bandolier concluded that "as both the prevalence [inidal risk] and ~e odds ratio increase, the error in the approximation quickly becomes unacceptable."" But is this the case? In what circumstances will interpreting an odds ratio as though it were a relative risk lead to serious errors in interpretation? Odds x-ado as an approximado_n of" reladve risk - When faced with an odds rado, we want to "know the discrepancy between that odds rado and the relative risk. Figures 1 and 2 show the extent to whi~h the reported odds rado underestimates or overestimates the reladve risk for different odds rados and a given level of initial risk (see appendix for calculations). Figure 1 shows the underestimation of the retadve risk by the odds ratio in studies,that report odds rados of less than one (,typically studies of benefit from treat- ment or exposure). Even with initial risks as high as 50'~h and ver)" large reductiom in this risk (odds ratios of about 0.1), the odds ratio is only 50% smaller than the relative risk (0.1 for the odds ratio compared with a u'ue value for the reladve risk of 0.20). In fact, the discrepancy between the odds rado and the true reladve risk will never be greater than the initial risk (see appendix for proof). . Figure 2 shows the discrepanq," between the od~is rado and the reladve risk for studies which report odds ratios of greater than one (typically studies showing harm). Although h~ge discrepandes between the odds ratio and the reladve risk are poss~le, the odds ratio overstates the reladve risk by less than 50% for a wide range of both initial risks and effect sizes. For initial risks of I0% or less. even odds ratios of up to eight can reasonably be interpreted as relative risks; for inidal ~.~ 90 Odds ratios ~_ 40 - . 30 20 0.7 1°0 0.9 0 20 40 60 80 100 Fig 1 Amount by w~ich odds ratios of <1 underestimate relative risk, for different odds ratios and different levels of initial risk 0 20 40 60 80 100 In~ial ask (%) Fig 2 Amount by which odds ratios of >t overestimate relative risk, for different odds ratios and different levels of initial risk 990 B.'~ VOLL.%IE 316 -08 ;'.L-LRCH I998

Information in practice Example o~ use of odds ratios The fortnightly review by Dennis and Langhome, stroke units save lives: where do we go from here?" (BMf 1994"~09:1273-7) reported outcomes after stroke (death or living in an insrlmtion) for patients managed in SlX~t~t s~oke units compared with patients ~ on general medical watch. Specialist stroke units had the better outcomes, with a reported odds ratio of 0.66. The autimrs advised that an ~odds ratio of < 1.0 ind~r~ that outcome of care in a stroke unit is better," and concluded that "patients with stroke treated in sped~!i~t units were less likely to die than those treated in gener¢l medical wards." No further guidance was given on interpreting the quoted Be~-~e the fi~tuency of a poor outcon~e was very high (about 55%) there might be concern tha~ the odds rado is a poor estimatd ofth~ relative risk. In fact, the odds ratio of 0£6 corresponds to a rela~ risk of risk by just 19%. In other words, interpretng the odds ratio as a rehtive risk suggests a reduction in deleterious outcomes after stroke (death or living in an institution) of about a third compared with a more likely u-ue reduction of'about a fifth. CAearly, in either case this r~-preseats a substantial reduction in poor outcomes for a patient Stoup with a large initial risk. risks up to 30% the approximation breaks down when the effect size gives odds ratios of more than about three.As a conservative rule of thumb, i~the initial risk multiplied by the odds ratio is less than 10~0 then the odds ratio will overestimate the relative risk by les~ than twofold. Does the discrep .ancy influence our interpretation? [ The figures show that the odds ratio will always exaggerate the size of the effect compared with a rela- tive risL That is, if the odds ratio is less than one then it is always smaller than the relative risL Conversely, if the odds ratio is greater than one then it is always bigger than the relative risL Thus interpreting an odds ratio as though it were a relative risk could mislead us into believing that an effect size is bigger than is actually the Crucially, however, large discrepancies are seen for only large effect sizes. Suppose an odds ratio of. say. 0.2 reflects a true relative risk of 0.4. Such a discrepancy, is unlikely to alter your view: this is a large reduction in risk whichever way you look at it. This is particularly so as large discrepancies occur only when the initial risk is high and thus even modest changes in the relative risk will mean substantial gains. So, for studies which show reductions in risk. the odds ratio is unlikely to mislead: either it ~¢ilI be dose in value to the relative risk or it represents a substantial effect for ~oups at high initial risk. Thus any qualitative judgment is unaltered by the discrepancy benveen the odds ratio and the relative risk (see box). The same logic holds for studies which show increases in risk. The discrep,-mcy bet~veen the odds ratio and the relative risk becomes large only when there are large effects (a twotbid or threetbid hncrease in risk) tbr groups ,-already at a large initial ris'k. Although the odds ratio may diverge quite sharply from the reladve risL by the time it does so the message conveyed by the different measures is the same." these are large effecm Of course, although qualitative judgments may be unaltered by the odds ratio deviating from the relative risk, quantitatively we can still be led ashy. Thus it" we are interested in assessing the impact of'interventioaas quantitatively (for example, for a cost effectiveness analysis) then, for larger initial risks and substantial odds ratios, the actual relative risk should still be calculated. Conclusion The difference between the odds rado and the relative risk depends on the risks (or odds) in both groups. So for any reported odds ratio, the discrepancy between that odds ratio and the relative risk depends on both the initial risk and the odds ratio itself. This is possibly why textbooks are coy about giving a single figure for risk beneath which it is acceptable to interpret odds ratios as though they, were relative risks. Odds ratios may be non-intuitive in interpretation, but in almost all _renlistic cases interpreting them as though they, were relative risks is unlikely to change any qualitative assessment of the study findings. The odds ratio will always overstate the case when interpreted as a relative risk, and the degree of overstatement will increase as both the initial risk increases and the size of any u-eamaent effect increases. However, there is no point at which the degree ofover- statement is likely to lead to qualitatively different judgments about the study.. Substantial discrepancies between the odds ratio and the relative risk are seen only when the effect sizes are large and the initial risk is high. Whether a large increase or a large decrease in risk is indicated, our judgments are likely to be the Appendix: Calculation of discrepancy between odda ratios and relative risks If the profa)rtions of subjects ~xperiencing an event in groups are P~ (initial risk) and P._, (post-intervention risk) then the relative risk is P,.~/P~ and t_he odds ratio is ( 1 - P ~ )/ ( I - P.,.) x relative risk. Simple algebra leads this multiplier to be recast as 1 - Pi ~'(Pt xodds ratio). However. it is conven- ient m ~pres,s the discrepancy, between the odds rado ,and the rela~'~ risk as a proportion of the relative risL Therefore. for smdie~ in which the odds ratio is < 1. I minus this multiplier is the d/screpanq. (Pt- (P~ x odds ratio)). For studies in which the odds ratio is > 1, the multiplier minus gives the discrepancy. ((P~ x odds ratio)- P0. Figures I and 2 plot these discrepancy, wa.lues (as percentages) for s-arious inithl risks and odds ratios. Conu'ibutors: The ideas contained in this paper arose tixmx dis- cussions between HTOD and IKC and were daritied in debate with .~FE HTOD wrote the first draft of the manuscript, which ~us edited by LKC and .~f'K HTOD is guarantor/br the article. Conflict of interest: None. 1 Sim~air JC. Bracke~ .',lB. Clinically m~-.fial me'asure~ of effect in I~inarv armls~"~ of randomized ~] Clin E~i~ H~:47k~8 I 2" D~ J. S~ co~ w~t is ~ ~ mfio~ ~di~ i996:3CB:6-7. ,M~ ~. ~iral ,t~ut:~ l~ mr~cal r~ea~ London: Chapmaa and H~L ltgJ I. ~Acc~ted 24 Fe&ua~ l BMJ VOLUME 31~; 2,q .MARCH 1998 ~.)~,~ I

• L~6rmation m pracuce. Netlines Lest we forget ... • Andrmv Bamji has placed the Plastic Surgery Archives--a collection of m~_~al that documents the development of pl~dc surge~t at the beginning of the 20th cenm.--y, parti~11~rly aft~ the first world war-on the web on h0m,ps0~.l~tm. The site h~ links to other online ~ a~out the first world war, including a medical bibliography of the war F_.R online • As ER is probably the best medical drama on British television, it is nice to see so much ER ~!~ed stuffon the interact. A good starting place for exploring it all is the AIt.TV.ER site - (lfllp://www.dlgls~ve.c~m/er/erdex.Mml), wher~ you ~ pick up episode listings, sumrrmries and reviews, and aho commentaries on the medical conditions featured in each show. There is also an exhaustive set of links to other FaR pages and sites. British viewers can discuss the show on the newsgroup uk.medi~tv.er (newsazLmedia.rv.er). Evidence based medicine • There are ever more sources of evidence based medicine appearing on the web. The full text of the evidence based medicine journal B~o/~ is ava~hle free on a¢,al011~011~r/, the Internes Database of Evidence-Based Abswacts and Articles (IDEA) can 0bm_~ and the NHS Centre for Reviews Netting the Evidence a~lml0R-~da~r/Nlt~.l~nl), an index of online sources of evidence based medicine, complete ~th cornmmGIT~S~ ~C~IUC~ ~ A~td~-'w Booth at the School cffHealth and Reded Research (ScHARR), She~dd. O.line journals: Highwire Press *, With production of the BMJ website all set to change over to Highwire Press next month, it is htC'.01~.hi~.0~ in Europe) to see how many oniinejoumals they are m.n.~ag now-- everything from the ~ magazine (Im~#~.~amma¢.0~). Future titles will include the Annual Reviews series and the journal', of Se American Society for Microbiology and the American Heart Association. AH the journals are available as full text online both in HTML and Adobe Acrobat format and come with fully searchable archives of past issues. The ordy snag is that, for most of them, you must have a subscription. In the near future the I-Hghwire Press site will allow you to search all its journals in one go, and will ~ featm-e a Medline servic~ He@lth Inf~ on the Internet • He@lthInf~ont~L.rna (htlp://v~w.wollc~mo.a¢.uk/hoalthlnlo/) is a new bimonthly newsletter from the Wdlcome Trust and the Royal Society of Medicine, containing a range of contffbuted arddes and regular features. The first issue is avalhhle in full on the web at editorial board. Index to Theses • The Index to Theses ~ite (~l¢:t/mm.tim~.ma/) allows you to search an online database of theses accept~ for higher degrees by the Universities of Great Britain and Ireland. Abstracts are axrMl~hle for recent these~ To use the site you must be in an institution that mb~ to the "dead-tree" version of the databas~ Laparoscopy online - _ • The laparoscopy.com website laparosco~.com) features a feast of virtual laparoscopy, including mulfimedi~ walk-throughs of procedures, images, an online radio channel, and discussion forums. The V'm~le Embryo • The Wmible Embryo (http://vlsembry0.uc~.edu/) is an hnpressive oniine tour of the first four weeks of human life. For full appreO'~rlon of the site, however, you must have the Shockwave plug-in (avaihhle from ldJp://wmv.msctomedia.com) and plenty of memory allocated to your web browser. Compiled by Mark PaUen email m.patlen@qmw.ac.uk web page http://www.medmicro.mds.qmw.ac.uW-mpallen 0 O~ 0 992 BMJ VOLL.'ME 316 28 ,%-k.RCH 199~