J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.52.7.817 on 1 July 1989. Downloaded from

Journal ofNeurology, Neurosurgery, and Psychiatry 1989;52:817-820 Occasional review

The Ratio: a useful tool in neurosciences

PETER SANDERCOCK From the Department ofClinical Neurosciences, Western General Hospital, Edinburgh, UK

SUMMARY The odds ratio is very useful; when used as a measure of association (or to describe the results of a randomised controlled treatment trial) it expresses both the strength and direction of the association (or size and direction of treatment effect). Its 95% estimates the likely degree of error and provides a test of significance at the 5% level. Authors, when reporting the results of descriptive studies, case-control studies and randomised controlled trials, should consider presenting their results this way rather than by simple significance testing with a chi squared test. The odds ratio (an unlikely name for a useful tool) has greater in smokers than non-smokers, the been around for a long time.' Although it was is said to be five.'34

originally used to analyse case-control studies, it has However, cohort studies are expensive, take a long Protected by copyright. more recently been applied to other research designs- time to generate results and, though many individuals descriptive studies, randomised trials and overviews of are recruited, a relatively small number will develop randomised trials. Despite its widespread use in the under study. Case-control studies can be medicine, many of the basic textbooks of medical mounted much more quickly and cheaply; two groups mention the odds ratio only briefly, if at all. of subjects are assembled-one known to have the The aim ofthis article is to describe the usefulness and disease (cases) and the other who are, as far as practical application of the odds ratio simply and possible, known to be free of the disease but come clearly and with minimal use of the mathematical from the same general population (controls). Both formulae that appear in articles written by statistician- groups are examined for evidence of previous s'-3 (whose complexity may induce sleep, diplopia or exposure. The methods used to identify, collect and tension headache in the average neurological reader). examine cases and controls are much more important The first problem to overcome is terminology. The sources of error and than the statistical method- terms rate, risk, , relative risk (or risk s.134 Since the cases were assembled after they had ratio), odds and odds ratio each have precise developed signs of disease the rate is unkn- epidemiological definitions, and cannot be used inter- own, so it is not possible to calculate relative risk. changeably.'34 However, the ratio of the odds that a case had been http://jnnp.bmj.com/ Briefly, a relative risk is measured in prospective exposed (a:c in the example in table 1) to the odds that longitudinal . A group of disease free a control had been exposed (b:d), algebraically the individuals are identified, the presence or absence of same as ad/bc, is a close approximation to the relative the relevant exposure (for example, smoking) is deter- risk that would be obtained from a comparable cohort mined at baseline, and then the subjects are followed and the development of disease (for example, lung Table 1 Case-control study: multiple sclerosis andsheep cancer) in exposed and non-exposed subjects is exposure measured and compared. If, after correcting for any on September 26, 2021 by guest. two in the of Multiple Sckrosis difference between the groups length cases Healthy Controls follow up, the incidence rate oflung cancer is five times n = 200 n = 200 Shepherds 20' 4b Address for reprint requests: Dr Peter Sandercock, Department of No sheep exposure 180' 196d Clinical Sciences, Western General Hospital, Crewe Road, Edinburgh EH4 2XU, UK. Odds ratio = ad/bc = 5.4. X2 = 9.97. 95% confidence interval* 1-8 to 16 2. 2p = 0-0015. Received 30 January 1989. Accepted 6 February 1989 *See Appendix. 817 J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.52.7.817 on 1 July 1989. Downloaded from

818 Sandercock study,'3 provided the incidence ofdisease is fairly low association, nor of the likely sampling error. The in exposed and non-exposed subjects (or, in randomis- confidence intervals in this example indicate that a ed trials, only a small proportion of ;patients have larger sample would be required to determine the outcome events). The calculation olf confidence strength and directon ofthe association more precise- intervals for odds ratios is much simpler than for risk ly. ratios. The advantage ofthe odds ratio over the x2 becomes The odds ratio is therefore-like the re]lative risk-a clearer if one considers a smaller study with, say, only measure of association. In the example, t;he odds ratio 50 cases and 50 controls. If there were five shepherds was 5 3, which that the odds of occupational amongst the cases and one shepherd amongst the exposure to sheep were 5 3 times higher in cases than controls, the odds ratio would be almost identical controls. The odds ratio thus indicates thte direction of (5.4). The confidence intervals would be much wider the association; an odds ratio great(er than one (0-6 to 48.3), indicating that there might be an indicates a positive association between (exposure and association as strong as, and in the same direction as, disease, odds ratio equal to one indicates no associa- the first example, but with a much larger sampling tion, and the odds ratio less than on(e indicates a error because of a small sample size. The x2 negative association. Furthermore, thte odds ratio (1.60) is uninformative and merely shows no sig- indicates the strength of the associationi; odds ratios nificant association. close to one indicate only weak associatiions, whereas The odds ratio and its 95% confidence interval are ratios over about three for positive ass;ociations, or also useful in summarising the results of randomised near zero for negative associations in(dicate strong controlled trials oftreatments. In table 2, the results of associations. Of course, the presence of a strong a randomised controlled trial are given. In this case, association is not proof of causation. Pr(oof of causa- the convention is to arrange the layout of the 2 x 2 tion generally depends on other facttors such as table so that ifactive treatment is effective, and reduces consistency across several studies, spec-ificity of the the odds of the outcome event, in this case death or

association, biological plausibility, and (evidence of a disability related to stroke, then the odds ratio (that is, Protected by copyright. dose-response relationship between expc)sure and the the odds of being dead or disabled if allocated active risk of developing the disease.'34 treatment divided by the odds of being dead or It is a relatively simple procedure to calculate the disabled ifallocated placebo) is less than one. An odds approximate 95% confidence intervals for an odds ratio of one would indicate no treatment effect: (1- ratio (Appendix).'`3 The confidencce intervals 1) x 100 = 0. In the example, the odds ratio is 0*61; in immediately allow a test of statistical siignificance (at other words treatment reduced the odds of death or the 5% level for 95% confidence inteirvals); if the disability by (1-0.61) x 100 = 39%. The confidence interval includes an odds ratio of one, th(e result is not interval in this example is narrow, so the most significant. The interval also gives a fe,eling for the pessimistic estimate of treatment effect is that it only likely degree of sampling error; in the ex[ample given, reduces the odds of death or disability by (1- although the confidence intervals do nolt include one 0.72) x 100 = 28% or alternatively, the reduction and the difference between cases andI controls is might be as much as (-0 51) x 100 = 49%. In either therefore significant, there is still a wi(de degree of event the result is clearly statistically significant, since uilcertainty. The results would not only be compatible the confidence interval does not include one, but more with only a very weak association (lowetr confidence important than mere statistical significance, the reduc- http://jnnp.bmj.com/ limit odds ratio 1-8) but also a very stronig one (upper tion is clinically significant too, even assuming the confidence limit odds ratio 16 2). worst case ofthe upper confidence limit, ofa 28% odds The more traditional measure, the x2 statistic, reduction. confirms a statistically significant result, but does not The odds ratio also is useful in a statistical overview give any indication of the strength or dirrection of the of several different randomised trials, because of its statistical robustness and simplicity ofmanipulation.' Table 2 Randomised controlled study: Cerebr,'o protective The odds ratio and its 95% confidence interval are a agent in acute stroke

summary of the results of a single trial and when on September 26, 2021 by guest. Treated Placebo displayed graphically alongside the other trials n = 2000 n = 2000 included in the overview, allow a rapid visual assess- ment of the results of each separate trial, any Dead or disabled from stroke 264 (13.2%) 400(20%) heterogeneity of treatment effect between trials and Alive and not disabled 1736 1600 also the extent of random error. Thus, in an overview Odds ratio = 0-61. X2 = 32 9. oftrials ofantiplatelet agents in patients with vascular 95% confidence interval 0 51-072. < 000001. Odds reduction = 39%. 2p disease, such a graphical display made it clear that 95% confidence interval 28% to 49%. many of the apparent differences between trials (gen- J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.52.7.817 on 1 July 1989. Downloaded from

The Odds Ratio-a useful tool in neurosciences 819 erally the smaller ones with wide confidence intervals) was more to do with the play of chance than real differences in treatment effect.5This interpretation would not be possible in an analysis based on simple A significance testing.5 The odds ratio may also be used in descriptive studies in which a group of cases has been assembled B without any controls; the cases in the example, a study of patients with cerebral aneurysms (table 3a) have been divided into those with and without epilepsy. The occurs 2-7 odds ratio suggests post-operative epilepsy 0 1 2 4 6 8 times more often amongst patients with middle Epilepsy Epilepsy commoner Odds ratio commoner cerebral artery (MCA) aneurysms but the width ofthe in other in MCA confidence intervals, coming close to one, emphasises aneurysms aneurysms the uncertainty about the strength ofthe association; it Fig The graph illustrates the data in tables 3a and 3b. An is, however, conventionally significant at the 5% level. odds ratio ofone would suggest no association between If one of the 40 cases without post-operative epilepsy aneurysm location and development ofpost-operative epilepsy. The square gives thepoint estimate ofthe odds ratio with a non-MCA aneurysm were to be excluded from and the horizontal lines give the 95% confidence intervals. In the analysis (say because the angiogram had been re- this example, the odds ratio suggests that epilepsy occurs 2-6 examined and did not show an aneurysm) then the times more often in patients with MCA aneurysms compared odds ratio and confidence intervals would change very with other aneurysm sites. A and B correspond to tables 3a little (table 3b and figure), yet the X2statistic would no and 3b. In A, the confidence interval does (not quite) include longer be significant. This illustrates the absurdity of one, and the result is significant at the 5% level. In B, the choosing a single criterion ofp < 0-05 to differentiate confidence interval includes one and the result is non- "significant" from"non-significant". Is p = 0 0495 so significant. Protected by copyright. very different from p = 0 057? Of course not, yet in many papers, the former would be reported as "sig- One word of caution. In a graphical display, wide nificant" and the latter "non-significant"; by implica- confidence intervals are more visually striking than tion "clinically relevant" and "clinically irrelevant" narrow confidence intervals, yet greater emphasis respectively. Inspection of the confidence intervals should be placed on studies with narrow intervals reveals a clearer but statistically more uncertain because the result is more precise (that is, less sampling picture, and graphical display reveals how similar the error).5 Peto has suggested that the point estimate of two results really are (fig). the odds ratio should be drawn as a black square whose size is proportional to the statistical power of the study; the bigger the power, the larger the square.5 Table 3 (a) Descriptive study: association between middle This then visually offsets the effects of the associated cerebral artery aneurysms andpost-operative epilepsy in short confidence interval in a powerful study. patients with subarachnoid haemorrhage In summary, the odds ratio has many properties which make it in MCA Aneurysms useful expressing the results of a

aneurysm elsewhere variety of different research designs employed in the http://jnnp.bmj.com/ n = 50 n = 50 neurosciences. The time for simple significance testing and tabulation of Epilepsy post-op 20 10 X2has long passed; estimation of the No epilepsy post-op 30 40 size and direction of effects, taking account of sam- pling error and the statistical power of the study are Odds ratio = 2-7. X2 = 386. now necessary, and these requirements are fulfilled by 95% confidence interval 11 to 6-5. 2p = 0-0495. the odds ratio and its 95% confidence interval.

(b) Reanalysis ofdata in (a) after one patient without Appendix on September 26, 2021 by guest. epilepsy excludedfrom analysis One fairly simple way to calculate a good approxima- MCA Aneurysms tion to the 95% confidence interval2 is: aneurysms elsewhere First, calculate the of the natural the odds ratio Epilepsy post-op 20 10 logarithm of (lo& OR) No epilepsy post-op 30 39 SE Odds ratio = 2-6. X2 = 362. (loge OR)= + b + c +d 95% confidence interval 1 0 to 6-4. 2p = 0 057. J Neurol Neurosurg Psychiatry: first published as 10.1136/jnnp.52.7.817 on 1 July 1989. Downloaded from

820 Sandercock

1 Instruments TI-66 programmable calculator; please = 4 18 19 send a self-addressed envelope to the author, if a copy 120+4+ 180+196=056 is required. Next, calculate Y = lo& OR - (196 x SE (lo& OR)) References Z = log OR + (I 96 x SE (lo& OR)) Y = log 5-4 - 1-96 x 0 56) = 0X60 1 Feinstein AR. Clinical . The Architecture Z = loge 5 4 + (1 96 x 0 56) = 279 of . Philadelphia: Saunders, 1985:420-34. The lower 95% confidence interval for the odds ratio 2 Morris J, Gardner M. Calculating confidence intervals is: for relative risks (odds ratios) and standardised ratios eY = e"OW = 1[8 and rates. Br MedJ 1988;296:1313-36. 3 Breslow NE, Day NE. Statistical Methods in Cancer And the upper 95% confidence interval is: Research Vol I-The Analysis of case-control studies. ez = e279 = 162 IARC Publication no 32. Lyon: International Agency This approximation is less for Research on Cancer, 1980:42-81. accurate in very small 4 Sackett D, Haynes RB, Tugwell P. Clinical Epidemiology. samples, especially ifthe odds ratio is close to unity, in A Sciencefor ClinicalMedicine. Toronto: Little Brown, which case, exact methods have to be used. 1985. This formula is available as a programme listing, 5 Antiplatelet trialists collaboration. Secondary prevention written in GW-BASIC, which runs on IBM-compati- of vascular disease by prolonged antiplatelet treat- ble computers or as a similar programme for the Texas ment. Br Med J 1988;296:320-31. Protected by copyright. http://jnnp.bmj.com/ on September 26, 2021 by guest.