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Using Bayes’ nomogram to help interpret odds ratios
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1. John Page, MBBS, MSc1,
2. John Attia, MD, PhD, FRCPC2
1. 1Harvard University School of Public Health  Boston, Massachusetts, USA
2. 2University of Newcastle  Newcastle, New South Wales, Australia

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## Introduction

In certain scenarios, the odds ratio (OR) provides an unbiased estimate of the rate ratio in case control studies.1 However, the OR is also frequently used to estimate the risk ratio (relative risk) (RR) of an outcome in the presence of a risk factor. The degree of error in this estimate is frequently small, but can sometimes be substantial. The OR as an estimate of the RR always overestimates the effect of the exposure (results in an estimate further away from 1). The degree of divergence between the OR and the RR depends on the size of the OR and the probability of the outcome of interest (table).2–4 Given the value of the baseline risk and the estimate of the OR, the RR can be estimated by the use of a formula.3,5 However, the formula may be inconvenient and cumbersome for readers and users of epidemiological information. A nomogram is a graphical calculator that is a useful and convenient way to perform common calculations without the need to remember formulae. The use of the Bayes’ nomogram6 has simplified the use of diagnostic test information7–8 and is now frequently used by physicians who may be unaware of the formula involved in the conversion. In this editorial, we show that the Bayes’ nomogram, typically associated with likelihood ratios, can also be used to calculate the RR given the OR and the baseline risk.

Degree of divergence between odds ratios (ORs) and relative risks (RRs)*

 *Approximately 60 additional journals are reviewed. This list is available on request. Acta Obstet Gynecol Scand  Age Ageing  Am J Cardiol  Am J Med  Am J Obstet Gynecol  Am J Psychiatry  Am J Public Health  Am J Respir Crit Care Med  Ann Emerg Med  Ann Intern Med  Ann Surg  Arch Dis Child  Arch Gen Psychiatry  Arch Intern Med  Arch Neurol Arch Pediatr Adolesc Med  Arch Surg  Arthritis Rheum  BJOG  BMJ  Br J Gen Pract  Br J Psychiatry  Br J Surg  CMAJ  Chest  Circulation  Cochrane Library  Crit Care Med  Diabetes Care  Gastroenterology Gut  Heart  Hypertension  JAMA  J Am Coll Cardiol  J Am Coll Surg  J Am Geriatr Soc  J Clin Epidemiol  J Fam Pract  J Gen Intern Med  J Infect Dis  J Intern Med  J Neurol Neurosurg Psychiatry  J Pediatr J Vasc Surg  Lancet  Med Care  Med J Aust  N Engl J Med  Neurology  Obstet Gynecol  Pain  Pediatrics  Rheumatology  Spine  Stroke  Surgery  Thorax

Journals reviewed for this issue*

## Method

Our method uses 2 steps to convert from OR to RR, given a baseline risk. The first step uses Bayes’ nomogram (figure).6 Using a straight edge on the nomogram, line up the baseline probability of an event on axis A, with the OR on axis B, and read off the postexposure probability on axis C. The postexposure probability divided by the baseline probability then yields the RR. Thus, with available information on the OR from epidemiological studies and the baseline risk, Bayes’ nomogram calculates the postexposure risk in the presence of the risk factor. Knowledge of the postexposure risk also allows easy and accurate calculation of the absolute risk difference and the number needed to treat (NNT)9 or the number needed to harm (NNH).10

Nomogram to calculate postexposure probability given estimates of the odds ratio and baseline probability. This nomogram is equivalent to the Bayes’ nomogram, but with different labels.

We present 2 examples to show the use of Bayes’ nomogram to calculate postexposure probability, RR, absolute risk difference, and NNH.

### EXAMPLE 1

We are interested in estimating the risk for precipitating heart failure in an older man who has started taking nonsteroidal anti-inflammatory drugs (NSAIDs) for arthritis. Our search reveals a recent case control study11 suggesting an OR of 10.5 for developing heart failure associated with the use of NSAIDs by patients with a history of heart disease. To apply this information, we need to estimate our patient’s baseline risk of heart failure. To do this, we use the equations derived by Kannel et al based on the Framingham database.12 Using the example given in that article of the 60 year old man with documented coronary disease who had a vital capacity of 2.5 l, systolic blood pressure of 160 mm Hg, heart rate of 85 beats/min, and evidence of left ventricular hypertrophy on electrocardiogram and cardiomegaly on chest radiogram, this patient’s 4 year risk of heart failure is 34%. His 1 year risk is thus approximately 8.5%. Using Bayes’ nomogram (figure), we anchor a straight edge at 0.085 (baseline risk) on axis A and direct it through axis B at 10.5 (OR). The postexposure risk can be read off axis C as 0.49, or a 49% chance of developing heart failure over 1 year after starting NSAIDs. The RR is then estimated by dividing the post-test probability, 49%, by the pretest probability, 8.5%, to get the RR of 5.8 (not an RR of 10.5 as some would misinterpret the OR). The absolute risk difference is 0.49 − 0.085 = 0.405. The NNH is the reciprocal of the absolute risk difference of 0.405, which is approximately 2.5. Thus, 5 such patients exposed to NSAIDs for a year would be expected to result in 2 new cases of heart failure.

### EXAMPLE 2

A meta-analysis compared endoscopic ligation with sclerotherapy for the treatment of esophageal variceal bleeding.13 The overall rebleeding risk with sclerotherapy in the 7 included studies was 47%; the OR was 0.52 (95% CI 0.37 to 0.74) in favour of ligation therapy. Although it might be tempting to interpret this as a 48% relative risk reduction (RRR), this is not accurate. Using Bayes’ nomogram and anchoring the straight edge at 0.47 (baseline risk) on axis A and 0.52 on axis B (OR), we read 0.32 on axis C, which is the probability of rebleeding with ligation (postexposure risk). To determine the RR associated with ligation compared with sclerotherapy, we divide 0.32 by 0.47, giving an answer of 0.68. This means that the RR is 0.68 and the RRR is 32% (1 − 0.68), not the 48% we would erroneously get if we equated the OR and RR without regard for the baseline risk and magnitude of the OR.

## Discussion

ORs are frequently interpreted as RRs. Although the 2 are often very close, if the baseline risk is >10–20% and the magnitude of the OR is far from 1, the divergence can be substantial. In these cases, we have shown how a Bayes’ nomogram can be used to conveniently calculate more accurate estimates of the RRs. Please note, however, that since the nomogram axes are on the logarithmic scale, interpolation requires some care. Numbers greater than a given mark on the scale will be further away than would be predicted by using a linear scale. Given the fact that the likelihood ratio is a form of OR, and indeed that the positive likelihood ratio divided by the negative likelihood ratio gives the OR, it is not surprising that the nomogram should be suitable for this purpose. However, in our experience with teaching evidence-based medicine, it is an application of Bayes’ nomogram that is not commonly known or used.

## References

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