Background
Odds ratio and risk ratio are measures of association used to describe the efficacy of interventions and disease determinates; however, they are not interchangeable measures of association.
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Remove maintenance messageCorresponding author: A.M. O'Connor, Veterinary Diagnostic and Production Animal Medicine, College of Veterinary Medicine, Iowa State University, 1600 S. 16th Street, Ames, IA 50010; email: oconnor@iastate.edu
Odds ratio and risk ratio are measures of association used to describe the efficacy of interventions and disease determinates; however, they are not interchangeable measures of association.
To illustrate that interpretation of the odds ratio as a riskbased measure of efficacy can be misleading.
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A metaanalysis reported, the odds ratio and the risk ratio as measures of vaccine effect. Example data were obtained from a metaanalysis of the risk of infection with Tritrichomonas fetus (T. fetus), in trials that assessed wholecell killed T. fetus vaccination in beef heifers.
When risk was used as the measure of disease frequency, the summary risk ratio was 0.82 (95% CI = 0.7–1.01), a 18% decrease in risk of infection. When odds were used as the measure of disease frequency and the summary odds ratio was 0.41 (95% CI = 0.2–0.84), a 59% decrease in odds of infection.
Problems arise for clinicians or authors when they interpret the odds ratio as a risk ratio. In the example provided, the efficacy of protective interventions was overestimated. In the case of disease determinates that increase the occurrence of disease, the interpretation of the odds ratio as a risk ratio would also lead to overestimation of the effect. It is important not to use the terms risk or probability of disease when the odds are the measure of disease frequency.
Problems arise for clinicians or authors when they interpret the odds ratio as a risk ratio, as the efficacy of protective interventions or the strength of disease determinate associations will be overestimated. Both the odds ratio and the risk ratio are valid measures of association; however, as they use different measures of disease frequency (odds versus risk), it is important they are not treated as interchangeable. Risk refers to the probability of being diseased in a specified time period. Odds refer to the ratio of the probability of being diseased to the probability of not being diseased in a specified time period. The purpose of this communication is to provide an empirical example of the different interpretations of the odds ratio and risk ratio with the aim of improving interpretation of these common measures of association.
To illustrate the different interpretations of the odds ratio and risk ratio, data from a metaanalysis on the risk of infection with Tritrichomonas fetus (T. fetus) in a group of trials that assessed vaccination with wholecell killed T. fetus in beef heifers (Fig 1) are used. Each study in the metaanalysis is labeled and the number of events and number enrolled in vaccinated and unvaccinated groups are reported in the 4 columns of numerical data on the lefthand side. On the righthand side is a forest plot that contains a new line of data for each study. Each box in the forest plot represents the point estimate of the effect of the vaccine, either an odds ratio or a risk ratio. The horizontal line through each box represents the 95% confidence interval for the vaccine effect. At the bottom of each forest plot is a large diamond, which is the summary effect measure, a weighted combination of the estimates from each study. The top and bottom points of the diamond represent the point estimate and the lateral points represent the bounds of the 95% confidence interval. When the vaccine has no effect the numerator and denominator in the risk ratio or odds ratio will be the same, and therefore the ratio will be one.
In the upper panel of Figure 1, risk has been used as the measure of disease frequency and the summary risk ratio is 0.82 (95% CI = 0.7–1.01). In the lower panel, odds are used as the measure of disease frequency and the summary odds ratio is 0.41 (95% CI = 0.2–0.84). A risk ratio of 0.82 signifies that the magnitude of risk of disease in the numerator is approximately 82% of the magnitude of risk of disease in the denominator (ie, that the numerator is approximately 18% lower than the denominator). Therefore, the metaanalysis results in Figure 1 suggest that vaccination proportionally reduced the risk of infection with T. fetus by only 18% (vaccine efficacy = 1–RR, 1–0.82), but the same studies estimated that vaccination reduced the odds of infection by 60% (1–0.4). Clearly, it would be incorrect to say vaccination reduced the risk of infection by 60%, as it is the odds that are reduced by 60%. If the end user wants to interpret and evaluate the intervention based on risk, which is far more intuitive than odds, then they must restrict their interpretation to the risk, in this situation, an 18% decrease in risk.
Problems arise if the odds ratio is misinterpreted as a risk ratio. For protective interventions such as illustrated here, incorrectly interpreting the odds ratio as a risk ratio will always lead to an overestimation of the protective effect. Although our example relies upon a clinical trial with a protective intervention, the issue of misinterpretation will occur for any hypothesis testing studies were the odds ratio and risk ratio are valid measures of the association. If the study addresses a disease determinate that increases disease occurrence, then the odds and risk ratios would be greater than one. The impact of misinterpretation of the odds ratio as a risk would be the same, overestimation of the risk of disease associated with exposure. This happens because the difference between the odds ratio and risk ratio increases as the baseline risk of disease increases.
Consider hypothetical data from three populations (Table 1). The 1st 2 sets of data present data from a study of a protective intervention as might occur in a vaccine trial. In a vaccine trial the effect of the protective intervention could be reported as either the risk ratio or the odds ratio as both can be calculated from the trial data. The odds ratio is obtained by dividing the odds of disease in 1 group by the odds of disease in another. The risk ratio is obtained by dividing the risk of disease in 1 group by the risk of disease in another. The odds are the ratio of 2 simple proportions (Table 2: Formula 1). The risk is a simple proportion (Table 2: Formula 2). Notice that the risk of disease is only part of the formula for the odds; ie, the numerator of the odds is the risk of disease.
Disease Present  Disease Absent  Odds  Risk  

 
High baseline risk population for a protective factor (40%)  
Vaccinated  20  80  20 ÷ 80 = 0.25a  20 ÷ 100 = 0.20b 
Not vaccinated  40  60  40 ÷ 60 = 0.33a  40 ÷ 100 = 0.40b 
Relative measure of association  Odds ratio = 0.25 ÷ 0.33 = 0.37c  Risk ratio = 0.2 ÷ 0.4 = 0.50d  
Low baseline risk population for a protective factor (4%)  
Vaccinated  2  98  2 ÷ 98 = 0.02a  2 ÷ 100 = 0.02b 
Not vaccinated  4  96  4 ÷ 96 = 0.042a  4 ÷ 100 = 0.04b 
Relative measure of association  Odds ratio = 0.02 ÷ 0.041 = 0.48c  Risk ratio = 0.02 ÷ 0.04 = 0.50d  
High baseline risk population for a factor that increases disease (40%)  
Exposure present  80  20  80 ÷ 20 = 4a  80 ÷ 100 = 0.80b 
Exposure absent  60  40  60 ÷ 40 = 1.5a  60 ÷ 100 = 0.60b 
Relative measure of association  Odds ratio = 4 ÷ 1.5 = 2.67c  Risk ratio = 0.8 ÷ 0.6 = 1.33d 
Formula 1 

Formula 2 

Formula 3 

Formula 4 

Formula 5 

Formula 6 

Using the data from Table 1 to illustrate the difference, the odds of disease in the vaccinated animals are 0.25 (Table 2: Formula 1) while the risk of disease in the vaccinated animals is 0.20 or 20%. The difficulty or confusion arises if clinicians or authors incorrectly conclude there is a 25% risk of disease in the vaccinated animals in Table 1, rather than a 0.25 odds of disease in the vaccinates. This difficulty in translation is further compounded when summary measures of association such as the odds ratio or risk ratio are used. In Table 1, the odds ratio is 0.375, meaning the odds of disease were 62.5% lower in the vaccinated animals compared to the unvaccinated animals (Table 2: Formula 3). However, the risk ratio of 0.5 means the risk of disease was 50% lower in the vaccinated animals compared to the unvaccinated animals (Table 2: Formula 4). The difference of 12.5% might not seem important, but since the application of these ratios is dependent upon the baseline level of disease the difference in the odds ratio and risk ratio can become large.
The difference between the odds ratio and the risk ratio increases as the baseline level of disease increases, as shown in Table 1. When the baseline prevalence of disease in the control group changes from 40% to 4%, the risk ratio does not change; however, the odds ratio does and becomes more similar to the risk ratio as the disease becomes rarer. This is because, when the disease is rare, the denominators of the odds and the risk are very similar. For example, in Table 1 when the baseline risk is 4%, the odds of disease in vaccinated animals is given by 2 ÷ 98, whereas the risk of disease in vaccinated animals is given by 2 ÷ 100.
The third set of data in Table 1 contains hypothetical data where the factor being studied increases the occurrence of disease. The odds ratio is 2.67, whereas the risk ratio is 1.3. Incorrect interpretation of the odds ratio as suggesting a 167% increase in risk ratio is clearly an overestimate of the increased risk, which is 30%.
Difficulty interpreting odds and the odds ratio is common among clinicians and researchers; and numerous publications are available with a detailed discussion of the appropriate interpretation of the odds ratio.[13] As suggested by Prasad et al, “Probably, no one (with the possible exception of certain statisticians) intuitively understands the ratio of odds”.[4] The issue is that most people, including clinicians, think in terms of probability (risk) rather than odds.
The solution is for authors to use riskbased measures of disease, which are not difficult to interpret. When prospective trials are conducted with binary outcomes, calculation of risk is possible.[3] However, for metaanalysis, clinical trials, or observational studies that require adjustment for covariates, the odds ratio might be preferred because it has desirable mathematical properties. If metaanalysis, clinical trials, or observational studies are conducted using odds ratio, it is possible to convert it back to the risk ratio, provided a sensible assumed risk of disease in the control group can be specified (Table 2: Formula 6).[5, 6] When clinicians are unsure of the assumed risk of disease to use, a low, moderate, and high risk estimate can be used to obtain several risk ratios. Such an approach would facilitate interpretation and clarify the impact of uncertainty about the most appropriate assumed risk of disease.
Some groups recommend translating the relative measures such as the odds ratio and risk ratio to absolute risk differences to further facilitate translation of the research findings for clinicians.[7] To do this, it is necessary to specify a sensible assumed risk of disease in the control group. For example, using the data from Figure 1, a sensible assumed risk of the infection in the control group may be 80/102 (78%), which can be converted to 78 cases in 100 unvaccinated animals. Using the summary risk ratio of 0.82, this would translate to an expectation of 64 (78 cases times 0.82) infected animals infected animals (95% CI = 55–79) in 100 vaccinated animals (Table 2: Formula 5). If authors begin with the odds ratio, Formula 6 in Table 2 can be used to convert this to the risk ratio. The assumed control risk can be a median or mean risk, though it should be justified.[6] Again, if there is uncertainty about the sensible assumed risk, a low, moderate, and high estimate might be used.
The non equivalence of the risk ratio and odds ratio does not indicate that either is wrong or both estimates are entirely valid ways of describing an intervention effect. Clinicians and authors must avoid the easy error of interpreting the odds as a risk or probability. In particular, it is important not to use the terms risk or probability of disease when the odds are the measure of disease frequency.
Conflict of Interest Declaration: Author discloses no conflict of interest.
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