When to Use this Tip
This tip is suitable for all clinicians and clinical trainees who already understand risk and risk ratios (RR) but who do not have a clear understanding of the concept of odds ratios (OR). The objective is to foster this understanding. This tip takes 30 minutes to complete.
By the end of this tip, learners should be able to:
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Understand what an odds ratio (OR) is.
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Understand that OR is an alternative to RR as an effect measure in treatment studies.
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Understand that OR = 1 when there is no treatment effect
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Understand that, compared to RR, OR makes the effect appear larger.
This tip is useful for learners who are critically appraising an article or a meta-analysis about a therapy or a harmful association and encounter results reported as odds ratios. Characteristically, clinical learners find risk and ratios of risk intuitive but find odds and ratios of odds unfamiliar, non-intuitive, and mystifying. Their confusion revolves around 2 stumbling blocks. Firstly, they do not understand how odds ratios are calculated. Secondly, they do not understand how to interpret the clinical importance of results when they are reported as odds ratios. If learners have worked through tip 1, they are comfortable with the notion of odds as an alternative measure of frequency and its relationship to the more familiar measure, risk. This tip uses that familiarity as a point of departure to demystify the concept of ratios of odds in relationship to the more familiar ratios of risk.
The Script
The first step is to translate what the learners have already absorbed in tip 1 into the framework of a 2 × 2 table such as in Figure 2. Tell the learners to imagine that the figure represents the results of a therapeutic trial comparing an outcome such as mortality between treatment and control groups. They are already familiar with this kind of a table by virtue of understanding risk and risk ratios.6 Complete the 2 × 2 grid with the numbers as shown, telling the group that they correspond to the number of study subjects in each cell. Ask the group: What is the risk of the outcome in the treatment group? After a moment, a participant offers “40 over 100.” Create a new column labeled ‘Risk’ to the right of the 2 × 2 grid and write 0.40 across from the ‘treatment’ row. Now ask, what is the odds of the outcome in the treatment group? After a little thought, someone suggests “40 over 60.” Create a new column, labeled ‘Odds’, to the right of the ‘Risk’ column and write 0.67 across from the ‘treatment’ row.
The learners now work through the same process for the control group, and you write 0.50 and 1 under the risk and odds columns, respectively. Now ask: What is the relative risk of the outcome? Someone proposes “40% over 50%.” You ask the group to verify the correctness of this response and then write the result in a column below the 2 × 2 grid. You now ask, what is the ratio of the odds of the outcome between the treatment and control groups? After some thought, someone catches on and suggests “0.67 over 1.” You congratulate the group for having grasped the gist of the demonstration and write 0.67 in the bottom row under the ‘odds’ column.
At this point, it is usually appropriate to acknowledge the common inconsistencies in terminology that one encounters in the literature. For example, one characteristically encounters ‘relative risk’ and ‘odds ratio’, although it would be equally appropriate to use the terms ‘risk ratio’ and ‘relative odds’, respectively.
At this stage, some learners may ask, “what does a relative odds of 0.67 mean, and why is this different from the relative risk?” The demonstration has partially demystified odds ratios for the group, but it has not made them any more intuitive.
Returning to the punch line of tip 1, point out that there are two things that a clinician needs to understand about the relationship between risk ratios and odds ratios: (1) when are odds ratios sufficiently close to risk ratios to be considered to be equivalent and (2) when they are not the same, in what direction is the difference? To address the second principle, you point to the two values in Figure 2 and ask the group “Are these two numbers close enough to be considered to be the same?” Most of the group will say no, and after further deliberation, someone will usually suggest that the odds ratio of 0.67 corresponds to an importantly larger treatment effect. Someone else may suggest that, taken as an approximation of the risk ratio, it would imply a risk reduction of 33%, in contrast to the true risk reduction of 20%. You agree with these observations and suggest that, as a general rule, when the two measures yield different numbers, the odds ratio, if taken as identical to the risk ratio, overestimates the treatment effect, sometimes substantially.
To approach the second issue of when can the two measures be taken as interchangeable, you can either give the group a ‘rule of thumb’ that this is appropriate when the event rate in the control group is not higher than 30% or you can demonstrate it quantitatively by working through an additional demonstration using a control event rate of 20%, 25%, and 30%.
Figure 3 illustrates the relationship between RR and OR when the control event rate is 20%.
Even with risk reduction as high as 50%, the OR of 0.44 is closer to the RR of 0.50 than when the control event rate was 50%. If a smaller effect size of 25% risk reduction had been used in this second example, the two estimates become even closer. The resulting relative risk of 0.75 is now quite close to the OR of 0.71.
To understand odds ratios in terms of relative risk, you can use a non-quantitative approach. The rules are:
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1.
The RR will always be closer to 1.0 than the OR.
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If the baseline risk (the risk of adverse events in the control group) is low (say, less than 30%), the difference between RR and OR is unlikely to be important. Therefore, the two may be used interchangeably for interpreting results.
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If the OR is near 1.0, the difference between RR and OR is unlikely to be important.
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Following directly from the above, the only time you are likely to run into trouble treating the OR as an RR is when the baseline risk is high (over 30%) and the OR is not close to 1.0 (say less than 0.67 or greater than 1.3). Under these circumstances, the RR is likely to be appreciably closer to 1.0 than the OR.
For those mathematically inclined, one can be much more precise using the formula below.
$${\text{RR}} = \frac{{{\text{OR}}}}{{1 + {\text{CER}}\left( {{\text{OR}} - 1} \right)}};\,{\text{where}}\,{\text{CER = control}}\,{\text{event}}\,{\text{rate}}\left( {{\text{same}}\,{\text{as}}\,{\text{control}}\,{\text{group}}\,{\text{risk}}} \right).$$
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As you can see, here too you need to know the event rate in the control group to be able to do this conversion. There are also published tables that convert OR to RR.1,2
Throughout this demonstration, we have assumed that learners are comfortable with the clinical interpretation of relative risk and that understanding the relationship between odds ratios and relative risk renders the former comprehensible. Helping learners grasp the clinical significance of relative risk requires different demonstrations, which have been described elsewhere.6
The Bottom Line
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–Stumbling block: learners are unfamiliar with how odds ratios are calculated and how to interpret their clinical importance.
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–Odds ratios are calculated from a 2 × 2 table in a fashion completely analogous to the calculation of risk ratios.
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–If the baseline risk is less than 30%, the difference between odds ratios and risk ratios is unlikely to be important.
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–When odds ratios are different from the corresponding risk ratios, they are further from 1, implying a greater treatment effect.
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–When interpreting the clinical importance of results expressed in OR, treat OR as RR.
See Appendix 2 for the summary card for this tip.
Addendum to Tip 2
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–You may want to point out that when the risk of death (or any outcome) with the experimental treatment is the same as that with the control treatment, say 20% in each arm, then RR = 1 and also OR = 1. You may like to take the learners through Figure 2 using 20 in the Outcome + cells and 80 in the Outcome − cells for both treatment and control groups.
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–Some learners may mention that they have encountered the use of ‘relative odds reductions’ as measures of therapeutic effectiveness. This is most commonly found in reports of stroke trials8 but also may be encountered in evidence-based summaries.9 To address this, first point out that ‘relative odds reduction’ is an alternative measure of effect to relative risk reduction. Then, remind learners of what they just learned in tip 2, i.e., that, when RR and OR are different, relative risk will always be closer to 1 than OR. From this and learners’ prerequisite knowledge of the relationship between relative risk and relative risk reduction, learners should immediately grasp that, when different, relative odds reduction will always be numerically larger than relative risk reduction. You may also point out that, for this reason, authors seeking to inflate the apparent effectiveness of a study drug may be motivated to report the results as ROR instead of as RRR.
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–Interpretation of OR for a case control study and logistic regression follows on similar lines; however, it may be worth noting that case control studies usually address association of a disease with etiologic agents, and a causal association in such studies is usually reported with OR more than 1. The larger the OR, the stronger is the association.