As clinicians, we often want to know by how much a patient’s risk of having a health outcome is increased or decreased by the presence of some risk factor or exposure. Investigators assist us in this manner by studying a group of people who have some risk factor (exposed group) and comparing them with a group of people who do not have the risk factor (unexposed group). After following both groups for some time, usually years, the investigators can determine how many times more likely it was that the exposed group developed the outcome than the unexposed group. This number is called the risk ratio, or “relative risk.” In certain study designs or analytic techniques, however, the relative risk cannot be determined directly, and investigators report relative odds. This number is commonly called the odds ratio, and even authors of research articles may interpret this ratio incorrectly.^{1} As readers, it is important to understand the difference between odds ratios and risk ratios because their meanings and interpretations are quite different.^{2,3}

#### The Risk Ratio

In a cohort study, investigators begin by identifying the presence or absence of an exposure (eg, cigarette smoking) in two groups. They then follow the two groups over time (ie, prospectively) to determine the number in each group who develop the outcome of interest (eg, lung cancer). The number of people who develop the outcome divided by the total number in the group is called the incidence (Table 1). The incidence is what we call the *risk* of developing the outcome in that group. The incidence (risk) in the exposed is then divided by the incidence (risk) in the unexposed to determine the ratio of the two risks: the risk ratio (RR), or relative risk. This RR tells us how many times more likely the outcome occurs among people with the risk factor (or exposure). If the RR = 1, then the risk is the same in the two groups. If the RR is >1, the risk of the outcome is greater in those with the exposure; and if the RR <1, the risk of the outcome is lower in those with the exposure. For example, a cohort study presenting a RR of 15 for the association between cigarette smoking and lung cancer tells us that the incidence of lung cancer in the smokers was 15 times that of the incidence of lung cancer in the nonsmokers.

#### The Odds Ratio

In the calculation of risk (incidence) in each of the two groups described above, the denominator in the ratio *includes* that which is mathematically represented in the numerator. This fraction is called a probability (and is what we mean when we talk about the “chance” of something). A ratio in which the denominator *does not include* that which is mathematically represented in the numerator is called “odds.” Note in the calculations accompanying Figure 1 that the odds are greater than the probability.

Equation (Uncited) Image Tools |
Fig. 1 Image Tools |

##### Cohort Studies

Let us return to thinking about the cohort study. If investigators simply divide the number of people exposed to the risk factor who developed the outcome (eg, smokers who developed lung cancer) by the number of people exposed to the risk factor who did not develop the outcome (eg, smokers who did not develop lung cancer), they have determined the odds of developing the outcome in an exposed person (represented by a/b in Table 1). Investigators could similarly determine the odds of developing the outcome in an unexposed person by dividing the number of people not exposed to the risk factor who developed the outcome (eg, nonsmokers who developed lung cancer) by the number of people not exposed to the risk factor who did not develop the outcome (eg, nonsmokers who did not develop lung cancer) (represented by c/d in Table 1). The relative odds are simply the first odds divided by the second: the odds ratio (OR). This OR tells us whether the odds of developing the outcome are greater if a person is exposed to a risk factor. If the OR = 1, then the odds are the same in the two groups. If the OR is >1, the odds of the outcome are greater in those with the exposure; and if the OR <1, the odds of the outcome are lower in those with the exposure. For example, a cohort study presenting an OR of 15 for the association between cigarette smoking and lung cancer tells us that the odds of developing lung cancer in the smokers was 15 times the odds of developing lung cancer in the nonsmokers.

##### Case Control Studies

To appreciate the value of the odds ratio, let us now consider the design of a case control study. In this type of study, investigators begin by identifying a group of people who have the outcome of interest (eg, lung cancer) as well as a select group of people who do not have the outcome but who are otherwise, except for the exposure of interest, similar to the people who do. Investigators then look back in time—often through chart review or interviews—to assess for the presence or absence of the potential risk factor/exposure (eg, cigarette smoking). The investigators cannot calculate incidence (risk) in the two groups because the overall prevalence of the outcome is not known. The investigators can, however, calculate the odds that a person with the outcome had the risk factor (represented by a/c in Table 2) and the odds that a person without the outcome had the risk factor (represented by b/d in Table 2). The ratio of these two odds tells us whether the odds of having had an exposure (risk factor) are greater if a person has the outcome. For example, a case control study presenting an OR of 15 for the association between cigarette smoking and lung cancer tells us that the odds of having been a smoker among people who had lung cancer was 15 times the odds of having been a smoker among people who did not have lung cancer.

Note that the OR in this case does not directly tell us the odds of having the outcome. It turns out, though, that the calculation of the OR in either a cohort or case control study simplifies to the same formula (Tables 1 and 2), so that mathematically the OR for exposure and the OR for outcome are equivalent. Because of this equivalence, it is legitimate to say, based on the example above, that the odds of having lung cancer among people who smoked was 15 times the odds of having lung cancer among people who did not smoke.

#### When Does the Odds Ratio Approximate the Risk Ratio?

You may recall that case control studies are particularly useful for studying rare outcomes.^{4} You can imagine how difficult and time-consuming it would be to conduct a cohort study of a disease that occurs very infrequently or takes a long time to develop. By starting the study with a group of people who already have the disease (outcome), the case control study design is much more efficient. A potential disadvantage would be that a risk ratio cannot be calculated. However, when the outcome is rare, an odds ratio does provide a close estimate of the risk ratio. The reason is that small numbers for the outcome will not affect the calculations very much because they exert little influence on the denominators in the RR calculation (see text below Table 1).

#### Why Is Use of the Odds Ratio so Common?

Risk ratios can be calculated directly only for cohort studies. Odds ratios, as already discussed, can be calculated not only for cohort studies but also for case control studies. Odds ratios can also be calculated for cross-sectional (prevalence) studies. In addition to the fact that ORs can be calculated for many types of study designs, there is another reason why they are so often reported.^{2} In observational studies, certain factors associated with both the outcome and the exposure can distort the association between the exposure and the outcome. When investigators are aware of and measure these factors—called confounders—they can use certain analytic techniques to adjust for their effects to provide a better estimate of the effect of the exposure itself. Techniques (such as logistic regression) that are commonly used to adjust for confounders (see Sonis^{5} for a review of confounding) yield odds ratios (rather than risk ratios) between each confounder variable and the outcome as well as between the exposure of interest and the outcome. An assumption of the reader may be that the OR gives a close approximation of the RR in all of these situations. The potential problem is that in some situations the OR may exaggerate the measure of association that would be determined by a RR.

#### An Example

Although an occasional paper will present interpretations of ORs as relative risks,^{1} it is not correct to do so. For example: if an OR is 3.5, it is not correct to say that the chance of the outcome is 3.5 times more likely in the exposed group compared with the unexposed group. One can say that the odds are 3.5 times greater. What is commonly done is to simply say there is an association and present the OR for the reader to interpret. Let me illustrate in a brief example why it is important to understand what the OR means.

Using data from the Behavior Risk Factor Surveillance Survey, some colleagues and I recently examined the associations between people’s recollection of being given advice to make various lifestyle changes to lower blood pressure and their reports of whether they were actually making the lifestyle changes.^{6} Using eating habits as an example, it turned out that (by logistic regression) recalling being given advice to change eating habits was associated (adjusted OR: 4.2, 95% CI: 3.8–4.7) with reporting actually changing eating habits. It would not have been correct to say that compared with respondents who did not receive advice to change their eating habits, those who did receive such advice were 4.2 times more likely to report making changes in their eating habits.

A look at the percentages makes this clear. The percentage who reported changing eating habits among people recalling advice was 82% compared with 51% among those who did not recall advice (*P* < 0.001).^{6} While significant, 82% is certainly not 4.2 times greater than 51%. It would not have been wrong for us to report the odds ratio (as long as no incorrect interpretations were presented). However, because our outcome was so common, we decided to use an alternative method^{7} to provide an estimate of the RR, and reported that instead, which turned out to be 1.6. (We used Poisson regression with robust error variance. Since this was a cross-sectional study, this is really a prevalence ratio, but the computation is mathematically identical to the RR^{7}).

Had we reported ORs, it is quite possible that readers would assume that advice had a much stronger association than it really did, even after taking into account the other limitations of the study, including recall bias.

#### Estimating the RR When the Outcome Is Common

Sometimes called the “rarity assumption,” a take-home point is that the OR provides a reasonable estimation of the RR when the outcome is rare (in the study population). This statement begs the question, “What is rare?” To answer that question, it is helpful to examine the mathematical relationship between the OR and RR at varying frequencies of the outcome in the unexposed group (Table 3 and Fig. 2). It has been suggested that “rare” is approximately 10% or less.^{8} In situations in which the incidence is >10%, if the OR is <0.5 or >2.5, the OR starts to notably exaggerate the association one would see with the RR.

One proposed method to estimate the RR from the OR in studies in which the outcome is common is to use the following formula^{8}:

where *P*_{0} represents the incidence of the outcome in the unexposed group. This method is not perfect, but it can give you an idea of what the magnitude of the association might really be. Other more sophisticated issues and statistical techniques have been described,^{7,9,10} but they are more mathematically complex. Note also that there is at least one occasion when the rarity assumption is irrelevant. That is, an OR from a case-cohort study is a direct estimate of the RR irrespective of the frequency of the outcome. In a case-cohort study, cases (those who develop an outcome) are sampled from all incidence cases while controls are sampled from a cohort (at risk for the outcome) that is formed at the beginning of the study regardless of their future outcome status.^{11}

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Fig. 2 Image Tools |

#### Effect Size

Another term for “magnitude of association” mentioned above is *effect size*. That is, ORs and RRs give you an idea of how large (or small) the exposure (or intervention’s) effect on the outcome might be. Obviously, the greater the RR (or OR), the greater the effect size (assuming statistical significance). Small effect sizes, even when statistically significant, are more difficult to accept at face value because of the possibility of confounding factors for which the investigators have not accounted.^{12,13}

Furthermore, keep in mind that ORs and RRs are relative as opposed to absolute measures, and a relative risk can appear large even though the absolute risk is quite small. For example, an observational study published in 2003 showed regular nonsteroidal anti-inflammatory drug (NSAID) use for ≥5 years to be inversely associated with breast cancer with a RR of 0.81.^{14} This represents a relative risk reduction of 19%. However, the absolute risk reduction turns out to be only 0.09%. (Based on Table 2 of the article, the exposure event rate is 0.004 (404.6/100,000 person-years) and the control event rate is 0.0049 (490/100,000 person-years) for a difference of 0.0009.)

In contrast to the relative perspective offered by ORs and RRs, the number needed to treat (NNT), with which you may already be familiar, is an example of an effect size measure that is based on absolute risk reduction.^{15} In the breast cancer and NSAIDs example, 1111 women (1/0.0009) would need to consume NSAIDs regularly for over 5 years to prevent one case of breast cancer.

#### Confidence Intervals

Ratio measures (OR and RR) should always be presented with a confidence interval (usually a 95% confidence interval, CI). The confidence interval gives the reader an idea of the statistical significance as well as the precision of the ratio estimate. A 95% CI that crosses 1 is not significant because 1 is the “null value.” That is, an exposure that has a risk ratio (or odds ratio) of 1 has no association with the outcome. Let’s say you read a study that tells you that residents’ attendance at clinical epidemiology lectures is associated with falling asleep on hospital rounds (OR: 3.72; 95% CI: 1.02–6.40). This means that the odds of a resident falling asleep on rounds are 3.72 times greater if the resident attended the clinical epidemiology lecture than if the resident had not. The CI tells you that this OR is significant (remember that when the null value is 1, a CI beginning or ending on exactly 1.00 is equivalent to a *P*-value of 0.05). The CI also tells you that if the investigators repeated the study 100 times, the estimate would be somewhere between 1.02 and 6.40 in 95 of those studies. In other words, the association may really be as low as 1.02 (a very small magnitude) or as high as 6.40 (a much greater magnitude).

#### Relative to Whom?

When you are interpreting relative odds (as represented in ORs) and relative risks (as represented in RRs), it is imperative that you know the referent group. For dichotomous exposures (eg, attendance at a clinical epidemiology lecture: yes or no), it is generally straightforward. In the example above, the referent group is the residents who did *not* attend the lecture. We could have made the reference group the residents who *did* attend the lecture. The OR would be 0.27 (the inverse of 3.72), meaning that residents who did not attend the lecture had 0.27 times the odds of falling asleep on hospital rounds compared with (or relative to) those who did attend the lecture (ie, they stayed awake). When the exposure is not dichotomous, the referent group may be one of a number of groups. Race/ethnicity is a common “exposure” that often has 3 or more categories (eg, white, black, Hispanic, Asian, other). Authors must indicate the referent group so that readers can interpret what the OR (or RR) means. This indication is often placed in the results table(s) that displays the ORs or RRs. The referent category may be marked by the word “referent,” but sometimes is only indicated by showing the ratio measure as “1.0.”

#### Summary

It is hoped that this brief paper provided an overview and greater appreciation and understanding of two commonly used measures of association—the risk ratio and the odds ratio. It was not my intent to convey that odds ratios are inferior, only that they are different from risk ratios in ways that can be important when interpreting the literature. There are many great reference books^{4,11,16–21} to which interested readers can refer to learn more.

#### Acknowledgments

The author thanks the anonymous reviewers who provided helpful suggestions and references for the revised manuscript.