Risk, Odds and Effect Measures
Much of clinical research comes down to a deceptively simple question: how does the probability of an outcome — a disease, a complication, a cure — change in the presence of some exposure, treatment, or test result? To answer that question rigorously, medicine uses a family of measures: risk, odds, relative risk, odds ratio, and risk difference. These terms appear constantly in clinical papers, yet they are frequently confused, and the distinction between them matters enormously for how results are interpreted and communicated to patients.
The 2×2 Contingency Table
Most measures of association in epidemiology and clinical research start from a 2×2 contingency table, which cross-classifies observations by exposure (or treatment) and outcome. The four cells are conventionally labelled $a$, $b$, $c$, $d$:
| Event present | Event absent | Total | |
|---|---|---|---|
| Exposed | $a$ | $b$ | $n_1 = a + b$ |
| Unexposed | $c$ | $d$ | $n_0 = c + d$ |
| Total | $a + c$ | $b + d$ | $n$ |
To make the subsequent calculations concrete, consider a cohort study following 1,000 adults for ten years to investigate whether smoking is associated with myocardial infarction (MI). Five hundred are smokers and 500 are non-smokers.
| MI | No MI | Total | |
|---|---|---|---|
| Smokers | 100 | 400 | 500 |
| Non-smokers | 20 | 480 | 500 |
| Total | 120 | 880 | 1,000 |
This table will serve as the running example throughout the following sections.
Risk
In epidemiology, risk is simply the probability that an event will occur in a defined group over a defined period of time. It is calculated as the number of people who experience the event divided by the total number of people who were at risk of experiencing it at the start of the follow-up period.
$$ R = \frac{\text{number of events}}{\text{number at risk}} $$Risk is bounded between 0 and 1 (or equivalently, 0% and 100%), which makes it directly interpretable: a risk of 0.20 means that 20 out of every 100 people in that group will experience the event. In the example above:
- Risk of MI in smokers: $R_1 = 100/500 = 0.20$ (20%)
- Risk of MI in non-smokers: $R_0 = 20/500 = 0.04$ (4%)
These are the fundamental quantities from which all subsequent comparative measures are derived. The terms , , and are used interchangeably in this context.
Odds
Odds express the same underlying information as risk, but in a different form. Instead of comparing the number of events to the total number of people at risk, odds compare the number of events to the number of non-events:
$$ \text{Odds} = \frac{p}{1 - p} = \frac{\text{number of events}}{\text{number of non-events}} $$The odds of MI in smokers are $100/400 = 0.25$, which is read as "1 to 4" — for every four smokers who do not have an MI, one does. The odds of MI in non-smokers are $20/480 \approx 0.042$.
Odds range from 0 to infinity, and they are not directly interpretable as a probability without conversion. This is a persistent source of confusion: in everyday speech, "the odds are 50–50" usually means a probability of 0.5, whereas statistically, odds of 1 (expressed as "1 to 1") correspond to a probability of 0.5. To convert between the two:
$$ p = \frac{\text{odds}}{1 + \text{odds}} \qquad \text{odds} = \frac{p}{1-p} $$One important property is that when the event is — say, a risk below 10% — the odds and the risk are numerically very close to each other. A risk of 0.05 corresponds to odds of $0.05/0.95 \approx 0.053$. As the risk increases, however, the two diverge markedly: a risk of 0.50 gives odds of 1.0, and a risk of 0.80 gives odds of 4.0.
If odds are not directly interpretable as probabilities, why use them at all? The reason is partly historical (the odds ratio has desirable mathematical properties), partly practical (certain study designs, especially case-control studies, only allow the odds ratio to be estimated), and partly because odds arise naturally from logistic regression, covered in the Logistic Regression chapter.
Risk Difference
The risk difference (RD), also called the absolute risk reduction (ARR) in treatment contexts, is the simplest way to compare two risks: subtract one from the other.
$$ \text{RD} = R_1 - R_0 $$In the smoking example: $\text{RD} = 0.20 - 0.04 = 0.16$. Smokers have a 16 percentage point higher absolute risk of MI over ten years than non-smokers. The risk difference has the same units as risk itself (a proportion), and it is the most clinically meaningful measure of the magnitude of an effect in absolute terms.
The risk difference is often the most honest summary of a treatment effect. Consider two treatments that each halve the risk of an outcome: one reduces the risk from 40% to 20% (RD = 0.20), and the other reduces the risk from 2% to 1% (RD = 0.01). Both have the same relative effect, but the first treatment prevents 20 events per 100 patients while the second prevents only 1. A patient — or a health system deciding where to allocate resources — needs the absolute figures to make sense of this.
Number Needed to Treat
Closely related to the risk difference is the number needed to treat (NNT): the average number of patients who must receive the treatment for one additional patient to benefit. It is the reciprocal of the absolute risk reduction:
$$ \text{NNT} = \frac{1}{|\text{RD}|} $$In the smoking example, $\text{NNT} = 1/0.16 \approx 6$: approximately 6 people would need to stop smoking for one MI to be prevented over ten years. A low NNT signals a highly effective intervention; a high NNT signals a modest absolute benefit even if the relative effect appears impressive.
Relative Risk
The relative risk (RR), also called the risk ratio, compares the risk in the exposed group to the risk in the unexposed group by taking their ratio:
$$ \text{RR} = \frac{R_1}{R_0} = \frac{a / n_1}{c / n_0} $$In the example: $\text{RR} = 0.20 / 0.04 = 5.0$. Smokers are five times as likely to develop MI as non-smokers over the study period. An RR of 1 indicates no association; an RR greater than 1 indicates increased risk; an RR less than 1 indicates a protective association. The relative risk is the natural measure of association in cohort studies and randomised controlled trials, because both designs follow participants forward in time from a known starting point, making it possible to estimate incidence directly in each group. A confidence interval and a $p$-value are always reported alongside the point estimate.
The principal limitation of relative measures is that they can make small absolute effects look large, and vice versa. An RR of 5.0 is striking — but if the baseline risk is 0.2%, the exposed group has a risk of only 1.0%, an absolute difference of 0.8 percentage points. Context — specifically the baseline risk — is always needed to translate a relative risk into a clinically meaningful statement.
Odds Ratio
The odds ratio (OR) compares the odds of the event in the exposed group to the odds in the unexposed group:
$$ \text{OR} = \frac{a/b}{c/d} = \frac{a \cdot d}{b \cdot c} $$In the example: $\text{OR} = (100 \times 480) / (400 \times 20) = 48{,}000 / 8{,}000 = 6.0$. The odds of MI are six times higher in smokers than in non-smokers. The same directional interpretation as the RR applies: OR = 1 means no association, OR > 1 means increased odds, OR < 1 means decreased odds.
Why is the odds ratio so important in medical research? The key reason lies in study design. In a case-control study, investigators select people who already have the outcome (cases) and a comparison group without the outcome (controls), and then look backwards to measure prior exposure. Because the numbers of cases and controls are chosen by the investigator — not determined by the natural incidence of disease — it is impossible to estimate the risk in the exposed group directly from the table. The odds ratio, however, can be estimated validly from a case-control design. This is one of the most important and counterintuitive results in epidemiology: even though absolute risk cannot be estimated from a case-control study, the odds ratio can. The odds ratio is also the natural output of logistic regression (see the Logistic Regression chapter), where each regression coefficient $b$ corresponds to a log odds ratio, and $e^b$ is the adjusted odds ratio controlling for other covariates in the model.
Comparing Relative Risk and Odds Ratio
In the example, the RR is 5.0 and the OR is 6.0. They point in the same direction and both indicate a strong association, but they are not the same number. This discrepancy is not an error — it is an inherent mathematical property, and understanding it is essential for reading medical literature correctly.
When the outcome is rare (conventionally, a risk below approximately 10% in both groups), the odds and the risk are numerically similar, so the odds ratio closely approximates the relative risk. This is known as the rare disease assumption. When the outcome is common, however, the odds ratio will always be further from 1 than the relative risk: an OR > 1 will overestimate the RR, and an OR < 1 will underestimate it.
In our example, the outcome is common (20% in smokers, 4% in non-smokers). The OR of 6.0 overstates the risk ratio of 5.0. Misreporting an odds ratio as if it were a risk ratio — which happens frequently in both research articles and media reporting — leads to an inflated perception of effect size. A paper that reports "patients who smoked had 6 times the odds of MI" is literally correct, but a reader who interprets it as "smokers are 6 times as likely to have an MI" has misread an odds ratio as a risk ratio. In randomised trials and cohort studies where the relative risk can be calculated directly, it should generally be preferred over the odds ratio for communicating results to a clinical audience.
MedCalc computes the relative risk, odds ratio, and risk difference — each with a confidence interval — in its Relative risk & Odds ratio procedure. How a test result changes the probability of disease — through the likelihood ratio — is covered in the Likelihood ratios chapter.