Medical studies often compare how often an outcome happens in one group versus another. A 2 by 2 table organizes the data by exposure or treatment status and by whether the outcome occurred. From this table, students can calculate absolute risk, relative risk, and odds ratio.
These measures help researchers describe whether an exposure is linked to higher, lower, or unchanged chance of disease.
Understanding Statistics: Risk, Relative Risk, and Odds Ratio
The first skill is checking what each group total means. A risk needs a clear starting group. If fifty people out of one hundred treated people become ill, the risk is fifty out of one hundred.
If five people out of one hundred untreated people become ill, the risk is five out of one hundred. The relative risk is ten because the first risk is ten times the second. Yet the absolute difference is forty extra cases per one hundred people.
Both descriptions matter. A large relative change can come from a very small starting risk. For example, a change from one case per ten thousand people to two cases per ten thousand doubles the risk, but only adds one case per ten thousand.
Relative risk uses risk directly, so it is often easiest to explain to patients and the public. A value of one means the groups have the same measured risk. A value greater than one points to a higher measured risk in the exposed group.
A value below one points to a lower measured risk. These values do not show how many people are affected.
Students should always calculate or report the two original risks before focusing on their ratio. This prevents misleading conclusions when a headline reports a large percentage increase without stating that the outcome was rare.
Odds describe a different comparison. They compare the number with an outcome to the number without it within the same group. Odds ratios are especially useful in case control studies.
In those studies, researchers may begin by selecting people who already have a disease, then find out which exposures they had. Because the total number of exposed people in the population is not known from that design, direct risks cannot usually be calculated. An odds ratio can still be found.
When an outcome is rare, an odds ratio is close to the relative risk. When an outcome is common, the odds ratio can look much more extreme than the relative risk. An odds ratio of three does not always mean the risk is three times as high.
None of these measures proves that an exposure caused an outcome. Groups can differ in important ways before the study starts. Age, smoking, income, diet, access to care, and existing illness can affect results.
This problem is called confounding. Randomized trials reduce confounding by assigning treatments by chance. Observational studies need careful matching or statistical adjustment, though adjustment cannot fix missing information.
Students should check the study design, group sizes, length of follow up, and how the outcome was defined. Small groups can produce unstable estimates, especially when only a few outcomes occur. A result should be read as evidence with uncertainty, not as a final certainty.
Key Facts
- For a 2 by 2 table with exposed disease = a, exposed no disease = b, unexposed disease = c, unexposed no disease = d.
- Risk in exposed group = a / (a + b).
- Risk in unexposed group = c / (c + d).
- Relative risk = [a / (a + b)] / [c / (c + d)].
- Odds in exposed group = a / b, and odds in unexposed group = c / d.
- Odds ratio = (a / b) / (c / d) = ad / bc.
Vocabulary
- Absolute risk
- Absolute risk is the probability that an outcome occurs in a specific group.
- Relative risk
- Relative risk is the ratio of the risk in an exposed or treated group to the risk in an unexposed or control group.
- Odds
- Odds compare the number of times an outcome occurs to the number of times it does not occur.
- Odds ratio
- An odds ratio compares the odds of an outcome in one group with the odds of the outcome in another group.
- 2 by 2 contingency table
- A 2 by 2 contingency table is a four-cell table that counts subjects by group status and outcome status.
Common Mistakes to Avoid
- Using odds when the problem asks for risk. Risk uses outcome cases divided by the total number in that group, while odds use outcome cases divided by non-cases in that group.
- Flipping the comparison groups. Relative risk and odds ratio depend on which group is in the numerator, so reversing the groups changes the value to its reciprocal.
- Interpreting a relative risk of 2 as a 2 percentage point increase. A relative risk of 2 means the risk is twice as large, not that the absolute risk increased by 2 percent.
- Treating odds ratio and relative risk as always identical. They are close when the outcome is rare, but they can differ greatly when the outcome is common.
Practice Questions
- 1 In a study, 30 of 200 exposed people develop a disease, while 10 of 200 unexposed people develop it. Find the risk in each group and the relative risk.
- 2 A 2 by 2 table has a = 24, b = 76, c = 12, and d = 88. Calculate the odds in each group and the odds ratio.
- 3 A treatment study reports a relative risk of 0.60 for a bad outcome. Explain in words what this means and whether it suggests the treatment is associated with higher or lower risk.