# Odds Ratio

The odds ratio (OR) is a measure of how strongly an event is associated with exposure. The odds ratio is a ratio of two sets of odds: the odds of the event occurring in an exposed group versus the odds of the event occurring in a non-exposed group. Odds ratios commonly are used to report case-control studies. The odds ratio helps identify how likely an exposure is to lead to a specific event. The larger the odds ratio, the higher odds that the event will occur with exposure. Odds ratios smaller than one imply the event has fewer odds of happening with the exposure.

## Mathematics

Odds Ratio = odds of the event in the exposed group/odds of the event in the non-exposed group

If the data is set up in a 2 x 2 table as shown in the figure then the odds ratio is (a/b) / (c/d) = ad/bc. The following is an example to demonstrate calculating the odds ratio (OR).

**Example 1**

If we have a hypothetical group of smokers (exposed) and non-smokers (not exposed), then we can look for the rate of lung cancer (event). If 17 smokers have lung cancer, 83 smokers do not have lung cancer, one non-smoker has lung cancer, and 99 non-smokers do not have lung cancer, the odds ratio is calculated as follows.

First, we calculate the odds in the exposed group.

- Odds in exposed group = (smokers with lung cancer) / (smokers without lung cancer) = 17/83 = 0.205

Next we calculate the odds for the non-exposed group.

- Odds in not exposed group = (non-smokers with lung cancer) / (non-smokers without lung cancer) = 1/99 = 0.01

Finally we can calculate the odds ratio.

- Odds ratio = (odds in exposed group) / (odds in not exposed group) = 0.205 / 0.01 = 20.5

Thus using the odds ratio, this hypothetical group of smokers has 20 times the odds of having lung cancer than non-smokers. The question then arises: is this significant?

**Odds Ratio Confidence Interval**

To answer if this finding is significant, the confidence interval is calculated. The confidence interval gives an expected range for the true odds ratio for the population to fall within. If estimating the odds of lung cancer in smokers versus non-smokers of the general population based on a smaller sample, the true population odds ratio may be different than the odds ratio found in the sample. In order to calculate the confidence interval, the alpha, or our level of significance, is specified. An alpha of 0.05 means the confidence interval is 95% (1 โ alpha) the true odds ratio of the overall population is within range. A 95% confidence is traditionally chosen in the medical literature (but other confidence intervals can be used). The following formula is used for a 95% confidence interval (CI).

- Upper 95% CI = e ^ [ln(OR) + 1.96 sqrt(1/a + 1/b + 1/c + 1/d)]
- Lower 95% CI = e ^ [ln(OR) - 1.96 sqrt(1/a + 1/b + 1/c + 1/d)]

Where 'e' is the mathematical constant for the natural log, 'ln' is the natural log, 'OR' is the odds ratio calculated, 'sqrt' is the square root function and a, b, c and d are the values from the 2 x 2 table. Calculating the 95% confidence interval for our previous hypothetical population we get:

Upper 95% CI =e ^ [ln(OR) + 1.96 sqrt(1/a + 1/b + 1/c + 1/d)] =e ^ [ln(20.5) + 1.96 sqrt(1/17 + 1/83 + 1/1 + 1/99] =e ^ [3.02 + 1.96 sqrt(1.08)] =e ^ [3.02 + 1.96 (1.04)] =e ^ [3.02 + 2.04] =e ^ [5.06] = 158

Lower 95% CI =

e ^ [ln(OR) - 1.96 sqrt(1/a + 1/b + 1/c + 1/d)] =e ^ [ln(20.5) - 1.96 sqrt(1/17 + 1/83 + 1/1 + 1/99)] =e ^ [3.02 - 1.96 sqrt(1.08)] =e ^ [3.02 - 1.96 (1.04)] =e ^ [3.02 - 2.04] =e ^ [0.98] = 2.7

Thus the odds ratio in this example is 20.5 with a 95% confidence interval of [2.7, 158]. (Note: If no rounding is performed when doing the above calculations, the odds ratio is 20.28 with 95% CI of [2.64, 155.6] which is fairly close to the rounded calculations.)

**Confidence Interval Interpretation**

If the confidence interval for the odds ratio includes the number 1 then the calculated odds ratio would not be considered statistically significant. This can be seen from the interpretation of the odds ratio. An odds ratio greater than 1 implies there are greater odds of the event happening in the exposed versus the non-exposed group. An odds ratio of less than 1 implies the odds of the event happening in the exposed group are less than in the non-exposed group. An odds ratio of exactly 1 means the odds of the event happening are the exact same in the exposed versus the non-exposed group. Thus, if the confidence interval includes 1 (eg, [0.01, 2], [0.99, 1.01], or [0.99, 100] all include one in the confidence interval), then the expected true population odds ratio may be above or below 1, so it is uncertain whether the exposure increases or decreases the odds of the event happening with our specified level of confidence.

## Issues of Concern

The odds ratio can be confused with Relative Risk. As stated above, the odds ratio is a ratio of 2 odds. As odds of an event are always positive, the odds ratio is always positive and ranges from zero to very large. The relative risk is a ratio of probabilities of the event occurring in all exposed individuals versus the event occurring in all non-exposed individuals. In a 2-by-2 table with cells a, b, c, and d (see figure), the odds ratio is odds of the event in the exposure group (a/b) divided by the odds of the event in the control or non-exposure group (c/d). Thus the odds ratio is (a/b) / (c/d) which simplifies to ad/bc. This is compared to the relative risk which is (a / (a+b)) / (c / (c+d)). If the disease condition (event) is rare, then the odds ratio and relative risk may be comparable, but the odds ratio will overestimate the risk if the disease is more common. In such cases, the odds ratio should be avoided, and the relative risk will be a more accurate estimation of risk. Commonly, odds ratios will be reported in case-control studies, in which relative risks cannot be calculated.

The relative risk for the above hypothetical example of smokers versus non-smokers developing lung cancer is calculated as:

- Relative Risk = (17/100) / (1/100) = 0.17 / 0.01 = 17

Thus in our example, the odds ratio is 20.5 (smokers have 20 times the odds of having lung cancer than non-smoker); whereas the relative risk is 17 (smokers have 17 times the relative risk to have lung cancer than non-smokers).

## Clinical Significance

The odds ratio is the ratio of the odds of the event happening in an exposed group versus a non-exposed group. The odds ratio is commonly used to report the strength of association between exposure and an event. The larger the odds ratio, the more likely the event is to be found with exposure. The smaller the odds ratio is than 1, the less likely the event is to be found with exposure. It is important to look at the confidence interval for the odds ratio, and if the odds ratio confidence interval includes 1, then the odds ratio did not reach statistical significance.

## References

*Part or all of this article or section is derived from Odds Ratio by Steven Tenny; Mary R. Hoffman., used under CC BY 4.0*