# How to calculate an adjusted odds ratio

Written by stephen byron cooper
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An odds ratio is the likelihood of an even occurring given a specific condition. Mathematicians often refer to an odds ratio as “OR.” The OR covers the probability of something happening divided by the probability of it not happening. Adjusted odds ratios take account of confounding factors. These are characteristics of the sources in the base data for the calculation that are likely to skew the results of the OR calculation.

## Source data

The starting point for calculating an adjusted odds ratio is a set of observations of an occurrence. These figures usually come from observations of behaviour and are can include factors such as people’s eating habits, fat intake, smoking rate or exercise regime. The proposal of an adjusted OR calculation is likely to be an examination of the likelihood of some illness occurring in the group of people being studied. As a result, the source data for the calculation will be a group of people who indulge in a particular habit compared against a second control group of people who were chosen at random from the general population.

## Comparison

The control group in the study provides the second set of data against which to compare the study group. For example, if you are trying to find whether smoking causes cancer of the mouth, you will have a group of smokers and a group of non smokers. You record how many of each group got cancer of the mouth and how many didn’t. Adding the results from the two groups together, you will have a total number of people in the study who developed the cancer and a total number of people who did not.

## Probability

The first step in calculating the OR is to find the probability of people getting the condition in each group as a proportion of the total number of people who got that condition. If you have smokers and non-smokers, a possible outcome could be that out of 1000 smokers, three got cancer of the mouth and of the 1000 non-smokers two got cancer of the mouth -- giving a total of five people who got the cancer. Therefore the proportion of smokers among cancer sufferers is 3/5 and the proportion of non-smokers developing it is 2/5, which is 0.6 and 0.4 respectively.

## Odds ratio

The odds ratio is the probability of group one getting the condition divided by the probability of group two getting the condition. In this example, that gives (3/5) / (2/5). That results in an OR of 1.5. However, as the total number of sufferers applies to each group it can be taken out of the OR calculation. The OR is derived by dividing the study group’s results by the control groups result. In this example, the OR would be is 3/2 = 1.5.

Scientists conducting studies have to be careful about their choice of subjects. Many studies do not gather their own data, but use data from studies carried out by other groups. The study group data and the control group data may have come from two different sources. It is very easy to get the results you want to see by choosing subject groups that have different circumstances. For example, if you wanted to show that smoking causes cancer of the mouth you could select a group of pensioners who smoke for the study group and a group of teenagers for the control group. In this case you are more likely to get more sufferers of cancer in the study group than in the control group whether or not any of them smoked because age is an influence on the development of cancer, which is called a confounding factor. If it discovered that pensioners are 50% more likely to get cancer of the mouth than teenagers, then the probability of of the study group should be reduced by that amount before including it in the OR calculation. The resulting figure will be an adjusted odds ratio.

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