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 Interpreting results: Odds ratio

## Odds ratio

Here are the sample data for a case-control study (the first study to link smoking to lung cancer). The investigators chose to study a group of cases with lung cancer and a group of controls without lung cancer. They then asked whether each person had smoked or not  (Doll and Hill, British Med. J, 1950, 739-748). The results were:

Cases (lung cancer)

Control

Smoked

688

650

Never smoked

21

59

With a retrospective case-control data, direct calculations of the relative risk or the difference between proportions should not be performed, as the results are not meaningful. When designing this kind of study, you decide how many cases and controls to study. Those numbers don't have to be equal. Changing the ratio of cases to controls would also change the computed values for the relative risk and difference between proportions. For that reason, it makes no sense to compute or try to interpret these values from case-control data.

In contrast, changing the ratio of cases to controls does not change the expected value of the odds ratio. If the disease or condition you are studying is rare, you can interpret the Odds ratio as an approximation of the relative risk

For the sample data above, the odds of a case being a smoker is 688/21 or 32.8. The odds of a control being a smoker is 650/59 or 11.0. The odds ratio is 32.8/11.0, which is 3.0. Prism reports the value more precisely as 2.974 with a 95% confidence interval ranging from 1.787 to 4.950. You can interpret this odds ratio as a relative risk. The risk of a smoker getting lung cancer is about three times the risk of a nonsmoker getting lung cancer.

## How Prism computes the confidence interval of the odds ratio

Prism computes the confidence interval of the odds ratio using computed either using the Woolf logit method (reference 1; the only method used by Prism 6 and earlier) or the Baptista-Pike method (2) which we recommend. Choose on the Options tab of the Contingency table dialog.  Fagerland (3) reviews the various methods available to compute this confidence interval.

If any cell has a zero and you choose the Woolf method, Prism adds 0.5 to all cells before calculating the odds ratio and its confidence interval. In this case, we suggest you switch to the Baptista-Pike method.

## References

1.Woolf B. On estimating the relation between blood group and disease. Ann Human Gene 1955; 19: 251–253.

2. Baptista J and Pike MC. Exact two-sided confidence limits for the odds ratio in a 2  2 table. J R Stat Soc C Appl Stat 1977; 26: 214–220.

3.Fagerland MW, Lydersen S, Laake P. Recommended confidence intervals for two independent binomial proportions. Stat Methods Med Res. SAGE Publications; 2011 Oct 13.

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