The O’Brien-Fleming method is a statistical procedure used to control the false discovery rate (FDR) in multiple hypothesis testing. It is a method for controlling the number of false positives that occur when testing multiple hypotheses simultaneously.

The O’Brien-Fleming method involves setting a predetermined threshold for the FDR, and then adjusting the critical values for each hypothesis test based on this threshold. This allows for a more stringent control of the FDR, as the critical values are adjusted in a way that takes into account the number of hypotheses being tested.

An example of the O’Brien-Fleming method in action is a study that is testing the effectiveness of a new drug. The researchers have a hypothesis that the drug will be effective in treating a particular medical condition, and they want to test this hypothesis in a clinical trial. The trial involves administering the drug to a group of patients and measuring the change in their symptoms.

To control the FDR in this situation, the researchers might set a predetermined threshold for the FDR of 0.05. This means that they want to keep the probability of a false positive (i.e., a result that indicates the drug is effective when it is not) to less than 5%.

To adjust the critical values for the hypothesis test based on this threshold, the researchers might use the O’Brien-Fleming method. This involves calculating the critical value for the hypothesis test using a formula that takes into account the predetermined threshold for the FDR and the number of hypotheses being tested.

For example, if the researchers are conducting a single hypothesis test (i.e., testing the effectiveness of the drug in a single clinical trial), the critical value might be set at p < 0.05. This means that if the p-value (i.e., the probability of obtaining a result as extreme or more extreme than the observed result if the null hypothesis is true) is less than 0.05, the researchers would reject the null hypothesis and conclude that the drug is effective.

However, if the researchers are conducting multiple hypothesis tests (e.g., testing the effectiveness of the drug in multiple clinical trials), the critical value would be adjusted using the O’Brien-Fleming method to take into account the number of tests being conducted. This would involve using a formula that adjusts the critical value based on the predetermined threshold for the FDR and the number of hypotheses being tested.

For example, if the researchers are conducting two hypothesis tests (i.e., two clinical trials), the critical value for the O’Brien-Fleming method might be set at p < 0.025 (i.e., half the threshold for a single hypothesis test). This means that if the p-value is less than 0.025 in either of the two clinical trials, the researchers would reject the null hypothesis and conclude that the drug is effective.

In this way, the O’Brien-Fleming method allows researchers to control the FDR in multiple hypothesis testing by adjusting the critical values for each test based on a predetermined threshold. This helps to reduce the likelihood of false positives and ensures that the results of the study are more reliable and trustworthy.