In statistics, when performing multiple comparisons, a false positive ratio (also known as fall-out or false alarm ratio) is the probability of falsely rejecting the null hypothesis for a particular test. The false positive rate is calculated as the ratio between the number of negative events wrongly categorized as positive (false positives) and the total number of actual negative events (regardless of classification). The false positive rate (or "false alarm rate") usually refers to the expectancy of the false positive ratio.

Definition

The false positive rate is where \mathrm{FP} is the number of false positives, \mathrm{TN} is the number of true negatives and is the total number of ground truth negatives. The level of significance that is used to test each hypothesis is set based on the form of inference (simultaneous inference vs. selective inference) and its supporting criteria (for example FWER or FDR), that were pre-determined by the researcher. When performing multiple comparisons in a statistical framework such as above, the false positive ratio (also known as the false alarm ratio, as opposed to false positive rate / false alarm rate ) usually refers to the probability of falsely rejecting the null hypothesis for a particular test. Using the terminology suggested here, it is simply V/m_0. Since V is a random variable and m_0 is a constant (V \leq m_0), the false positive ratio is also a random variable, ranging between 0–1. The false positive rate (or "false alarm rate") usually refers to the expectancy of the false positive ratio, expressed by E(V/m_0). It is worth noticing that the two definitions ("false positive ratio" / "false positive rate") are somewhat interchangeable. For example, in the referenced article V/m_0 serves as the false positive "rate" rather than as its "ratio".

Classification of multiple hypothesis tests

Comparison with other error rates

While the false positive rate is mathematically equal to the type I error rate, it is viewed as a separate term for the following reasons: The false positive rate should also not be confused with the family-wise error rate, which is defined as. As the number of tests grows, the familywise error rate usually converges to 1 while the false positive rate remains fixed. Lastly, it is important to note the profound difference between the false positive rate and the false discovery rate: while the first is defined as E(V/m_0), the second is defined as E(V/R).