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Concept
False positive rate
Summary
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.
The false positive rate is
where is the number of false positives, 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 .
Since V is a random variable and is a constant (), 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 .
It is worth noticing that the two definitions ("false positive ratio" / "false positive rate") are somewhat interchangeable. For example, in the referenced article serves as the false positive "rate" rather than as its "ratio".
Classification of multiple hypothesis tests
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 type I error rate is often associated with the a-priori setting of the significance level by the researcher: the significance level represents an acceptable error rate considering that all null hypotheses are true (the "global null" hypothesis).
Official source
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