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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).

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False positives and false negatives

A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test result incorrectly indicates the absence of a condition when it is actually present. These are the two kinds of errors in a binary test, in contrast to the two kinds of correct result (a and a ).

Multiple comparisons problem

In statistics, the multiple comparisons, multiplicity or multiple testing problem occurs when one considers a set of statistical inferences simultaneously or infers a subset of parameters selected based on the observed values. The more inferences are made, the more likely erroneous inferences become. Several statistical techniques have been developed to address that problem, typically by requiring a stricter significance threshold for individual comparisons, so as to compensate for the number of inferences being made.

Statistical significance

In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by , is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true; and the p-value of a result, , is the probability of obtaining a result at least as extreme, given that the null hypothesis is true. The result is statistically significant, by the standards of the study, when .

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