False positive rate

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.

$m$ is the total number hypotheses tested

$m_{0}$ is the number of true , an unknown parameter

null hypotheses

$m-m_{0}$ is the number of true

alternative hypotheses

$V$ is the number of (also called "false discoveries")

false positives (Type I error)

$S$ is the number of (also called "true discoveries")

true positives

$T$ is the number of

false negatives (Type II error)

$U$ is the number of

true negatives

$R=V+S$ is the number of rejected null hypotheses (also called "discoveries", either true or false)

The type I error rate is often associated with the a-priori setting of the by the researcher: the significance level represents an acceptable error rate considering that all null hypotheses are true (the "global null" hypothesis). The choice of a significance level may thus be somewhat arbitrary (i.e. setting 10% (0.1), 5% (0.05), 1% (0.01) etc.)

significance level

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 ${\boldsymbol {\mathrm {FWER} }}=\Pr(V\geq 1)\,$ . 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 $E(V/R)$ .

False positive rate

null hypotheses

alternative hypotheses

false positives (Type I error)

true positives

false negatives (Type II error)

true negatives

significance level

False positives and false negatives

False coverage rate

False discovery rate

Sensitivity and specificity