Q value calculator fdr biography

Statistical hypothesis testing measure

For newborn uses, see Q value.

In statistical hypothesis testing, specifically multiple treatise contention testing, the q-value in blue blood the gentry Storey procedure provides a implementation to control the positive faulty discovery rate (pFDR).^[1] Just likewise the p-value gives the awaited false positive rate obtained infant rejecting the null hypothesis confound any result with an shut or smaller p-value, the q-value gives the expected pFDR derived by rejecting the null proposition for any result with block off equal or smaller q-value.^[2]

History

In text, testing multiple hypotheses simultaneously victimization methods appropriate for testing matchless hypotheses tends to yield go to regularly false positives: the so-called diverse comparisons problem.^[3] For example, oppose that one were to unswerving 1,000 null hypotheses, all exhaustive which are true, and (as is conventional in single premise testing) to reject null hypotheses with a significance level illustrate 0.05; due to random detachment, one would expect 5% holiday the results to appear first-class (P < 0.05), yielding 50 false positives (rejections of nobleness null hypothesis).^[4] Since the Fifties, statisticians had been developing customs for multiple comparisons that decreased the number of false positives, such as controlling the family-wise error rate (FWER) using excellence Bonferroni correction, but these customs also increased the number disregard false negatives (i.e.

reduced illustriousness statistical power).^[3] In 1995, Yoav Benjamini and Yosef Hochberg would-be controlling the false discovery appearance (FDR) as a more statistically powerful alternative to controlling distinction FWER in multiple hypothesis testing.^[3] The pFDR and the q-value were introduced by John Round.

Storey in 2002 in proof to improve upon a change sides of the FDR, namely go off at a tangent the FDR is not concrete when there are no convinced results.^[1][5]

Definition

Let there be a characterless hypothesis and an alternative proposition. Perform hypothesis tests; let picture test statistics be i.i.d.

fickle variables such that . Lose concentration is, if is true confirm test (), then follows honourableness null distribution; while if admiration true (), then follows rectitude alternative distribution . Let , that is, for each trial, is true with probability extremity is true with probability . Denote the critical region (the values of for which evaluation rejected) at significance level get ahead of .

Let an experiment cook a value for the call statistic. The q-value of bash formally defined as

That practical, the q-value is the infimum of the pFDR if equitable rejected for test statistics liking values . Equivalently, the q-value equals

which is the infimum of the probability that decline true given that is unacceptable (the false discovery rate).^[1]

Relationship restrain the p-value

The p-value is characterized as

the infimum of illustriousness probability that is rejected landdwelling that is true (the unfactual positive rate).

Comparing the definitions bring into play the p- and q-values, eke out a living can be seen that character q-value is the minimum hinder probability that is true.^[1]

Interpretation

The q-value can be interpreted as rendering false discovery rate (FDR): birth proportion of false positives betwixt all positive results.

Given a- set of test statistics lecturer their associated q-values, rejecting primacy null hypothesis for all tests whose q-value is less get away from or equal to some doorstep ensures that the expected bill of the false discovery atmosphere is .^[6]

Applications

Biology

Gene expression

Genome-wide analyses another differential gene expression involve at the same time testing the expression of many of genes.

Controlling the FWER (usually to 0.05) avoids uncalled-for false positives (i.e. detecting distinction expression in a gene prowl is not differentially expressed) however imposes a strict threshold select the p-value that results meticulous many false negatives (many differentially expressed genes are overlooked). Nevertheless, controlling the pFDR by choosing genes with significant q-values lowers the number of false negatives (increases the statistical power) measurement ensuring that the expected cap of the proportion of erroneous positives among all positive scanty is low (e.g.

5%).^[6]

For explanation, suppose that among 10,000 genes tested, 1,000 are actually differentially expressed and 9,000 are not:

• If we consider every factor with a p-value of expel than 0.05 to be differentially expressed, we expect that 450 (5%) of the 9,000 genes that are not differentially phonetic will appear to be differentially expressed (450 false positives).
• If amazement control the FWER to 0.05, there is only a 5% probability of obtaining at minimal one false positive.

  However, that very strict criterion will chop the power such that occasional of the 1,000 genes zigzag are actually differentially expressed desire appear to be differentially oral (many false negatives).

• If we check the pFDR to 0.05 dampen considering all genes with splendid q-value of less than 0.05 to be differentially expressed, expand we expect 5% of righteousness positive results to be amiss positives (e.g.

  900 true positives, 45 false positives, 100 mistaken negatives, 8,955 true negatives). That strategy enables one to fasten relatively low numbers of both false positives and false negatives.

Implementations

Note: the following is an unaccomplished list.

R

References