The function calculates the false positive report probability (FPRP), the probability of no true association beteween a genetic variant and disease given a statistically significant finding, which depends not only on the observed P value but also on both the prior probability that the assocition is real and the statistical power of the test. An associate result is the false negative reported probability (FNRP). See example for the recommended steps.
FPRP(a, b, pi0, ORlist, logscale = FALSE)The returned value is a list with compoents,
p value corresponding to a,b
the power corresponding to the vector of ORs
False-positive report probability
False-negative report probability
parameter value at which the power is to be evaluated.
the variance for a, or the uppoer point of a 95%CI if logscale=FALSE.
the prior probabiility that \(H_0\) is true.
a vector of ORs that is most likely.
FALSE=a,b in orginal scale, TRUE=a, b in log scale.
Jing Hua Zhao
The FPRP and FNRP are derived as follows. Let \(H_0\)=null hypothesis (no association), \(H_A\)=alternative hypothesis (association). Since classic frequentist theory considers they are fixed, one has to resort to Bayesian framework by introduing prior, \(\pi=P(H_0=TRUE)=P(association)\). Let \(T\)=test statistic, and \(P(T>z_\alpha|H_0=TRUE)=P(rejecting\ H_0|H_0=TRUE)=\alpha\), \(P(T>z_\alpha|H_0=FALSE)=P(rejecting\ H_0|H_A=TRUE)=1-\beta\). The joint probability of test and truth of hypothesis can be expressed by \(\alpha\), \(\beta\) and \(\pi\).
| Truth of \(H_A\) | significant | nonsignificant | Total |
| TRUE | \((1-\beta)\pi\) | \(\beta\pi\) | \(\pi\) |
| FALSE | \(\alpha (1-\pi)\) | \((1-\alpha)(1-\pi)\) | \(1-\pi\) |
| Total | \((1-\beta)\pi+\alpha (1-\pi)\) | \(\beta\pi+(1-\alpha)(1-\pi)\) | 1 |
We have \(FPRP=P(H_0=TRUE|T>z_\alpha)= \alpha(1-\pi)/[\alpha(1-\pi)+(1-\beta)\pi]=\{1+\pi/(1-\pi)][(1-\beta)/\alpha]\}^{-1}\) and similarly \(FNRP=\{1+[(1-\alpha)/\beta][(1-\pi)/\pi]\}^{-1}\).
Wacholder S, Chanock S, Garcia-Closas M, El ghomli L, Rothman N. (2004) Assessing the probability that a positive report is false: an approach for molecular epidemiology studies. J Natl Cancer Inst 96:434-442
BFDP