False-positive report probability
FPRP(a, b, pi0, ORlist, logscale = FALSE)The returned value is a list with compoents, p p value corresponding to a,b. power the power corresponding to the vector of ORs. FPRP False-positive report probability. FNRP 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 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.
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 \(\pi\).
Joint probability of significance of test and truth of hypothesis
| 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 \(FNRP=\{1+[(1-\alpha)/\beta][(1-\pi)/\pi]\}^{-1}\).
wacholder04gap
BFDP