FPRP {gap}

False-positive report probability

Description

False-positive report probability

Usage

FPRP(a, b, pi0, ORlist, logscale = FALSE)

Arguments

a

parameter value at which the power is to be evaluated.

b

the variance for a, or the uppoer point of a 95%CI if logscale=FALSE.

pi0

the prior probabiility that H_0 is true.

ORlist

a vector of ORs that is most likely.

logscale

FALSE=a,b in orginal scale, TRUE=a, b in log scale.

Details

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}.

Value

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.

Author(s)

Jing Hua Zhao

References

Wacholder S, Chanock S, Garcia-Closas M, El Ghormli L, Rothman N (2004). “Assessing the probability that a positive report is false: an approach for molecular epidemiology studies.” J Natl Cancer Inst, 96(6), 434-42. ISSN 0027-8874 (Print) 0027-8874, doi:10.1093/jnci/djh075.

See Also

BFDP

Examples

## Not run: 
# Example by Laure El ghormli & Sholom Wacholder on 25-Feb-2004
# Step 1 - Pre-set an FPRP-level criterion for noteworthiness

T <- 0.2

# Step 2 - Enter values for the prior that there is an association

pi0 <- c(0.25,0.1,0.01,0.001,0.0001,0.00001)

# Step 3 - Enter values of odds ratios (OR) that are most likely, assuming that
#          there is a non-null association

ORlist <- c(1.2,1.5,2.0)

# Step 4 - Enter OR estimate and 95% confidence interval (CI) to obtain FPRP 												

OR <- 1.316
ORlo <- 1.08
ORhi <- 1.60

logOR <- log(OR)
selogOR <- abs(logOR-log(ORhi))/1.96
p <- ifelse(logOR>0,2*(1-pnorm(logOR/selogOR)),2*pnorm(logOR/selogOR))
p
q <- qnorm(1-p/2)
POWER <- ifelse(log(ORlist)>0,1-pnorm(q-log(ORlist)/selogOR),
                pnorm(-q-log(ORlist)/selogOR))
POWER
FPRPex <- t(p*(1-pi0)/(p*(1-pi0)+POWER\
row.names(FPRPex) <- pi0
colnames(FPRPex) <- ORlist
FPRPex
FPRPex>T

## now turn to FPRP
OR <- 1.316
ORhi <- 1.60
ORlist <- c(1.2,1.5,2.0)
pi0 <- c(0.25,0.1,0.01,0.001,0.0001,0.00001)
z <- FPRP(OR,ORhi,pi0,ORlist,logscale=FALSE)
z

## End(Not run)