Estimating Replication Rate for Genome-Wide Association Studies

Description

Replication Rate (RR) is the probability of replicating a statistically significant association in genome-wide association studies. This R-package provide the estimation method for replication rate which makes use of the summary statistics from the primary study. We can use the estimated RR to determine the sample size of the replication study, and to check the consistency between the results of the primary study and those of the replication study.

Details

The goal of genome-wide association studies (GWAS) is to discover genetic variants associated with diseases/traits. Replication is a common validation method in GWAS. We regard an association as true finding when it shows significance in both the primary and replication studies. A worth pondering question is: what is the probability of a primary association (i.e. statistically significant association in the primary study) being validated in the replication study?

We refer the Bayesian replication probability as the replication rate (RR). Here we implement the estimation method for RR which makes use of the summary statistics from the primary study. We can use the estimated RR to determine the sample size of the replication study, and to check the consistency between the results of the primary study and those of the replication study.

The principal component of RRate package is repRateEst. Also we implement sample size determination method (repSampleSizeRR and repSampleSizeRR2) and consistency checking method (Hosmer-Lemeshow test, HLtest).

1. To estimate the RR, we need obtain the summary statistics of each genotyped SNPs in the primary study. We have put a example summary statistics (smryStats1) in the package. You can use data(smryStats1) to load the example data. You can also obtain the ground-truth parameters (allele frequencies, odds ratios) of the example data using data(param). We also put the corresponding summary statistics of the replicaition study in the package (smryStats2).

2. You can use SEest to estimate the standard error of the observed log-odds ratio.

SEest(n0,n1,fU,fA)

Details about the function can be seen using help(SEest).

3. You can use repRateEst to estimate the RR for each associations discovered from the primary study (i.e. primary associations).

repRateEst(MUhat,SE, SE2,zalpha2,zalphaR2,boot=100,output=TRUE,idx=TRUE,dir='output',info=T)

Details about the function can be seen using help(repRateEst).

4. You can use repSampleSizeRR and repSampleSizeRR2 to determine the sample size of the replication study.

repSampleSizeRR(RR, n, MUhat,SE,zalpha2,zalphaR2,idx=TRUE)

repSampleSizeRR2(RR,CCR2, MUhat,SE,fU,fA,zalpha2,zalphaR2, idx=TRUE)

Details about these functions can be seen using help(repSampleSizeRR) and help(repSampleSizeRR2).

5. You can use HLtest to check the consistency between the results of the primary study and those of the replication study.

HLtest(x,p,g=10,null='all',boot=1000,info=T,dir='.')

Details about the function can be seen using help(HLtest)

Author(s)

Wei Jiang, Jing-Hao Xue and Weichuan Yu

Maintainer: Wei Jiang <wjiangaa@connect.ust.hk>

References

Jiang, W., Xue, J-H, and Yu, W. What is the probability of replicating a statistically significant association in genome-wide association studies?. Submitted.

See Also

repRateEst, SEest, repSampleSizeRR, repSampleSizeRR2, HLtest

Examples

alpha<-5e-6               #Significance level in the primary study
alphaR<-5e-3              #Significance level in the replication study
zalpha2<-qnorm(1-alpha/2)
zalphaR2<-qnorm(1-alphaR/2)

##Load data
data('smryStats1')        #Example of summary statistics in 1st study
z1<-smryStats1$Z          #Z values in 1st study
n2.0<-2000                #Number of individuals in control group
n2.1<-2000                #Number of individuals in case group

SE2<-SEest(n2.0, n2.1, smryStats1$F_U, smryStats1$F_A) #SE in replication study
######  RR estimation  ######
RRresult<-repRateEst(log(smryStats1$OR),smryStats1$SE, SE2,zalpha2,zalphaR2, output=TRUE,dir='.')
RR<-RRresult$RR           #Estimated RR

Hosmer-Lemeshow test

Description

Test whether each element of x is sampled with the probability specified by the corrsponding element in p.

Usage

HLtest(x, p, g = 10, null = "all", boot = 1000, info = T, dir = ".")

Arguments

Details

Null Hypothesis: Each element of x is sampled with a probability which is the corresponding element of p. We group x to g groups according to p. Then we compare the success proportion with the mean value of p in each group.

Value

A list is returned:

Author(s)

Wei Jiang, Jing-Hao Xue and Weichuan Yu

Maintainer: Wei Jiang <wjiangaa@connect.ust.hk>

References

Hosmer, D. W., & Lemesbow, S. (1980). Goodness of fit tests for the multiple logistic regression model. Communications in statistics-Theory and Methods, 9(10), 1043-1069.

Jiang, W., Xue, J-H, and Yu, W. What is the probability of replicating a statistically significant association in genome-wide association studies?. Submitted.

See Also

RRate repRateEst, SEest, repSampleSizeRR, repSampleSizeRR2,

Examples

alpha<-5e-6               #Significance level in the primary study
alphaR<-5e-3              #Significance level in the replication study
zalpha2<-qnorm(1-alpha/2)
zalphaR2<-qnorm(1-alphaR/2)

##Load data
data('smryStats1')        #Example of summary statistics in 1st study
n2.0<-2000                #Number of individuals in control group
n2.1<-2000                #Number of individuals in case group

SE2<-SEest(n2.0, n2.1, smryStats1$F_U, smryStats1$F_A) #SE in replication study
######  RR estimation  ######
RRresult<-repRateEst(log(smryStats1$OR),smryStats1$SE, SE2,zalpha2,zalphaR2, output=TRUE,dir='.')

#### Hosmer-Lemeshow test  ####
data('smryStats2')        #Example of summary statistics in 2nd study
sigIdx<-(smryStats1$P<alpha)
repIdx<-(sign(smryStats1$Z[sigIdx])*smryStats2$Z[sigIdx]>zalphaR2)
groupNum<-10 
HLresult<-HLtest(repIdx,RRresult$RR,g=groupNum,dir='.')

Estimating Replication Rate for primary associations

Description

repRateEst implements a replication rate estimation method. Two-component mixture prior is used in the estimation.

Usage

repRateEst(MUhat, SE, SE2, zalpha2, zalphaR2, boot = 100, output = TRUE, 
idx = TRUE, dir = "output", info = TRUE)

Arguments

Details

The RR estimation is based on the following two-component mixture model: mu=pi_0 delta_0+(1-pi_0) N(0, sigma_0^2).

Details can be seen the following reference paper.

Value

repRateEst returns the RR, lfdr, prediction power and infered parameters. The returened value is a LIST:

Author(s)

Wei Jiang, Jing-Hao Xue and Weichuan Yu

Maintainer: Wei Jiang <wjiangaa@connect.ust.hk>

References

Jiang, W., Xue, J-H, and Yu, W. What is the probability of replicating a statistically significant association in genome-wide association studies?. Submitted.

See Also

RRate, SEest, repSampleSizeRR, repSampleSizeRR2, HLtest

Examples

alpha<-5e-6               #Significance level in the primary study
alphaR<-5e-3              #Significance level in the replication study
zalpha2<-qnorm(1-alpha/2)
zalphaR2<-qnorm(1-alphaR/2)

##Load data
data('smryStats1')        #Example of summary statistics in 1st study
n2.0<-2000                #Number of individuals in control group
n2.1<-2000                #Number of individuals in case group

SE2<-SEest(n2.0, n2.1, smryStats1$F_U, smryStats1$F_A) #SE in replication study
######  RR estimation  ######
RRresult<-repRateEst(log(smryStats1$OR),smryStats1$SE, SE2,zalpha2,zalphaR2, output=TRUE,dir='.')
RR<-RRresult$RR           #Estimated RR

Estimating standard error of the observed log-odds ratio

Description

SEest implements the Woolf's method to estimate the standard error of the observed log-odds ratio.

Usage

SEest(n0,n1,fU,fA)

Arguments

Details

The Woolf's method to estimate the standard error of log(OR) is based on the following formula:

se(log(OR))=sqrt(1/(n0 fU(1-fU))+1/(n1 fA(1-fA))).

Value

The estimated standard error is returned.

Author(s)

Wei Jiang, Jing-Hao Xue and Weichuan Yu

Maintainer: Wei Jiang <wjiangaa@connect.ust.hk>

References

Woolf, B. (1955). On estimating the relation between blood group and disease. Ann Hum Genet, 19(4), 251-253.

Jiang, W., Xue, J-H, and Yu, W. What is the probability of replicating a statistically significant association in genome-wide association studies?. Submitted.

See Also

RRate, repRateEst, repSampleSizeRR, repSampleSizeRR2, HLtest

Examples

##Load data
data('smryStats1')        #Example of summary statistics in 1st study
n2.0<-2000                #Number of individuals in control group
n2.1<-2000                #Number of individuals in case group

SE2<-SEest(n2.0, n2.1, smryStats1$F_U, smryStats1$F_A) #SE in replication study

Sample size determination for the replication study based on RR

Description

repSampleSizeRR and repSampleSizeRR2 implement the RR-based sample size determination method for the replication study. If the replication study has the same control-to-case ratio with the primary study, then repSampleSizeRR can be used. Otherwise, repSampleSize2 is more suitable.

Usage

repSampleSizeRR(GRR, n, MUhat, SE, zalpha2, zalphaR2, idx = TRUE)

repSampleSizeRR2(GRR,CCR2, MUhat,SE,fU,fA,zalpha2,zalphaR2, idx=TRUE)

Arguments

Value

The determined sample size of the replication study is returned.

Author(s)

Wei Jiang, Jing-Hao Xue and Weichuan Yu

Maintainer: Wei Jiang <wjiangaa@connect.ust.hk>

References

Jiang, W., Xue, J-H, and Yu, W. What is the probability of replicating a statistically significant association in genome-wide association studies?. Submitted.

See Also

RRate repRateEst, SEest, HLtest

Examples

alpha<-5e-6               #Significance level in the primary study
alphaR<-5e-3              #Significance level in the replication study
zalpha2<-qnorm(1-alpha/2)
zalphaR2<-qnorm(1-alphaR/2)

##Load data
data('smryStats1')        #Example of summary statistics in 1st study
#### Sample size determination ###
n1<-4000                  #Sample size of the primary study
n2_1<-repSampleSizeRR(0.8, n1, log(smryStats1$OR),smryStats1$SE,zalpha2,zalphaR2)

CCR2<-2                   #Control-to-case ration in the replication study
n2_2<-repSampleSizeRR2(0.8, CCR2, log(smryStats1$OR),smryStats1$SE,smryStats1$F_U, 
smryStats1$F_A,zalpha2,zalphaR2)

Example summary statistics

Description

Example summary statistics from the primary and replication studies in GWAS. These summary statistics are generated from a simulated study. The primary study (smryStats1) has 2000 cases and 2000 controls. The replication study (smryStats2) has 1000 cases and 1000 controls. The SNP number is 10,000. The disease pravalence is 1%. Minor allele frequencies are from U(0.05, 0.5), and log-odds ratio are from 0.95N(0,1)+0.05N(0, 0.04). The ground-truth parameters for each SNP are in param.

Usage

data(smryStats1)

data(smryStats2)

data(param)

Details

The data sets are described as

SNP: SNP ID

F_A: Allele frequency in case group

F_U: Allele frequency in control group

Z: Z value from log-odds ratio test

P: P value

OR: Odds ratio

SE: Standard error of log(OR)

Author(s)

Wei Jiang, Jing-Hao Xue and Weichuan Yu

Maintainer: Wei Jiang <wjiangaa@connect.ust.hk>

References

Jiang, W., Xue, J-H, and Yu, W. What is the probability of replicating a statistically significant association in genome-wide association studies?. Submitted.

See Also

RRate repRateEst, SEest, repSampleSizeRR, repSampleSizeRR2, HLtest