pfa:
Estimates False Discovery Proportion Under Arbitrary Covariance Dependence

Description

This package contains functions for performing multiple testing and estimating the false discovery proportion (FDP). pfa.test(X,...) finds the false nulls in p hypotheses; pfa.test(X,Y,...) tests the difference of two multiple-dimensional population means; pfa.gwas(X,Y,...) performs the genome-wise association study(GWAS).

Usage

pfa.test(X,Y,tval,Sigma,reg="L2",K,e=0.05,gamma,mat_est="poet",plot="-log")
pfa.gwas(X,Y,tval,v,reg="L1",e=0.05,gamma,K,plot="-log")

Arguments

X,Y

In pfa.test, either X and Y are data matrices of two different samples, or X is a vector of test statistics and Y is missing. In pfa.gwas, X is the design matrix and Y contains observations of the response variable.

Sigma

the covariance matrix.

standard deviation of noises. By default, it is estimated by refitted cross-validation.

tval

a sequence of thresholding level for p-values. By default, tval is chosen automatically.

reg

method used to estimate factors. If reg="L1", the method is least absolute value regression; if reg="L2", the method is least-squares (with large outliers filtered out). Default is "L1".

number of factors. By default, given the covariance matrix, K is the smallest integer such that the sum of squares of the smallest (p-K) eigenvalues is no larger than e=0.01 times its trace.

a parameter used to choose the number of factors in PFA. Default value is 0.01.

gamma

a parameter used to estimate the true Null proportion: pi0 = (percentage of (p-values > gamma)) /(1- gamma). By default, it is chosen automatically.

mat_est

method used to estimate the covariance matrix. If mat_est="sample", the estimate is the (pooled) sample covariance matrix; if mat_est="poet", the estimate comes from the poet package. Default is "poet".

plot

plotting mode. If plot="-log", in the FDP plot, the x axis is -log(t); if plot="log", the x axis is log(t); if plot="linear", the x axis is t; if plot="none", no graph is generated. Default is "-log".

Value

Pvalue: Sorted p-values.
adjPvalue: Sorted adjusted p-values.
FDP: Estimated FDPs.
pi0: Estimated true null proportion.
K: Number of factors used in the PFA method.
sigma: Estimated standard deviation of noises. This component is NULL except in the return of pfaRegress.

Details

pfa.test(X,Sigma=Sigma,...): X is a vector of test statistics. Suppose it has a multivariate Gaussian distribution N(mu, Sigma). We would like to test: mu(i)=0, i=1,...,p. Given a threshold t, we reject hypothesis i if and only if P(i)=X(i)/sqrt(Sigma(i,i)),i=1,...,p. We apply the PFA method [Fan, Han and Gu (2012)] to estimate the false discovery proportion (FDP) for arbitrary thresholds. In this case, the covariance matrix Sigma is required.

pfa.test(X,...): X is an n-by-p matrix containing i.i.d. samples of the multivariate Gaussian distrubtion N(mu, Sigma). Again, we would like to test: mu(i)=0, i=1,...,p. The test statistics is the vector of sample mean. When Sigma is unknown, we instead use the sample covariance matrix or the estimate from POET-PFA method [Fan and Han (2016)]. The number of factors determined for POET and for PFA is based on the eigenvalue ratio test in Ahn and Horenstein (2013).

pfa.test(X,Y,...): X and Y are i.i.d. samples from the distributions N(mu1, Sigma) and N(mu2, Sigma). We would like to test: mu1(i)=mu2(i), i=1,...,p. The test statistics is the vector of sample mean difference. When Sigma is unknown, we instead use the pooled sample covariance matrix or the estimate from POET-PFA method.

pfa.gwas(X,Y,...): X is the data matrix of p covariates (e.g. SNP measurements) and Y is the data vector of response variables (e.g. indicator of a trait). We would like to test: whether covariate i is marginally associated with the response, i=1,...,p. The test statistics is the maginal regression coefficients. We suppose Y=X_1*beta_1+...+X_p*beta_p+epsilon, where epsilon's are i.i.d. samples from N(0,sigma^2). Then the covariane matrix of the test statistics is nothing but the sample covariance matrix of X.

References

Fan, Han and Gu (2012) "Estimating False Discovery Proportion Under Arbitrary Covariance Dependence" (with discussion) JASA.

Fan and Han (2016) "Estimation of False Discovery Proportion with Unknown Dependence", Manuscript.

Examples

Run this code

# Example 1: multiple testing with known covariance

require(MASS)
p <- 100
Sigma <- matrix(0.4,p,p)
diag(Sigma)<- 1
mu <- as.vector(c(rep(3,5), rep(0, p-5)))
Z <- mvrnorm(1, mu, Sigma)
RE1 <- pfa.test(Z, Sigma=Sigma,reg="L1")
summary(RE1)

# Example 2: multiple testing with unknown covariance 

n <- 200
p <- 300
K <- 3
mu <- as.vector(c(rep(2,10),rep(0,p-10)))
B <- matrix(runif(K*p, min=-1, max=1), nrow=K)
f <- matrix(rnorm(K*n), nrow=n)
Bf <- f %*% B
X <- matrix(rep(0, n*p), nrow=n)
for (i in 1:n) 
   X[i,] <- mu + Bf[i,] + rnorm(p)
## Not run: RE2 <- pfa.test(X, tval="pval")
## Not run: summary(RE2)

# Example 3: testing the marginal regression coefficients

n <- 100
p <- 300
beta <- as.matrix(c(rep(2, 10), rep(0, p-10)))
X <- matrix(rep(0, n*p), nrow=n)
X[,1:10] <- matrix(rnorm(n*10), nrow=n)
z <- as.matrix(rnorm(n))
y <- as.matrix(rnorm(n))
X[,11:p] <- as.matrix(rnorm(n*(p-10)), nrow=n)
for (i in 11:p) {
    rho1 <- runif(1,min=-0.2,max=0.2)
    rho2 <- runif(1,min=-0.2,max=0.2)
    X[,i] <- X[,i] + z*rho1 + y*rho2 
}
eps <- as.matrix(rnorm(n))
Y <- X %*% beta + eps
## Not run: RE3 <- pfa.gwas(X,Y)
## Not run: summary(RE3)

# Example 4: GWAS on the CCT8 gene

data(CEU)
## Not run: RE4 <- pfa.gwas(CEU$x, CEU$y, t=exp(-seq(1.8,3.6,0.1)), reg="L2")
## Not run: summary(RE4)

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