Methods for combining p-values.
| Parameters | |
|---|---|
| p | Vector of p-values |
| K | Number of smallest p-values in RTP |
| tau | Threshold p-value for TFisher methods |
| stepscale | Multiplier for default integration step size |
p.fisher(p)
Standard Fisher's method using all p-values.
p.art(K, p)
Augmented truncated product function from Vsevolozhskaya et al.
p.tfisher.soft(tau, p)
Soft thresholding TFisher method from Zhang et al.
This is implemented by a fast C++ TFisher function.
p.tfisher.tpm(tau, p)
Truncated product method (tpm) from Zaykin et al.
solved by TFisher function.
p.rtp(K, p, stepscale = 1)
Rank truncated product p-value by integrand gamma PDF x beta CDF or beta PDF x (1-gamma CDF) and Riemann sum integral over [0, inf) or [0, 1].
The second integrand is used for K / number of all p-values proportion greater than 1/3.
Parameter stepscale affects the integration step size.
E.g. stepscale = 0.5 halves the default step size and increases accuracy (which mostly is unnecessary).
p.rtp.simulated(K, p, rounds = 100000, seed = 0)
Rank truncated product p-value by Monte Carlo simulation. This is a slow function and not good for actual use, but the program code in
function rtpSimulated in
the rtp.cpp file demonstrates what rtp p-values are really measuring.
Parameter seed is used to initialize random number generator. If 0, randomly seeded.
p.rtp.mutoss(K, p)
Rank truncated product p-value from
https://rdrr.io/cran/mutoss/src/R/Rank_Truncated.R. This implements exactly equations from Dudbridge and Koeleman. Integrated over [0, 1] by 1000 equidistant function evaluations.
p.rtp.qbeta(K, p, abstol = 1e-4, reltol = 1e-2)
Rank truncated product p-value by inverse beta CDF method from Vsevolozhskaya et al.
Integrated over [0, 1] by R integrate.
p.rtp.dbeta.cuba(K, p)
Rank truncated product p-value by integrand beta PDF x (1 - gamma CDF) integrated by package cubature function pcubature over [0, 1] with very small tolerances.
The integrand implements Dudbridge and Koeleman equations.
This gives high accuracy "exact" reference values very reliably, but quite slow.
R-style documentation may come soon or later. Statistical and computational methods are somewhat commented in the program code.
Package rtp is built and tested with R version 4.0.4, Rcpp version 1.0.6 and g++ compiler version 8.3.0 in Rtools 4.0 on Windows system.
install.packages("devtools") # if not installed
library(devtools)
install_github("pekkizen/rtp")
library(rtp)# Generate 199 random p-values, add one small value 1e-7 and
# calculate combined p-value by different methods.
K <- 10 # 10 smallest p-values
L <- 200
p <- c(1e-7, runif(L - 1))
tau <- K / L # p-values less than 0.05
p.art(K, p)
p.rtp(K, p)
p.tfisher.soft(tau, p)
p.tfisher.tpm(tau, p)
p.fisher(p)
# Comparing methods has also a dedicated function.
pvalues.methods(K = 10, L = 200, small = 1e-7)
# Comparing accuracy of the rtp functions.
# rounds are for p.rtp.simulated.
pvalues.rtp(K=10, L=200, small=1e-8, seed=0, rounds=100000)
# The four integrand functions can be examined by
plot.quantile.integrands(K = 10, L = 200, small = 1e-1)
plot.BxG.integrand(K = 10, L = 200, small = 1e-3)
plot.GxB.integrand(K = 10, L = 200, small = 1e-3)
# Integrand and p-value functions benchmarks.
bench.integrands(K = 10, L = 200, small = 1e-5)
bench.pvalues(K = 10, L = 200, small = 1e-5)Zaykin DV, Zhivotovsky LA, Westfall PH and Weir BS. (2002). Truncated product method for combining p-values Genetic Epidemiology 22: 170-185. ftp://statgen.ncsu.edu/pub/zaykin/tpm
Dudbridge F. and Koeleman B. P. (2003). Rank truncated product of p-values, with application to genomewide association scans. Genet. Epidemiol. 25, 360–366. doi: 10.1002/gepi.10264
Olga A. Vsevolozhskaya, Fengjiao Hu and Dmitri V. Zaykin. (2019). Detecting Weak Signals by Combining Small P-Values in Genetic Association Studies. Front. Genet. 10:1051. doi: 10.3389/fgene.2019.01051. https://github.com/dmitri-zaykin/Total_Decor.
Zhang, Hong; Tong, Tiejun; Landers, John; Wu, Zheyang. TFisher: A powerful truncation and weighting procedure for combining p-values. Ann. Appl. Stat. 14 (2020), no. 1, 178--201. doi:10.1214/19-AOAS1302. https://CRAN.R-project.org/package=TFisher
https://en.wikipedia.org/wiki/Order_statistic
Order statistics sampled from a uniform distribution.
https://en.wikipedia.org/wiki/Gamma_distribution
Characterization using shape α and rate β.
The probability density function.
The cumulative distribution function for positive
integers α and β = 1.
https://en.wikipedia.org/wiki/Beta_distribution
Probability density function.
https://en.wikipedia.org/wiki/Beta_function
https://dlmf.nist.gov/8.17#E5. Formula 8.17.5
Regularized incomplete beta function = beta CDF function.