The correlated \(N(0, 1)\) figure in cashr

Introduction

Document the correlated \(N(0, 1)\) figure in the cashr paper.

source("../code/gdash_lik.R")

Loading required package: EQL

Loading required package: ttutils

Loading required package: SQUAREM

Loading required package: REBayes

Loading required package: Matrix

Warning: package 'Matrix' was built under R version 3.4.4

Loading required package: CVXR

Attaching package: 'CVXR'

The following object is masked from 'package:stats':

    power

Loading required package: PolynomF

Loading required package: Rmosek

Loading required package: ashr

source("../code/gdfit.R")
z.mat <- readRDS("../output/z_null_liver_777.rds")
Z.gtex <- readRDS("../output/paper/simulation/Z.gtex.rds")

sel = c(32, 327, 23, 459)
z.sel <- z.mat[sel, ]
z.sel[3, ] <- Z.gtex[[4503]]

gd.ord <- 10

x.plot = seq(- max(abs(z.sel)) - 2, max(abs(z.sel)) + 2, length = 1000)
hermite = Hermite(gd.ord)
gd0.std = dnorm(x.plot)
matrix_lik_plot = cbind(gd0.std)
for (j in 1 : gd.ord) {
  gd.std = (-1)^j * hermite[[j]](x.plot) * gd0.std / sqrt(factorial(j))
  matrix_lik_plot = cbind(matrix_lik_plot, gd.std)
}

z = z.sel[4, ]
w <- gdfit(z, gd.ord, w.lambda = 10, w.rho = 0.5)$w
y.plot = matrix_lik_plot %*% w
z.hist = hist(z, breaks = 100, plot = FALSE)
y.max = max(z.hist$density, y.plot, dnorm(0))

setEPS()
postscript("../output/paper/cor_z_hist.eps", width = 8, height = 6)
#pdf("../output/paper/cor_z_hist.pdf", width = 8, height = 6)

par(mfrow = c(2, 2)) # 2-by-2 grid of plots
par(oma = c(0.5, 2.5, 0, 0)) # make room (i.e. the 4's) for the overall x and y axis titles
par(mar = c(2, 2, 3.5, 1)) # make the plots be closer together

# now plot the graphs with the appropriate axes removed (via xaxt and yaxt),
# remove axis labels (so that they are not redundant with overall labels,
# and set some other nice choices for graphics parameters
for (i in 1 : 4) {
  z = z.sel[i, ]
  w <- gdfit(z, gd.ord)$w
  y.plot = matrix_lik_plot %*% w
  z.hist = hist(z, breaks = 100, plot = FALSE)
  hist(z, breaks = seq(-10, 10, by = 0.1), prob = TRUE, ylim = c(0, y.max), main = NULL, xlab = "", xlim = range(c(abs(z.sel), -abs(z.sel))))
  lines(x.plot, dnorm(x.plot), col = "blue", lwd = 2)
  lines(x.plot, y.plot, col = "red", lwd = 2)
  legend("topleft", bty = "n", paste0('(', letters[i], ')'), cex = 1.25)
}

# print the overall labels
mtext('Density', side = 2, outer = TRUE, line = 1)
mtext(latex2exp::TeX('Histograms of $10^4$ Correlated N(0,1) z-scores'), line = -2, outer = TRUE)

legend("topleft", inset = c(-0.65, -0.25), legend = c("N(0, 1)", "Gaussian Derivatives"), lty = 1, lwd = 2, xpd = NA, col = c("blue", "red"), ncol = 2)

dev.off()

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ashr::plot_diagnostic on these correlated noise

for (i in 1 : 4) {
  par(mfrow = c(1, 1))
  z = z.sel[i, ]
  hist(z, breaks = 100, prob = TRUE, ylim = c(0, y.max), main = NULL, xlab = "", xlim = range(c(abs(z.sel), -abs(z.sel))))
  lines(x.plot, dnorm(x.plot), col = "green", lwd = 2)
  par(mfrow = c(2, 2))
  fit.ash.n <- ashr::ash(z, 1, mixcompdist = "normal", method = "fdr")
  cat("mixcompdist = normal")
  ashr::plot_diagnostic(fit.ash.n, breaks = 100, plot.hist = TRUE)
  par(mfrow = c(2, 2))
  fit.ash.u <- ashr::ash(z, 1, mixcompdist = "uniform", method = "fdr")
  cat("mixcompdist = uniform")
  ashr::plot_diagnostic(fit.ash.u, breaks = 100, plot.hist = TRUE)
  par(mfrow = c(2, 2))
  fit.ash.hu <- ashr::ash(z, 1, mixcompdist = "halfuniform", method = "fdr")
  cat("mixcompdist = halfuniform")
  ashr::plot_diagnostic(fit.ash.hu, breaks = 100, plot.hist = TRUE)
}

mixcompdist = normal

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mixcompdist = uniform

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mixcompdist = halfuniform

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mixcompdist = normal

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mixcompdist = uniform

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mixcompdist = halfuniform

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mixcompdist = normal

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mixcompdist = uniform

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mixcompdist = halfuniform

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mixcompdist = normal

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mixcompdist = uniform

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mixcompdist = halfuniform

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different methods applied to (c)

q <- 0.1
z <- z.sel[3, ]
p <- pnorm(-abs(z)) * 2
## under 0.005
sum(p <= 0.005)

[1] 809

pnorm(qnorm(0.0025), 0, 1.6) * 2 * 1e4

[1] 793.6266

p.bh <- p.adjust(p, method = "BH")
## BHq at FDR 0.05
sum(p.bh <= q)

[1] 1822

fit.q <- qvalue::qvalue(p)
## pi0 by qvalue
fit.q$pi0

[1] 0.433538

## qvalue at FDR 0.05
sum(fit.q$qvalues <= q)

[1] 3818

## pi0 by ashr
fit.a <- ashr::ash(z, 1, mixcompdist = "normal", method = "fdr")
ashr::get_pi0(fit.a)

[1] 0.01512495

## ashr at FDR 0.05
sum(ashr::get_qvalue(fit.a) <= q)

[1] 10000

shoulder inflation figure

## the image is 7.5 * 3
setEPS()
postscript("../output/paper/cor_z_cdf.eps", width = 8, height = 2.5)
#pdf("../output/paper/cor_z_cdf.pdf", width = 8, height = 2.5)

par(mfrow = c(1, 3))
par(oma = c(1, 2.5, 0, 8)) # make room (i.e. the 4's) for the overall x and y axis titles
par(mar = c(2, 2, 2.5, 1)) # make the plots be closer together

plot(ecdf(z), xlab = "", ylab = "", lwd = 2, main = expression("panel (c) z-scores"), cex.main = 1.5)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01)), col = "blue", lwd = 2)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01), 0, 1.6), col = "green", lwd = 2)
rect(xleft = c(-5, 2.5),
     xright = c(-2.5, 5),
     ytop = c(0.05, 1),
     ybottom = c(0, 0.95), border = "red", lty = c(1, 5))

plot(ecdf(z), xlab = "", ylab = "", main = expression("left tail"), lwd = 2, xlim = c(-5, -2.5), ylim = c(0, 0.05), cex.main = 1.5, bty = "n")
box(col = "red")
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01)), col = "blue", lwd = 2)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01), 0, 1.6), col = "green", lwd = 2)

plot(ecdf(z), xlab = "", ylab = "", main = expression("right tail"), lwd = 2, xlim = c(2.5, 5), ylim = c(0.95, 1), cex.main = 1.5, bty = "n")
box(col = "red", lty = 5)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01)), col = "blue", lwd = 2)
lines(seq(-6, 6, by = 0.01), pnorm(seq(-6, 6, by = 0.01), 0, 1.6), col = "green", lwd = 2)

mtext('CDF', side = 2, outer = TRUE, line = 1)

legend("topright", inset = c(-0.68, 0.3), legend = c('panel (c)', 'N(0, 1)', expression(N(0, 1.6^2))), lty = 1, lwd = 2, xpd = NA, col = c('black', "blue", "green"), ncol = 1, cex = 1.25, bty = 'n')

dev.off()

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# 7.5 * 3
setEPS()
postscript("../output/paper/cor_z_pval.eps", width = 12, height = 3)
#pdf("../output/paper/cor_z_pval.pdf", width = 12, height = 3)

thresh.color <- c("maroon", "purple", "orange")
#thresh.color <- scales::hue_pal()(10)[5 : 7]

par(mfrow = c(1, 4))
par(oma = c(0, 0, 0, 11)) # make room (i.e. the 4's) for the overall x and y axis titles

par(mar = c(4.5, 4, 4.5, 1)) # make the plots be closer together
p.hist <- hist(p, breaks = seq(0, 1, by = 0.01), plot = FALSE)
plot(0, 0, xlab = "p-values", ylab = "", type = "n", xlim = c(0, 1), ylim = c(0, max(p.hist$density)), main = expression(atop("Histogram of p-val of", 'panel (c) z-scores')), cex.main = 1.5, cex.lab = 1.5)
title(ylab = "Density", line = 2.5, cex.lab = 1.5)
abline(v = c(0.05 / 1e4, pnorm(-sqrt(2 * log(1e4))) * 2, 0.005), lwd = 2, col = thresh.color[3 : 1], lty = c(4, 2, 1))
hist(p, prob = TRUE, breaks = seq(0, 1, by = 0.01), xlab = "", add = TRUE, col = rgb(0, 0, 0, 0.75))

Warning in rect(x$breaks[-nB], 0, x$breaks[-1L], y, col = col, border =
border, : semi-transparency is not supported on this device: reported only
once per page

set.seed(5)
p.norm.1 <- pnorm(-abs(rnorm(1e4))) * 2
set.seed(25)
p.norm.1.6 <- pnorm(-abs(rnorm(1e4, 0, 1.6))) * 2
y.max <- -log(min(p.norm.1, p, p.norm.1.6))
y.max <- 20

par(mar = c(4.5, 4, 4.5, 1)) # make the plots be closer together

plot(sample(-log(p)), ylim = c(0, y.max), ylab = "", main = expression(atop('-log(p-val) of', "panel (c) z-scores")), cex.main = 1.5, cex.lab = 1.5)
title(ylab = '-log(p)', cex.lab = 1.5, line = 2.5)
abline(h = -log(c(
  0.005,
  pnorm(-sqrt(2 * log(1e4))) * 2,
  0.05 / 1e4
)), lwd = 2, col = thresh.color, lty = c(1, 2, 4))

plot(-log(p.norm.1), ylim = c(0, y.max), ylab = "", main = expression(atop('-log(p-val) of', "indep N(0,1) samples")), col = "blue", cex.main = 1.5, cex.lab = 1.5)
title(ylab = '-log(p)', cex.lab = 1.5, line = 2.5)
abline(h = -log(c(
  0.005,
  pnorm(-sqrt(2 * log(1e4))) * 2,
  0.05 / 1e4
)), lwd = 2, col = thresh.color, lty = c(1, 2, 4))

plot(-log(p.norm.1.6), ylim = c(0, y.max), ylab = "", main = expression(atop('-log(p-val) of', paste("indep ", N(0, 1.6^2), " samples"))), col = "green", cex.main = 1.5, cex.lab = 1.5)
title(ylab = '-log(p)', cex.lab = 1.5, line = 2.5)
abline(h = -log(c(
  0.005,
  pnorm(-sqrt(2 * log(1e4))) * 2,
  0.05 / 1e4
)), lwd = 2, col = thresh.color, lty = c(1, 2, 4))

#legend("topright", inset = c(-0.5, 0), legend = c("p = 0.005", "Universal Threshold", "Bonferroni"), lty = 1, lwd = 2, xpd = NA, col = c("red", "orange", "yellow"), ncol = 1, cex = 1.25)

legend("topright", inset = c(-0.82, 0.3),
       legend = c(
         latex2exp::TeX('p-val = $0.05 / 10^4$'),
         'Univ Thresh',
         "p-val = 0.005"
         ), lty = c(4, 2, 1), lwd = 2, xpd = NA,
       col = thresh.color[3 : 1], ncol = 1, cex = 1.25, bty = "n")

dev.off()

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Session information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.14

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] ashr_2.2-7        Rmosek_8.0.69     PolynomF_1.0-2    CVXR_0.95        
[5] REBayes_1.3       Matrix_1.2-14     SQUAREM_2017.10-1 EQL_1.0-0        
[9] ttutils_1.0-1    

loaded via a namespace (and not attached):
 [1] latex2exp_0.4.0   gmp_0.5-13.1      Rcpp_0.12.16     
 [4] pillar_1.2.2      plyr_1.8.4        compiler_3.4.3   
 [7] git2r_0.21.0      workflowr_1.1.1   R.methodsS3_1.7.1
[10] R.utils_2.6.0     iterators_1.0.9   tools_3.4.3      
[13] digest_0.6.15     bit_1.1-13        tibble_1.4.2     
[16] gtable_0.2.0      evaluate_0.10.1   lattice_0.20-35  
[19] rlang_0.2.0       foreach_1.4.4     yaml_2.1.19      
[22] parallel_3.4.3    Rmpfr_0.7-0       ECOSolveR_0.4    
[25] stringr_1.3.1     knitr_1.20        rprojroot_1.3-2  
[28] bit64_0.9-7       grid_3.4.3        qvalue_2.10.0    
[31] R6_2.2.2          rmarkdown_1.9     reshape2_1.4.3   
[34] ggplot2_2.2.1     magrittr_1.5      whisker_0.3-2    
[37] splines_3.4.3     scales_0.5.0      MASS_7.3-50      
[40] backports_1.1.2   codetools_0.2-15  htmltools_0.3.6  
[43] scs_1.1-1         assertthat_0.2.0  colorspace_1.3-2 
[46] stringi_1.2.2     lazyeval_0.2.1    munsell_0.4.3    
[49] pscl_1.5.2        doParallel_1.0.11 truncnorm_1.0-8  
[52] R.oo_1.22.0