Gaussian derivatives applied to Smemo’s data

Introduction

Re-analyze Smemo et al 2014’s mouse heart RNA-seq data after discussion with Matthew.

counts.mat = read.table("../data/smemo.txt", header = T, row.name = 1)
counts.mat = counts.mat[, -5]

Gene selection

Only use genes with total counts of \(4\) samples \(\geq 5\).

counts = counts.mat[rowSums(counts.mat) >= 5, ]
design = model.matrix(~c(0, 0, 1, 1))

Number of selected genes: 17191

Summary statistics

source("../code/count_to_summary.R")
summary <- count_to_summary(counts, design)
betahat <- summary$betahat
sebetahat <- summary$sebetahat
z <- summary$z

Fitting \(z\) with Gaussian derivatives

With stretch GD can fit \(z\) scores, but it seems there should be signals.

GD Coefficients:

0 : 1 ; 1 : 0.0119430018126296 ; 2 : 1.61071078428823 ; 3 : 0.366170906281441 ; 4 : 1.70110410088524 ; 5 : 0.676196157715947 ; 6 : 0.938754567209366 ; 7 : 0.550191966323002 ; 8 : 0.238942600379678 ; 9 : 0.161306266269737 ; 10 : 0.0430996146907588 ;

Discovered by BH and ASH

Feeding summary statistics to BH and ASH, both give thousands of discoveries.

fit.BH = p.adjust((1 - pnorm(abs(z))) * 2, method = "BH")
## Number of discoveries by BH
sum(fit.BH <= 0.05)

[1] 2541

fit.ash = ashr::ash(betahat, sebetahat, method = "fdr")
## Number of discoveries by ASH
sum(get_svalue(fit.ash) <= 0.05)

[1] 6440

Fitting ASH first or Gaussian derivatives first

Using default setting \(L = 10\), \(\lambda = 10\), \(\rho = 0.5\), compare the GD-ASH results by fitting ASH first vs fitting GD first. They indeed arrive at different local minima.

fit.gdash.ASH <- gdash(betahat, sebetahat,
                       gd.priority = FALSE)
## Regularized log-likelihood by fitting ASH first
fit.gdash.ASH$loglik

[1] -12483.86

fit.gdash.GD <- gdash(betahat, sebetahat)
## Regularized log-likelihood by fitting GD first
fit.gdash.GD$loglik

[1] -22136.92

GD-ASH with larger penalties on \(w\)

Using \(\lambda = 50\), \(\rho = 0.1\), fitting ASH first and GD first give the same result, and produce 1400+ discoveries with \(q\) values \(\leq 0.05\), all of which are discovered by BH.

L = 10
lambda = 50
rho = 0.1
fit.gdash.ASH <- gdash(betahat, sebetahat,
                       gd.ord = L, w.lambda = lambda, w.rho = rho,
                       gd.priority = FALSE)
## Regularized log-likelihood by fitting ASH first
fit.gdash.ASH$loglik

[1] -13651.59

## Number of discoveries
sum(fit.gdash.ASH$qvalue <= 0.05)

[1] 1431

fit.gdash.GD <- gdash(betahat, sebetahat,
                      gd.ord = L, w.lambda = lambda, w.rho = rho,
                      gd.priority = TRUE)
## Regularized log-likelihood by fitting GD first
fit.gdash.GD$loglik

[1] -13651.59

## Number of discoveries
sum(fit.gdash.GD$qvalue <= 0.05)

[1] 1431

GD Coefficients:

0 : 1 ; 1 : -0.0475544308510135 ; 2 : 0.707888470469342 ; 3 : 0.149489828947119 ; 4 : -8.97499076623316e-14 ; 5 : 0.109281416075664 ; 6 : -3.00530934822662e-13 ; 7 : 0.0783545592042359 ; 8 : -2.99572304462426e-13 ; 9 : 0.0911488252640105 ; 10 : -2.99578347875936e-13 ;

Fitting CASH

source("../code/gdash_lik.R")
source("../code/gdfit.R")
library(edgeR)

Loading required package: limma

library(limma)
library(locfdr)

counts.mat = read.table("../data/smemo.txt", header = T, row.name = 1)
counts.mat = counts.mat[, -5]
counts = counts.mat[rowSums(counts.mat) >= 5, ]
design = model.matrix(~c(0, 0, 1, 1))
dgecounts = edgeR::calcNormFactors(edgeR::DGEList(counts = counts, group = design[, 2]))
v = limma::voom(dgecounts, design, plot = FALSE)
lim = limma::lmFit(v)
r.ebayes = limma::eBayes(lim)
p = r.ebayes$p.value[, 2]
t = r.ebayes$t[, 2]
z = -sign(t) * qnorm(p/2)

fit.locfdr <- locfdr(z)

fit.qvalue <- qvalue::qvalue(p)

betahat = lim$coefficients[, 2]
sebetahat = betahat / z
fit.cash <- gdash(betahat, sebetahat, gd.ord = 10)

x.plot <- seq(-10, 10, length = 1000)
gd.ord <- 10
hermite = Hermite(gd.ord)
gd0.std = dnorm(x.plot)
matrix_lik_plot = cbind(gd0.std)
for (i in 1 : gd.ord) {
  gd.std = (-1)^i * hermite[[i]](x.plot) * gd0.std / sqrt(factorial(i))
  matrix_lik_plot = cbind(matrix_lik_plot, gd.std)
}
y.plot = matrix_lik_plot %*% fit.cash$w * fit.cash$fitted_g$pi[1]

method.col <- scales::hue_pal()(5)
# method.col <- c("#377eb8", "#984ea3", "#4daf4a", "#ff7f00", "#e41a1c")

setEPS()
postscript("../output/fig/mouseheart.eps", height = 5, width = 12)

par(mfrow = c(1, 2))
hist(z, prob = TRUE, main = "", xlab = expression(paste(z, "-scores")), cex.lab = 1.25)
lines(x.plot, y.plot, col = method.col[5], lwd = 2)
lines(x.plot, dnorm(x.plot), col = 
       "orange"
      #  method.col[2]
      , lty = 2, lwd = 2)
lines(x.plot, dnorm(x.plot, fit.locfdr$fp0[3, 1], fit.locfdr$fp0[3, 2]) * fit.locfdr$fp0[3, 3], col = method.col[3], lty = 2, lwd = 2)
legend("topleft", col = c("orange", method.col[3], method.col[5]), lty = c(2, 2, 1), legend = c("N(0, 1)", "Empirical null", expression(pi[0]~hat(f))), bty = "n", cex = 1.25)

par(mar = par("mar") + c(0, 1, 0, 0))
g1 <- fit.cash$fitted_g
g1.plot.x <- seq(-0.5, 0.5, length = 1000)
g1.plot.y <- rowSums(sapply(2 : length(g1$pi), function (i) {g1$pi[i] * dnorm(g1.plot.x, g1$mean[i], g1$sd[i])}))
plot(g1.plot.x, g1.plot.y, xlim = c(-0.35, 0.35), type = "l", xlab = expression(paste(theta, " (", log[2], " fold change)")), ylab = expression(hat(g)[1](theta)), cex.lab = 1.25)

dev.off()

quartz_off_screen 
                2 

Session information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

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] locfdr_1.1-8      edgeR_3.20.9      limma_3.34.9     
 [4] ashr_2.2-7        Rmosek_8.0.69     PolynomF_1.0-2   
 [7] CVXR_0.95         REBayes_1.3       Matrix_1.2-14    
[10] SQUAREM_2017.10-1 EQL_1.0-0         ttutils_1.0-1    

loaded via a namespace (and not attached):
 [1] qvalue_2.10.0     locfit_1.5-9.1    reshape2_1.4.3   
 [4] splines_3.4.3     lattice_0.20-35   colorspace_1.3-2 
 [7] htmltools_0.3.6   yaml_2.1.19       gmp_0.5-13.1     
[10] rlang_0.2.0       R.oo_1.22.0       pillar_1.2.2     
[13] Rmpfr_0.7-0       R.utils_2.6.0     bit64_0.9-7      
[16] scs_1.1-1         foreach_1.4.4     plyr_1.8.4       
[19] stringr_1.3.1     munsell_0.4.3     gtable_0.2.0     
[22] workflowr_1.1.1   R.methodsS3_1.7.1 codetools_0.2-15 
[25] evaluate_0.10.1   knitr_1.20        doParallel_1.0.11
[28] pscl_1.5.2        parallel_3.4.3    Rcpp_0.12.16     
[31] backports_1.1.2   scales_0.5.0      truncnorm_1.0-8  
[34] bit_1.1-13        ggplot2_2.2.1     digest_0.6.15    
[37] stringi_1.2.2     grid_3.4.3        rprojroot_1.3-2  
[40] ECOSolveR_0.4     tools_3.4.3       magrittr_1.5     
[43] lazyeval_0.2.1    tibble_1.4.2      whisker_0.3-2    
[46] MASS_7.3-50       assertthat_0.2.0  rmarkdown_1.9    
[49] iterators_1.0.9   R6_2.2.2          git2r_0.21.0     
[52] compiler_3.4.3