DE analysis using a topic model: evaluation using simulated data

Last updated: 2021-10-15

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File	Version	Author	Date	Message
Rmd	afaa55d	Peter Carbonetto	2021-10-15	workflowr::wflow_publish(“de_analysis_detailed_look.Rmd”)
Rmd	13d2f7d	Peter Carbonetto	2021-10-14	Added pvalue/lfsr histograms for deseq2 and fastTopics.
Rmd	09089ff	Peter Carbonetto	2021-10-14	Revising de_analysis_detailed_look after fixing z-score calculations in de_analysis.
html	f61da64	Peter Carbonetto	2021-10-14	Fixed a bug in the power vs fdr plot in de_analysis_detailed_look.
Rmd	cb60f4c	Peter Carbonetto	2021-10-14	workflowr::wflow_publish(“de_analysis_detailed_look.Rmd”)
html	3d62d9e	Peter Carbonetto	2021-10-14	Revised plots and explanatory text in de_analysis_detailed_look.
Rmd	5d5c6a6	Peter Carbonetto	2021-10-14	workflowr::wflow_publish(“de_analysis_detailed_look.Rmd”)
html	468f4d9	Peter Carbonetto	2021-10-13	Added draft plots to de_analysis_detailed_look investigation.
Rmd	eceb899	Peter Carbonetto	2021-10-13	workflowr::wflow_publish(“de_analysis_detailed_look.Rmd”, verbose = TRUE)
html	7236187	Peter Carbonetto	2021-10-13	Revised the data simulation and added some “shrink vs. noshrink” plots
Rmd	6d63993	Peter Carbonetto	2021-10-13	Made another simplification to simulate_twotopic_umi_data and added plots to de_analysis_detailed_look analysis comparing results with and without shrinkage.
Rmd	157f7d3	Peter Carbonetto	2021-10-12	Working on improved script driving_genes_better.R.
html	af856a2	Peter Carbonetto	2021-10-12	Fixed the plots showing properties of the simulated data in the
Rmd	2726a86	Peter Carbonetto	2021-10-12	workflowr::wflow_publish(“de_analysis_detailed_look.Rmd”)
html	b8eadee	Peter Carbonetto	2021-10-11	Added some basic histograms to the de_analysis_detailed_look analysis.
Rmd	e2577f1	Peter Carbonetto	2021-10-11	Working on the data simulation step in the de_analysis_detailed_look
html	cd17ccf	Peter Carbonetto	2021-10-09	Initial build of the de_analysis_detailed_look workflowr page.
Rmd	b8e1fe8	Peter Carbonetto	2021-10-09	workflowr::wflow_publish(“de_analysis_detailed_look.Rmd”)
Rmd	888ea7d	Peter Carbonetto	2021-10-08	I have an initial implementation of function simulate_twotopic_scrnaseq_data used to evaluate the de_analysis methods in fastTopics.

Add summary of the analysis here.

Load the packages used in the analysis below, and some additional functions for simulating the data.

library(Matrix)
library(scran)
# Warning: package 'S4Vectors' was built under R version 3.6.3
# Warning: package 'GenomeInfoDb' was built under R version 3.6.3
# Warning: package 'DelayedArray' was built under R version 3.6.3
library(DESeq2)
library(fastTopics)
library(ggplot2)
library(cowplot)
source("../code/de_eval_functions.R")

Simulate UMI counts

We begin by simulating counts from a rank-2 Poisson NMF model with parameters chosen to roughly mimic the UMI counts from a single-cell RNA sequencing experiment. For now, we focus our attention on the rank-2 (i.e., two topics) case to simplify the evaluation of the DE methods; comparing gene expression between three or topics brings some additional complications which aren’t necessary for assessing basic properties of the new DE methods. For more details on how the data are simulated, see function simulate_twotopic_umi_data.

set.seed(1)
dat <- simulate_twotopic_umi_data()
X <- dat$X
F <- dat$F
L <- dat$L

Before fitting a topic model and running a DE analysis, we first inspect some basic properties of the simulated data.

The sample sizes (total counts for each cell) are roughly normal on the (base-10) log scale:

s <- rowSums(X)
p1 <- ggplot(data.frame(s = log10(s)),aes(x = s)) +
  geom_histogram(color = "white",fill = "black",bins = 32) +
  labs(x = "log10 size",y = "cells") +
  theme_cowplot()
print(p1)

Version	Author	Date
7236187	Peter Carbonetto	2021-10-13
af856a2	Peter Carbonetto	2021-10-12
b8eadee	Peter Carbonetto	2021-10-11

The expression rates were simulated so that they are normally distributed on the log scale:

p2 <- ggplot(data.frame(f = as.vector(F)),aes(x = log10(f))) +
  geom_histogram(color = "white",fill = "black",bins = 32) +
  labs(x = "log10 expression rate",y = "genes") +
  theme_cowplot()
print(p2)

Version	Author	Date
7236187	Peter Carbonetto	2021-10-13
af856a2	Peter Carbonetto	2021-10-12
b8eadee	Peter Carbonetto	2021-10-11

About half of the genes have nonzero differences in expression between the two topics. Among the nonzero gene expression differences, the log-fold changes (LFCs) were simulated from the normal distribution:

nonzero_lfc <- abs(F[,1] - F[,2]) > 1e-8
lfc <- log2(F[,2]/F[,1])
p3 <- ggplot(data.frame(lfc = lfc[nonzero_lfc]),aes(x = lfc)) +
  geom_histogram(color = "white",fill = "black",bins = 32) +
  labs(x = "LFC",y = "genes") +
  theme_cowplot()
print(p3)
mean(nonzero_lfc)
# [1] 0.496

Version	Author	Date
7236187	Peter Carbonetto	2021-10-13
af856a2	Peter Carbonetto	2021-10-12
b8eadee	Peter Carbonetto	2021-10-11

Fit multinomial topic model to UMI counts

Here we fit a multinomial topic model to the simulated UMI count data. To simplify evaluation of the DE methods, we assume that the topic proportions are known, and set them to their ground truth values (that is, the values used to simulate the data). In this way, the only error that can arise is in the estimates of the expression rates $f_{ij}$.

fit0 <- init_poisson_nmf(X,F = dat$F,L = with(dat,s*L))
fit <- fit_poisson_nmf(X,fit0 = fit0,numiter = 40,method = "scd",
                       update.loadings = NULL,verbose = "none")
fit <- poisson2multinom(fit)
summary(fit)
# Model overview:
#   Number of data rows, n: 200
#   Number of data cols, m: 10000
#   Rank/number of topics, k: 2
# Evaluation of model fit (40 updates performed):
#   Poisson NMF log-likelihood: -7.169037374429e+05
#   Multinomial topic model log-likelihood: -7.158926302076e+05
#   Poisson NMF deviance: +8.564731575439e+05
#   Max KKT residual: +4.123695e-06

Warning: The above code chunk cached its results, but it won’t be re-run if previous chunks it depends on are updated. If you need to use caching, it is highly recommended to also set knitr::opts_chunk$set(autodep = TRUE) at the top of the file (in a chunk that is not cached). Alternatively, you can customize the option dependson for each individual chunk that is cached. Using either autodep or dependson will remove this warning. See the knitr cache options for more details.

DE analysis with and without shrinkage

First we perform a DE analysis without shrinking the LFC estimates:

set.seed(1)
de.noshrink <- de_analysis(fit,X,shrink.method = "none",
                           control = list(ns = 1e4,nc = 2))
# Fitting 10000 Poisson models with k=2 using method="scd".
# Computing log-fold change statistics from 10000 Poisson models with k=2.

Now we perform a second DE analysis using adaptive shrinkage to shrink (and hopefully improve accuracy of) the LFC estimates:

set.seed(1)
de <- de_analysis(fit,X,shrink.method = "ash",control = list(ns = 1e4,nc = 2))
# Fitting 10000 Poisson models with k=2 using method="scd".
# Computing log-fold change statistics from 10000 Poisson models with k=2.
# Stabilizing posterior log-fold change estimates using adaptive shrinkage.

In this scatterplot, we compare the posterior mean LFC estimates with and without performing the shrinkage step. As expected, the smaller LFC estimates, or the LFC estimates corresponding to genes with lower expression, tend to be more strongly shrunk toward zero.

pdat <- data.frame(noshrink  = de.noshrink$postmean[,2],
                   shrink    = de$postmean[,2],
                   log10mean = log10(de$f0))
p4 <- ggplot(pdat,aes(x = noshrink,y = shrink,fill = log10mean)) +
  geom_point(shape = 21,color = "white",size = 2) +
  geom_abline(intercept = 0,slope = 1,color = "black",linetype = "dotted") +
  scale_fill_gradient2(low = "deepskyblue",mid = "gold",high = "orangered",
                       midpoint = -4) +
  labs(x = "original LFC estimates",
       y = "shrunken LFC estimates") +
  theme_cowplot()
print(p4)

Version	Author	Date
3d62d9e	Peter Carbonetto	2021-10-14
468f4d9	Peter Carbonetto	2021-10-13
7236187	Peter Carbonetto	2021-10-13

These next two bar charts show the overall impact of the shrinkage on the distribution of the $z$-scores; dark blue bars are for diifferentially expressed genes, and orange bars are for non-differentially expressed genes.

pdat <- data.frame(noshrink = clamp(de.noshrink$z[,2],-8,+8),
                   shrink   = clamp(de$z[,2],-8,+8),
                   de       = factor(nonzero_lfc))
p5 <- ggplot(pdat,aes(x = noshrink,color = de,fill = de)) +
  geom_histogram(bins = 64) +
  scale_color_manual(values = c("darkorange","darkblue")) +
  scale_fill_manual(values = c("darkorange","darkblue")) +
  labs(x = "z-score",y = "genes",title = "without shrinkage") +
  theme_cowplot()
p6 <- ggplot(pdat,aes(x = shrink,color = de,fill = de)) +
  geom_histogram(bins = 64) +
  scale_color_manual(values = c("darkorange","darkblue")) +
  scale_fill_manual(values = c("darkorange","darkblue")) +
  labs(x = "z-score",y = "genes",title = "with shrinkage") +
  theme_cowplot()
print(plot_grid(p5,p6))

Version	Author	Date
468f4d9	Peter Carbonetto	2021-10-13
7236187	Peter Carbonetto	2021-10-13

And the next two bar charts show the overall impact of the shrinkage on the $p$-values.

pdat <- data.frame(noshrink = 10^(-de.noshrink$lpval[,2]),
                   shrink   = 10^(-de$lpval[,2]),
                   de       = factor(nonzero_lfc))
p7 <- ggplot(pdat,aes(x = noshrink,color = de,fill = de)) +
  geom_histogram(bins = 64) +
  scale_color_manual(values = c("darkorange","darkblue")) +
  scale_fill_manual(values = c("darkorange","darkblue")) +
  labs(x = "p-value",y = "genes",title = "without shrinkage") +
  theme_cowplot()
p8 <- ggplot(pdat,aes(x = shrink,color = de,fill = de)) +
  geom_histogram(bins = 64) +
  scale_color_manual(values = c("darkorange","darkblue")) +
  scale_fill_manual(values = c("darkorange","darkblue")) +
  labs(x = "p-value",y = "genes",title = "with shrinkage") +
  theme_cowplot()
print(plot_grid(p7,p8))

Version	Author	Date
468f4d9	Peter Carbonetto	2021-10-13
7236187	Peter Carbonetto	2021-10-13

Quantifying uncertainty in the LFC estimates

Above, we assessed the usefulness of applying the adaptive shrinkage method to improve the accuracy of the LFC estimates. This idea of shrinking the LFC estimates is not new, and is already implemented in DESeq2, also using adaptive shrinkage.

A more important aspect of the DE analysis methods in fastTopics is that it quantifies uncertainty in the LFC estimates by computing posterior HPD intervals. To see why this is important, consider this plot compare the point estimates (specifically, MLEs) of the LFCs (X axis) against the shrunken posterior mean estimates (Y axis). What is striking is that even very large (in magnitude) point estimates can be shrunk to zero; this happens because although the MLEs can be far from zero, the HPD intervals are also very large.

pdat <- data.frame(est       = de$est[,2],
                   shrink    = de$postmean[,2],
                   log10mean = log10(de$f0))
p9 <- ggplot(pdat,aes(x = est,y = shrink,fill = log10mean)) +
  geom_point(shape = 21,color = "white",size = 2) +
  geom_abline(intercept = 0,slope = 1,color = "black",linetype = "dotted") +
  scale_fill_gradient2(low = "deepskyblue",mid = "gold",high = "orangered",
                       midpoint = -4) +
  labs(x = "point estimates",
       y = "shrunken estimates") +
  theme_cowplot()
print(p9)

Version	Author	Date
f61da64	Peter Carbonetto	2021-10-14
3d62d9e	Peter Carbonetto	2021-10-14

To illustrate the impact of this on the DE analysis, consider the K-L divergence measure used in Dey, Hsiao & Stephens (2017) to rank genes, which doesn’t fully account for uncertainty in the LFCs. As a result, sometimes genes with very large LFCs get highly ranked, but after a full accounting for uncertainty, they are no longer ranked highly.

i     <- which(fit$F[,2] - fit$F[,1] > -1e-8)
D     <- fastTopics:::min_kl_poisson(fit$F)
pdat1 <- data.frame(kl = pmin(D[,2],0.00025),de = factor(nonzero_lfc))
pdat2 <- data.frame(z = clamp(de$z[,2],-4,+4),de = factor(nonzero_lfc))
pdat1 <- pdat1[i,]
pdat2 <- pdat2[i,]
p10 <- ggplot(pdat1,aes(x = kl,color = de,fill = de)) +
  geom_histogram(bins = 64) +
  scale_x_continuous(breaks = c(0,0.0001,0.0002)) +
  scale_color_manual(values = c("darkorange","darkblue")) +
  scale_fill_manual(values = c("darkorange","darkblue")) +
  labs(x = "K-L divergence",y = "genes") +
  theme_cowplot()
p11 <- ggplot(pdat2,aes(x = z,color = de,fill = de)) +
  geom_histogram(bins = 64) +
  scale_color_manual(values = c("darkorange","darkblue")) +
  scale_fill_manual(values = c("darkorange","darkblue")) +
  labs(x = "z-score",y = "genes") +
  theme_cowplot()
print(plot_grid(p10,p11))

Version	Author	Date
3d62d9e	Peter Carbonetto	2021-10-14
468f4d9	Peter Carbonetto	2021-10-13

To drive home this point, suppose we used the KL-divergences and the p-values returned by de_analysis(shrink.method = "ash") to discover differentially expressed genes. The power vs. FDR curves show that the rankings accounting for uncertainty in the LFC estimates (with or without shrinkage) perform better than the ranking based on the KL divergence measure.

pdat1 <- create_fdr_vs_power_curve(-D[,2],nonzero_lfc)
pdat2 <- create_fdr_vs_power_curve(-de.noshrink$lpval[,2],nonzero_lfc)
pdat3 <- create_fdr_vs_power_curve(-de$lpval[,2],nonzero_lfc)
pdat  <- rbind(cbind(pdat1,method = "kl"),
               cbind(pdat2,method = "noshrink"),
               cbind(pdat3,method = "shrink"))
p12 <- ggplot(pdat,aes(x = fdr,y = power,color = method)) +
  geom_point(size = 0.75)  +
  scale_color_manual(values = c("royalblue","limegreen","darkorange")) +
  theme_cowplot()
print(p12)

Version	Author	Date
f61da64	Peter Carbonetto	2021-10-14
468f4d9	Peter Carbonetto	2021-10-13

Comparison with DESeq2

Add text and code here.

i       <- which(apply(L,1,max) > 0.99)
Y       <- t(X[i,])
cluster <- factor(apply(L,1,which.max))
deseq   <- DESeqDataSetFromMatrix(Y,data.frame(cluster = cluster[i]),~cluster)
# converting counts to integer mode
sizeFactors(deseq) <- calculateSumFactors(Y)
deseq <- DESeq(deseq,test = "LRT",reduced=~1,useT = TRUE,minmu = 1e-6,
               minReplicatesForReplace = Inf)
# using pre-existing size factors
# estimating dispersions
# gene-wise dispersion estimates
# mean-dispersion relationship
# -- note: fitType='parametric', but the dispersion trend was not well captured by the
#    function: y = a/x + b, and a local regression fit was automatically substituted.
#    specify fitType='local' or 'mean' to avoid this message next time.
# final dispersion estimates
# fitting model and testing
deseq <- lfcShrink(deseq,coef = "cluster_2_vs_1",type = "ashr")
# using 'ashr' for LFC shrinkage. If used in published research, please cite:
#     Stephens, M. (2016) False discovery rates: a new deal. Biostatistics, 18:2.
#     https://doi.org/10.1093/biostatistics/kxw041
z.deseq <- with(deseq,log2FoldChange/lfcSE)

Add text here.

plot(deseq$log2FoldChange,de$postmean[,2],pch = 20,col = "royalblue")
abline(a = 0,b = 1,col = "black",lty = "dotted")

plot(z.deseq,de$z[,2],pch = 20,col = "royalblue")
abline(a = 0,b = 1,col = "black",lty = "dotted")