In const-ae/glmGamPoi: Fit a Gamma-Poisson Generalized Linear Model

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Fit Gamma-Poisson Generalized Linear Models Reliably.

The core design aims of gmlGamPoi are:

Fit the Gamma-Poisson models on arbitrarily large or small datasets
Be faster than alternative methods, such as DESeq2 or edgeR
Calculate exact or approximate results based on user preference
Support in memory or on-disk data
Follow established conventions around tools for RNA-seq analysis
Present a simple user-interface
Avoid unnecessary dependencies
Make integration into other tools easy

Installation

You can install the release version of r BiocStyle::Biocpkg("glmGamPoi") from BioConductor:

if (!requireNamespace("BiocManager", quietly = TRUE))
    install.packages("BiocManager")

BiocManager::install("glmGamPoi")

For the latest developments, see the r BiocStyle::Githubpkg("const-ae/glmGamPoi", "GitHub") repo.

Example

Load the glmGamPoi package

library(glmGamPoi)

To fit a single Gamma-Poisson GLM do:

# overdispersion = 1/size
counts <- rnbinom(n = 10, mu = 5, size = 1/0.7)

# design = ~ 1 means that an intercept-only model is fit
fit <- glm_gp(counts, design = ~ 1)
fit

# Internally fit is just a list:
as.list(fit)[1:2]

The glm_gp() function returns a list with the results of the fit. Most importantly, it contains the estimates for the coefficients β and the overdispersion.

Fitting repeated Gamma-Poisson GLMs for each gene of a single cell dataset is just as easy:

I will first load an example dataset using the TENxPBMCData package. The dataset has 33,000 genes and 4340 cells. It takes roughly 1.5 minutes to fit the Gamma-Poisson model on the full dataset. For demonstration purposes, I will subset the dataset to 300 genes, but keep the 4340 cells:

library(SummarizedExperiment)
library(DelayedMatrixStats)

# The full dataset with 33,000 genes and 4340 cells
# The first time this is run, it will download the data
pbmcs <- TENxPBMCData::TENxPBMCData("pbmc4k")

# I want genes where at least some counts are non-zero
non_empty_rows <- which(rowSums2(assay(pbmcs)) > 0)
pbmcs_subset <- pbmcs[sample(non_empty_rows, 300), ]
pbmcs_subset

I call glm_gp() to fit one GLM model for each gene and force the calculation to happen in memory.

fit <- glm_gp(pbmcs_subset, on_disk = FALSE)
summary(fit)

Benchmark

I compare my method (in-memory and on-disk) with r BiocStyle::Biocpkg("DESeq2") and r BiocStyle::Biocpkg("edgeR"). Both are classical methods for analyzing RNA-Seq datasets and have been around for almost 10 years. Note that both tools can do a lot more than just fitting the Gamma-Poisson model, so this benchmark only serves to give a general impression of the performance.

# Explicitly realize count matrix in memory so that it is a fair comparison
pbmcs_subset <- as.matrix(assay(pbmcs_subset))
model_matrix <- matrix(1, nrow = ncol(pbmcs_subset))


bench::mark(
  glmGamPoi_in_memory = {
    glm_gp(pbmcs_subset, design = model_matrix, on_disk = FALSE)
  }, glmGamPoi_on_disk = {
    glm_gp(pbmcs_subset, design = model_matrix, on_disk = TRUE)
  }, DESeq2 = suppressMessages({
    dds <- DESeq2::DESeqDataSetFromMatrix(pbmcs_subset,
                        colData = data.frame(name = seq_len(4340)),
                        design = ~ 1)
    dds <- DESeq2::estimateSizeFactors(dds, "poscounts")
    dds <- DESeq2::estimateDispersions(dds, quiet = TRUE)
    dds <- DESeq2::nbinomWaldTest(dds, minmu = 1e-6)
  }), edgeR = {
    edgeR_data <- edgeR::DGEList(pbmcs_subset)
    edgeR_data <- edgeR::calcNormFactors(edgeR_data)
    edgeR_data <- edgeR::estimateDisp(edgeR_data, model_matrix)
    edgeR_fit <- edgeR::glmFit(edgeR_data, design = model_matrix)
  }, check = FALSE, min_iterations = 3
)

On this dataset, glmGamPoi is more than 5 times faster than edgeR and more than 18 times faster than DESeq2. glmGamPoi does not use approximations to achieve this performance increase. The performance comes from an optimized algorithm for inferring the overdispersion for each gene. It is tuned for datasets typically encountered in single RNA-seq with many samples and many small counts, by avoiding duplicate calculations.

To demonstrate that the method does not sacrifice accuracy, I compare the parameters that each method estimates. The means and β coefficients are identical, but that the overdispersion estimates from glmGamPoi are more reliable:

# Results with my method
fit <- glm_gp(pbmcs_subset, design = model_matrix, on_disk = FALSE)

# DESeq2
dds <- DESeq2::DESeqDataSetFromMatrix(pbmcs_subset, 
                        colData = data.frame(name = seq_len(4340)),
                        design = ~ 1)
sizeFactors(dds)  <- fit$size_factors
dds <- DESeq2::estimateDispersions(dds, quiet = TRUE)
dds <- DESeq2::nbinomWaldTest(dds, minmu = 1e-6)

#edgeR
edgeR_data <- edgeR::DGEList(pbmcs_subset, lib.size = fit$size_factors)
edgeR_data <- edgeR::estimateDisp(edgeR_data, model_matrix)
edgeR_fit <- edgeR::glmFit(edgeR_data, design = model_matrix)

par(mfrow = c(2, 4), cex.main = 2, cex.lab = 1.5)
plot(fit$Beta[,1], coef(dds)[,1] / log2(exp(1)), pch = 16, 
     main = "Beta Coefficients", xlab = "glmGamPoi", ylab = "DESeq2")
abline(0,1)
plot(fit$Beta[,1], edgeR_fit$unshrunk.coefficients[,1], pch = 16,
     main = "Beta Coefficients", xlab = "glmGamPoi", ylab = "edgeR")
abline(0,1)

plot(fit$Mu[,1], assay(dds, "mu")[,1], pch = 16, log="xy",
     main = "Gene Mean", xlab = "glmGamPoi", ylab = "DESeq2")
abline(0,1)
plot(fit$Mu[,1], edgeR_fit$fitted.values[,1], pch = 16, log="xy",
     main = "Gene Mean", xlab = "glmGamPoi", ylab = "edgeR")
abline(0,1)

plot(fit$overdispersions, rowData(dds)$dispGeneEst, pch = 16, log="xy",
     main = "Overdispersion", xlab = "glmGamPoi", ylab = "DESeq2")
abline(0,1)
plot(fit$overdispersions, edgeR_fit$dispersion, pch = 16, log="xy",
     main = "Overdispersion", xlab = "glmGamPoi", ylab = "edgeR")
abline(0,1)

I am comparing the gene-wise estimates of the coefficients from all three methods. Points on the diagonal line are identical. The inferred Beta coefficients and gene means agree well between the methods, however the overdispersion differs quite a bit. DESeq2 has problems estimating most of the overdispersions and sets them to 1e-8. edgeR only approximates the overdispersions which explains the variation around the overdispersions calculated with glmGamPoi.

Scalability

The method scales linearly, with the number of rows and columns in the dataset. For example: fitting the full pbmc4k dataset with subsampling on a modern MacBook Pro in-memory takes ~1 minute and on-disk a little over 4 minutes. Fitting the pbmc68k (17x the size) takes ~73 minutes (17x the time) on-disk.

Differential expression analysis

glmGamPoi provides an interface to do quasi-likelihood ratio testing to identify differentially expressed genes. To demonstrate this feature, we will use the data from Kang et al. (2018) provided by the MuscData package. This is a single cell dataset of 8 Lupus patients for which 10x droplet-based scRNA-seq was performed before and after treatment with interferon beta. The SingleCellExperiment object conveniently provides the patient id (ind), treatment status (stim) and cell type (cell):

sce <- muscData::Kang18_8vs8()
colData(sce)

For demonstration purpose, I will work on a subset of the genes and cells:

set.seed(1)
# Take highly expressed genes and proper cells:
sce_subset <- sce[rowSums(counts(sce)) > 100, 
                  sample(which(sce$multiplets == "singlet" & 
                              ! is.na(sce$cell) &
                              sce$cell %in% c("CD4 T cells", "B cells", "NK cells")), 
                         1000)]
# Convert counts to dense matrix
counts(sce_subset) <- as.matrix(counts(sce_subset))
# Remove empty levels because glm_gp() will complain otherwise
sce_subset$cell <- droplevels(sce_subset$cell)

We will identify which genes in CD4 positive T-cells are changed most by the treatment. We will fit a full model including the interaction term stim:cell. The interaction term will help us identify cell type specific responses to the treatment:

fit <- glm_gp(sce_subset, design = ~ cell + stim +  stim:cell - 1,
              reference_level = "NK cells")
summary(fit)

To see how the coefficient of our model are called, we look at the colnames(fit$Beta):

colnames(fit$Beta)

In our example, we want to find the genes that change specifically in T cells. Finding cell type specific responses to a treatment is a big advantage of single cell data over bulk data. To get a proper estimate of the uncertainty (cells from the same donor are not independent replicates), we create a pseudobulk for each sample:

# The contrast argument specifies what we want to compare
# We test the expression difference of stimulated and control T-cells
#
# There is no sample label in the colData, so we create it on the fly
# from `stim` and `ind` columns in colData(fit$data).
de_res <- test_de(fit, contrast = `stimstim` + `cellCD4 T cells:stimstim`, 
                  pseudobulk_by = paste0(stim, "-", ind)) 

# The large `lfc` values come from groups were nearly all counts are 0
# Setting them to Inf makes the plots look nicer
de_res$lfc <- ifelse(abs(de_res$lfc) > 20, sign(de_res$lfc) * Inf, de_res$lfc)

# Most different genes
head(de_res[order(de_res$pval), ])

The test is successful and we identify interesting genes that are differentially expressed in interferon-stimulated T cells: IFI6, IFIT3, and IRF7 literally stand for Interferon Induced/Regulated Protein.

To get a more complete overview of the results, we can make a volcano plot that compares the log2-fold change (LFC) vs the logarithmized p-values.

library(ggplot2)
ggplot(de_res, aes(x = lfc, y = -log10(pval))) +
  geom_point(size = 0.6, aes(color = adj_pval < 0.1)) +
  ggtitle("Volcano Plot", "Genes that change most through interferon-beta treatment in T cells")

Another important task in single cell data analysis is the identification of marker genes for cell clusters. For this we can also use our Gamma-Poisson fit.

Let's assume we want to find genes that differ between T cells and the B cells. We can directly compare the corresponding coefficients and find genes that differ in the control condition:

marker_genes <- test_de(fit, `cellCD4 T cells` - `cellB cells`, sort_by = pval)
head(marker_genes)

If we want find genes that differ in the stimulated condition, we just include the additional coefficients in the contrast:

marker_genes2 <- test_de(fit, (`cellCD4 T cells` + `cellCD4 T cells:stimstim`) - 
                               (`cellB cells` + `cellB cells:stimstim`), 
                        sort_by = pval)

head(marker_genes2)

We identify many genes related to the human leukocyte antigen (HLA) system that is important for antigen presenting cells like B-cells, but are not expressed by T helper cells. The plot below shows the expression differences.

A note of caution: applying test_de() to single cell data without the pseudobulk gives overly optimistic p-values. This is due to the fact that cells from the same sample are not independent replicates! It can still be fine to use the method for identifying marker genes, as long as one is aware of the difficulties interpreting the results.

# Create a data.frame with the expression values, gene names, and cell types
tmp <- data.frame(gene = rep(marker_genes$name[1:6], times = ncol(sce_subset)),
                  expression = c(counts(sce_subset)[marker_genes$name[1:6], ]),
                  celltype = rep(sce_subset$cell, each = 6))

ggplot(tmp, aes(x = celltype, y = expression)) +
  geom_jitter(height = 0.1) +
  stat_summary(geom = "crossbar", fun = "mean", color = "red") +
  facet_wrap(~ gene, scales = "free_y") +
  ggtitle("Marker genes of B vs. T cells")