2.1 Overview

HVGs are defined as genes with biological components that are significantly greater than zero. These genes are interesting as they drive differences in the expression profiles between cells, and should be prioritized for further investigation. Formal detection of HVGs allows us to avoid genes that are highly variable due to technical factors such as sampling noise during RNA capture and library preparation. This adds another level of statistical rigour to our previous analyses, in which we only modelled the technical component.

2.2 Setting up the data

library(BiocFileCache)
bfc <- BiocFileCache("raw_data", ask=FALSE)
wilson.fname <- bfcrpath(bfc, file.path("ftp://ftp.ncbi.nlm.nih.gov/geo/series",
    "GSE61nnn/GSE61533/suppl/GSE61533_HTSEQ_count_results.xls.gz"))

library(R.utils)
wilson.name2 <- "GSE61533_HTSEQ_count_results.xls"
gunzip(wilson.fname, destname=wilson.name2, remove=FALSE, overwrite=TRUE)

library(readxl)
all.counts <- read_excel(wilson.name2)
gene.names <- all.counts$ID
all.counts <- as.matrix(all.counts[,-1])
rownames(all.counts) <- gene.names

library(SingleCellExperiment)
sce.hsc <- SingleCellExperiment(list(counts=all.counts))
dim(sce.hsc)

## [1] 38498    96

is.spike <- grepl("^ERCC", rownames(sce.hsc))
isSpike(sce.hsc, "ERCC") <- is.spike
summary(is.spike)

##    Mode   FALSE    TRUE 
## logical   38406      92

library(scater)
sce.hsc <- calculateQCMetrics(sce.hsc)
libsize.drop <- isOutlier(sce.hsc$total_counts, nmads=3, type="lower", log=TRUE)
feature.drop <- isOutlier(sce.hsc$total_features_by_counts, nmads=3, type="lower", log=TRUE)
spike.drop <- isOutlier(sce.hsc$pct_counts_ERCC, nmads=3, type="higher")
sce.hsc <- sce.hsc[,!(libsize.drop | feature.drop | spike.drop)]
data.frame(ByLibSize=sum(libsize.drop), ByFeature=sum(feature.drop),
    BySpike=sum(spike.drop), Remaining=ncol(sce.hsc))

##   ByLibSize ByFeature BySpike Remaining
## 1         2         2       3        92

to.keep <- nexprs(sce.hsc, byrow=TRUE) > 0
sce.hsc <- sce.hsc[to.keep,]
summary(to.keep)

##    Mode   FALSE    TRUE 
## logical   17229   21269

library(scran)
sce.hsc <- computeSumFactors(sce.hsc)
summary(sizeFactors(sce.hsc))

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.4075  0.8058  0.9604  1.0000  1.1503  2.0414

sce.hsc <- computeSpikeFactors(sce.hsc, type="ERCC", general.use=FALSE)
summary(sizeFactors(sce.hsc, "ERCC"))

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##  0.2562  0.6198  0.8623  1.0000  1.2122  3.0289

sce.hsc <- normalize(sce.hsc)

2.3 Testing for significantly positive biological components

We fit a mean-variance trend to the spike-in transcripts to quantify the technical component of the variance, as previously described. The biological component for each gene is defined as the difference between its total variance and the fitted value of the trend (Figure 1).

var.fit <- trendVar(sce.hsc, parametric=TRUE, loess.args=list(span=0.3))
var.out <- decomposeVar(sce.hsc, var.fit)
plot(var.out$mean, var.out$total, pch=16, cex=0.6, xlab="Mean log-expression", 
    ylab="Variance of log-expression")
curve(var.fit$trend(x), col="dodgerblue", lwd=2, add=TRUE)
cur.spike <- isSpike(sce.hsc)
points(var.out$mean[cur.spike], var.out$total[cur.spike], col="red", pch=16)

Figure 1: Variance of normalized log-expression values for each gene in the HSC dataset, plotted against the mean log-expression. The blue line represents the mean-dependent trend fitted to the variances of the spike-in transcripts (red).

We define HVGs as those genes that have a biological component that is significantly greater than zero. We use a false discovery rate (FDR) of 5% after correcting for multiple testing with the Benjamini-Hochberg method.

hvg.out <- var.out[which(var.out$FDR <= 0.05),]
nrow(hvg.out)

## [1] 511

We rank the results to focus on genes with larger biological components. This highlights an interesting aspect of the underlying hypothesis test, which is based on the ratio of the total variance to the expected technical variance. Ranking based on p-value tends to prioritize HVGs that are more likely to be true positives but, at the same time, less likely to be interesting. This is because the ratio can be very large for HVGs that have very low total variance and do not contribute much to the cell-cell heterogeneity.

hvg.out <- hvg.out[order(hvg.out$bio, decreasing=TRUE),] 
write.table(file="hsc_hvg.tsv", hvg.out, sep="\t", quote=FALSE, col.names=NA)
head(hvg.out)

## DataFrame with 6 rows and 6 columns
##                      mean            total              bio             tech
##                 <numeric>        <numeric>        <numeric>        <numeric>
## Fos      6.46229730082989 19.5774140760543 12.8221002879735 6.75531378808077
## Dusp1    6.82314467206793 15.6360357795109 10.1162290203638 5.51980675914714
## Rgs1     5.31345464503953 20.3107030102909 10.0119450815048  10.298757928786
## Ppp1r15a 6.66579727019324 14.5266651189811 8.47596930729373 6.05069581168739
## Ly6a     8.40354443058819   10.05833414214 8.05800100918705 2.00033313295293
## Egr1     6.71592505528811  13.857027821927 7.97752724423428 5.87950057769273
##                       p.value                  FDR
##                     <numeric>            <numeric>
## Fos      1.01066135594786e-18 7.14571267367002e-16
## Dusp1    7.24908882891304e-18 4.15568711216418e-15
## Rgs1      9.4342535672312e-08 1.10557984759415e-05
## Ppp1r15a 1.73432258509514e-12 4.90489551366039e-10
## Ly6a     2.99530600782523e-50  9.0762051045687e-47
## Egr1     5.70569022066076e-12 1.49411599099303e-09

We check the distribution of expression values for the genes with the largest biological components. We see that the variance estimate is not driven by one or two outlier cells (Figure 2).

fontsize <- theme(axis.text=element_text(size=12), axis.title=element_text(size=16))
plotExpression(sce.hsc, features=rownames(hvg.out)[1:10]) + fontsize

Figure 2: Violin plots of normalized log-expression values for the top 10 genes with the largest biological components in the HSC dataset. Each point represents the log-expression value in a single cell.

2.4 Further comments

There are many other strategies for defining HVGs, based on a variety of metrics:

• the coefficient of variation, using the technicalCV2() function (Brennecke et al. 2013) or the DM() function (Kim et al. 2015) in scran.
• the dispersion parameter in the negative binomial distribution, using the estimateDisp() function in edgeR (McCarthy, Chen, and Smyth 2012).
• a proportion of total variability, using methods in the BASiCS package (Vallejos, Marioni, and Richardson 2015).

Here, we used the variance of the log-expression values because the log-transformation protects against genes with strong expression in only one or two cells. This reduces the risk that the set of top HVGs is not dominated by genes with (mostly uninteresting) outlier expression patterns.

We also save the HSC dataset to file for later use, using saveRDS() as previously described.

saveRDS(sce.hsc, file="hsc_data.rds")

3.1 Details of trend fitting parameters

Fitting of the mean-variance trend with trendVar() is complicated by the small number of spike-in transcripts and the uneven distribution of their abundances. For low numbers of cells, these issues are exacerbated by the low precision of the variance estimates. Thus, some tuning of trend parameters such as span may be required to achieve a suitable fit - see ?trendVar for more details. Setting parametric=TRUE is especially useful for modelling the expected wave-like shape of the mean-variance relationship. (This is not the default setting as it is not robust for arbitrary trend shapes.)

The trendVar function will also automatically filter out low-abundance genes prior to trend fitting. This ensures that low-abundance genes do not interfere with the fit due to discreteness, which biases the estimate of variability of the variances around the trend; or due to the frequency of low-abundance genes, which reduces the sensitivity of span-based smoothing algorithms at higher abundances. The internal choice of filtering strategy involves a number of considerations:

• Filtering uses the average of log-expression values rather than the (library size-adjusted) average count. The mean log-expression is independent of the variance estimate in a linear modelling framework (Bourgon, Gentleman, and Huber 2010), which ensures that the filter does not introduce spurious trends in the variances at the filter boundary.
• The filter threshold is specified with the min.mean argument in trendVar. We use the default threshold of 0.1 (min.mean) based on the appearance of discrete patterns in the variance estimates for simulated Poisson-distributed counts. Lower thresholds of 0.001-0.01 may be more suitable for very sparse data, e.g., from droplet-based protocols.
• The filter used in trendVar is not applied in decomposeVar by default. Retention of all genes ensures that weak biological signal from rare subpopulations is not discarded. To apply the filter in decomposeVar, users should set subset.row=rowMeans(logcounts(sce)) > 0.1 in the function call.

Comments from Aaron:

• On occasion, users may observe a warning from trendVar() about the lack of centering in the size factors. Recall that the trend is fitted to the mean and variances of the spike-in transcripts, and the technical component for each endogenous gene is estimated by interpolation. This assumes that an endogenous gene is comparable to a spike-in transcript of the same abundance. In particular, we assume that variation is primarily driven by the magnitude of the counts, based on the well-known mean-variance relationships in count models. Thus, we need to ensure that similarities in the average counts are preserved in the normalized expression values. This is achieved by centering the gene- and spike-in-based size factors in normalize(), such that features with similar average counts will also have similar normalized abundances. However, if the SingleCellExperiment object was manipulated (e.g., subsetted) after normalize() and before trendVar(), centering may not be preserved - hence the warning.

3.2 When spike-ins are unavailable

In some datasets, spike-in RNA may not have been added in appropriate quantities (or indeed at all). It may also be inappropriate to assume Poisson technical noise with makeTechTrend(), especially for read count data where amplification noise is non-negligible. In such cases, an alternative approach is to fit the trend to the variance estimates of the endogenous genes. This is done using the use.spikes=FALSE setting in trendVar, as shown below for the HSC dataset.

var.fit.nospike <- trendVar(sce.hsc, parametric=TRUE, 
    use.spikes=FALSE, loess.args=list(span=0.2))
var.out.nospike <- decomposeVar(sce.hsc, var.fit.nospike)

The simplest interpretation of the results assumes that the majority of genes are not variably expressed. This means that the technical component dominates the total variance for most genes, such that the fitted trend can be treated as an estimate of the technical component. In Figure 3, the trend passes through or close to most of the spike-in variances, indicating that our assumption is valid.

plot(var.out.nospike$mean, var.out.nospike$total, pch=16, cex=0.6, 
    xlab="Mean log-expression", ylab="Variance of log-expression")
curve(var.fit.nospike$trend(x), col="dodgerblue", lwd=2, add=TRUE)
points(var.out.nospike$mean[cur.spike], var.out.nospike$total[cur.spike], col="red", pch=16)

Figure 3: Variance of normalized log-expression values for each gene in the 416B dataset, plotted against the mean log-expression. The blue line represents the mean-dependent trend fitted to the variances of the endogenous genes (black), with spike-in transcripts shown in red.

If our assumption does not hold, the output of decomposeVar is more difficult to interpret. The fitted value of the trend can no longer be generally interpreted as the technical component, as it contains some biological variation as well. Instead, recall that the biological component reported by decomposeVar represents the residual for each gene over the majority of genes with the same abundance. One could assume that the variabilities of most genes are driven by constitutive “house-keeping” processes, which are biological in origin but generally uninteresting. Any gene with an increase in its variance is relatively highly variable and can be prioritized for further study.

4.1 Using the block= argument

Our previous analysis of the 416B dataset specified block= in trendVar() to ensure that systematic differences between plates do not inflate the variance. This involves estimating the mean and variance of the log-expression separately in each plate, followed by fitting a single trend to the plate-specific means and variances of all spike-in transcripts. In doing so, we implicitly assume that the trend is the same between plates, which is reasonable for this dataset (Figure 4).

# Loading the saved object.
sce.416B <- readRDS("416B_data.rds") 

# Repeating the trendVar() call.
var.fit <- trendVar(sce.416B, parametric=TRUE, block=sce.416B$Plate,
    loess.args=list(span=0.3))

matplot(var.fit$means, var.fit$vars, col=c("darkorange", "forestgreen"),
    xlab="Mean log-expression", ylab="Variance of log-expression")
curve(var.fit$trend(x), add=TRUE, col="red")

Figure 4: Plate-specific variance estimates for all spike-in transcripts in the 416B dataset, plotted against the plate-specific means. Each point represents a spike-in transcript, numbered by the plate from which the values were estimated. The red line denotes the fitted mean-variance trend.

The use of block= also assumes that the average size factor within each plate is close to unity for both endogenous genes and spike-in transcripts. This means that scaling normalization will preserve the magnitude of the counts, allowing genes of the same average abundance to be compared within and across plates. Here, the distributions of size factors exhibit only modest deviations from unity in the averages for the endogenous genes and spike-in transcripts (Figure 5), indicating that our assumption is again reasonable.

tmp.416B <- sce.416B
tmp.416B$log_size_factor <- log(sizeFactors(sce.416B))
tmp.416B$log_size_factor_ERCC <- log(sizeFactors(sce.416B, "ERCC"))
p1 <- plotColData(tmp.416B, x="Plate", y="log_size_factor")
p2 <- plotColData(tmp.416B, x="Plate", y="log_size_factor_ERCC")
multiplot(p1, p2, cols=2)

Figure 5: Plate-specific distribution of the size factors for endogenous genes (left) and spike-in transcripts (right).

The most obvious violation of the above assumption occurs when more spike-in RNA is added in a particular batch. Spike-in size factors would then be systematically larger than unity in that batch, meaning that average abundances after normalization would not be comparable between spike-in transcripts and endogenous genes. This would compromise the accuracy of estimation of the technical component by interpolation from the trend. More generally, batches that are processed separately may not exhibit to the same technical variation, such that the use of a single trend would be inappropriate.

4.2 Fitting batch-specific trends

For datasets containing multiple batches, an alternative strategy is to perform trend fitting and variance decomposition separately for each batch. This accommodates differences in the mean-variance trends between batches, especially if a different amount of spike-in RNA was added to the cells in each batch. We demonstrate this approach by treating each plate in the 416B dataset as a different batch, using the multiBlockVar() function. This yields plate-specific estimates of the biological and technical components for each gene.

sce.416B.2 <- multiBlockNorm(sce.416B, sce.416B$Plate)
comb.out <- multiBlockVar(sce.416B.2, block=sce.416B.2$Plate,
    trend.args=list(parametric=TRUE, loess.args=list(span=0.4)))

Statistics are combined across multiple batches using the combineVar() function within multiBlockVar(). This function computes a weighted average across batches for the means and variances, and applies Fisher’s method for combining the \(p\)-values. The weighting is based to the number of cells in each batch, which improves precision by favouring batches with more cells and more information. These results can be used in downstream functions such as denoisePCA, or for detecting highly variable genes (see below).

head(comb.out[,1:6])

## DataFrame with 6 rows and 6 columns
##                                   mean               total
##                              <numeric>           <numeric>
## ENSMUSG00000103377 0.00807160215928894   0.011921865486065
## ENSMUSG00000103147  0.0346526072192529  0.0722196162535234
## ENSMUSG00000103161 0.00519472222570747 0.00493857699521053
## ENSMUSG00000102331   0.018666093059853   0.032923591860573
## ENSMUSG00000102948   0.059057000132083  0.0881371257735823
## Rp1                 0.0970243712569606    0.45233813529556
##                                    bio               tech           p.value
##                              <numeric>          <numeric>         <numeric>
## ENSMUSG00000103377 -0.0277229724196103 0.0396448379056754                 1
## ENSMUSG00000103147 -0.0764556948275004  0.148675311081024 0.999999999962404
## ENSMUSG00000103161 -0.0173491251755451 0.0222877021707557                 1
## ENSMUSG00000102331 -0.0540089343559534 0.0869325262165264 0.999999999999991
## ENSMUSG00000102948  -0.169989420924752  0.258126546698334                 1
## Rp1                0.00813766887883806  0.444200466416722 0.163473096421542
##                                  FDR
##                            <numeric>
## ENSMUSG00000103377                 1
## ENSMUSG00000103147                 1
## ENSMUSG00000103161                 1
## ENSMUSG00000102331                 1
## ENSMUSG00000102948                 1
## Rp1                0.538786121423717

We visualize the quality of the batch-specific trend fits by extracting the relevant statistics from comb.out (Figure 6).

par(mfrow=c(1,2))
is.spike <- isSpike(sce.416B.2)
for (plate in levels(sce.416B.2$Plate)) {
    cur.out <- comb.out$per.block[[plate]]
    plot(cur.out$mean, cur.out$total, pch=16, cex=0.6, xlab="Mean log-expression", 
        ylab="Variance of log-expression", main=plate)
    curve(metadata(cur.out)$trend(x), col="dodgerblue", lwd=2, add=TRUE)
    points(cur.out$mean[is.spike], cur.out$total[is.spike], col="red", pch=16)
}

Figure 6: Variance of normalized log-expression values for each gene in each plate of the 416B dataset, plotted against the mean log-expression. The blue line represents the mean-dependent trend fitted to the variances of the spike-in transcripts (red).

By fitting separate trends, we avoid the need to assume that a single trend is present across batches. However, this also reduces the precision of each trend fit, as less information is available within each batch. We recommend using block= as the default unless there is clear evidence for differences in the trends between batches.

Comments from Aaron:

• We run multiBlockNorm() to adjust the size factors within each level of the blocking factor. Specifically, the spike-in size factors across cells in a given batch is scaled so that the mean is equal to that of the gene-based size factors for the same set of cells. Log-normalized expression values are then recalculated using these centred size factors. This procedure ensures that the average abundances of the spike-in transcripts are comparable to the endogenous genes, avoiding problems due to differences in the quantity of spike-in RNA between batches. Otherwise, if the globally-centred size factors (from normalize()) were used, there would be a systematic difference in the scaling of spike-in transcripts compared to endogenous genes in batches with more or less spike-in RNA than the dataset average. The fitted trend would then be shifted along the x-axis and fail to accurately capture the technical component for each gene.

4.3 Using the design= argument

For completeness, it is worth mentioning the design= argument in trendVar(). This will estimate the residual variance from a linear model fitted to the log-normalized expression values for each gene. The linear model can include blocking factors for known unwanted factors of variation, ensuring that they do not inflate the variance estimate. The technical component for each gene is obtained at the average abundance across all cells.

lfit <- trendVar(sce.416B, design=model.matrix(~sce.416B$Plate))

We do not recommend using this approach for categorical blocking factors in one-way layouts. This is because it does not consider the mean of each blocking level, resulting in an inaccurate estimate of the technical component in the presence of a strong blocking effect. However, it is the only choice for dealing with real covariates or multiple blocking factors in an additive model.