modelGeneVarWithSpikes: Model the per-gene variance with spike-ins
In scran: Methods for Single-Cell RNA-Seq Data Analysis

Description Usage Arguments Details Value Default size factor choices Explanation for size factor rescaling Computing p-values Handling uninteresting factors Author(s) See Also Examples

Model the variance of the log-expression profiles for each gene, decomposing it into technical and biological components based on a mean-variance trend fitted to spike-in transcripts.

modelGeneVarWithSpikes(x, ...)

## S4 method for signature 'ANY'
modelGeneVarWithSpikes(
  x,
  spikes,
  size.factors = NULL,
  spike.size.factors = NULL,
  block = NULL,
  design = NULL,
  subset.row = NULL,
  pseudo.count = 1,
  ...,
  equiweight = TRUE,
  method = "fisher",
  BPPARAM = SerialParam()
)

## S4 method for signature 'SummarizedExperiment'
modelGeneVarWithSpikes(x, ..., assay.type = "counts")

## S4 method for signature 'SingleCellExperiment'
modelGeneVarWithSpikes(
  x,
  spikes,
  size.factors = NULL,
  spike.size.factors = NULL,
  ...,
  assay.type = "counts"
)

`x`	A numeric matrix of counts where rows are (usually endogenous) genes and columns are cells. Alternatively, a SummarizedExperiment or SingleCellExperiment containing such a matrix.
`...`	For the generic, further arguments to pass to each method. For the ANY method, further arguments to pass to `fitTrendVar`. For the SummarizedExperiment method, further arguments to pass to the ANY method. For the SingleCellExperiment method, further arguments to pass to the SummarizedExperiment method.
`spikes`	A numeric matrix of counts where each row corresponds to a spike-in transcript. This should have the same number of columns as `x`. Alternatively, for the SingleCellExperiment method, this can be a string or an integer scalar specifying the `altExp` containing the spike-in count matrix.
`size.factors`	A numeric vector of size factors for each cell in `x`, to be used for scaling gene expression.
`spike.size.factors`	A numeric vector of size factors for each cell in `spikes`, to be used for scaling spike-ins.
`block`	A factor specifying the blocking levels for each cell in `x`. If specified, variance modelling is performed separately within each block and statistics are combined across blocks.
`design`	A numeric matrix containing blocking terms for uninteresting factors of variation.
`subset.row`	See `?"scran-gene-selection"`, specifying the rows for which to model the variance. Defaults to all genes in `x`.
`pseudo.count`	Numeric scalar specifying the pseudo-count to add prior to log-transformation.
`equiweight`	A logical scalar indicating whether statistics from each block should be given equal weight. Otherwise, each block is weighted according to its number of cells. Only used if `block` is specified.
`method`	String specifying how p-values should be combined when `block` is specified, see `combinePValues`.
`BPPARAM`	A BiocParallelParam object indicating whether parallelization should be performed across genes.
`assay.type`	String or integer scalar specifying the assay containing the counts. For the SingleCellExperiment method, this is used to retrieve both the endogenous and spike-in counts.

For each gene and spike-in transcript, we compute the variance and mean of the log-expression values. A trend is fitted to the variance against the mean for spike-in transcripts using fitTrendVar. The technical component for each gene is defined as the value of the trend at that gene's mean abundance. The biological component is then defined as the residual from the trend.

This function is almost the same as modelGeneVar, with the only difference being that the trend is fitted on spike-in variances rather than using the means and variances of endogenous genes. It assumes that a constant amount of spike-in RNA was added to each cell, such that any differences in observed expression of the spike-in transcripts can be wholly attributed to technical noise; and that endogenous genes and spike-in transcripts follow the same mean-variance relationship.

Unlike modelGeneVar, modelGeneVarWithSpikes starts from a count matrix (for both genes and spike-ins) and requires size factors and a pseudo-count specification to compute the log-expression values. This is because there are certain requirements for how normalization is performed when comparing spike-in transcripts with endogenous genes - see comments in “Explanation for size factor rescaling”. We enforce this by centering the size factors for both sets of features and recomputing the log-expression values prior to computing means and variances.

A DataFrame is returned where each row corresponds to a gene in x (or in subset.row, if specified). This contains the numeric fields:

mean:: Mean normalized log-expression per gene.
total:: Variance of the normalized log-expression per gene.
bio:: Biological component of the variance.
tech:: Technical component of the variance.
p.value, FDR:: Raw and adjusted p-values for the test against the null hypothesis that bio<=0.

If block is not specified, the metadata of the DataFrame contains the output of running fitTrendVar on the spike-in transcripts, along with the mean and var used to fit the trend.

If block is specified, the output contains another per.block field. This field is itself a DataFrame of DataFrames, where each internal DataFrame contains statistics for the variance modelling within each block and has the same format as described above. Each internal DataFrame's metadata contains the output of fitTrendVar for the cells of that block.

If no size factors are supplied, they are automatically computed depending on the input type:

If size.factors=NULL for the ANY method, the sum of counts for each cell in x is used to compute a size factor via the librarySizeFactors function.
If spike.size.factors=NULL for the ANY method, the sum of counts for each cell in spikes is used to compute a size factor via the librarySizeFactors function.
If size.factors=NULL for the SingleCellExperiment method, sizeFactors(x) is used if available. Otherwise, it defaults to library size-derived size factors.
If spike.size.factors=NULL for the SingleCellExperiment method and spikes is not a matrix, sizeFactors(altExp(x, spikes) is used if available. Otherwise, it defaults to library size-derived size factors.

If size.factors or spike.size.factors are supplied, they will override any size factors present in x.

The use of a spike-in-derived trend makes several assumptions. The first is that a constant amount of spike-in RNA was added to each cell, such that any differences in observed expression of the spike-in transcripts can be wholly attributed to technical noise. The second is that endogenous genes and spike-in transcripts follow the same mean-variance relationship, i.e., a spike-in transcript captures the technical noise of an endogenous gene at the same mean count.

Here, the spike-in size factors across all cells are scaled so that their mean is equal to that of the gene-based size factors for the same set of cells. This ensures that the average normalized abundances of the spike-in transcripts are comparable to those of the endogenous genes, allowing the trend fitted to the former to be used to determine the biological component of the latter. Otherwise, differences in scaling of the size factors would shift the normalized expression values of the former away from the latter, violating the second assumption.

If block is specified, rescaling is performed separately for all cells in each block. This aims to avoid problems from (frequent) violations of the first assumption where there are differences in the quantity of spike-in RNA added to each batch. Without scaling, these differences would lead to systematic shifts in the spike-in abundances from the endogenous genes when fitting a batch-specific trend (even if there is no global difference in scaling across all batches).

The p-value for each gene is computed by assuming that the variance estimates are normally distributed around the trend, and that the standard deviation of the variance distribution is proportional to the value of the trend. This is used to construct a one-sided test for each gene based on its bio, under the null hypothesis that the biological component is equal to zero. The proportionality constant for the standard deviation is set to the std.dev returned by fitTrendVar. This is estimated from the spread of variance estimates for spike-in transcripts, so the null hypothesis effectively becomes “is this gene more variable than spike-in transcripts of the same abundance?”

Setting block will estimate the mean and variance of each gene for cells in each level of block separately. The trend is fitted separately for each level, and the variance decomposition is also performed separately. Per-level statistics are then combined to obtain a single value per gene:

For means and variance components, this is done by averaging values across levels. If equiweight=FALSE, a weighted average is used where the value for each level is weighted by the number of cells. By default, all levels are equally weighted when combining statistics.
Per-level p-values are combined using combinePValues according to method. By default, Fisher's method is used to identify genes that are highly variable in any batch. Whether or not this is responsive to equiweight depends on the chosen method.
Blocks with fewer than 2 cells are completely ignored and do not contribute to the combined mean, variance component or p-value.

Use of block is the recommended approach for accounting for any uninteresting categorical factor of variation. In addition to accounting for systematic differences in expression between levels of the blocking factor, it also accommodates differences in the mean-variance relationships.

Alternatively, uninteresting factors can be used to construct a design matrix to pass to the function via design. In this case, a linear model is fitted to the expression profile for each gene and the residual variance is calculated. This approach is useful for covariates or additive models that cannot be expressed as a one-way layout for use in block. However, it assumes that the error is normally distributed with equal variance for all observations of a given gene.

Use of block and design together is currently not supported and will lead to an error.

Aaron Lun

fitTrendVar, for the trend fitting options.

modelGeneVar, for modelling variance without spike-in controls.

library(scuttle)
sce <- mockSCE()
sce <- logNormCounts(sce)

# Using spike-ins.
spk <- modelGeneVarWithSpikes(sce, "Spikes")
spk

plot(spk$mean, spk$total, log="xy")
points(metadata(spk)$mean, metadata(spk)$cv2, col="red", pch=16)
curve(metadata(spk)$trend(x), add=TRUE, col="dodgerblue")

# With blocking (and spike-ins).
block <- sample(LETTERS[1:2], ncol(sce), replace=TRUE)
blk <- modelGeneVarWithSpikes(sce, "Spikes", block=block)
blk

par(mfrow=c(1,2))
for (i in colnames(blk$per.block)) {
    current <- blk$per.block[[i]]
    plot(current$mean, current$total)
    points(metadata(current)$mean, metadata(current)$cv2, col="red", pch=16)
    curve(metadata(current)$trend(x), add=TRUE, col="dodgerblue")
}