# correlatePairs: Test for significant correlations In scran: Methods for Single-Cell RNA-Seq Data Analysis

## Description

Identify pairs of genes that are significantly correlated in their expression profiles, based on Spearman's rank correlation.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```correlatePairs(x, ...) ## S4 method for signature 'ANY' correlatePairs( x, null.dist = NULL, ties.method = c("expected", "average"), iters = 1e+06, block = NULL, design = NULL, equiweight = TRUE, use.names = TRUE, subset.row = NULL, pairings = NULL, BPPARAM = SerialParam() ) ## S4 method for signature 'SummarizedExperiment' correlatePairs(x, ..., assay.type = "logcounts") ```

## Arguments

 `x` A numeric matrix-like object of log-normalized expression values, where rows are genes and columns are cells. Alternatively, a SummarizedExperiment object containing such a matrix. `...` For the generic, additional arguments to pass to specific methods. For the SummarizedExperiment method, additional methods to pass to the ANY method. `null.dist` A numeric vector of rho values under the null hypothesis. `ties.method` String specifying how tied ranks should be handled. `iters` Integer scalar specifying the number of iterations to use in `correlateNull` to build the null distribution. `block` A factor specifying the blocking level for each cell in `x`. If specified, correlations are computed separately within each block and statistics are combined across blocks. `design` A numeric design matrix containing uninteresting factors to be ignored. `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. `use.names` A logical scalar specifying whether the row names of `x` should be used in the output. Alternatively, a character vector containing the names to use. `subset.row` See `?"scran-gene-selection"`. `pairings` A `NULL` value indicating that all pairwise correlations should be computed; or a list of 2 vectors of genes between which correlations are to be computed; or a integer/character matrix with 2 columns of specific gene pairs - see below for details. `BPPARAM` A BiocParallelParam object that specifies the manner of parallel processing to use. `assay.type` A string specifying which assay values to use.

## Details

The `correlatePairs` function identifies significant correlations between all pairs of genes in `x`. This allows prioritization of genes that are driving systematic substructure in the data set. By definition, such genes should be correlated as they are behaving in the same manner across cells. In contrast, genes driven by random noise should not exhibit any correlations with other genes.

We use Spearman's rho to quantify correlations robustly based on ranked gene expression. To identify correlated gene pairs, the significance of non-zero correlations is assessed using a permutation test. The null hypothesis is that the ranking of normalized expression across cells should be independent between genes. This allows us to construct a null distribution by randomizing the ranks within each gene. A pre-computed empirical distribution can be supplied as `null.dist`, which saves some time by avoiding repeated calls to `correlateNull`.

The p-value for each gene pair is defined as the tail probability of this distribution at the observed correlation (with some adjustment to avoid zero p-values). Correction for multiple testing is done using the BH method. The lower bound of the p-values is determined by the value of `iters`. If the `limited` field is `TRUE` in the returned dataframe, it may be possible to obtain lower p-values by increasing `iters`. This should be examined for non-significant pairs, in case some correlations are overlooked due to computational limitations. The function will automatically raise a warning if any genes are limited in their significance at a FDR of 5%.

For the SingleCellExperiment method, correlations should be computed for normalized expression values in the specified `assay.type`.

## Value

A DataFrame is returned with one row per gene pair and the following fields:

`gene1, gene2`:

Character or integer fields specifying the genes in the pair. If `use.names=FALSE`, integers are returned representing row indices of `x`, otherwise gene names are returned.

`rho`:

A numeric field containing the approximate Spearman's rho.

`p.value, FDR`:

Numeric fields containing the permutation p-value and its BH-corrected equivalent.

`limited`:

A logical scalar indicating whether the p-value is at its lower bound, defined by `iters`.

Rows are sorted by increasing `p.value` and, if tied, decreasing absolute size of `rho`. The exception is if `subset.row` is a matrix, in which case each row in the dataframe correspond to a row of `subset.row`.

## Accounting for uninteresting variation

If the experiment has known (and uninteresting) factors of variation, these can be included in `design` or `block`. `correlatePairs` will then attempt to ensure that these factors do not drive strong correlations between genes. Examples might be to block on batch effects or cell cycle phase, which may have substantial but uninteresting effects on expression.

The approach used to remove these factors depends on whether `design` or `block` is used. If there is only one factor, e.g., for plate or animal of origin, `block` should be used. Each level of the factor is defined as a separate group of cells. For each pair of genes, correlations are computed within each group, and a weighted mean based on the group size) of the correlations is taken across all groups. The same strategy is used to generate the null distribution where ranks are computed and shuffled within each group.

For experiments containing multiple factors or covariates, a design matrix should be passed into `design`. The `correlatePairs` function will fit a linear model to the (log-normalized) expression values. The correlation between a pair of genes is then computed from the residuals of the fitted model. Similarly, to obtain a null distribution of rho values, normally-distributed random errors are simulated in a fitted model based on `design`; the corresponding residuals are generated from these errors; and the correlation between sets of residuals is computed at each iteration.

We recommend using `block` wherever possible. While `design` can also be used for one-way layouts, this is not ideal as it assumes normality of errors and deals poorly with ties. Specifically, zero counts within or across groups may no longer be tied when converted to residuals, potentially resulting in spuriously large correlations.

If any level of `block` has fewer than 3 cells, it is ignored. If all levels of `block` have fewer than 3 cells, all output statistics are set to `NA`. Similarly, if `design` has fewer than 3 residual d.f., all output statistics are set to `NA`.

## Gene selection

The `pairings` argument specifies the pairs of genes to compute correlations for:

• By default, correlations will be computed between all pairs of genes with `pairings=NULL`. Genes that occur earlier in `x` are labelled as `gene1` in the output DataFrame. Redundant permutations are not reported.

• If `pairings` is a list of two vectors, correlations will be computed between one gene in the first vector and another gene in the second vector. This improves efficiency if the only correlations of interest are those between two pre-defined sets of genes. Genes in the first vector are always reported as `gene1`.

• If `pairings` is an integer/character matrix of two columns, each row is assumed to specify a gene pair. Correlations will then be computed for only those gene pairs, and the returned dataframe will not be sorted by p-value. Genes in the first column of the matrix are always reported as `gene1`.

If `subset.row` is not `NULL`, only the genes in the selected subset are used to compute correlations - see `?"scran-gene-selection"`. This will interact properly with `pairings`, such that genes in `pairings` and not in `subset.row` will be ignored.

We recommend setting `subset.row` and/or `pairings` to contain only the subset of genes of interest. This reduces computational time and memory usage by only computing statistics for the gene pairs of interest. For example, we could select only HVGs to focus on genes contributing to cell-to-cell heterogeneity (and thus more likely to be involved in driving substructure). There is no need to account for HVG pre-selection in multiple testing, because rank correlations are unaffected by the variance.

## Handling tied values

By default, `ties.method="expected"` which uses the expectation of the pairwise correlation from randomly broken ties. Imagine obtaining unique ranks by randomly breaking ties in expression values, e.g., by addition of some very small random jitter. The reported value of the correlation is simply the expected value across all possible permutations of broken ties.

With `ties.method="average"`, each set of tied observations is assigned the average rank across all tied values. This yields the standard value of Spearman's rho as computed by `cor`.

We use the expected rho by default as avoids inflated correlations in the presence of many ties. This is especially relevant for single-cell data containing many zeroes that remain tied after scaling normalization. An extreme example is that of two genes that are only expressed in the same cell, for which the standard rho is 1 while the expected rho is close to zero.

Note that the p-value calculations are not accurate in the presence of ties, as tied ranks are not considered by `correlateNull`. With `ties.method="expected"`, the p-values are probably a bit conservative. With `ties.method="average"`, they will be very anticonservative.

Aaron Lun

## References

Lun ATL (2019). Some thoughts on testing for correlations. https://ltla.github.io/SingleCellThoughts/software/correlations/corsim.html

Phipson B and Smyth GK (2010). Permutation P-values should never be zero: calculating exact P-values when permutations are randomly drawn. Stat. Appl. Genet. Mol. Biol. 9:Article 39.

Compare to `cor` for the standard Spearman's calculation.
Use `correlateGenes` to get per-gene correlation statistics.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15``` ```library(scuttle) sce <- mockSCE() sce <- logNormCounts(sce) # Basic pairwise application (turning down iters for speed). out <- correlatePairs(sce, subset.row=1:100, iters=1e5) head(out) # Computing between specific subsets of genes: out <- correlatePairs(sce, pairings=list(1:10, 110:120), iters=1e5) head(out) # Computing between specific pairs: out <- correlatePairs(sce, pairings=rbind(c(1,10), c(2, 50)), iters=1e5) head(out) ```
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