In shazanfar/scHOT: single-cell higher order testing

knitr::opts_chunk$set(
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Introduction

Single cell Higher Order Testing (scHOT) is an R package that facilitates testing changes in higher order structure along either a developmental trajectory or across space. In this vignette, we go through two example analyses: 1) Testing variability changes along liver trajectory, and 2) Testing differential correlation across the mouse olfactory bulb.

library(SingleCellExperiment)
library(ggplot2)
library(scHOT)
library(scater)
library(matrixStats)

Testing variability changes along liver trajectory

The liver dataset contains data from two branches of a developmental trajectory, starting from immature hepatoblasts and bifurcates into either the hepatocyte or cholangiocyte lineages. For file size reasons we only load up the hepatocyte lineage for a small number of genes. In this example, we take the cells that belong to the initial part of the trajectory, i.e. from hepatoblast to the bifurcation point, and test for differential variability along this trajectory for a few genes.

data(liver)

liver_pseudotime_hep <- liver$liver_pseudotime_hep
liver_branch_hep <- liver$liver_branch_hep
first_branch_cells <- liver$first_branch_cells

gene_to_test <- as.matrix(c("Birc5", "H2afz", "Tacc3"))

Build the `scHOT` object

First we build the scHOT object, which is based on the SingleCellExperiment object class. scHOT objects can be built either from a matrix format or from an existing SingleCellExperiment object. In this case, we have matrix data so we build the scHOT object using scHOT_buildFromMatrix. Since the liver data represents a trajectory, we set the positionType as "trajectory", and provide the column name of the cell metadata (argument cellData) for which the cells should be ordered.

scHOT_traj <- scHOT_buildFromMatrix(
  mat = liver_branch_hep[,first_branch_cells],
  cellData = list(pseudotime = liver_pseudotime_hep[first_branch_cells]),
  positionType = "trajectory",
  positionColData = "pseudotime")
scHOT_traj

scHOT_traj is a scHOT object, but methods associated with SingleCellExperiment can also be used. For example, we use the scater package to plot the expression of the hepatoblast marker Sall4 along pseudotime, and note that this decreases as pseudotime increases.

scater::plotExpression(scHOT_traj, c("Sall4"),
                       exprs_values = "expression", x = "pseudotime")

scHOT wrapper function

Now using the scHOT wrapper function, we can perform higher order testing on the selected genes, provided as a one-column matrix. To do this, we also need to set the underlying higher order function, which in this case we use weighed variance as implemented in the matrixStats package. Since this function has a weight parameter, we set higherOrderFunctionType = "weighted". For basic implementation, no other parameters need to be specified (for speed, we set numberPermutations to a small value).

scHOT_traj_wrap = scHOT(scHOT_traj,
                        testingScaffold = gene_to_test,
                        higherOrderFunction = matrixStats::weightedVar,
                        higherOrderFunctionType = "weighted",
                        numberPermutations = 50)

Output is saved as a DataFrame in the scHOT_output slot, accessible either using the slot function, or using the @ accessor. In particular, we can interrogate the higher order sequence, the sequence of locally weighted variances along the trajectory. We can see from the plot that each of these genes increases in variability along pseudotime. Note that the plots are based on ggplot2 and so can be customised as desired.

slot(scHOT_traj_wrap, "scHOT_output")
scHOT_traj_wrap@scHOT_output

slot(scHOT_traj_wrap, "scHOT_output")$higherOrderSequence

plotHigherOrderSequence(scHOT_traj_wrap, gene_to_test)

plotOrderedExpression(scHOT_traj_wrap, gene_to_test) + 
  facet_wrap(~gene, scales = "free_y")

scHOT step-by-step

Now, we can perform the same testing but step-by-step, with description of the parameter selection at each step.

First, we add the testing scaffold. This is the set of genes for which we wish to perform higher order testing.

scHOT_traj@testingScaffold
scHOT_traj <- scHOT_addTestingScaffold(scHOT_traj, gene_to_test)
scHOT_traj@testingScaffold

Next, we provide parameters to build a weighting scheme. The most important parameter here is span, a value between 0 and 1 (default 0.25) which determines how large or small a window we wish to use for higher order testing. To distinguish between either ranked samples in a trajectory or in a spatial setting, we set positionType = "trajectory" and instruct to extract the trajectory ordering from the "pseudotime" column from colData(scHOT_traj).

scHOT_traj@weightMatrix
scHOT_traj <- scHOT_setWeightMatrix(scHOT_traj,
                                    positionType = "trajectory",
                                    positionColData = c("pseudotime"),
                                    nrow.out = NULL,
                                    span = 0.25)
dim(scHOT_traj@weightMatrix)
class(scHOT_traj@weightMatrix)
plot(scHOT_traj@weightMatrix[50,])

By default the weight matrix is square, with as many rows as the number of cells (columns), but if you have especially large data, you may want to reduce the weight matrix rows for faster runtimes. Here we reduce the weight matrix to roughly 50 rows using the nrow.out arguemnt. Note you can provide your own pre-prepared matrix using the weightMatrix argument.

scHOT_traj <- scHOT_setWeightMatrix(scHOT_traj,
                                    positionType = "trajectory",
                                    positionColData = c("pseudotime"),
                                    nrow.out = 50,
                                    span = 0.25)
dim(scHOT_traj@weightMatrix)
class(scHOT_traj@weightMatrix)
plot(scHOT_traj@weightMatrix[50,])

Now we calculate the global higher order function, in this case is simply the sample variance giving equal weight to all cells. This becomes important when you wish to test many genes and use a fast p-value estimation approach to speed up computation.

scHOT_traj <- scHOT_calculateGlobalHigherOrderFunction(
  scHOT_traj, 
  higherOrderFunction = matrixStats::weightedVar,
  higherOrderFunctionType = "weighted")

slot(scHOT_traj, "scHOT_output")
apply(assay(scHOT_traj, "expression")[c(gene_to_test),],1,var)

scHOT allows for a lot of customisation. In particular, you can set which tests you wish to perform permutation testing for, and if so, what number of permutations can be used, and if they should be stored for later use. Here, we set the number of permutations to a mix of 20 and 50. Allowing different permutation numbers is useful if we wish to perform a lot of permutations for one test, and few for another.

Note if you wish to explicitly set the number of permutations for all tests, then ensure that numberScaffold is set above the number of tests (default 100).

scHOT_setPermutationScaffold only gives the instructions for running permutation testing, it doesn't actually perform the testing.

scHOT_traj <- scHOT_setPermutationScaffold(scHOT_traj, 
                                           numberPermutations = c(20,50,20),
                                           storePermutations = c(TRUE, FALSE, TRUE))
slot(scHOT_traj, "scHOT_output")

Now we can calculate the observed test statistic. This is calculated as a summary function (default standard deviation) of the local higher order sequence, which is parametrised using the higher order function (here weighted variance) and weighting scheme (here thinned triangular weight matrix with span 0.25).

scHOT_traj <- scHOT_calculateHigherOrderTestStatistics(
  scHOT_traj,
  higherOrderSummaryFunction = sd)

slot(scHOT_traj, "scHOT_output")

Once the test statistic is calculated, we perform permutation testing, using the instructions we provided earlier.

system.time(scHOT_traj <- scHOT_performPermutationTest(
  scHOT_traj, 
  verbose = TRUE,
  parallel = FALSE))

slot(scHOT_traj, "scHOT_output")

To avoid P-values identically zero, we rescale zero P-values to 1/(1+numberPermutations).

We could also use the existing stored permutations to estimate P-values, since genes with a similar global higher order function are likely to have a similar null distribution. In this example case it does not show much gain computationally, but this can significantly reduce the number of permutation tests needed for large datasets and testing strategies.

Running scHOT_estimatePvalues results in more columns in the scHOT_output slot, corresponding to the number of permutations used in estimating, the range of the global higher order function used for estimating, as well as the estimated P-value itself and FDR adjusted P-value. Here, we set a maximum of 10,000 permutations to be used, for genes with at most a difference of 5 in the global higher order function (the variance).

scHOT_traj <- scHOT_estimatePvalues(scHOT_traj,
                                    nperm_estimate = 10000,
                                    maxDist = 5)
slot(scHOT_traj, "scHOT_output")

Note that having performed the testing using the thinned weight matrix, we can still plot, but beware that not every position is sampled.

plotHigherOrderSequence(scHOT_traj, gene_to_test)

Spatial differential correlation in Mouse Olfactory Bulb

In this example, we look at the Spatial Transcriptomics mouse olfactory bulb data. This data is provided in the form of a SingleCellExperiment object, with the spatial coordinates provided in the colData slot. For file size reasons we only load up a small number of genes, corresponding to highly variable genes (HVGs) that are not found to be significantly differentially expressed in space.

data(MOB_subset)
sce_MOB_subset <- MOB_subset$sce_MOB_subset

sce_MOB_subset

We build the scHOT object using scHOT_buildFromSCE. Note that scHOT only takes in a single assay slot, for which testing is based on.

scHOT_spatial <- scHOT_buildFromSCE(sce_MOB_subset,
                                    assayName = "logcounts",
                                    positionType = "spatial",
                                    positionColData = c("x", "y"))

scHOT_spatial

Perform scHOT step by step - skip ahead for wrapper using `scHOT`

In this example, we want to perform spatial differential correlation testing between distinct pairs of genes. We build up our testing scaffold by taking all pairs of the non-differentially expressed HVGs. For practicality, we only consider a small set of pairs for testing here.

pairs <- t(combn(rownames(sce_MOB_subset),2))
rownames(pairs) <- apply(pairs,1,paste0,collapse = "_")
head(pairs)

set.seed(2020)
pairs <- pairs[sample(nrow(pairs), 20), ]
if (!"Arrb1_Mtor" %in% rownames(pairs)) {
pairs <- rbind(pairs, "Arrb1_Mtor" = c("Arrb1", "Mtor"))
}
if (!"Dnm1l_Fam63b" %in% rownames(pairs)) {
pairs <- rbind(pairs, "Dnm1l_Fam63b" = c("Dnm1l", "Fam63b"))
}

scHOT_spatial <- scHOT_addTestingScaffold(scHOT_spatial, pairs)
scHOT_spatial@testingScaffold

Note that since we are performing differential correlation testing, our testing scaffold is a matrix with two columns. If you wish to use some higher order function with more than two genes, you can simply add more columns to the testing scaffold, and ensure there are more arguments in the provided higher order function.

Now we set the weight matrix, using the positional coordinates in the scHOT object. The span parameter here corresponds to the proportion of cells that have nonzero values around a radius of the central cell.

This can also be thinned for faster computation by setting the nrow.out argument to the number of samples to roughly thin to.

scHOT_spatial <- scHOT_setWeightMatrix(scHOT_spatial,
                                       positionColData = c("x","y"),
                                       positionType = "spatial",
                                       nrow.out = NULL,
                                       span = 0.05)

dim(slot(scHOT_spatial, "weightMatrix"))

We can visualise the weighting scheme for each row of the weight matrix.

cellID = 75
ggplot(as.data.frame(colData(scHOT_spatial)), aes(x = -x, y = y)) + 
  geom_point(aes(colour = slot(scHOT_spatial, "weightMatrix")[cellID,],
                 size = slot(scHOT_spatial, "weightMatrix")[cellID,])) + 
  scale_colour_gradient(low = "black", high = "purple") + 
  scale_size_continuous(range = c(1,5)) +
  theme_classic() +
  guides(colour = guide_legend(title = "Spatial Weight"),
         size = guide_legend(title = "Spatial Weight")) + 
    ggtitle(paste0("Central cell: ", cellID))

Now we can calculate the global higher order function. In this case, we use weighted Spearman correlation as our weighted higher order function, and so this is simply equivalent to calculating Spearman correlation for the gene pairs of interest.

scHOT_spatial <- scHOT_calculateGlobalHigherOrderFunction(
  scHOT_spatial, 
  higherOrderFunction = weightedSpearman,
  higherOrderFunctionType = "weighted")

slot(scHOT_spatial, "scHOT_output")

head(diag(cor(t(assay(scHOT_spatial, "expression")[pairs[,1],]),
         t(assay(scHOT_spatial, "expression")[pairs[,2],]),
         method = "spearman")))

Now we can set the permutation parameters. In this case, we only perform 50 permutations for around 10 randomly selected tests.

scHOT_spatial <- scHOT_setPermutationScaffold(scHOT_spatial, 
                                              numberPermutations = 50,
                                              numberScaffold = 10)

slot(scHOT_spatial, "scHOT_output")

Now we calculate the observed higher order test statistics, which in this case correspond to the summary of local correlation vectors.

scHOT_spatial <- scHOT_calculateHigherOrderTestStatistics(scHOT_spatial)

slot(scHOT_spatial, "scHOT_output")

Now we perform permutation testing for those tests which we provided with a nonzero value for number of permutations. This takes about a minute to run.

system.time(scHOT_spatial <- scHOT_performPermutationTest(
  scHOT_spatial, 
  verbose = TRUE,
  parallel = FALSE))

slot(scHOT_spatial, "scHOT_output")

With the above, we calculated P-values for some of the tests, but we have not performed permutation testing for all tests. Here, we employ a permutation sharing approach to estimate significance for the other tests. Ideally this would be performed with around a hundred tests each with around 1,000 permutations to ensure accurate P-value estimation. You can check how good an estimate you can expect by plotting the global higher order statistic against the permuted test statistics, there should be good representation along the x-axis and the fitted curve should appear quite smooth. Here the fitted curve is bumpy, so we would suggest selecting more genes and more permutations.

We estimate P-values by borrowing 100 permutations from closest tests with a difference in global higher order statistic of at most 0.1

scHOT_plotPermutationDistributions(scHOT_spatial)

scHOT_spatial <- scHOT_estimatePvalues(scHOT_spatial,
                                       nperm_estimate = 100,
                                       maxDist = 0.1)
slot(scHOT_spatial, "scHOT_output")

ggplot(as.data.frame(slot(scHOT_spatial, "scHOT_output")),
       aes(x = -log10(pvalPermutations), y = -log10(pvalEstimated))) + 
  geom_point() + 
  theme_classic() + 
  geom_abline(slope = 1, intercept = 0) +
  xlab("Permutation -log10(p-value)") +
  ylab("Estimated -log10(p-value)") +
  NULL

In this example we can still see fairly good concordance between the estimated and direct permutation tests, even with a very small number of permutations.

Once testing is done, we can interrogate the results with various plots. The plotHigherOrderSequence function will plot the points in space, coloured by the local correlation estimates, and plotOrderedExpression will plot the points in space, coloured by expression of each gene.

colData(scHOT_spatial)[, "-x"] <- -colData(scHOT_spatial)[, "x"]
plotHigherOrderSequence(scHOT_spatial, c("Dnm1l_Fam63b"),
                        positionColData = c("-x", "y"))

plotOrderedExpression(scHOT_spatial, c("Dnm1l", "Fam63b"),
                      positionColData = c("-x", "y"),
                      assayName = "expression")

Perform scHOT using `scHOT` wrapper function

Strip the existing scHOT output using scHOT_stripOutput before rerunning scHOT in a new context.

scHOT_spatial <- scHOT_stripOutput(scHOT_spatial, force = TRUE)

scHOT_spatial
slot(scHOT_spatial, "scHOT_output")

We can perform scHOT in a single wrapper function, which will perform the steps as described above, with all parameters given at once.

scHOT_spatial <- scHOT(scHOT_spatial,
                       testingScaffold = pairs,
                       positionType = "spatial",
                       positionColData = c("x", "y"),
                       nrow.out = NULL,
                       higherOrderFunction = weightedSpearman,
                       higherOrderFunctionType = "weighted",
                       numberPermutations = 50,
                       numberScaffold = 10,
                       higherOrderSummaryFunction = sd,
                       parallel = FALSE,
                       verbose = FALSE,
                       span = 0.05)

slot(scHOT_spatial, "scHOT_output")

Again, we can examine the results in terms of the spatial expression and the local higher order sequences.

plotOrderedExpression(scHOT_spatial, c("Arrb1", "Mtor"),
                      positionColData = c("-x", "y"),
                      assayName = "expression")

plotHigherOrderSequence(scHOT_spatial, "Arrb1_Mtor",
                        positionColData = c("-x", "y"))