# Getting started: scHOT

# 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
    #> class: scHOT
    #> dim: 568 176
    #> metadata(0):
    #> assays(1): expression
    #> rownames(568): 2810474O19Rik Abca1 ... Ahsg Epcam
    #> rowData names(0):
    #> colnames(176): E10.5D_3_02 E10.5D_2_01 ... C1A_E16.5 E11.5D_2_12
    #> colData names(1): pseudotime
    #> reducedDimNames(0):
    #> mainExpName: NULL
    #> altExpNames(0):
    #> testingScaffold dim: 0 0
    #> weightMatrix dim: 0 0
    #> scHOT_output colnames (0):
    #> param names (0):
    #> position type: trajectory

`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)
    #> Adding testing scaffold
    #> Set weight matrix
    #> Calculate gobal higher order function
    #> Calculate Higher Order Test Statistics
    #> Perform Permutation Test
    #> Permutation testing combination 1 of 3...
    #> Permutation testing combination 2 of 3...
    #> Permutation testing combination 3 of 3...
    #> Estimating p-values
    #> 1
    #> 2
    #> 3

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")
    #> DataFrame with 3 rows and 14 columns
    #>            gene_1 globalHigherOrderFunction numberPermutations
    #>       <character>                  <matrix>          <numeric>
    #> Birc5       Birc5                  1.991674                 50
    #> H2afz       H2afz                  0.238446                 50
    #> Tacc3       Tacc3                  2.346598                 50
    #>       storePermutations               higherOrderSequence higherOrderStatistic
    #>               <logical>                     <NumericList>            <numeric>
    #> Birc5              TRUE    0.326368,0.323759,0.322789,...             1.634854
    #> H2afz              TRUE 0.0622930,0.0619537,0.0615607,...             0.216771
    #> Tacc3              TRUE    0.803192,0.801351,0.820289,...             0.930598
    #>                            permutations pvalPermutations FDRPermutations
    #>                           <NumericList>        <numeric>       <numeric>
    #> Birc5    0.243933,0.316105,0.696767,...        0.0196078       0.0196078
    #> H2afz 0.0709209,0.0866192,0.0600563,...        0.0196078       0.0196078
    #> Tacc3    0.560326,0.640757,0.601707,...        0.0196078       0.0196078
    #>       numberPermutationsEstimated globalLowerRangeEstimated
    #>                         <integer>                 <numeric>
    #> Birc5                          50                  1.991674
    #> H2afz                          50                  0.238446
    #> Tacc3                          50                  2.346598
    #>       globalUpperRangeEstimated pvalEstimated FDREstimated
    #>                       <numeric>     <numeric>    <numeric>
    #> Birc5                  1.991674     0.0196078    0.0196078
    #> H2afz                  0.238446     0.0196078    0.0196078
    #> Tacc3                  2.346598     0.0196078    0.0196078
    scHOT_traj_wrap@scHOT_output
    #> DataFrame with 3 rows and 14 columns
    #>            gene_1 globalHigherOrderFunction numberPermutations
    #>       <character>                  <matrix>          <numeric>
    #> Birc5       Birc5                  1.991674                 50
    #> H2afz       H2afz                  0.238446                 50
    #> Tacc3       Tacc3                  2.346598                 50
    #>       storePermutations               higherOrderSequence higherOrderStatistic
    #>               <logical>                     <NumericList>            <numeric>
    #> Birc5              TRUE    0.326368,0.323759,0.322789,...             1.634854
    #> H2afz              TRUE 0.0622930,0.0619537,0.0615607,...             0.216771
    #> Tacc3              TRUE    0.803192,0.801351,0.820289,...             0.930598
    #>                            permutations pvalPermutations FDRPermutations
    #>                           <NumericList>        <numeric>       <numeric>
    #> Birc5    0.243933,0.316105,0.696767,...        0.0196078       0.0196078
    #> H2afz 0.0709209,0.0866192,0.0600563,...        0.0196078       0.0196078
    #> Tacc3    0.560326,0.640757,0.601707,...        0.0196078       0.0196078
    #>       numberPermutationsEstimated globalLowerRangeEstimated
    #>                         <integer>                 <numeric>
    #> Birc5                          50                  1.991674
    #> H2afz                          50                  0.238446
    #> Tacc3                          50                  2.346598
    #>       globalUpperRangeEstimated pvalEstimated FDREstimated
    #>                       <numeric>     <numeric>    <numeric>
    #> Birc5                  1.991674     0.0196078    0.0196078
    #> H2afz                  0.238446     0.0196078    0.0196078
    #> Tacc3                  2.346598     0.0196078    0.0196078

    slot(scHOT_traj_wrap, "scHOT_output")$higherOrderSequence
    #> NumericList of length 3
    #> [["1"]] 0.32636792198 0.323758922618346 ... 5.13350080711913 5.1463258826534
    #> [["2"]] 0.0622930368665992 0.0619537046607012 ... 0.752024900446298
    #> [["3"]] 0.803192064768597 0.801351141908489 ... 3.96122878375288

    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
    #> <0 x 0 matrix>
    scHOT_traj <- scHOT_addTestingScaffold(scHOT_traj, gene_to_test)
    scHOT_traj@testingScaffold
    #>      gene_1
    #> [1,] "Birc5"
    #> [2,] "H2afz"
    #> [3,] "Tacc3"

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
    #> <0 x 0 matrix>
    scHOT_traj <- scHOT_setWeightMatrix(scHOT_traj,
                                        positionType = "trajectory",
                                        positionColData = c("pseudotime"),
                                        nrow.out = NULL,
                                        span = 0.25)
    dim(scHOT_traj@weightMatrix)
    #> [1] 176 176
    class(scHOT_traj@weightMatrix)
    #> [1] "dgCMatrix"
    #> attr(,"package")
    #> [1] "Matrix"
    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)
    #> [1]  59 176
    class(scHOT_traj@weightMatrix)
    #> [1] "dgCMatrix"
    #> attr(,"package")
    #> [1] "Matrix"
    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")
    #> DataFrame with 3 rows and 2 columns
    #>        gene_1 globalHigherOrderFunction
    #>   <character>                  <matrix>
    #> 1       Birc5                  1.991674
    #> 2       H2afz                  0.238446
    #> 3       Tacc3                  2.346598
    apply(assay(scHOT_traj, "expression")[c(gene_to_test),],1,var)
    #>     Birc5     H2afz     Tacc3
    #> 1.9916742 0.2384462 2.3465976

`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")
    #> DataFrame with 3 rows and 4 columns
    #>        gene_1 globalHigherOrderFunction numberPermutations storePermutations
    #>   <character>                  <matrix>          <numeric>         <logical>
    #> 1       Birc5                  1.991674                 20              TRUE
    #> 2       H2afz                  0.238446                 50             FALSE
    #> 3       Tacc3                  2.346598                 20              TRUE

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")
    #> DataFrame with 3 rows and 6 columns
    #>        gene_1 globalHigherOrderFunction numberPermutations storePermutations
    #>   <character>                  <matrix>          <numeric>         <logical>
    #> 1       Birc5                  1.991674                 20              TRUE
    #> 2       H2afz                  0.238446                 50             FALSE
    #> 3       Tacc3                  2.346598                 20              TRUE
    #>                 higherOrderSequence higherOrderStatistic
    #>                       <NumericList>            <numeric>
    #> 1    0.326368,0.322652,0.312199,...             1.644648
    #> 2 0.0622930,0.0611996,0.0600874,...             0.217727
    #> 3    0.803192,0.838504,0.873359,...             0.940253

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))
    #> Permutation testing combination 1 of 3...
    #> Permutation testing combination 2 of 3...
    #> Permutation testing combination 3 of 3...
    #>    user  system elapsed
    #>   0.968   0.043   1.010

    slot(scHOT_traj, "scHOT_output")
    #> DataFrame with 3 rows and 9 columns
    #>        gene_1 globalHigherOrderFunction numberPermutations storePermutations
    #>   <character>                  <matrix>          <numeric>         <logical>
    #> 1       Birc5                  1.991674                 20              TRUE
    #> 2       H2afz                  0.238446                 50             FALSE
    #> 3       Tacc3                  2.346598                 20              TRUE
    #>                 higherOrderSequence higherOrderStatistic
    #>                       <NumericList>            <numeric>
    #> 1    0.326368,0.322652,0.312199,...             1.644648
    #> 2 0.0622930,0.0611996,0.0600874,...             0.217727
    #> 3    0.803192,0.838504,0.873359,...             0.940253
    #>                     permutations pvalPermutations FDRPermutations
    #>                    <NumericList>        <numeric>       <numeric>
    #> 1 0.711001,0.762682,0.574904,...        0.0476190        0.047619
    #> 2                             NA        0.0196078        0.047619
    #> 3 0.174131,0.344524,0.562641,...        0.0476190        0.047619

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")
    #> DataFrame with 3 rows and 14 columns
    #>        gene_1 globalHigherOrderFunction numberPermutations storePermutations
    #>   <character>                  <matrix>          <numeric>         <logical>
    #> 1       Birc5                  1.991674                 20              TRUE
    #> 2       H2afz                  0.238446                 50             FALSE
    #> 3       Tacc3                  2.346598                 20              TRUE
    #>                 higherOrderSequence higherOrderStatistic
    #>                       <NumericList>            <numeric>
    #> 1    0.326368,0.322652,0.312199,...             1.644648
    #> 2 0.0622930,0.0611996,0.0600874,...             0.217727
    #> 3    0.803192,0.838504,0.873359,...             0.940253
    #>                     permutations pvalPermutations FDRPermutations
    #>                    <NumericList>        <numeric>       <numeric>
    #> 1 0.711001,0.762682,0.574904,...        0.0476190        0.047619
    #> 2                             NA        0.0196078        0.047619
    #> 3 0.174131,0.344524,0.562641,...        0.0476190        0.047619
    #>   numberPermutationsEstimated globalLowerRangeEstimated
    #>                     <integer>                 <numeric>
    #> 1                          40                   1.99167
    #> 2                          40                   1.99167
    #> 3                          40                   1.99167
    #>   globalUpperRangeEstimated pvalEstimated FDREstimated
    #>                   <numeric>     <numeric>    <numeric>
    #> 1                    2.3466     0.0243902    0.0365854
    #> 2                    2.3466     0.9250000    0.9250000
    #> 3                    2.3466     0.0243902    0.0365854

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
    #> class: SingleCellExperiment
    #> dim: 43 262
    #> metadata(1): log.exprs.offset
    #> assays(2): counts logcounts
    #> rownames(43): Abat Actb ... Vdac3 Wdfy3
    #> rowData names(0):
    #> colnames(262): 16.92x9.015 16.945x11.075 ... 27.018x20.088
    #>   17.964x10.137
    #> colData names(3): x y sizeFactor
    #> reducedDimNames(0):
    #> mainExpName: NULL
    #> altExpNames(0):

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
    #> class: scHOT
    #> dim: 43 262
    #> metadata(0):
    #> assays(1): expression
    #> rownames(43): Abat Actb ... Vdac3 Wdfy3
    #> rowData names(0):
    #> colnames(262): 16.92x9.015 16.945x11.075 ... 27.018x20.088
    #>   17.964x10.137
    #> colData names(3): x y sizeFactor
    #> reducedDimNames(0):
    #> mainExpName: NULL
    #> altExpNames(0):
    #> testingScaffold dim: 0 0
    #> weightMatrix dim: 0 0
    #> scHOT_output colnames (0):
    #> param names (0):
    #> position type: 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)
    #>               [,1]   [,2]
    #> Abat_Actb     "Abat" "Actb"
    #> Abat_Arrb1    "Abat" "Arrb1"
    #> Abat_Atp6v1c1 "Abat" "Atp6v1c1"
    #> Abat_Atrnl1   "Abat" "Atrnl1"
    #> Abat_Bai1     "Abat" "Bai1"
    #> Abat_Calm1    "Abat" "Calm1"

    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
    #>                gene_1    gene_2
    #> Dnm1l_Gucy1b3  "Dnm1l"   "Gucy1b3"
    #> Bai1_Vdac3     "Bai1"    "Vdac3"
    #> Igf2_Tuba1a    "Igf2"    "Tuba1a"
    #> Gm15421_Myo10  "Gm15421" "Myo10"
    #> Mtor_Tuba1a    "Mtor"    "Tuba1a"
    #> Cldn11_Luzp2   "Cldn11"  "Luzp2"
    #> Calm1_Wdfy3    "Calm1"   "Wdfy3"
    #> Atrnl1_Calm1   "Atrnl1"  "Calm1"
    #> Atrnl1_Dnm3    "Atrnl1"  "Dnm3"
    #> Mlc1_Uchl1     "Mlc1"    "Uchl1"
    #> Ildr2_Tmem47   "Ildr2"   "Tmem47"
    #> Actb_Gm15421   "Actb"    "Gm15421"
    #> Atrnl1_Wdfy3   "Atrnl1"  "Wdfy3"
    #> Gucy1b3_Ildr2  "Gucy1b3" "Ildr2"
    #> Fam63b_Gm15421 "Fam63b"  "Gm15421"
    #> Scd2_Wdfy3     "Scd2"    "Wdfy3"
    #> Abat_Prrc2b    "Abat"    "Prrc2b"
    #> Ildr2_Nfib     "Ildr2"   "Nfib"
    #> Cst3_Mlc1      "Cst3"    "Mlc1"
    #> Cldn11_Mlc1    "Cldn11"  "Mlc1"
    #> Arrb1_Mtor     "Arrb1"   "Mtor"
    #> Dnm1l_Fam63b   "Dnm1l"   "Fam63b"

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"))
    #> [1] 262 262

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")
    #> DataFrame with 22 rows and 3 columns
    #>                    gene_1      gene_2 globalHigherOrderFunction
    #>               <character> <character>                  <matrix>
    #> Dnm1l_Gucy1b3       Dnm1l     Gucy1b3                 0.0635577
    #> Bai1_Vdac3           Bai1       Vdac3                 0.0382758
    #> Igf2_Tuba1a          Igf2      Tuba1a                 0.1976297
    #> Gm15421_Myo10     Gm15421       Myo10                 0.0348911
    #> Mtor_Tuba1a          Mtor      Tuba1a                 0.0810291
    #> ...                   ...         ...                       ...
    #> Ildr2_Nfib          Ildr2        Nfib                -0.0164664
    #> Cst3_Mlc1            Cst3        Mlc1                 0.4294438
    #> Cldn11_Mlc1        Cldn11        Mlc1                 0.0168096
    #> Arrb1_Mtor          Arrb1        Mtor                 0.0762560
    #> Dnm1l_Fam63b        Dnm1l      Fam63b                 0.1251182

    head(diag(cor(t(assay(scHOT_spatial, "expression")[pairs[,1],]),
             t(assay(scHOT_spatial, "expression")[pairs[,2],]),
             method = "spearman")))
    #> [1]  0.06355767  0.03827582  0.19762970  0.03489105  0.08102912 -0.07465933

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")
    #> DataFrame with 22 rows and 5 columns
    #>                    gene_1      gene_2 globalHigherOrderFunction
    #>               <character> <character>                  <matrix>
    #> Dnm1l_Gucy1b3       Dnm1l     Gucy1b3                 0.0635577
    #> Bai1_Vdac3           Bai1       Vdac3                 0.0382758
    #> Igf2_Tuba1a          Igf2      Tuba1a                 0.1976297
    #> Gm15421_Myo10     Gm15421       Myo10                 0.0348911
    #> Mtor_Tuba1a          Mtor      Tuba1a                 0.0810291
    #> ...                   ...         ...                       ...
    #> Ildr2_Nfib          Ildr2        Nfib                -0.0164664
    #> Cst3_Mlc1            Cst3        Mlc1                 0.4294438
    #> Cldn11_Mlc1        Cldn11        Mlc1                 0.0168096
    #> Arrb1_Mtor          Arrb1        Mtor                 0.0762560
    #> Dnm1l_Fam63b        Dnm1l      Fam63b                 0.1251182
    #>               numberPermutations storePermutations
    #>                        <numeric>         <logical>
    #> Dnm1l_Gucy1b3                  0              TRUE
    #> Bai1_Vdac3                     0              TRUE
    #> Igf2_Tuba1a                    0              TRUE
    #> Gm15421_Myo10                 50              TRUE
    #> Mtor_Tuba1a                    0              TRUE
    #> ...                          ...               ...
    #> Ildr2_Nfib                    50              TRUE
    #> Cst3_Mlc1                     50              TRUE
    #> Cldn11_Mlc1                    0              TRUE
    #> Arrb1_Mtor                     0              TRUE
    #> Dnm1l_Fam63b                   0              TRUE

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")
    #> DataFrame with 22 rows and 7 columns
    #>                    gene_1      gene_2 globalHigherOrderFunction
    #>               <character> <character>                  <matrix>
    #> Dnm1l_Gucy1b3       Dnm1l     Gucy1b3                 0.0635577
    #> Bai1_Vdac3           Bai1       Vdac3                 0.0382758
    #> Igf2_Tuba1a          Igf2      Tuba1a                 0.1976297
    #> Gm15421_Myo10     Gm15421       Myo10                 0.0348911
    #> Mtor_Tuba1a          Mtor      Tuba1a                 0.0810291
    #> ...                   ...         ...                       ...
    #> Ildr2_Nfib          Ildr2        Nfib                -0.0164664
    #> Cst3_Mlc1            Cst3        Mlc1                 0.4294438
    #> Cldn11_Mlc1        Cldn11        Mlc1                 0.0168096
    #> Arrb1_Mtor          Arrb1        Mtor                 0.0762560
    #> Dnm1l_Fam63b        Dnm1l      Fam63b                 0.1251182
    #>               numberPermutations storePermutations
    #>                        <numeric>         <logical>
    #> Dnm1l_Gucy1b3                  0              TRUE
    #> Bai1_Vdac3                     0              TRUE
    #> Igf2_Tuba1a                    0              TRUE
    #> Gm15421_Myo10                 50              TRUE
    #> Mtor_Tuba1a                    0              TRUE
    #> ...                          ...               ...
    #> Ildr2_Nfib                    50              TRUE
    #> Cst3_Mlc1                     50              TRUE
    #> Cldn11_Mlc1                    0              TRUE
    #> Arrb1_Mtor                     0              TRUE
    #> Dnm1l_Fam63b                   0              TRUE
    #>                                higherOrderSequence higherOrderStatistic
    #>                                      <NumericList>            <numeric>
    #> Dnm1l_Gucy1b3    0.1619819,0.0935048,0.3071218,...             0.276894
    #> Bai1_Vdac3     0.0173442,-0.6709547,-0.1768493,...             0.368065
    #> Igf2_Tuba1a         0.841093,0.751830,0.817549,...             0.311370
    #> Gm15421_Myo10  0.1327957,-0.0544596,-0.0364751,...             0.285366
    #> Mtor_Tuba1a    0.4097516,-0.0465381, 0.0754223,...             0.345202
    #> ...                                            ...                  ...
    #> Ildr2_Nfib     0.0684620,-0.1097042, 0.0142888,...             0.281278
    #> Cst3_Mlc1        -0.320555, 0.234777,-0.228425,...             0.289625
    #> Cldn11_Mlc1      -0.504075,-0.197786,-0.346225,...             0.321270
    #> Arrb1_Mtor          0.290687,0.420980,0.301204,...             0.323072
    #> Dnm1l_Fam63b  -0.0531722, 0.2088504, 0.0133499,...             0.364277

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))
    #> Permutation testing combination 4 of 22...
    #> Permutation testing combination 8 of 22...
    #> Permutation testing combination 14 of 22...
    #> Permutation testing combination 18 of 22...
    #> Permutation testing combination 19 of 22...
    #>    user  system elapsed
    #>  15.060   0.002  15.065

    slot(scHOT_spatial, "scHOT_output")
    #> DataFrame with 22 rows and 10 columns
    #>                    gene_1      gene_2 globalHigherOrderFunction
    #>               <character> <character>                  <matrix>
    #> Dnm1l_Gucy1b3       Dnm1l     Gucy1b3                 0.0635577
    #> Bai1_Vdac3           Bai1       Vdac3                 0.0382758
    #> Igf2_Tuba1a          Igf2      Tuba1a                 0.1976297
    #> Gm15421_Myo10     Gm15421       Myo10                 0.0348911
    #> Mtor_Tuba1a          Mtor      Tuba1a                 0.0810291
    #> ...                   ...         ...                       ...
    #> Ildr2_Nfib          Ildr2        Nfib                -0.0164664
    #> Cst3_Mlc1            Cst3        Mlc1                 0.4294438
    #> Cldn11_Mlc1        Cldn11        Mlc1                 0.0168096
    #> Arrb1_Mtor          Arrb1        Mtor                 0.0762560
    #> Dnm1l_Fam63b        Dnm1l      Fam63b                 0.1251182
    #>               numberPermutations storePermutations
    #>                        <numeric>         <logical>
    #> Dnm1l_Gucy1b3                  0              TRUE
    #> Bai1_Vdac3                     0              TRUE
    #> Igf2_Tuba1a                    0              TRUE
    #> Gm15421_Myo10                 50              TRUE
    #> Mtor_Tuba1a                    0              TRUE
    #> ...                          ...               ...
    #> Ildr2_Nfib                    50              TRUE
    #> Cst3_Mlc1                     50              TRUE
    #> Cldn11_Mlc1                    0              TRUE
    #> Arrb1_Mtor                     0              TRUE
    #> Dnm1l_Fam63b                   0              TRUE
    #>                                higherOrderSequence higherOrderStatistic
    #>                                      <NumericList>            <numeric>
    #> Dnm1l_Gucy1b3    0.1619819,0.0935048,0.3071218,...             0.276894
    #> Bai1_Vdac3     0.0173442,-0.6709547,-0.1768493,...             0.368065
    #> Igf2_Tuba1a         0.841093,0.751830,0.817549,...             0.311370
    #> Gm15421_Myo10  0.1327957,-0.0544596,-0.0364751,...             0.285366
    #> Mtor_Tuba1a    0.4097516,-0.0465381, 0.0754223,...             0.345202
    #> ...                                            ...                  ...
    #> Ildr2_Nfib     0.0684620,-0.1097042, 0.0142888,...             0.281278
    #> Cst3_Mlc1        -0.320555, 0.234777,-0.228425,...             0.289625
    #> Cldn11_Mlc1      -0.504075,-0.197786,-0.346225,...             0.321270
    #> Arrb1_Mtor          0.290687,0.420980,0.301204,...             0.323072
    #> Dnm1l_Fam63b  -0.0531722, 0.2088504, 0.0133499,...             0.364277
    #>                                 permutations pvalPermutations FDRPermutations
    #>                                <NumericList>        <numeric>       <numeric>
    #> Dnm1l_Gucy1b3                             NA               NA              NA
    #> Bai1_Vdac3                                NA               NA              NA
    #> Igf2_Tuba1a                               NA               NA              NA
    #> Gm15421_Myo10 0.311944,0.312378,0.324099,...             0.94            0.98
    #> Mtor_Tuba1a                               NA               NA              NA
    #> ...                                      ...              ...             ...
    #> Ildr2_Nfib    0.310644,0.347605,0.368242,...             0.98            0.98
    #> Cst3_Mlc1     0.249664,0.273539,0.280372,...             0.22            0.70
    #> Cldn11_Mlc1                               NA               NA              NA
    #> Arrb1_Mtor                                NA               NA              NA
    #> Dnm1l_Fam63b                              NA               NA              NA

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")
    #> DataFrame with 22 rows and 15 columns
    #>                    gene_1      gene_2 globalHigherOrderFunction
    #>               <character> <character>                  <matrix>
    #> Dnm1l_Gucy1b3       Dnm1l     Gucy1b3                 0.0635577
    #> Bai1_Vdac3           Bai1       Vdac3                 0.0382758
    #> Igf2_Tuba1a          Igf2      Tuba1a                 0.1976297
    #> Gm15421_Myo10     Gm15421       Myo10                 0.0348911
    #> Mtor_Tuba1a          Mtor      Tuba1a                 0.0810291
    #> ...                   ...         ...                       ...
    #> Ildr2_Nfib          Ildr2        Nfib                -0.0164664
    #> Cst3_Mlc1            Cst3        Mlc1                 0.4294438
    #> Cldn11_Mlc1        Cldn11        Mlc1                 0.0168096
    #> Arrb1_Mtor          Arrb1        Mtor                 0.0762560
    #> Dnm1l_Fam63b        Dnm1l      Fam63b                 0.1251182
    #>               numberPermutations storePermutations
    #>                        <numeric>         <logical>
    #> Dnm1l_Gucy1b3                  0              TRUE
    #> Bai1_Vdac3                     0              TRUE
    #> Igf2_Tuba1a                    0              TRUE
    #> Gm15421_Myo10                 50              TRUE
    #> Mtor_Tuba1a                    0              TRUE
    #> ...                          ...               ...
    #> Ildr2_Nfib                    50              TRUE
    #> Cst3_Mlc1                     50              TRUE
    #> Cldn11_Mlc1                    0              TRUE
    #> Arrb1_Mtor                     0              TRUE
    #> Dnm1l_Fam63b                   0              TRUE
    #>                                higherOrderSequence higherOrderStatistic
    #>                                      <NumericList>            <numeric>
    #> Dnm1l_Gucy1b3    0.1619819,0.0935048,0.3071218,...             0.276894
    #> Bai1_Vdac3     0.0173442,-0.6709547,-0.1768493,...             0.368065
    #> Igf2_Tuba1a         0.841093,0.751830,0.817549,...             0.311370
    #> Gm15421_Myo10  0.1327957,-0.0544596,-0.0364751,...             0.285366
    #> Mtor_Tuba1a    0.4097516,-0.0465381, 0.0754223,...             0.345202
    #> ...                                            ...                  ...
    #> Ildr2_Nfib     0.0684620,-0.1097042, 0.0142888,...             0.281278
    #> Cst3_Mlc1        -0.320555, 0.234777,-0.228425,...             0.289625
    #> Cldn11_Mlc1      -0.504075,-0.197786,-0.346225,...             0.321270
    #> Arrb1_Mtor          0.290687,0.420980,0.301204,...             0.323072
    #> Dnm1l_Fam63b  -0.0531722, 0.2088504, 0.0133499,...             0.364277
    #>                                 permutations pvalPermutations FDRPermutations
    #>                                <NumericList>        <numeric>       <numeric>
    #> Dnm1l_Gucy1b3                             NA               NA              NA
    #> Bai1_Vdac3                                NA               NA              NA
    #> Igf2_Tuba1a                               NA               NA              NA
    #> Gm15421_Myo10 0.311944,0.312378,0.324099,...             0.94            0.98
    #> Mtor_Tuba1a                               NA               NA              NA
    #> ...                                      ...              ...             ...
    #> Ildr2_Nfib    0.310644,0.347605,0.368242,...             0.98            0.98
    #> Cst3_Mlc1     0.249664,0.273539,0.280372,...             0.22            0.70
    #> Cldn11_Mlc1                               NA               NA              NA
    #> Arrb1_Mtor                                NA               NA              NA
    #> Dnm1l_Fam63b                              NA               NA              NA
    #>               numberPermutationsEstimated globalLowerRangeEstimated
    #>                                 <integer>                 <numeric>
    #> Dnm1l_Gucy1b3                         100                -0.0164664
    #> Bai1_Vdac3                            150                -0.0422096
    #> Igf2_Tuba1a                            50                 0.1789213
    #> Gm15421_Myo10                         150                -0.0422096
    #> Mtor_Tuba1a                           150                -0.0164664
    #> ...                                   ...                       ...
    #> Ildr2_Nfib                            150                -0.0422096
    #> Cst3_Mlc1                              50                 0.4294438
    #> Cldn11_Mlc1                           150                -0.0422096
    #> Arrb1_Mtor                            100                -0.0164664
    #> Dnm1l_Fam63b                          100                 0.0348911
    #>               globalUpperRangeEstimated pvalEstimated FDREstimated
    #>                               <numeric>     <numeric>    <numeric>
    #> Dnm1l_Gucy1b3                 0.0348911     1.0000000     1.000000
    #> Bai1_Vdac3                    0.0348911     0.0533333     0.513333
    #> Igf2_Tuba1a                   0.1789213     0.6800000     0.997333
    #> Gm15421_Myo10                 0.0348911     0.9466667     0.998730
    #> Mtor_Tuba1a                   0.1789213     0.1933333     0.605000
    #> ...                                 ...           ...          ...
    #> Ildr2_Nfib                    0.0348911      0.953333     0.998730
    #> Cst3_Mlc1                     0.4294438      0.220000     0.605000
    #> Cldn11_Mlc1                   0.0348911      0.426667     0.782222
    #> Arrb1_Mtor                    0.0348911      0.410000     0.782222
    #> Dnm1l_Fam63b                  0.1789213      0.070000     0.513333

    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
    #> class: scHOT
    #> dim: 43 262
    #> metadata(0):
    #> assays(1): expression
    #> rownames(43): Abat Actb ... Vdac3 Wdfy3
    #> rowData names(0):
    #> colnames(262): 16.92x9.015 16.945x11.075 ... 27.018x20.088
    #>   17.964x10.137
    #> colData names(4): x y sizeFactor -x
    #> reducedDimNames(0):
    #> mainExpName: NULL
    #> altExpNames(0):
    #> testingScaffold dim: 22 2
    #> weightMatrix dim: 262 262
    #> scHOT_output colnames (0):
    #> param names (3): higherOrderFunctionType higherOrderFunction
    #>   higherOrderSummaryFunction
    #> position type: spatial
    slot(scHOT_spatial, "scHOT_output")
    #> DataFrame with 0 rows and 0 columns

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")
    #> DataFrame with 22 rows and 15 columns
    #>                    gene_1      gene_2 globalHigherOrderFunction
    #>               <character> <character>                  <matrix>
    #> Dnm1l_Gucy1b3       Dnm1l     Gucy1b3                 0.0635577
    #> Bai1_Vdac3           Bai1       Vdac3                 0.0382758
    #> Igf2_Tuba1a          Igf2      Tuba1a                 0.1976297
    #> Gm15421_Myo10     Gm15421       Myo10                 0.0348911
    #> Mtor_Tuba1a          Mtor      Tuba1a                 0.0810291
    #> ...                   ...         ...                       ...
    #> Ildr2_Nfib          Ildr2        Nfib                -0.0164664
    #> Cst3_Mlc1            Cst3        Mlc1                 0.4294438
    #> Cldn11_Mlc1        Cldn11        Mlc1                 0.0168096
    #> Arrb1_Mtor          Arrb1        Mtor                 0.0762560
    #> Dnm1l_Fam63b        Dnm1l      Fam63b                 0.1251182
    #>               numberPermutations storePermutations
    #>                        <numeric>         <logical>
    #> Dnm1l_Gucy1b3                  0              TRUE
    #> Bai1_Vdac3                     0              TRUE
    #> Igf2_Tuba1a                    0              TRUE
    #> Gm15421_Myo10                  0              TRUE
    #> Mtor_Tuba1a                   50              TRUE
    #> ...                          ...               ...
    #> Ildr2_Nfib                     0              TRUE
    #> Cst3_Mlc1                      0              TRUE
    #> Cldn11_Mlc1                    0              TRUE
    #> Arrb1_Mtor                    50              TRUE
    #> Dnm1l_Fam63b                   0              TRUE
    #>                                higherOrderSequence higherOrderStatistic
    #>                                      <NumericList>            <numeric>
    #> Dnm1l_Gucy1b3    0.1619819,0.0935048,0.3071218,...             0.276894
    #> Bai1_Vdac3     0.0173442,-0.6709547,-0.1768493,...             0.368065
    #> Igf2_Tuba1a         0.841093,0.751830,0.817549,...             0.311370
    #> Gm15421_Myo10  0.1327957,-0.0544596,-0.0364751,...             0.285366
    #> Mtor_Tuba1a    0.4097516,-0.0465381, 0.0754223,...             0.345202
    #> ...                                            ...                  ...
    #> Ildr2_Nfib     0.0684620,-0.1097042, 0.0142888,...             0.281278
    #> Cst3_Mlc1        -0.320555, 0.234777,-0.228425,...             0.289625
    #> Cldn11_Mlc1      -0.504075,-0.197786,-0.346225,...             0.321270
    #> Arrb1_Mtor          0.290687,0.420980,0.301204,...             0.323072
    #> Dnm1l_Fam63b  -0.0531722, 0.2088504, 0.0133499,...             0.364277
    #>                                 permutations pvalPermutations FDRPermutations
    #>                                <NumericList>        <numeric>       <numeric>
    #> Dnm1l_Gucy1b3                             NA               NA              NA
    #> Bai1_Vdac3                                NA               NA              NA
    #> Igf2_Tuba1a                               NA               NA              NA
    #> Gm15421_Myo10                             NA               NA              NA
    #> Mtor_Tuba1a   0.314568,0.334639,0.297245,...             0.26             0.7
    #> ...                                      ...              ...             ...
    #> Ildr2_Nfib                                NA               NA              NA
    #> Cst3_Mlc1                                 NA               NA              NA
    #> Cldn11_Mlc1                               NA               NA              NA
    #> Arrb1_Mtor    0.299674,0.280243,0.321495,...             0.42             0.7
    #> Dnm1l_Fam63b                              NA               NA              NA
    #>               numberPermutationsEstimated globalLowerRangeEstimated
    #>                                 <integer>                 <numeric>
    #> Dnm1l_Gucy1b3                         200                 0.0434098
    #> Bai1_Vdac3                            200                 0.0434098
    #> Igf2_Tuba1a                            50                 0.1789213
    #> Gm15421_Myo10                         200                 0.0434098
    #> Mtor_Tuba1a                           250                 0.0434098
    #> ...                                   ...                       ...
    #> Ildr2_Nfib                            150                 0.0434098
    #> Cst3_Mlc1                             250                 0.0434098
    #> Cldn11_Mlc1                           200                 0.0434098
    #> Arrb1_Mtor                            200                 0.0434098
    #> Dnm1l_Fam63b                          250                 0.0434098
    #>               globalUpperRangeEstimated pvalEstimated FDREstimated
    #>                               <numeric>     <numeric>    <numeric>
    #> Dnm1l_Gucy1b3                 0.0861924         0.970     0.970000
    #> Bai1_Vdac3                    0.0861924         0.045     0.469333
    #> Igf2_Tuba1a                   0.1789213         0.540     0.913846
    #> Gm15421_Myo10                 0.0861924         0.955     0.970000
    #> Mtor_Tuba1a                   0.1789213         0.192     0.616000
    #> ...                                 ...           ...          ...
    #> Ildr2_Nfib                    0.0810291         0.960     0.970000
    #> Cst3_Mlc1                     0.1789213         0.920     0.970000
    #> Cldn11_Mlc1                   0.0861924         0.510     0.913846
    #> Arrb1_Mtor                    0.0861924         0.485     0.913846
    #> Dnm1l_Fam63b                  0.1789213         0.064     0.469333

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"))

# Misc

    sessionInfo()
    #> R version 4.6.0 (2026-04-24)
    #> Platform: x86_64-pc-linux-gnu
    #> Running under: Ubuntu 24.04.4 LTS
    #>
    #> Matrix products: default
    #> BLAS:   /usr/lib/x86_64-linux-gnu/openblas-pthread/libblas.so.3
    #> LAPACK: /usr/lib/x86_64-linux-gnu/openblas-pthread/libopenblasp-r0.3.26.so;  LAPACK version 3.12.0
    #>
    #> locale:
    #>  [1] LC_CTYPE=en_US.UTF-8       LC_NUMERIC=C
    #>  [3] LC_TIME=en_US.UTF-8        LC_COLLATE=en_US.UTF-8
    #>  [5] LC_MONETARY=en_US.UTF-8    LC_MESSAGES=en_US.UTF-8
    #>  [7] LC_PAPER=en_US.UTF-8       LC_NAME=C
    #>  [9] LC_ADDRESS=C               LC_TELEPHONE=C
    #> [11] LC_MEASUREMENT=en_US.UTF-8 LC_IDENTIFICATION=C
    #>
    #> time zone: Etc/UTC
    #> tzcode source: system (glibc)
    #>
    #> attached base packages:
    #> [1] stats4    stats     graphics  grDevices utils     datasets  methods
    #> [8] base
    #>
    #> other attached packages:
    #>  [1] scater_1.41.1               scuttle_1.23.1
    #>  [3] scHOT_1.25.0                ggplot2_4.0.3
    #>  [5] SingleCellExperiment_1.35.1 SummarizedExperiment_1.43.0
    #>  [7] Biobase_2.73.1              GenomicRanges_1.65.0
    #>  [9] Seqinfo_1.3.0               IRanges_2.47.1
    #> [11] S4Vectors_0.51.2            BiocGenerics_0.59.3
    #> [13] generics_0.1.4              MatrixGenerics_1.25.0
    #> [15] matrixStats_1.5.0
    #>
    #> loaded via a namespace (and not attached):
    #>  [1] beeswarm_0.4.0      gtable_0.3.6        xfun_0.57
    #>  [4] bslib_0.11.0        ggrepel_0.9.8       lattice_0.22-9
    #>  [7] vctrs_0.7.3         tools_4.6.0         parallel_4.6.0
    #> [10] pkgconfig_2.0.3     BiocNeighbors_2.7.2 Matrix_1.7-5
    #> [13] RColorBrewer_1.1-3  S7_0.2.2            lifecycle_1.0.5
    #> [16] deldir_2.0-4        compiler_4.6.0      farver_2.1.2
    #> [19] ggforce_0.5.0       codetools_0.2-20    scattermore_1.2
    #> [22] vipor_0.4.7         htmltools_0.5.9     sys_3.4.3
    #> [25] buildtools_1.0.0    sass_0.4.10         yaml_2.3.12
    #> [28] jquerylib_0.1.4     MASS_7.3-65         BiocParallel_1.47.0
    #> [31] DelayedArray_0.39.2 cachem_1.1.0        viridis_0.6.5
    #> [34] abind_1.4-8         nlme_3.1-169        rsvd_1.0.5
    #> [37] tidyselect_1.2.1    digest_0.6.39       BiocSingular_1.29.0
    #> [40] splines_4.6.0       labeling_0.4.3      maketools_1.3.2
    #> [43] polyclip_1.10-7     fastmap_1.2.0       grid_4.6.0
    #> [46] cli_3.6.6           SparseArray_1.13.2  magrittr_2.0.5
    #> [49] S4Arrays_1.13.0     withr_3.0.2         scales_1.4.0
    #> [52] ggbeeswarm_0.7.3    rmarkdown_2.31      XVector_0.53.0
    #> [55] igraph_2.3.1        gridExtra_2.3       ScaledMatrix_1.21.0
    #> [58] beachmat_2.29.0     evaluate_1.0.5      knitr_1.51
    #> [61] viridisLite_0.4.3   irlba_2.3.7         mgcv_1.9-4
    #> [64] rlang_1.2.0         Rcpp_1.1.1-1.1      glue_1.8.1
    #> [67] tweenr_2.0.3        reshape_0.8.10      jsonlite_2.0.0
    #> [70] R6_2.6.1            plyr_1.8.9