---
title: "A framework to discover Spatially Variable genes via spatial clusters"
author:
- name: Peiying Cai
  affiliation:
  - &IMLS Institute for Molecular Life Sciences, University of Zurich, Switzerland
  - &SIB SIB Swiss Institute of Bioinformatics, University of Zurich, Switzerland
  email: peiying.cai@uzh.ch
- name: Simone Tiberi
  affiliation:
  - &Stats Departiment of Statistical Sciences, University of Bologna, Italy
  email: simone.tiberi@unibo.it
date: "`r format(Sys.Date(), '%m/%d/%Y')`"
bibliography: References.bib
vignette: >
  %\VignetteIndexEntry{A framework to discover spatially variable genes}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
output: 
  BiocStyle::html_document
editor_options: 
  chunk_output_type: console
  markdown: 
    wrap: sentence
---

------------------------------------------------------------------------

```{r setup, echo=FALSE, results="hide"}
knitr::opts_chunk$set(tidy=FALSE, cache=TRUE,
                        dev="png",
                        message=TRUE, error=FALSE, warning=TRUE)
library(utils)
```

# Introduction

*DESpace* is an intuitive framework for identifying spatially variable (SV) genes (SVGs) via *edgeR* [@edgeR], one of the most common methods for performing differential expression analyses.

Based on pre-annotated spatial clusters as summarized spatial information, *DESpace* models gene expression using a negative binomial (NB), via *edgeR* [@edgeR], with spatial clusters as covariates.
SV genes (SVGs) are then identified by testing the significance of spatial clusters.

Our approach assumes that the spatial structure can be summarized by spatial clusters, which should reproduce the key features of the tissue (e.g., white matter and layers in brain cortex).
A significant test of these covariates indicates that space influences gene expression, hence identifying spatially variable genes.

Our model is flexible and robust, and is significantly faster than the most SV methods.
Furthermore, to the best of our knowledge, it is the only SV approach that allows:
- performing a SV test on each individual spatial cluster, hence identifying the key regions affected by spatial variability;
- jointly fitting multiple samples, targeting genes with consistent spatial patterns across biological replicates.

Below, we illustrate en example usage of the package.

# Basics

`DESpace` is implemented as a R package within Bioconductor, which is the main venue for omics analyses, and we use various other Bioconductor packages (e.g., SpatialLIBD, and edgeR).

`DESpace` package is available on Bioconductor and can be installed with the following command:

```{r, eval=FALSE}
if (!requireNamespace("BiocManager", quietly = TRUE)) {
    install.packages("BiocManager")
}

BiocManager::install("DESpace")

## Check that you have a valid Bioconductor installation
BiocManager::valid()
```

The development version of `DESpace`can also be installed from the Bioconductor-devel branch or from GitHub.

To access the R code used in the vignettes, type:

```{r vignettes, eval=FALSE}
browseVignettes("DESpace")
```

Questions relative to *DESpace* should be reported as a new issue at [*BugReports*](https://github.com/peicai/DESpace/issues).

To cite *DESpace*, type:

```{r citation, eval=FALSE}
citation("DESpace")
```

Load R packages:
```{r package}
suppressMessages({
library(DESpace)
library(ggplot2)
library(SpatialExperiment)
})
```

# Data

As an example dataset, we consider a human dorsolateral pre-frontal cortex (DLPFC) spatial transcriptomics dataset from the 10x Genomics Visium platform, including three neurotypical adult donors (i.e., biological replicates), with four images per subject [@LIBD].
The full dataset consists of 12 samples, which can be accessed via [*spatialLIBD*](https://bioconductor.org/packages/release/data/experiment/vignettes/spatialLIBD/inst/doc/spatialLIBD.html) Bioconductor package.

## Input data

Here, we consider a subset of the original data, consisting of three biological replicates: 1 image for each of the three brain subjects.
Initially, in Section 3 *individual sample* , we fit our approach on a single sample, whose data is stored in `spe3` whereas all 3 samples will later be jointly used in Section 4 *Multiple samples*.

```{r load-example-data, message = FALSE}
# Connect to ExperimentHub
ehub <- ExperimentHub::ExperimentHub()
# Download the full real data (about 2.1 GB in RAM) use:
spe_all <- spatialLIBD::fetch_data(type = "spe", eh = ehub)

# Create three spe objects, one per sample:
spe1 <- spe_all[, colData(spe_all)$sample_id == '151507']
spe2 <- spe_all[, colData(spe_all)$sample_id == '151669']
spe3 <- spe_all[, colData(spe_all)$sample_id == '151673']
rm(spe_all)
# Select small set of random genes for faster runtime in this example
set.seed(123)
sel_genes <- sample(dim(spe1)[1],2000)
spe1 <- spe1[sel_genes,]
spe2 <- spe2[sel_genes,]
spe3 <- spe3[sel_genes,]
# For covenience, we use “gene names” instead of “gene ids”:
rownames(spe1) <- rowData(spe1)$gene_name
rownames(spe2) <- rowData(spe2)$gene_name
rownames(spe3) <- rowData(spe3)$gene_name
# Specify column names of spatial coordinates in colData(spe) 
coordinates <- c("array_row", "array_col")
# Specify column names of spatial clusters in colData(spe) 
spatial_cluster <- 'layer_guess_reordered'
```

The spatial tissues of each sample were manually annotated in the original manuscript [@LIBD], and spots were labeled into one of the following categories: white matter (WM) and layers 1 to 6.
The manual annotations are stored in column `layer_guess_reordered` of the `colData`, while columns `array_col` and `array_row` provide the spatial coordinates of spots.

```{r visualize colData}
# We select a subset of columns
keep_col <- c(coordinates,spatial_cluster,"expr_chrM_ratio","cell_count")
head(colData(spe3)[keep_col])
```

## Quality control/filtering

Quality control (QC) procedures at the spot and gene level aim to remove both low-quality spots, and lowly abundant genes.
For QC, we adhere to the instructions from "Orchestrating Spatially Resolved Transcriptomics Analysis with Bioconductor" ([*OSTA*](https://lmweber.org/OSTA-book/quality-control.html)).
The library size, UMI counts, ratio of mitochondrial chromosome (chM) expression, and number of cells per spot are used to identify low-quality spots.

```{r QC spots}
# Sample 1:
# Calculate per-spot QC metrics and store in colData
spe1 <- scuttle::addPerCellQC(spe1,)
# Remove combined set of low-quality spots
spe1 <- spe1[, !(colData(spe1)$sum < 10 |             # library size
                colData(spe1)$detected < 10 |         # number of expressed genes
                colData(spe1)$expr_chrM_ratio > 0.30| # mitochondrial expression ratio
                colData(spe1)$cell_count > 10)]       # number of cells per spot
# Sample 2:
# Calculate per-spot QC metrics and store in colData
spe2 <- scuttle::addPerCellQC(spe2,)
# Remove combined set of low-quality spots
spe2 <- spe2[, !(colData(spe2)$sum < 20 |
                colData(spe2)$detected < 15 |
                colData(spe2)$expr_chrM_ratio > 0.35|
                colData(spe2)$cell_count > 8)]
# Sample 3:
spe3 <- scuttle::addPerCellQC(spe3,)
# Remove combined set of low-quality spots
spe3 <- spe3[, !(colData(spe3)$sum < 25 |
                colData(spe3)$detected < 25 |
                colData(spe3)$expr_chrM_ratio > 0.3|
                colData(spe3)$cell_count > 15)]
```

Then, we discard lowly abundant genes, which were detected in less than 20 spots.

```{r QC genes}
# For each sample i:
for(i in seq_len(3)){
    spe_i <- eval(parse(text = paste0("spe", i)))
    # Select QC threshold for lowly expressed genes: at least 20 non-zero spots:
    qc_low_gene <- rowSums(assays(spe_i)$counts > 0) >= 20
    # Remove lowly abundant genes
    spe_i <- spe_i[qc_low_gene,]
    assign(paste0("spe", i), spe_i)
    message("Dimension of spe", i, ": ", dim(spe_i)[1], ", ", dim(spe_i)[2])
}
```

# Individual sample

We fit our approach to discover SVGs in an individual sample.
In Section 4 *Multiple samples*, we will show how to jointly embed multiple replicates.

## Clustering

This framework relies on spatial clusters being accessible and successfully summarizing the primary spatial characteristics of the data.
In most datasets, these spatial features are either accessible or can be easily generated with spatial clustering algorithms.

### Manual annotation

If manual annotations are provided (e.g., annotated by a pathologist), we can directly use those.
With the `spe` or `spe` object that contains coordinates of the spot-level data, we can visualize spatial clusters.

```{r view LIBD layers, fig.width=5,fig.height=6}
# View LIBD layers for one sample
CD <- as.data.frame(colData(spe3))
ggplot(CD, 
    aes(x=array_col,y=array_row, 
    color=factor(layer_guess_reordered))) +
    geom_point() + 
    theme_void() + scale_y_reverse() + 
    theme(legend.position="bottom") + 
    labs(color = "", title = paste0("Manually annotated spatial clusters"))
```

```{r free memory, include=FALSE}
# To reduce memory usage, we can remove the ehub object, which typically requires a large amount of memory. If needed, the spe_i object can be removed as well, as neither is needed for SV testing.
rm(ehub)
rm(spe_i)
gc()
```

### Spatially resolved clustering

If manual annotations are not available, we can use spatially resolved clustering tools.
These methods, by jointly employing spatial coordinates and gene expression data, enable obtaining spatial clusters.
Although, in this vignette we use pre-computed manually annotated clusters, below we provide links to two popular spatially resolved clustering tools: BayesSpace [@BayesSpace] and StLearn [@stLearn].

#### BayesSpace

BayesSpace is a Bioconductor package that provides a Bayesian statistical approach for spatial transcriptomics data clustering ([*BayesSpace*](https://edward130603.github.io/BayesSpace/index.html)).
There is a specific vignette for using BayesSpace on this dataset (human DLPFC) [*here*](https://edward130603.github.io/BayesSpace/articles/maynard_DLPFC.html).

After using BayesSpace, the spatial cluster assignments (`spatial.cluster`) are available in the `colData(spe)`.

#### StLearn

StLearn, a python-based package, is designed for spatial transciptomics data.
It allows spatially-resolved clustering based on Louvain or k-means ([*stLearn*](https://stlearn.readthedocs.io/en/latest/index.html)).
There is a tutorial for using StLearn on this dataset (human DLPFC) [*here*](https://stlearn.readthedocs.io/en/latest/tutorials/stSME_clustering.html#Human-Brain-dorsolateral-prefrontal-cortex-(DLPFC)).

After running stLearn, we can store results as a `csv` file.

```{python.reticulate = FALSE stLearn[py multi-sample], eval = FALSE}
# Save spatial results
obsm.to_csv("stLearn_clusters.csv") 
```

Then, we can load these results in R and store spatial clusters in the `spe` object.

```{r stLearn[r multi-sample], eval = FALSE}
stLearn_results <- read.csv("stLearn_clusters.csv", sep = ',', 
                            header = TRUE)
# Match colData(spe) and stLearn results
stLearn_results <- stLearn_results[match(rownames(colData(spe3)), 
                                    rownames(stLearn_results)), ]
colData(spe3)$stLearn_clusters <- stLearn_results$stLearn_pca_kmeans
```

## SV testing

Once we have spatial clusters, we can search for SVGs.

### Gene-level test

Fit the model via `DESpace_test` function.
Parameter `spe` specifies the input `SpatialExperiment` or `SingleCellExperiment` object, while `spatial_cluster` defines the column names of `colData(spe)` containing spatial clusters. To obtain all statistics, set `verbose` to `TRUE` (default value).

```{r DESpace}
set.seed(123)
results <- DESpace_test(spe = spe3,
                        spatial_cluster = spatial_cluster, 
                        verbose = TRUE)
```

A list of results is returned.
The main results of interest are stored in the `gene_results`: a `data.fame`, where columns contain gene names (`gene_id`), likelihood ratio test statistics (`LR`), average (across spots) log-2 counts per million (`logCPM`), raw p-values (`PValue`) and Benjamini-Hochberg adjusted p-values (`FDR`).

```{r}
head(results$gene_results, 3)
```

The second element of the results (a `DGEList` object `estimated_y`) contains the estimated common dispersion, which can later be used to speed-up calculation when testing individual clusters.

The third and forth element of the results (`DGEGLM` and `DGELRT` objects) contain full statistics from `edgeR::glmFit` and `edgeR::glmLRT`.

```{r}
class(results$estimated_y); class(results$glmLrt); class(results$glmFit)
```


Visualize the gene expression of the three most significant genes in `FeaturePlot()`.
Note that the gene names in vector `feature`, should also appear in the count matrix’s row names. Specifying the column names of spatial coordinates of spots is only necessary when they are not named `row` and `col`.

```{r top SVGs expression plot}
(feature <- results$gene_results$gene_id[seq_len(3)])
FeaturePlot(spe3, feature, 
            coordinates = coordinates, 
            ncol = 3, title = TRUE)
```

Additionally, function `FeaturePlot()` can draw an outline around each cluster.

```{r expression plots all cluster}
FeaturePlot(spe3, feature, 
            coordinates = coordinates, 
            Annotated_cluster = TRUE,
            spatial_cluster = spatial_cluster, 
            cluster = 'all', 
            legend_cluster = TRUE, title = TRUE)
```

### Individual cluster test

*DESpace* can also be used to reveal the specific areas of the tissue affected by spatial variability; i.e., spatial clusters that are particularly over/under abundant compared to the average.
Function `individual_test()` can be used to identify SVGs for each individual cluster.
Parameter `spatial_cluster` indicates the column names of `colData(spe)` containing spatial clusters.

For every spatial cluster we test, `edgeR` would normally re-compute the dispersion estimates based on the specific design of the test.
However, this calculation represents the majority of the overall computing time.
Therefore, to speed-up calculations, we propose to use the dispersion estimates which were previously computed for the gene-level tests.
Albeit this is an approximation, in our benchmarks, it leads to comparable performance in terms of sensitivity and specificity.
If you want to use pre-computed gene-level dispersion estimates, set `edgeR_y` to `estimated_y`.
Alternatively, if you want to re-compute dispersion estimates (significantly slower, but marginally more accurate option), leave `edgeR_y` empty.

```{r individual cluster test}
set.seed(123)
cluster_results <- individual_test(spe3, 
                                    edgeR_y = results$estimated_y,
                                    spatial_cluster = spatial_cluster)
```

`individual_test()` returns a list containing the results of the individual cluster tests.
Similarly to above, for each cluster, results are reported as a `data.fame`, where columns contain gene names (`gene_id`), likelihood ratio test statistics (`LR`), log2-fold changes (`logFC`), raw p-values (`PValue`) and Benjamini-Hochberg adjusted p-values (`FDR`).
NB: that the `logFC` compares each cluster to the rest of the tissue; e.g., a logFC of 2 for WM test indicates that the average gene expression in WM is (4 times) higher than the average gene expression in non-WM tissue.

Visualize results for WM.

```{r visualize results WM}
class(cluster_results)
names(cluster_results)
```

`top_results` function can be used to combine gene-and cluster-level results.
By default, results from `top_results()` report both adjusted p-values and log2-FC; however, users can also choose to only report either, by specifying `select = "FDR"` or `select = "logFC"`.
Below, `gene_PValue` and `gene_FDR` columns refer to the gene-level testing, while the subsequent columns indicate the cluster-specific results.

```{r top results all}
merge_res <- top_results(results$gene_results, cluster_results)
head(merge_res,3)
merge_res <- top_results(results$gene_results, cluster_results, 
                        select = "FDR")
head(merge_res,3)
```

We can further specify a cluster and check top genes detected by *DESpace*.

```{r top results WM}
# Check top genes for WM
results_WM <- top_results(cluster_results = cluster_results, 
                        cluster = "WM")
head(results_WM, 3)
```

With `high_low` parameter, we can further filter genes to visualize those with higher (`high_low = "high"`) or lower (`high_low = "low"`) average abundance in the specified cluster, compared to the average abundance in the rest of the tissue.
By default, `high_low = “both”` and all results are provided.

```{r top results high_low}
results_WM_both <- top_results(cluster_results = cluster_results, 
                                cluster = "WM", 
                                high_low = "both")
```

Here we present the highly abundant cluster SVGs; i.e., SVGs with higher expression in WM compared to the rest of the area.

```{r show top results high}
head(results_WM_both$high_genes, 3)
```

We visualize the lowly abundant cluster SVGs; i.e., SVGs with lower expression in WM compared to the rest of the area.

```{r show top results low}
head(results_WM_both$low_genes, 3)
```

Visualize the gene expression of the top genes for layer WM.
A cluster outline can be drawn by specifying the column names of clusters stored in `colData(spe)` and the vector of cluster names via `spatial_cluster` and `cluster`.

```{r expression plots high_low}
# SVGs with higher than average abundance in WM
feature <- rownames(results_WM_both$high_genes)[seq_len(3)]
FeaturePlot(spe3, feature, spatial_cluster = spatial_cluster, 
            coordinates = coordinates, cluster = 'WM', 
            legend_cluster = TRUE, Annotated_cluster = TRUE, 
            linewidth = 0.6, title = TRUE)

# SVGs with lower than average abundance in WM
feature <- rownames(results_WM_both$low_genes)[seq_len(3)]
FeaturePlot(spe3, feature, spatial_cluster = spatial_cluster, 
            coordinates = coordinates, cluster = 'WM', 
            legend_cluster = TRUE, Annotated_cluster = TRUE, 
            linewidth = 0.6,title = TRUE)
```

# Multiple samples

If biological replicates are available, our framework allows jointly modeling them to target SVGs with coherent spatial patterns across samples.
This approach may be particularly beneficial when data are characterized by a large degree of (biological and technical) variability, such as in cancer.
Importantly, only genes detected (above filtering thresholds) in all samples will be analyzed.

## Clustering

Similar to gene-level testing, the multi-sample extension requires pre-annotated spatial clusters.

### Manual annotation

If the manual annotation for each sample is available, we can combine all samples and use manual annotations directly.
Note that cluster labels must be consistent across samples (i.e., WM in sample 1 should represent the same tissue as WM in sample 2).
With the `spe.combined` object that contains coordinates of the spot-level data, we can visualize spatial clusters.

```{r view LIBD layers [multi]}
set.seed(123)
# Use common genes
a <- rownames(counts(spe1)); 
b <- rownames(counts(spe2)); 
c <- rownames(counts(spe3))
# find vector of common genes across all samples:
CommonGene <- Reduce(intersect, list(a,b,c))
spe1 <- spe1[CommonGene,]
spe2 <- spe2[CommonGene,]
spe3 <- spe3[CommonGene,]

# Combine three samples
spe.combined <- cbind(spe1, spe2, spe3, deparse.level = 1)
ggplot(as.data.frame(colData(spe.combined)), 
    aes(x=array_col, y=array_row,
    color=factor(layer_guess_reordered))) +
    geom_point() + 
    facet_wrap(~sample_id) +
    theme_void() + scale_y_reverse() +
    theme(legend.position="bottom") + 
    labs(color = "", title = "Manually annotated spatial clusters")
```

### Spatially resolved (multi-sample) clustering

Similarly to above, if manual annotations are not available, we can use spatially resolved clustering tools.
Both BayesSpace [@BayesSpace] and StLearn [@stLearn] allow jointly clustering multiple samples.
In particular each tool has a specific vignettes for multi-testing clustering: [*BayesSpace vignettes*](https://edward130603.github.io/BayesSpace/articles/joint_clustering.html), and [*stLearn vignettes*](https://stlearn.readthedocs.io/en/latest/tutorials/Integration_multiple_datasets.html).

#### Single sample clustering

In our benchmarks, we have noticed that, with both *BayesSpace* and *StLearn*, joint spatial clustering of multiple samples is more prone to failure and inaccurate results than spatial clustering of individual samples.
Therefore, if multi-sample clustering fails, we suggest trying to cluster individual samples (as in Section 3 *Individual sample*) and manually match cluster ids across samples, to ensure that "cluster 1" always refers to the same spatial region in all samples.

## SV testing

Once we have spatial clusters for multiple samples, we add them to `colData(spe.combined)` as the column `layer_guess_reordered` and fit the model with spatial clusters as covariates.

### Gene-level test

Fit the model via `DESpace_test()`.
Parameter `spe` specifies the input `SpatialExperiment` or `SingleCellExperiment` object, while `spatial_cluster` and `sample_col` define the column names of `colData(spe)` containing spatial clusters and sample ids.
With `replicates = TRUE`, we fit the multi-sample model.

The second element of the result (a `DGEList` object `estimated_y_multi`) contains the estimated common dispersion for the multi-sample case.

```{r DESpace [multi]}
set.seed(123)
multi_results <- DESpace_test(spe = spe.combined,
                                spatial_cluster = spatial_cluster,
                                sample_col = 'sample_id',
                                replicates = TRUE)
```

A list of results are returned.
The main results of interest are stored in the `gene_results`.

```{r}
head(multi_results$gene_results,3)
```

The second element of the results (a `DGEList` object `estimated_y`) contains the estimated common dispersion, which can later be used to speed-up calculation when testing individual clusters.

```{r}
class(multi_results$estimated_y)
```

For each sample, we can visualize the gene expression of the most significant SVGs.
Note that column names of spatial coordinates of spots should be `row` and `col`.

```{r top SVGs expression plot [multi]}
## Top three spatially variable genes
feature <- multi_results$gene_results$gene_id[seq_len(3)]; feature
## Sample names
samples <- unique(colData(spe.combined)$sample_id); samples
## Use purrr::map to combine multiple figures
spot_plots <- purrr::map(seq_along(samples), function(j) {
    ## Subset spe for each sample j
    spe_j <- spe.combined[, colData(spe.combined)$sample_id == samples[j] ]
    ## Store three gene expression plots with gene names in `feature` for spe_j
    spot_plots <- FeaturePlot(spe_j, feature, 
                            coordinates = coordinates,
                            spatial_cluster = spatial_cluster, title = TRUE,
                            Annotated_cluster = TRUE, legend_cluster = TRUE)
    return(spot_plots)
})
patchwork::wrap_plots(spot_plots, ncol=1)
```

### Individual cluster test

Similarly to what shown in Section 3 *Individual sample*, our framework can discover the key SV spatial clusters also when jointly fitting multiple samples.
For a multi-sample testing, set `replicates = TRUE` in `individual_test()`.

```{r individual cluster test [multiple-sample]}
set.seed(123)
cluster_results <- individual_test(spe.combined, 
                                edgeR_y = multi_results$estimated_y,
                                replicates = TRUE, 
                                spatial_cluster = spatial_cluster)
```

`individual_test()` returns a list containing the results of individual clusters, specified in `cluster` parameter.
In this case, logFC refers to the log2-FC between the average abundance, across all samples, of a spatial cluster and the average abundance of all remaining clusters (e.g., WM vs. non-WM tissue).

Visualize results for WM.

```{r}
class(cluster_results)
names(cluster_results)
```

As above, `top_results` function can be used to combine gene-level and cluster-level results.

```{r}
merge_res <- top_results(multi_results$gene_results, cluster_results, 
                        select = "FDR")
head(merge_res,3)
```

We can further select a cluster of interest, and check the top genes detected in that cluster.

```{r}
# Check top genes for WM
results_WM <- top_results(cluster_results = cluster_results, 
                        cluster = "WM")
# For each gene, adjusted p-values for each cluster
head(results_WM,3)
```

With `high_low = "both"`, we can further filter genes to visualize highly and lowly abundant SVGs.

```{r}
results_WM_both <- top_results(cluster_results = cluster_results, 
                            cluster = "WM", high_low = "both")
```

Here we present the highly abundant cluster SVGs; i.e., SVGs with higher expression in WM compared to the rest of the tissue.

```{r}
head(results_WM_both$high_genes,3)
```

We visualize the lowly abundant cluster SVGs; i.e., SVGs with lower expression in WM compared to the rest of the tissue.

```{r}
head(results_WM_both$low_genes,3)
```

Visualize the gene expression of top three genes for layer WM.

```{r, fig.width=10, fig.height=15}
# SVGs with higher abundance in WM, than in non-WM tissue
feature_high <- rownames(results_WM_both$high_genes)[seq_len(3)]
# SVGs with lower abundance in WM, than in non-WM tissue
feature_low <- rownames(results_WM_both$low_genes)[seq_len(3)]
plot_list_high <- list(); plot_list_low <- list()
## Sample names
samples <- unique(colData(spe.combined)$sample_id)
for(j in seq_along(samples)){
    ## Subset spe for each sample j
    spe_j <- spe.combined[, colData(spe.combined)$sample_id == samples[j]]
    ## Gene expression plots with top highly abundant cluster SVGs for spe_j
    plot_list_high[[j]] <- FeaturePlot(spe_j, feature_high, 
                                coordinates = coordinates,
                                spatial_cluster = spatial_cluster, 
                                linewidth = 0.6,
                                cluster = 'WM', Annotated_cluster = TRUE,
                                legend_cluster = TRUE, title = TRUE)
    ## Gene expression plots with top lowly abundant cluster SVGs for spe_j
    plot_list_low[[j]] <- FeaturePlot(spe_j, feature_low, 
                                coordinates = coordinates,
                                spatial_cluster = spatial_cluster, 
                                linewidth = 0.6,
                                cluster = 'WM', Annotated_cluster = TRUE,
                                legend_cluster = TRUE, title = TRUE)
}
# Expression plots for SVGs with higher abundance in WM, than in non-WM tissue
patchwork::wrap_plots(plot_list_high, ncol=1)
# Expression plots for SVGs with lower abundance in WM, than in non-WM tissue
patchwork::wrap_plots(plot_list_low, ncol=1)
```

### Sample-specific covariates (e.g., batch effects)
If sample-specific covariates, such as batch effects, are available, we can account for them in *DESpace*.
The adjustment works as in the *edgeR* original framework: the mean of the negative binomial model is expressed as a function of spatial clusters, and additional nuisance covariates; differential testing is then performed on spatial clusters only, to identify SVGs.
Note that sample-specific covariates can be used instead of samples, but a joint modelling of both samples and (sample-specific) covariates is not possible because the two variables are nested. 

To show an example application, we artificially separate samples in 2 batches:
```{r DESpace [add batch]}
spe.combined$batch_id = ifelse(spe.combined$sample_id == "151507", "batch_1", "batch_2")

table(spe.combined$batch_id, spe.combined$sample_id)
```

Analyses are performed, as explained above, in Section 5; yet, when running `DESpace_test`, we set the `sample_col` to `batch_id`:
```{r DESpace [batch test], eval = FALSE}
set.seed(123)
batch_results <- DESpace_test(spe = spe.combined,
                                spatial_cluster = spatial_cluster,
                                sample_col = 'batch_id',
                                replicates = TRUE)
```


# Session info

```{r, sessionInfo}
sessionInfo()
```

# References