Single-cell mRNA sequencing can uncover novel cell-to-cell heterogeneity in gene expression levels within seemingly homogeneous populations of cells. However, these experiments are prone to high levels of technical noise, creating new challenges for identifying genes that show genuine heterogeneous expression within the group of cells under study.
BASiCS (Bayesian Analysis of Single-Cell Sequencing data) is an integrated Bayesian hierarchical model that propagates statistical uncertainty by simultaneously performing data normalisation (global scaling), technical noise quantification and two types of supervised downstream analyses:
Regression = TRUE
as a function parameter in
BASiCS_MCMC
.In all cases, a probabilistic output is provided and a decision rule is calibrated using the expected false discovery rate (EFDR) (Newton et al. 2004).
A brief description for the statistical model implemented in BASiCS
is provided in Section @ref(methodology) of this document. The original
implementation of BASiCS (Vallejos, Marioni, and
Richardson 2015) requires the use of spike-in
molecules — that are artificially introduced to each cell’s lysate — to
perform these analyses. More recently, Eling et
al. (2018) extendeded BASiCS to also address datasets for which
spikes-ins are not available (see Section
@ref(alternative-implementation-modes)). To use this feature, please set
WithSpikes = FALSE
as a function parameter in
BASiCS_MCMC
.
Important: BASiCS has been designed in the context of supervised experiments where the groups of cells (e.g. experimental conditions, cell types) under study are known a priori (e.g. case-control studies). Therefore, we DO NOT advise the use of BASiCS in unsupervised settings where the aim is to uncover sub-populations of cells through clustering.
Parameter estimation is performed using the BASiCS_MCMC
function. Downstream analyses implemented in BASiCS rely on appropriate
post-processing of the output returned by BASiCS_MCMC
.
Essential parameters for running BASiCS_MCMC
are:
Data
: a SingleCellExperiment
object
created as in Section @ref(the-input-dataset)N
: total number of iterationsThin
: length of the thining period (i.e. only every
Thin
iterations will be stored in the output of the
BASiCS_MCMC
)Burn
: length of burn-in period (i.e. the initial
Burn
iterations that will be discarded from the output of
the BASiCS_MCMC
)Regression
: if this parameter is set equal to
TRUE
, the regression BASiCS model will be used (Eling et al. 2018). The latter infers a global
regression trend between mean expression and over-dispersion parameters.
This trend is subsequently used to derive a residual
over-dispersion measure that is defined as departures with respect
to this trend. This is now the recommended default setting for
BASiCS*.If the optional parameter PrintProgress
is set to
TRUE
, the R console will display the progress of the MCMC
algorithm. For other optional parameters refer to
help(BASiCS_MCMC)
.
Here, we illustrate the usage of BASiCS_MCMC
using a
built-in synthetic dataset. As the outcome of this function is
stochastic, users should expect small variations across different runs.
To ensure reproducible results, users may consider using a fixed seed
(as illustrated in the following code).
NOTE: WE USE A SMALL NUMBER OF ITERATIONS FOR ILLUSTRATION
PURPOSES ONLY. LARGER NUMBER OF ITERATIONS ARE USUALLY REQUIRED TO
ACHIEVE CONVERGENCE. OUR RECOMMENDED SETTING IS N=20000
,
Thin=20
and Burn=10000
.
set.seed(1)
Data <- makeExampleBASiCS_Data()
Chain <- BASiCS_MCMC(
Data = Data,
N = 1000, Thin = 10, Burn = 500,
PrintProgress = FALSE, Regression = TRUE
)
## -----------------------------------------------------
## MCMC sampler has been started: 1000 iterations to go.
## -----------------------------------------------------
## -----------------------------------------------------
## End of Burn-in period.
## -----------------------------------------------------
##
## -----------------------------------------------------
## -----------------------------------------------------
## All 1000 MCMC iterations have been completed.
## -----------------------------------------------------
## -----------------------------------------------------
##
## -----------------------------------------------------
## Please see below a summary of the overall acceptance rates.
## -----------------------------------------------------
##
## Minimum acceptance rate among mu[i]'s: 0.46
## Average acceptance rate among mu[i]'s: 0.61496
## Maximum acceptance rate among mu[i]'s: 0.816
##
##
## Minimum acceptance rate among delta[i]'s: 0.43
## Average acceptance rate among delta[i]'s: 0.52244
## Maximum acceptance rate among delta[i]'s: 0.608
##
##
## Acceptance rate for phi (joint): 0.882
##
##
## Minimum acceptance rate among nu[j]'s: 0.424
## Average acceptance rate among nu[j]'s: 0.541133
## Maximum acceptance rate among nu[j]'s: 0.726
##
##
## Minimum acceptance rate among theta[k]'s: 0.736
## Average acceptance rate among theta[k]'s: 0.736
## Maximum acceptance rate among theta[k]'s: 0.736
##
## -----------------------------------------------------
##
As a default, the code above runs the original implementation mode of
BASiCS (spikes without regression; see Section
@ref(alternative-implementation-modes)). To use the regression BASiCS
model (Eling et al. 2018), please set
Regression = TRUE
. To use the no-spikes implementation of
BASiCS, please add WithSpikes = FALSE
as an additional
parameter.
The Data
and Chain
(a
BASiCS_Chain
object) objects created by the code above can
be use for subsequent downstream analyses. See Section
@ref(analysis-for-a-single-group-of-cells) for highly/lowly variable
gene detection (single group of cells, see also functions
BASiCS_DetectHVG
and BASiCS_DetectLVG
) and
Section @ref(analysis-for-two-groups-of-cells) for differential
expression analyses (two groups of cells, see also function
BASiCS_TestDE
).
Important remarks:
Please ensure the acceptance rates displayed in the console
output of BASiCS_MCMC
are around 0.44. If they are too far
from this value, you should increase N
and
Burn
.
It is essential to assess the convergence of the MCMC algorithm before performing downstream analyses. For guidance regarding this step, refer to the ‘Convergence assessment’ section of this vignette
Typically, setting N=20000
, Thin=20
and
Burn=10000
leads to stable results.
The input dataset for BASiCS must be stored as an
SingleCellExperiment
object (see SingleCellExperiment
package).
The generation of the input SingleCellExperiment
object
requires a matrix of raw counts Counts
(columns: cells,
rows: genes) after quality control (e.g. removing low quality cells) and
filtering of lowly expressed genes. If spike-in molecules are contained
in Counts
, a logical vector Tech
is required
to indicate which rows contain technical spike-in molecules and a
data.frame
object SpikeInfo
containing the
names of the spike-in molecules in the first column and the absolute
number of molecules per well in the second column. More details are
provided in section @ref(the-input-dataset). If spike-ins are not
available, a vector BatchInfo
containing batch information
is required.
The newBASiCS_Data
function can be used to create the
input data object based on the following information:
Counts
: a matrix of raw expression counts with
dimensions q times n. Within this matrix, q0 rows must correspond
to biological genes and q − q0 rows must
correspond to technical spike-in genes. Gene names must be stored as
rownames(Counts)
.
Tech
: a logical vector
(TRUE
/FALSE
) with q elements. If
Tech[i] = FALSE
the gene i
is biological;
otherwise the gene is spike-in. This vector must be specified in the
same order of genes as in the Counts
matrix.
SpikeInfo
(optional): a data.frame
with
q − q0
rows. First column must contain the names associated to the spike-in
genes (as in rownames(Counts)
). Second column must contain
the input number of molecules for the spike-in genes (amount per cell).
If a value for this parameter is not provided when calling
newBASiCS_Data
, SpikeInfo
is set as
NULL
as a default value. In those cases, the
BatchInfo
argument has to be provided to allow using the
no-spikes implementation of BASiCS.
BatchInfo
(optional): vector of length n to indicate batch structure
(whenever cells have been processed using multiple batches). If a value
for this parameter is not provided when calling
newBASiCS_Data
, BASiCS will assume the data contains a
single batch of samples.
For example, the following code generates a synthetic dataset with 50 genes (40 biological and 10 spike-in) and 40 cells.
set.seed(1)
Counts <- matrix(rpois(50 * 40, 2), ncol = 40)
rownames(Counts) <- c(paste0("Gene", 1:40), paste0("Spike", 1:10))
colnames(Counts) <- paste0("Cell", 1:40)
Tech <- c(rep(FALSE, 40), rep(TRUE, 10))
set.seed(2)
SpikeInput <- rgamma(10, 1, 1)
SpikeInfo <- data.frame(
"SpikeID" = paste0("Spike", 1:10),
"SpikeInput" = SpikeInput
)
# No batch structure
DataExample <- newBASiCS_Data(Counts, Tech, SpikeInfo)
# With batch structure
DataExample <- newBASiCS_Data(
Counts, Tech, SpikeInfo,
BatchInfo = rep(c(1, 2), each = 20)
)
To convert an existing SingleCellExperiment
object
(Data
) into one that can be used within BASiCS,
meta-information must be stored in the object.
If spike-ins are in use, observed counts must be stored in
altExp(Data)
metadata(Data)
: the SpikeInfo
object is
stored in the metadata
slot of the
SingleCellExperiment
object:
metadata(Data) <- list(SpikeInput = SpikeInfo)
.
colData(Data)$BatchInfo
: the BatchInfo
object is stored in the colData
slot of the
SingleCellExperiment
object.
Once the additional information is included, the object can be used within BASiCS.
NOTE: Input number of molecules for spike-in should be calculated using experimental information. For each spike-in gene i, we use
$$ \mu_{i} = C_i \times 10^{-18} \times (6.022 \times 10^{23}) \times V \times D \hspace{0.5cm} \mbox{where,} $$
To run BASiCS without incorporating reads from technical spike-in
genes, and existing SingleCellExperiment
object can be
used. The only modification to the existing object is to assign the
colData(Data)$BatchInfo
slot.
set.seed(1)
CountsNoSpikes <- matrix(rpois(50 * 40, 2), ncol = 40)
rownames(CountsNoSpikes) <- paste0("Gene", 1:50)
colnames(CountsNoSpikes) <- paste0("Cell", 1:40)
# With batch structure
DataExampleNoSpikes <- SingleCellExperiment(
assays = list(counts = CountsNoSpikes),
colData = data.frame(BatchInfo = rep(c(1, 2), each = 20))
)
Note: BASiCS assumes that a pre-processing quality control step has been applied to remove cells with poor quality data and/or lowly expressed genes that were undetected through sequencing. When analysing multiple groups of cells, the gene filtering step must be jointly applied across all groups to ensure the same genes are retained.
The function BASiCS_Filter
can be used to perform this
task. For examples, refer to help(BASiCS_Filter)
. Moreover,
the scater
package provides enhanced functionality for the pre-processing of
scRNA-seq datasets.
We illustrate this analysis using a small extract from the MCMC chain
obtained in (Vallejos, Richardson, and Marioni
2016) when analysing the single cell samples provided in (Grün, Kester, and Oudenaarden 2014). This is
included within BASiCS
as the ChainSC
dataset.
The following code is used to identify highly variable genes
(HVG) and lowly variable genes (LVG) among
these cells. The VarThreshold
parameter sets a lower
threshold for the proportion of variability that is assigned to the
biological component (Sigma
). In the examples below:
For each gene, these functions return posterior probabilities as a measure of HVG/LVG evidence. A cut-off value for these posterior probabilities is set by controlling the EFDR (as a default option, EFDR is set as 0.10).
HVG <- BASiCS_DetectHVG(ChainSC, VarThreshold = 0.6)
LVG <- BASiCS_DetectLVG(ChainSC, VarThreshold = 0.2)
plot_grid(
BASiCS_PlotVG(HVG, "Grid"),
BASiCS_PlotVG(HVG, "VG")
)
To access the results of these tests, please use
as.data.frame
.
## GeneName GeneIndex Mu Delta Sigma Prob HVG
## 1 Amacr 21 7.164534 1.598386 0.6627267 0.6933333 TRUE
## 2 Phlda2 250 9.372955 2.171966 0.7445820 0.9600000 TRUE
## 3 Rhox13 286 9.138169 2.417712 0.7695457 0.9733333 TRUE
## GeneName GeneIndex Mu Delta Sigma Prob LVG
## 1 A630072M18Rik 8 76.74158 0.13842019 0.17424707 0.7333333 TRUE
## 2 Actn1 14 143.07299 0.08887830 0.11822042 0.9733333 TRUE
## 3 Ahsa1 20 167.46692 0.06774259 0.09373365 1.0000000 TRUE
## 4 Ass1 30 160.89463 0.09317097 0.12406375 0.9733333 TRUE
## 5 Atp1b1 36 182.63844 0.11706217 0.15493222 0.9200000 TRUE
## 6 Atp5g2 37 344.95180 0.06381609 0.09031420 1.0000000 TRUE
## 7 Btf3 55 263.29872 0.04880375 0.06858356 1.0000000 TRUE
## 8 Chchd2 69 576.15694 0.03078477 0.04702879 1.0000000 TRUE
## 9 Cops6 78 95.28245 0.11270865 0.14890886 0.8400000 TRUE
## 10 Dnajc19 102 69.75996 0.13088091 0.16615340 0.7200000 TRUE
## 11 Eef1d 112 164.97660 0.07334786 0.10212111 0.9866667 TRUE
## 12 Eif3e 115 172.85517 0.09311557 0.12818532 0.9866667 TRUE
## 13 Fkbp4 141 334.41897 0.07084192 0.09779841 1.0000000 TRUE
## 14 Ftl1 147 2296.21095 0.04504962 0.06656123 1.0000000 TRUE
## 15 Gars 150 144.21635 0.11851140 0.15426362 0.8533333 TRUE
## 16 Gm13251 159 214.95780 0.08542479 0.11957642 0.9866667 TRUE
## 17 Gm5506 161 2117.86447 0.06794125 0.09553558 1.0000000 TRUE
## 18 Gm5860 162 879.56308 0.07914106 0.11234069 1.0000000 TRUE
## 19 Hnrnpa3 176 698.74412 0.02606945 0.03976716 1.0000000 TRUE
## 20 Lrrc14b 191 82.26530 0.10443434 0.13761078 0.9200000 TRUE
## 21 Mcm7 199 171.91942 0.09402518 0.12149953 0.9466667 TRUE
## 22 Mlec 205 92.63264 0.09693363 0.12870287 0.9333333 TRUE
## 23 Myh9 214 75.00987 0.12209255 0.15676464 0.7733333 TRUE
## 24 Nedd8 218 140.92099 0.08071945 0.10778409 0.9600000 TRUE
## 25 Nmrk1 223 745.61731 0.13012128 0.16557363 0.8933333 TRUE
## 26 Nop56 225 237.46748 0.06437228 0.09231176 1.0000000 TRUE
## 27 Nop58 226 168.68657 0.13485637 0.16715635 0.7466667 TRUE
## 28 Npm3-ps1 230 167.04112 0.09636922 0.13345401 0.8933333 TRUE
## 29 Nucks1 232 334.69529 0.07277417 0.10074699 1.0000000 TRUE
## 30 Psmc3 269 131.08951 0.09918195 0.13530298 0.9333333 TRUE
## 31 Psmd8 270 106.57423 0.10824779 0.14710745 0.9200000 TRUE
## 32 Ptges3 272 297.56496 0.08525855 0.11596188 1.0000000 TRUE
## 33 Rad23b 275 157.73725 0.06005642 0.08739974 0.9866667 TRUE
## 34 Rbbp4 277 181.18047 0.09990468 0.13249099 0.9466667 TRUE
## 35 Rif1 287 121.96288 0.09407026 0.12419407 0.9200000 TRUE
## 36 Rpl15 290 1479.56150 0.02680944 0.03884788 1.0000000 TRUE
## 37 Rpl23a 291 1328.68382 0.02128793 0.03264723 1.0000000 TRUE
## 38 Rps11 293 1307.43808 0.02753237 0.04070404 1.0000000 TRUE
## 39 Slc23a1 310 1608.25591 0.14337288 0.18284658 0.7333333 TRUE
## 40 Slc25a3 311 371.72673 0.04075325 0.05901678 1.0000000 TRUE
## 41 Snx3 323 123.78433 0.11090234 0.14466506 0.9066667 TRUE
## 42 Snx5 324 152.55332 0.10992735 0.14451260 0.8933333 TRUE
## 43 Sod1 326 636.54334 0.03142035 0.04662697 1.0000000 TRUE
## 44 Sod2 327 222.74992 0.08470952 0.11801915 0.9866667 TRUE
## 45 Srp72 331 71.67213 0.13480074 0.17290929 0.6933333 TRUE
SummarySC <- Summary(ChainSC)
plot(SummarySC, Param = "mu", Param2 = "delta", log = "xy")
HTable <- as.data.frame(HVG)
LTable <- as.data.frame(LVG)
with(HTable, points(Mu, Delta))
with(LTable, points(Mu, Delta))
Note: this decision rule implemented in this
function has changed with respect to the original release of BASiCS
(where EviThreshold
was defined such that EFDR = EFNR).
However, the new choice is more stable (sometimes, it was not posible to
find a threshold such that EFDR = EFNR).
To illustrate the use of the differential mean expression and
differential over-dispersion tests between two cell populations, we use
extracts from the MCMC chains obtained in (Vallejos, Richardson, and Marioni 2016) when
analysing the (Grün, Kester, and Oudenaarden
2014) dataset (single cells vs pool-and-split samples). These
were obtained by independently running the BASiCS_MCMC
function for each group of cells.
Test <- BASiCS_TestDE(
Chain1 = ChainSC, Chain2 = ChainRNA,
GroupLabel1 = "SC", GroupLabel2 = "PaS",
EpsilonM = log2(1.5), EpsilonD = log2(1.5),
EFDR_M = 0.10, EFDR_D = 0.10,
Offset = TRUE, PlotOffset = TRUE, Plot = TRUE
)
In BASiCS_TestDE
, EpsilonM
sets the log2
fold change (log2FC) in expression (μ) and EpsilonD
the
log2FC in over-dispersion (δ).
As a default option: EpsilonM = EpsilonD = log2(1.5)
(i.e. the test is set to detect absolute increases of 50% or above). To
adjust for differences in overall mRNA content, an internal offset
correction is performed when OffSet=TRUE
. This is the
recommended default setting.
Previously, the output of this function was a set of plots and a
nested list structure. The new output is an object of S4 class
BASiCS_ResultsDE
.
## An object of class BASiCS_ResultsDE, containing:
## -------------------------------------------------------------
## An object of class BASiCS_ResultDE.
## -------------------------------------------------------------
## 0 genes with a change in mean expression
## - Higher mean expression in SC samples: 0
## - Higher mean expression in PaS samples: 0
## - Fold change tolerance = 150 %
## - Probability threshold = 0.666666666666667
## - EFDR = NaN %
## - EFNR = 19.64 %
## -------------------------------------------------------------
## An object of class BASiCS_ResultDE.
## -------------------------------------------------------------
## 0 genes with a change in over dispersion
## - Higher over dispersion in SC samples: 0
## - Higher over dispersion in PaS samples: 0
## - Fold change tolerance = 150 %
## - Probability threshold = 0.666666666666667
## - EFDR = NA %
## - EFNR = NA %
You can access the results of these tests using
as.data.frame
. You must specify which of the mean,
dispersion and residual overdispersion results you want using the
Which
argument, as follows.
## [1] GeneName MeanOverall Mean1 Mean2 MeanFC
## [6] MeanLog2FC ProbDiffMean ResultDiffMean
## <0 rows> (or 0-length row.names)
## [1] GeneName DispOverall Disp1 Disp2 DispFC
## [6] DispLog2FC ProbDiffDisp ResultDiffDisp
## <0 rows> (or 0-length row.names)
## This object doesn't contain residual overdispersion tests as the chains
## were run using the non-regression version of BASiCS
# head(as.data.frame(DE, Parameter = "Disp"))
There’s a rowData
field and accessor, so we can add gene
information that will be added when formatting each of the fields (eg
different gene identifiers).
## DataFrame with 350 rows and 1 column
## GeneName
## <character>
## 1 1700094D03Rik
## 2 1700097N02Rik
## 3 1810026B05Rik
## 4 2310008H04Rik
## 5 2410137M14Rik
## ... ...
## 346 Tef
## 347 Terf2
## 348 Tfeb
## 349 Timm22
## 350 Tinf2
rowData(Test) <- cbind(rowData(Test), Index = 1:nrow(rowData(Test)))
as.data.frame(Test, Parameter = "Mean")
## [1] GeneName MeanOverall Mean1 Mean2 MeanFC
## [6] MeanLog2FC ProbDiffMean ResultDiffMean Index
## <0 rows> (or 0-length row.names)
As of BASiCS v2, the differential expression plots have been re-done
in ggplot2 and can be generated using BASiCS_PlotDE
. The
default plots are “Grid” (EFDR vs EFNR), “MA”, and “Volcano”. These are
done for “Mean”, “Disp”, and “ResDisp”. We can choose to plot eg only
mean, or mean and overdispersion, etc, and only MA plot, MA plot and
volcano, etc.
Due to the confounding between mean and over-dispersion that is
typically observed in scRNA-seq datasets, the non-regression BASiCS
model (run using Regression = FALSE
as a function parameter
in BASiCS_MCMC
) can only be used to assess changes in
over-dispersion for those genes in which the mean expression does not
change between the groups. In this case, we recommend users to use
EpsilonM = 0
as a conservative option to avoid changes in
over-dispersion to be confounded by mean expression (the genes for which
mean expression changes are marked as ExcludedFromTesting
in the ResultDiffDisp
field).
Test <- BASiCS_TestDE(
Chain1 = ChainSC, Chain2 = ChainRNA,
GroupLabel1 = "SC", GroupLabel2 = "PaS",
EpsilonM = 0, EpsilonD = log2(1.5),
EFDR_M = 0.10, EFDR_D = 0.10,
Offset = TRUE, PlotOffset = FALSE, Plot = FALSE
)
Note: If the regression BASiCS model has been used
(Regression = TRUE
as a function parameter in
BASiCS_MCMC
), BASiCS_TestDE
will also report
changes in residual over-dispersion (not confounded by mean expression)
between the groups (see Section @ref(alternative-implementation-modes)
in this vignette).
Beyond its original implementation, BASiCS has been extended to address experimental designs in which spike-in molecules are not available as well as to address the confounding that is typically observed between mean and over-dispersion for scRNA-seq datasets (Eling et al. 2018). Alternative implementation modes are summarised below:
As a default, the BASiCS_MCMC
function uses
WithSpikes = TRUE
.
WithSpikes = FALSE
When technical spike-in genes are not available, BASiCS uses a
horizontal integration strategy which borrows information across
multiple technical replicates (Eling et al.
2018). Therefore, BASiCS_MCMC
will fail to run if a
single batch of samples is provided. Note: batch
information must be provided via the BatchInfo
argument
when using the newBASiCS_Data
function or
BatchInfo
must be stored as a slot in
colData(Data)
when using an existing
SingleCellExperiment
object.
DataNoSpikes <- newBASiCS_Data(
Counts, Tech, SpikeInfo = NULL,
BatchInfo = rep(c(1, 2), each = 20)
)
# Alternatively
DataNoSpikes <- SingleCellExperiment(
assays = list(counts = Counts),
colData = data.frame(BatchInfo = rep(c(1, 2), each = 20))
)
ChainNoSpikes <- BASiCS_MCMC(
Data = DataNoSpikes, N = 1000,
Thin = 10, Burn = 500,
WithSpikes = FALSE, Regression = TRUE,
PrintProgress = FALSE
)
## -----------------------------------------------------
## MCMC sampler has been started: 1000 iterations to go.
## -----------------------------------------------------
## Iteration: 0
## RefGene: 33
## RefGene before update: 2.01778
## sumAux (line 117): 34.019
## RefGene after update: 1.34108
## Iteration: 1
## RefGene: 17
## RefGene before update: 1.9953
## sumAux (line 117): 33.9223
## RefGene after update: 1.47719
## Iteration: 2
## RefGene: 9
## RefGene before update: 1.9953
## sumAux (line 117): 33.4081
## RefGene after update: 2.47039
## Iteration: 3
## RefGene: 48
## RefGene before update: 2.4174
## sumAux (line 117): 33.9961
## RefGene after update: 1.37205
## Iteration: 4
## RefGene: 33
## RefGene before update: 2.4174
## sumAux (line 117): 34.0968
## RefGene after update: 1.24067
## Iteration: 5
## RefGene: 17
## RefGene before update: 2.38188
## sumAux (line 117): 34.4132
## RefGene after update: 0.904141
## Iteration: 6
## RefGene: 15
## RefGene before update: 2.26451
## sumAux (line 117): 33.6945
## RefGene after update: 1.85513
## Iteration: 7
## RefGene: 15
## RefGene before update: 2.26451
## sumAux (line 117): 33.0558
## RefGene after update: 3.51348
## Iteration: 8
## RefGene: 25
## RefGene before update: 1.8864
## sumAux (line 117): 33.5089
## RefGene after update: 2.23352
## Iteration: 9
## RefGene: 16
## RefGene before update: 1.64623
## sumAux (line 117): 34.1356
## RefGene after update: 1.19343
## Iteration: 10
## RefGene: 15
## RefGene before update: 1.6502
## sumAux (line 117): 32.8664
## RefGene after update: 4.24621
## Iteration: 11
## RefGene: 20
## RefGene before update: 1.6502
## sumAux (line 117): 34.3556
## RefGene after update: 0.957799
## Iteration: 12
## RefGene: 15
## RefGene before update: 1.82135
## sumAux (line 117): 33.2928
## RefGene after update: 2.77219
## Iteration: 13
## RefGene: 3
## RefGene before update: 1.61719
## sumAux (line 117): 34.4318
## RefGene after update: 0.887469
## Iteration: 14
## RefGene: 15
## RefGene before update: 1.81402
## sumAux (line 117): 32.7807
## RefGene after update: 4.62634
## Iteration: 15
## RefGene: 9
## RefGene before update: 1.98949
## sumAux (line 117): 33.3532
## RefGene after update: 2.60984
## Iteration: 16
## RefGene: 9
## RefGene before update: 1.98949
## sumAux (line 117): 33.4522
## RefGene after update: 2.36371
## Iteration: 17
## RefGene: 16
## RefGene before update: 1.93262
## sumAux (line 117): 34.01
## RefGene after update: 1.35317
## Iteration: 18
## RefGene: 17
## RefGene before update: 1.78534
## sumAux (line 117): 33.2244
## RefGene after update: 2.96835
## Iteration: 19
## RefGene: 33
## RefGene before update: 1.78534
## sumAux (line 117): 33.4321
## RefGene after update: 2.41176
## Iteration: 20
## RefGene: 17
## RefGene before update: 2.45285
## sumAux (line 117): 33.1901
## RefGene after update: 3.07216
## Iteration: 21
## RefGene: 48
## RefGene before update: 2.15019
## sumAux (line 117): 32.6337
## RefGene after update: 5.35902
## Iteration: 22
## RefGene: 15
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## -----------------------------------------------------
## End of Burn-in period.
## -----------------------------------------------------
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##
## -----------------------------------------------------
## -----------------------------------------------------
## All 1000 MCMC iterations have been completed.
## -----------------------------------------------------
## -----------------------------------------------------
##
## -----------------------------------------------------
## Please see below a summary of the overall acceptance rates.
## -----------------------------------------------------
##
## Minimum acceptance rate among mu[i]'s: 0.372
## Average acceptance rate among mu[i]'s: 0.48584
## Maximum acceptance rate among mu[i]'s: 0.62
##
##
## Minimum acceptance rate among delta[i]'s: 0.722
## Average acceptance rate among delta[i]'s: 0.7918
## Maximum acceptance rate among delta[i]'s: 0.838
##
##
## Minimum acceptance rate among nu[jk]'s: 0.884
## Average acceptance rate among nu[jk]'s: 0.935
## Maximum acceptance rate among nu[jk]'s: 0.982
##
##
## Minimum acceptance rate among theta[k]'s: 0.772
## Average acceptance rate among theta[k]'s: 0.785
## Maximum acceptance rate among theta[k]'s: 0.798
##
##
## -----------------------------------------------------
##
Regression = TRUE
The BASiCS model uses a joint informative prior formulation to account for the relationship between mean and over-dispersion gene-specific parameters. The latter is used to infer a global regression trend between these parameters and, subsequently, to derive a residual over-dispersion measure that is defined as departures with respect to this trend.
DataRegression <- newBASiCS_Data(
Counts, Tech, SpikeInfo,
BatchInfo = rep(c(1, 2), each = 20)
)
ChainRegression <- BASiCS_MCMC(
Data = DataRegression, N = 1000,
Thin = 10, Burn = 500,
Regression = TRUE,
PrintProgress = FALSE
)
## -----------------------------------------------------
## MCMC sampler has been started: 1000 iterations to go.
## -----------------------------------------------------
## -----------------------------------------------------
## End of Burn-in period.
## -----------------------------------------------------
##
## -----------------------------------------------------
## -----------------------------------------------------
## All 1000 MCMC iterations have been completed.
## -----------------------------------------------------
## -----------------------------------------------------
##
## -----------------------------------------------------
## Please see below a summary of the overall acceptance rates.
## -----------------------------------------------------
##
## Minimum acceptance rate among mu[i]'s: 0.428
## Average acceptance rate among mu[i]'s: 0.49715
## Maximum acceptance rate among mu[i]'s: 0.576
##
##
## Minimum acceptance rate among delta[i]'s: 0.746
## Average acceptance rate among delta[i]'s: 0.78325
## Maximum acceptance rate among delta[i]'s: 0.82
##
##
## Acceptance rate for phi (joint): 0.836
##
##
## Minimum acceptance rate among nu[j]'s: 0.794
## Average acceptance rate among nu[j]'s: 0.8442
## Maximum acceptance rate among nu[j]'s: 0.97
##
##
## Minimum acceptance rate among theta[k]'s: 0.72
## Average acceptance rate among theta[k]'s: 0.73
## Maximum acceptance rate among theta[k]'s: 0.74
##
## -----------------------------------------------------
##
This implementation provides additional functionality when performing
downstream analyses. These are illustrated below using a small extract
from the MCMC chain obtained when analysing the dataset provided in
(Grün, Kester, and Oudenaarden 2014)
(single cell versus pool-and-split samples). These are included within
BASiCS
as the ChainSCReg
and
ChainRNAReg
datasets.
To visualize the fit between over-dispersion δi and mean expression $ _i$ the following function can be used.
The BASiCS_TestDE
test function will automatically
perform differential variability testing based on the residual
over-dispersion parameters ϵi when its
output includes two Chain
objects that were generated by
the regression BASiCS model.
data("ChainSCReg")
Test <- BASiCS_TestDE(
Chain1 = ChainSCReg, Chain2 = ChainRNAReg,
GroupLabel1 = "SC", GroupLabel2 = "PaS",
EpsilonM = log2(1.5), EpsilonD = log2(1.5),
EpsilonR = log2(1.5) / log2(exp(1)),
EFDR_M = 0.10, EFDR_D = 0.10,
Offset = TRUE, PlotOffset = FALSE, Plot = FALSE
)
This test function outputs an extra slot containing the results of
the differential testing residual over-dispersion test. Only genes that
are expressed in at least 2 cells (in both groups) are included in the
test. Genes that do not satisfy this condition are marked as
ExcludedFromRegression
in the
ResultDiffResDisp
field. By performing the regression, all
genes can be tested for changes in expression variability independent of
changes in mean expression.
## [1] GeneName ResDispOverall ResDisp1 ResDisp2
## [5] ResDispDistance ProbDiffResDisp ResultDiffResDisp MeanOverall
## <0 rows> (or 0-length row.names)
Note: Additional parameters for this sampler
include: k
number of regression components
(k
-2 radial basis functions, one intercept and one linear
component), Var
the scale parameter influencing the width
of the basis functions and eta
the degrees of freedom. For
additional details about these choices, please refer to Eling et al. (2018).
To externally store the output of BASiCS_MCMC
(recommended), additional parameters StoreChains
,
StoreDir
and RunName
are required. For
example:
Data <- makeExampleBASiCS_Data()
Chain <- BASiCS_MCMC(
Data,
N = 1000, Thin = 10, Burn = 500,
Regression = TRUE, PrintProgress = FALSE,
StoreChains = TRUE, StoreDir = tempdir(), RunName = "Example"
)
## -----------------------------------------------------
## MCMC sampler has been started: 1000 iterations to go.
## -----------------------------------------------------
## -----------------------------------------------------
## End of Burn-in period.
## -----------------------------------------------------
##
## -----------------------------------------------------
## -----------------------------------------------------
## All 1000 MCMC iterations have been completed.
## -----------------------------------------------------
## -----------------------------------------------------
##
## -----------------------------------------------------
## Please see below a summary of the overall acceptance rates.
## -----------------------------------------------------
##
## Minimum acceptance rate among mu[i]'s: 0.43
## Average acceptance rate among mu[i]'s: 0.63636
## Maximum acceptance rate among mu[i]'s: 0.772
##
##
## Minimum acceptance rate among delta[i]'s: 0.436
## Average acceptance rate among delta[i]'s: 0.52052
## Maximum acceptance rate among delta[i]'s: 0.616
##
##
## Acceptance rate for phi (joint): 0.906
##
##
## Minimum acceptance rate among nu[j]'s: 0.44
## Average acceptance rate among nu[j]'s: 0.5334
## Maximum acceptance rate among nu[j]'s: 0.696
##
##
## Minimum acceptance rate among theta[k]'s: 0.732
## Average acceptance rate among theta[k]'s: 0.732
## Maximum acceptance rate among theta[k]'s: 0.732
##
## -----------------------------------------------------
##
In this example, the output of BASiCS_MCMC
will be
stored as a BASiCS_Chain
object in the file
“chain_Example.Rds”, within the tempdir()
directory.
To load pre-computed MCMC chains,
To assess convergence of the chain, the convergence diagnostics
provided by the package coda
can be used. Additionally, the
chains can be visually inspected. For example:
See help(BASiCS_MCMC)
for example plots associated to
other model parameters. In the figure above:
?acf
)To access the MCMC chains associated to individual parameter use the
function displayChainBASiCS
. For example,
## Gene1 Gene2
## [1,] 5.35826 14.90511
## [2,] 12.75166 11.77155
As a summary of the posterior distribution, the function
Summary
calculates posterior medians and the High Posterior
Density (HPD) intervals for each model parameter. As a default option,
HPD intervals contain 0.95 probability.
The function displaySummaryBASiCS
extract posterior
summaries for individual parameters. For example
## median lower upper
## Gene1 9.692761 5.35826 5.057426
## Gene2 12.643747 14.84602 10.968432
The following figures display posterior medians and the corresponding HPD 95% intervals for gene-specific parameters μi (mean) and δi (over-dispersion)
par(mfrow = c(1, 2))
plot(ChainSummary, Param = "mu", main = "All genes", log = "y")
plot(ChainSummary, Param = "delta",
Genes = c(2, 5, 10, 50), main = "5 customized genes"
)
See help(BASiCS_MCMC)
for example plots associated to
other model parameters.
It is also possible to produce a matrix of normalised and denoised
expression counts for which the effect of technical variation is
removed. For this purpose, we implemented the function
BASiCS_DenoisedCounts
. For each gene i and cell j this function returns $$
x^*_{ij} = \frac{ x_{ij} } {\hat{\phi}_j \hat{\nu}_j},
$$ where xij is
the observed expression count of gene i in cell j, ϕ̂j denotes the
posterior median of ϕj and ν̂j is the
posterior median of νj.
## Cell1 Cell2
## Gene1 1.678073 20.53865
## Gene2 10.068436 13.69243
Note: the output of BASiCS_DenoisedCounts
requires no further data normalisation.
Alternativelly, the user can compute the normalised and denoised
expression rates underlying the expression of all genes across cells
using BASiCS_DenoisedRates
. The output of this function is
given by $$
\Lambda_{ij} = \hat{\mu_i} \hat{\rho}_{ij},
$$ where $\hat{\mu_i}$
represents the posterior median of μj and ρ̂ij is
given by its posterior mean (Monte Carlo estimate based on the MCMC
sample of all model parameters).
DenoisedRates <- BASiCS_DenoisedRates(
Data = Data, Chain = Chain,
Propensities = FALSE
)
DenoisedRates[1:2, 1:2]
## Cell1 Cell2
## Gene1 2.144986 17.52305
## Gene2 10.612366 13.60491
Alternative, denoised expression propensities ρ̂ij can also be extracted
DenoisedProp <- BASiCS_DenoisedRates(
Data = Data, Chain = Chain,
Propensities = TRUE
)
DenoisedProp[1:2, 1:2]
## Cell1 Cell2
## Gene1 0.2212977 1.807849
## Gene2 0.8393370 1.076019
We first describe the model introduced in [1], which relates to a single group of cells.
Throughout, we consider the expression counts of q genes, where q0 are expressed in the
population of cells under study (biological genes) and the remaining
q − q0 are
extrinsic spike-in (technical) genes. Let Xij be
a random variable representing the expression count of a gene i in cell j
(i = 1, …, q; j = 1, …, n). BASiCS is
based on the following hierarchical model: $$X_{ij} \big| \mu_i, \phi_j, \nu_j, \rho_{ij}
\sim \left\{ \begin{array}{ll} \mbox{Poisson}(\phi_j \nu_j \mu_i
\rho_{ij}), \mbox{ for }i=1,\ldots,q_0, j=1,\ldots,n \\
\mbox{Poisson}(\nu_j \mu_i), \mbox{ for }i=q_0+1,\ldots,q,
j=1,\ldots,n, \end{array} \right.$$
where νj and ρij are mutually independent random effects such that νj|sj, θ ∼ Gamma(1/θ, 1/(sjθ)) and ρij|δi ∼ Gamma(1/δi, 1/δi)1.
A graphical representation of this model is displayed below. This is based on the expression counts of 2 genes (i: biological and i′: technical) at 2 cells (j and j′). Squared and circular nodes denote known observed quantities (observed expression counts and added number of spike-in mRNA molecules) and unknown elements, respectively. Whereas black circular nodes represent the random effects that play an intermediate role in our hierarchical structure, red circular nodes relate to unknown model parameters in the top layer of hierarchy in our model. Blue, green and grey areas highlight elements that are shared within a biological gene, technical gene or cell, respectively.
In this setting, the key parameters to be used for downstream analyses are:
μi: mean expression parameter for gene i in the group of cells under study. In case of the spike-in technical genes, μi is assumed to be known and equal to the input number of molecules of the corresponding spike-in gene).
δi: over-dispersion parameter for gene i, a measure for the excess of variation that is observed after accounting for technical noise (with respect to Poisson sampling)
Additional (nuisance) parameters are interpreted as follows:
ϕj: cell-specific normalizing parameters related to differences in mRNA content (identifiability constrain: $\sum_{j=1}^n \phi_j = n$).
sj: cell-specific normalizing parameters related to technical cell-specific biases (for more details regarding this interpretation see (Vallejos et al. 2017)).
θ: technical over-dispersion parameter, controlling the strenght of cell-to-cell technical variability.
When cells from the same group are processed in multiple sequencing batches, this model is extended so that the technical over-dispersion parameter θ is batch-specific. This extension allows a different strenght of technical noise to be inferred for each batch of cells.
In Vallejos, Richardson, and Marioni (2016), this model has been extended to cases where multiple groups of cells are under study. This is achieved by assuming gene-specific parameters to be also group-specific. Based on this setup, evidence of differential expression is quantified through log2-fold changes of gene-specific parameters (mean and over-dispersion) between the groups. Moreover, Eling et al. (2018) further extended this model by addressing the mean/over-dispersion confounding that is characteristic of scRNA-seq datasets as well as experimental designs where spike-ins are not available.
More details regarding the model setup, prior specification and implementation are described in Vallejos, Marioni, and Richardson (2015), Vallejos, Richardson, and Marioni (2016) and Eling et al. (2018).
This work has been funded by the MRC Biostatistics Unit (MRC grant no. MRC_MC_UP_0801/1; Catalina Vallejos and Sylvia Richardson), EMBL European Bioinformatics Institute (core European Molecular Biology Laboratory funding; Catalina Vallejos, Nils Eling and John Marioni), CRUK Cambridge Institute (core CRUK funding; John Marioni) and The Alan Turing Institute (EPSRC grant no. EP/N510129/1; Catalina Vallejos).
We thank several members of the Marioni laboratory (EMBL-EBI; CRUK-CI) for support and discussions throughout the development of this R library. In particular, we are grateful to Aaron Lun (LTLA) for advise and support during the preparation the Bioconductor submission.
We also acknowledge feedback, bug reports and contributions from (Github aliases provided within parenthesis): Ben Dulken (bdulken), Chang Xu (xuchang116), Danilo Horta (Horta), Dmitriy Zhukov (dvzhukov), Jens Preußner (jenzopr), Joanna Dreux (Joannacodes), Kevin Rue-Albrecht (kevinrue), Luke Zappia (lazappi), Nitesh Turaga (nturaga), Mike Morgan (MikeDMorgan), Muad Abd El Hay (Cumol), Simon Anders (s-andrews), Shian Su (Shians), Yongchao Ge and Yuan Cao (yuancao90), among others.
## R version 4.4.1 (2024-06-14)
## Platform: x86_64-pc-linux-gnu
## Running under: Ubuntu 24.04 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=C
## [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] ggplot2_3.5.1 cowplot_1.1.3
## [3] BASiCS_2.17.0 SingleCellExperiment_1.27.2
## [5] SummarizedExperiment_1.35.1 Biobase_2.65.1
## [7] GenomicRanges_1.57.1 GenomeInfoDb_1.41.1
## [9] IRanges_2.39.2 S4Vectors_0.43.2
## [11] BiocGenerics_0.51.0 MatrixGenerics_1.17.0
## [13] matrixStats_1.3.0 knitr_1.48
## [15] BiocStyle_2.33.1
##
## loaded via a namespace (and not attached):
## [1] gridExtra_2.3 rlang_1.1.4 magrittr_2.0.3
## [4] compiler_4.4.1 reshape2_1.4.4 vctrs_0.6.5
## [7] stringr_1.5.1 pkgconfig_2.0.3 crayon_1.5.3
## [10] fastmap_1.2.0 backports_1.5.0 XVector_0.45.0
## [13] labeling_0.4.3 scuttle_1.15.4 utf8_1.2.4
## [16] promises_1.3.0 rmarkdown_2.28 UCSC.utils_1.1.0
## [19] xfun_0.47 bluster_1.15.0 zlibbioc_1.51.1
## [22] cachem_1.1.0 beachmat_2.21.5 jsonlite_1.8.8
## [25] highr_0.11 later_1.3.2 DelayedArray_0.31.11
## [28] BiocParallel_1.39.0 irlba_2.3.5.1 parallel_4.4.1
## [31] cluster_2.1.6 R6_2.5.1 RColorBrewer_1.1-3
## [34] stringi_1.8.4 bslib_0.8.0 limma_3.61.9
## [37] jquerylib_0.1.4 Rcpp_1.0.13 assertthat_0.2.1
## [40] httpuv_1.6.15 Matrix_1.7-0 igraph_2.0.3
## [43] tidyselect_1.2.1 abind_1.4-5 yaml_2.3.10
## [46] viridis_0.6.5 codetools_0.2-20 miniUI_0.1.1.1
## [49] plyr_1.8.9 lattice_0.22-6 tibble_3.2.1
## [52] withr_3.0.1 shiny_1.9.1 posterior_1.6.0
## [55] coda_0.19-4.1 evaluate_0.24.0 pillar_1.9.0
## [58] BiocManager_1.30.25 tensorA_0.36.2.1 checkmate_2.3.2
## [61] distributional_0.4.0 generics_0.1.3 munsell_0.5.1
## [64] scales_1.3.0 xtable_1.8-4 glue_1.7.0
## [67] metapod_1.13.0 maketools_1.3.0 tools_4.4.1
## [70] hexbin_1.28.3 BiocNeighbors_1.23.0 sys_3.4.2
## [73] ScaledMatrix_1.13.0 locfit_1.5-9.10 buildtools_1.0.0
## [76] scran_1.33.1 grid_4.4.1 edgeR_4.3.11
## [79] colorspace_2.1-1 GenomeInfoDbData_1.2.12 BiocSingular_1.21.2
## [82] cli_3.6.3 rsvd_1.0.5 fansi_1.0.6
## [85] viridisLite_0.4.2 S4Arrays_1.5.7 dplyr_1.1.4
## [88] gtable_0.3.5 sass_0.4.9 digest_0.6.37
## [91] SparseArray_1.5.31 dqrng_0.4.1 farver_2.1.2
## [94] htmltools_0.5.8.1 lifecycle_1.0.4 httr_1.4.7
## [97] statmod_1.5.0 mime_0.12 ggExtra_0.10.1
## [100] MASS_7.3-61
We parametrize the Gamma distribution such that if X ∼ Gamma(a, b), then E(X) = a/b and var(X) = a/b2.↩︎