- NEBULA v1.5.3
- Overview
- Installation
- Functions
- Basic usage
- Specifying scaling factors
- Using Seurat/SingleCellExperiment Objects
- Difference between NEBULA-LN and NEBULA-HL
- Filtering low-expression genes
- Checking convergence for the summary statistics and quality control
- Using other mixed models
- Special attention paid to testing subject-level variables
- Testing contrasts
- Extracting marginal and conditional Pearson residuals
- Parallel computing
- References

The *nebula* package is an R package that provides fast
algorithms for fitting negative binomial and Poisson mixed models for
analyzing large-scale, multi-subject single-cell data. The package
*nebula* accounts for the hierarchical structure of the data by
decomposing the total overdispersion into between-subject and
within-subject components using a negative binomial mixed model (NBMM).
Users can utilize the package for various tasks, such as identifying
marker genes, testing treatment effects, detecting genes with
differential expression, performing cell-level co-expression analysis,
and obtaining Pearson residuals for downstream analyses.

More details can be found in (He et al. 2021) (https://www.nature.com/articles/s42003-021-02146-6).

To install the latest version from github:

```
install.packages("devtools")
library(devtools)
install_github("lhe17/nebula")
```

During installation, the *nebula* package may first install
the *Rfast* package, which requires the presence of GSL in the
environment. The installation also requires Rcpp-1.0.7 and has been
tested on R-4.1.0. Starting from version 1.2.0, *nebula* is no
longer compatible with R-3.6 or earlier versions of R. Users who have
R-3.6 may install version 1.1.8 via R-forge (https://r-forge.r-project.org/R/?group_id=2407).
However, it is not recommended to use an older version of
*nebula*.

Please contact hyx520101@gmail.com for more information.

The current version provides the following functions.

`nebula`

: performs an association analysis using NBMMs given a count matrix and subject IDs.`group_cell`

: reorders cells to group them by the subject IDs.`nbresidual`

: extracts Pearson residuals from the fitted model.`scToNeb`

: retrieves data from`Seurat`

or`SingleCellExperiment`

for calling`nebula`

.

We use an example data set to illustrate how to use nebula to perform an association analysis of multi-subject single-cell data. The example data set attached to the R package can be loaded as follows.

```
library(nebula)
data(sample_data)
```

The example data set includes a count matrix of 6030 cells and 10 genes from 30 subjects.

```
dim(sample_data$count)
#> [1] 10 6176
```

The count matrix can be a matrix object or a sparse dgCMatrix object
(the same format as in `Seurat`

). The elements should be
integers.

```
sample_data$count[1:5,1:5]
#> 5 x 5 sparse Matrix of class "dgCMatrix"
#>
#> A . . . . .
#> B . . . . .
#> C . 1 2 . .
#> D . . . . .
#> E . . . . .
```

The subject IDs of each cell are stored in
`sample_data$sid`

. The subject IDs can be a character or
numeric vector, the length of which should equal the number of
cells.

```
head(sample_data$sid)
#> [1] "1" "1" "1" "1" "1" "1"
table(sample_data$sid)
#>
#> 1 10 11 12 13 14 15 16 17 18 19 2 20 21 22 23 24 25 26 27
#> 187 230 185 197 163 216 211 195 200 239 196 223 198 202 213 210 199 214 237 200
#> 28 29 3 30 4 5 6 7 8 9
#> 205 183 222 191 205 225 211 197 215 207
```

The next step is to build a design matrix for the predictors. The
example data set includes a data frame consisting of three predictors
stored in `sample_data$pred`

. To build the design matrix, we
can use the function `model.matrix`

. The intercept term must
be included in the design matrix. Each column in the design matrix
should have a unique variable name.

```
head(sample_data$pred)
#> X1 X2 cc
#> 1 0.6155094 0.9759191 control
#> 2 1.4608092 0.9759191 case
#> 3 1.6675054 0.9759191 control
#> 4 -0.1717715 0.9759191 case
#> 5 0.2277492 0.9759191 control
#> 6 -0.2635516 0.9759191 control
df = model.matrix(~X1+X2+cc, data=sample_data$pred)
head(df)
#> (Intercept) X1 X2 cccontrol
#> 1 1 0.6155094 0.9759191 1
#> 2 1 1.4608092 0.9759191 0
#> 3 1 1.6675054 0.9759191 1
#> 4 1 -0.1717715 0.9759191 0
#> 5 1 0.2277492 0.9759191 1
#> 6 1 -0.2635516 0.9759191 1
```

The association analysis between the gene expression and the
predictors can then be conducted using the `nebula`

function
as follows. The count matrix is an *M* by *N* matrix,
where *M* is the number of genes, and *N* is the number of
cells. The function by default fits the negative binomial gamma mixed
model (NBGMM) for each of the genes, and returns a list of summary
statistics including the fold change, p-values, and both subject-level
and cell-level overdispersions (\(\sigma^2\) and \(\phi^{-1}\)). The p-values returned by
`nebula`

are raw p-values (not adjusted for multiple
testing). Users can take advantage of a multicore CPU by specifying the
number of cores to use via the `ncore`

argument.

```
re = nebula(sample_data$count,sample_data$sid,pred=df,ncore=1)
#> Remove 0 genes having low expression.
#> Analyzing 10 genes with 30 subjects and 6176 cells.
#> Loading required package: foreach
#> Loading required package: future
#> Loading required package: rngtools
re
#> $summary
#> logFC_(Intercept) logFC_X1 logFC_X2 logFC_cccontrol se_(Intercept)
#> 1 -1.902455 -0.016755225 -0.097867225 0.047278197 0.06335820
#> 2 -2.046638 -0.002679074 -0.053812464 -0.022293899 0.06181112
#> 3 -2.033211 0.017954707 0.002398445 -0.048296661 0.08695028
#> 4 -2.008542 -0.005698984 -0.027780387 0.077357703 0.05509711
#> 5 -1.979437 0.011557090 -0.025198987 0.032890493 0.06155853
#> 6 -1.949991 0.013483039 -0.012548791 -0.031590577 0.07440949
#> 7 -1.969248 -0.003531361 0.075230699 -0.009075031 0.06185028
#> 8 -1.964371 0.013639930 -0.061302756 -0.059284665 0.07786361
#> 9 -2.072699 -0.017372176 -0.043828288 0.026624998 0.05737632
#> 10 -2.045646 0.030742876 0.022260805 -0.025516032 0.06842796
#> se_X1 se_X2 se_cccontrol p_(Intercept) p_X1 p_X2
#> 1 0.03534659 0.06449424 0.06879634 4.362617e-198 0.6354810 0.1291514
#> 2 0.03787429 0.06255849 0.07385888 2.052788e-240 0.9436079 0.3896819
#> 3 0.03696089 0.09238230 0.07258521 6.275230e-121 0.6271261 0.9792875
#> 4 0.03704556 0.05624824 0.07252600 5.822948e-291 0.8777381 0.6213846
#> 5 0.03750948 0.06101307 0.07331551 7.432319e-227 0.7579977 0.6795995
#> 6 0.03623477 0.07321208 0.07087566 2.257914e-151 0.7098168 0.8639067
#> 7 0.03631619 0.06068697 0.07133730 1.872102e-222 0.9225364 0.2151043
#> 8 0.03551903 0.07955877 0.06969748 1.957495e-140 0.7009654 0.4409831
#> 9 0.03816039 0.05767972 0.07453316 9.307495e-286 0.6489358 0.4473406
#> 10 0.03798694 0.06917485 0.07374591 2.292903e-196 0.4183419 0.7476005
#> p_cccontrol gene_id gene
#> 1 0.4919443 1 A
#> 2 0.7627706 2 B
#> 3 0.5058082 3 C
#> 4 0.2861434 4 D
#> 5 0.6537089 5 E
#> 6 0.6558008 6 F
#> 7 0.8987718 7 G
#> 8 0.3949916 8 H
#> 9 0.7209245 9 I
#> 10 0.7293432 10 J
#>
#> $overdispersion
#> Subject Cell
#> 1 0.08125256 0.8840821
#> 2 0.07102681 0.9255032
#> 3 0.17159404 0.9266395
#> 4 0.05026165 0.8124118
#> 5 0.07075366 1.2674146
#> 6 0.12086392 1.1096065
#> 7 0.07360445 0.9112956
#> 8 0.13571262 0.7549629
#> 9 0.05541398 0.8139652
#> 10 0.09496649 0.9410035
#>
#> $convergence
#> [1] 1 1 1 1 1 1 1 1 1 1
#>
#> $algorithm
#> [1] "NBGMM (LN)" "NBGMM (LN)" "NBGMM (LN)" "NBGMM (LN)" "NBGMM (LN)"
#> [6] "NBGMM (LN)" "NBGMM (LN)" "NBGMM (LN)" "NBGMM (LN)" "NBGMM (LN)"
#>
#> $covariance
#> NULL
#>
#> $random_effect
#> NULL
```

The cells in the count matrix need to be grouped by the subjects
(that is, the cells of the same subject should be placed consecutively)
before using as the input to the function `nebula`

. If the
cells are not grouped, the function `group_cell`

can be used
to first reorder the cells, as shown below. If a scaling factor is
specified by the user, it should also be included in
`group_cell`

. If the cells are already grouped,
`group_cell`

will return *NULL*.

```
data_g = group_cell(count=sample_data$count,id=sample_data$sid,pred=df)
re = nebula(data_g$count,data_g$id,pred=data_g$pred)
```

If `pred`

is not specified, `nebula`

will fit
the model with an intercept term by default. This can be used when only
the overdispersions are of interest.

The scaling factor for each cell is specified in `nebula`

using the argument `offset`

. The argument `offset`

has to be a vector of length *N* containing positive values. Note
that log(`offset`

) will be the offset term in the NBMM.
Common scaling factors can be the library size of a cell or a
normalizing factor adjusted using e.g., TMM. If not specified,
`nebula`

will set `offset`

as one by default,
which means that each cell is treated equally. If the input count matrix
is already normalized by another tool, e.g., scTransform, then you
should not specify `offset`

. However, since
`nebula`

directly models the raw counts, it is not
recommended to use a normalized count matrix for
`nebula`

.

```
library(Matrix)
# An example of using the library size of each cell as the scaling factor
re = nebula(sample_data$count,sample_data$sid,pred=df,offset=Matrix::colSums(sample_data$count))
```

If a single cell data processing package such as `Seurat`

or `SingleCellExperiment`

was used, nebula can be easily
implemented using the assistance of the helper function
`scToNeb`

. Assuming that the metadata relevant to subject IDs
and predictors are available in the object, `scToNeb`

can
retrieve and organize these objects and output a list that is similar to
the example data provided in this vignette. For a
`SingleCellExperiment`

object, `assay`

is not
required. For a `Seurat`

object, `assay`

can also
be specified to fit data from other assays. The `nebula`

package contains a sample Seurat object obtained from (Lab 2019) (https://github.com/satijalab/seurat-data) comprised of
pancreatic cells across eight samples.

```
library(nebula)
data("sample_seurat")
seuratdata <- scToNeb(obj = sample_seurat, assay = "RNA", id = "replicate", pred = c("celltype","tech"), offset="nCount_RNA")
## Make sure that the variables do not contain NA; Otherwise, df would have fewer rows.
df = model.matrix(~celltype+tech, data=seuratdata$pred)
## include only the first two cell types in the model to avoid separation due to too many binary variables
data_g = group_cell(count=seuratdata$count,id=seuratdata$id,pred=df[,c("(Intercept)","celltypeactivated_stellate","techcelseq2","techfluidigmc1","techindrop", "techsmartseq2")],offset=seuratdata$offset)
re = nebula(data_g$count,data_g$id,pred=data_g$pred,offset=data_g$offset)
```

The output will be a list with the first element containing
`counts`

, the second containing a `data.frame`

with all listed predictors, the third containing a character vector with
all subject IDs, and the fourth containing the normalizing factor. Users
can also use other scaling factors that may be stored within the
object’s metadata as a string in the `offset`

argument. If
subject ids are un-ordered, `group_cell`

can be used.

In *nebula*, a user can choose one of the two algorithms to
fit an NBGMM. NEBULA-LN uses an approximated likelihood based on the law
of large numbers, and NEBULA-HL uses an h-likelihood. A user can select
these methods through `method='LN'`

or
`method='HL'`

. NEBULA-LN is faster and performs particularly
well when the number of cells per subject (CPS) is large. In addition,
NEBULA-LN is much more accurate in estimating a very large subject-level
overdispersion. In contrast, NEBULA-HL is slower but more accurate in
estimating the cell-level overdispersion.

In the following analysis of the example data set comprising ~200 cells per subject, the difference of the estimated cell-level overdispersions between NEBULA-LN and NEBULA-HL is ~5% for most genes.

```
re_ln = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,method='LN',ncore=1)
#> Remove 0 genes having low expression.
#> Analyzing 10 genes with 30 subjects and 6176 cells.
re_hl = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,method='HL',ncore=1)
#> Remove 0 genes having low expression.
#> Analyzing 10 genes with 30 subjects and 6176 cells.
## compare the estimated overdispersions
cbind(re_hl$overdispersion,re_ln$overdispersion)
#> Subject Cell Subject Cell
#> 1 0.08432318 0.9284699 0.08125256 0.8840821
#> 2 0.07455464 0.9726513 0.07102681 0.9255032
#> 3 0.17403263 0.9817569 0.17159404 0.9266395
#> 4 0.05352153 0.8516679 0.05026165 0.8124118
#> 5 0.07480033 1.3254379 0.07075366 1.2674146
#> 6 0.12372424 1.1653129 0.12086392 1.1096065
#> 7 0.07724825 0.9578169 0.07360445 0.9112956
#> 8 0.13797645 0.7991948 0.13571262 0.7549629
#> 9 0.05879495 0.8568850 0.05541398 0.8139652
#> 10 0.09782333 0.9940222 0.09496649 0.9410035
```

Such difference has little impact on testing fixed-effects predictors under this sample size.

```
## compare the p-values for testing the predictors using NEBULA-LN and NEBULA-HL
cbind(re_hl$summary[,10:12],re_ln$summary[,10:12])
#> p_X1 p_X2 p_cccontrol p_X1 p_X2 p_cccontrol
#> 1 0.6373036 0.1346298 0.4950795 0.6354810 0.1291514 0.4919443
#> 2 0.9444825 0.3977109 0.7626827 0.9436079 0.3896819 0.7627706
#> 3 0.6282384 0.9787881 0.5087304 0.6271261 0.9792875 0.5058082
#> 4 0.8786074 0.6278827 0.2868256 0.8777381 0.6213846 0.2861434
#> 5 0.7596198 0.6872259 0.6544751 0.7579977 0.6795995 0.6537089
#> 6 0.7134192 0.8656686 0.6576835 0.7098168 0.8639067 0.6558008
#> 7 0.9216994 0.2230964 0.8977251 0.9225364 0.2151043 0.8987718
#> 8 0.7017082 0.4443604 0.3955342 0.7009654 0.4409831 0.3949916
#> 9 0.6505414 0.4561470 0.7238323 0.6489358 0.4473406 0.7209245
#> 10 0.4199828 0.7510837 0.7308108 0.4183419 0.7476005 0.7293432
```

The bias of NEBULA-LN in estimating the cell-level overdispersion
gets larger when the CPS value becomes lower or the gene expression is
more sparse. If the CPS value is <30, `nebula`

will set
`method='HL'`

regardless of the user’s input.

When NEBULA-LN is used, the user can opt for better accuracy of estimating a smaller subject-level overdispersion through the argument \(\kappa\). NEBULA first fits the data using NEBULA-LN. If the estimated \(\kappa\) for a gene is smaller than the user-defined value, NEBULA-HL will be used to estimate the subject-level overdispersion for the gene. The default value of \(\kappa\) is 800, which can provide a good estimate of the subject-level overdispersion as low as ~0.005. Our simulation results suggest that \(\kappa=200\) is often sufficient for achieving a well controlled false positive rate of testing a cell-level predictor. We do not recommend using a smaller \(\kappa\) than 200. Specifying a larger \(\kappa\) can obtain a more accurate estimate of a smaller subject-level overdispersion when the cell-level overdispersion is large, but will be computationally slower. On the other hand, testing a subject-level predictor (i.e., a variable whose values are shared across all cells from a subject, such as age, sex, treatment, genotype, etc) is more sensitive to the accuracy of the estimated subject-level overdispersion. So we recommend using \(\kappa=800\) (as default) or even larger when testing a subject-level predictor. Another option to testing a subject-level predictor is to use a Poisson gamma mixed model, which is extremely fast (>50x faster than NEBULA-LN) and will be described below.

NEBULA-HL automatically uses a higher-order Laplace approximation for
low-expressed genes of which the average count per subject is less than
3. The higher-order Laplace approximation substantially increases the
accuracy for estimating the subject-level overdispersion for
low-expressed genes and controls the false positive rate. Nevertheless,
we recommend removing genes with very low expression from the analysis
because there is little statistical power for these genes. Filtering out
low-expressed genes can be specified by `cpc=0.005`

(i.e.,
counts per cell<0.5%). The argument `cpc`

is defined by
the ratio between the total count of the gene and the number of
cells.

*nebula* reports convergence information about the estimation
algorithm for each gene along with the summary statistics. This is
useful and important information for quality control to filter out genes
of which the estimation procedure potentially does not converge.
Generally, a convergence code \(\leq\)
-20 suggests that the algorithm does not converge well. The results
should be interpreted with caution in these cases. The detailed
information about the convergence codes is listed below. The failure of
convergence may occur when the sample size is very small, there are too
few positive counts, or the gene has huge overdispersions. In these
cases, the likelihood can be flat, might reach the maximum at the
infinity, or the optimization is sensitive to the initial values. For
those genes that have a bad convergence code, in many cases, trying a
different negative binomial mixed model (e.g., NBLMM, see below for more
details) may solve the problem.

- Information about the convergence code:
- 1: The convergence is reached due to a sufficiently small improvement of the function value.
- -10: The convergence is reached because the gradients are close to zero (i.e., the critical point) and no improvement of the function value can be found.
- (!) -20: The optimization algorithm stops before the convergence because the maximum number of iterations is reached.
- (!) -25: The Hessian matrix is either almost singular or not positive definite.
- (!) -30: The convergence fails because the likelihood function
returns NaN.

- (!) -40: The convergence fails because the critical point is not reached and no improvement of the function value can be found.
- (!) -50: Only used for the PMM, indicating a failure of convergence.
- (!) -60: At least one of the estimated overdispersions reaches its upper bound.

Depending on the concrete application, the estimated gene-specific overdispersions can also be taken into consideration in quality control. For example, when testing differential expression for a variable, genes with a very large estimated cell-level overdispersion should be filtered out because such genes have huge unexplained noises. A large cell-level overdispersion is generally rare in UMI-based single cell data, especially among abundantly expressed genes, but more common in e.g., SMART-seq2 as PCR duplicates introduce substantial noises. It might be hard to give a precise cut-off for a large overdispersion because it also depends on the sample size of the data. Based on the empirical simulation study in (He et al. 2021), genes with an estimated cell-level overdispersion >100 should be removed for a data set with at least 50 cells per subject. On the other hand, if the purpose is to extract residuals for downstream analysis such as clustering, genes with a large cell-level overdispersion might be preferable because they have large variations. If the variable of interest is subject-level, genes with a very large subject-level overdispersion (>1) should be removed or interpreted cautiously as well.

In addition to the NBGMM, the *nebula* package provides
efficient estimation implementation for a Poisson gamma mixed model and
a negative binomial lognormal mixed model (NBLMM). This can be specified
through `model="PMM"`

and `model="NBLMM"`

,
respectively. The NBLMM is the same model as that adopted in the
`glmer.nb`

function in the *lme4* R package, but is
computationally much more efficient by setting `method='LN'`

.
The only difference between NBGMM and NBLMM is that NBGMM uses a gamma
distribution for the random effects while the NBLMM uses a lognormal
distribution. The PMM is the fastest among these models. Note that the
Poisson mixed model (PMM) should not be used to test a cell-level
predictor because it only estimates the subject-level overdispersion.
Here is an example of using the PMM to fit the example data set.

```
re = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,model='PMM',ncore=1)
#> Remove 0 genes having low expression.
#> Analyzing 10 genes with 30 subjects and 6176 cells.
```

logFC_(Intercept) | logFC_X1 | logFC_X2 | logFC_cccontrol | se_(Intercept) | se_X1 | se_X2 | se_cccontrol | p_(Intercept) | p_X1 | p_X2 | p_cccontrol | gene_id | gene |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

-1.903559 | -0.0155807 | -0.0976567 | 0.0511051 | 0.0661288 | 0.0329114 | 0.0655551 | 0.0642298 | 0 | 0.6359176 | 0.1363061 | 0.4262299 | 1 | A |

-2.047864 | -0.0032670 | -0.0536887 | -0.0189269 | 0.0644332 | 0.0355074 | 0.0635450 | 0.0694853 | 0 | 0.9266904 | 0.3981703 | 0.7853239 | 2 | B |

-2.032603 | 0.0179868 | 0.0009264 | -0.0505295 | 0.0908162 | 0.0345496 | 0.0932444 | 0.0676704 | 0 | 0.6026385 | 0.9920732 | 0.4552440 | 3 | C |

-2.009743 | -0.0055009 | -0.0278638 | 0.0782083 | 0.0573206 | 0.0350744 | 0.0574457 | 0.0686939 | 0 | 0.8753743 | 0.6276435 | 0.2549097 | 4 | D |

-1.980527 | 0.0106340 | -0.0248788 | 0.0312191 | 0.0644293 | 0.0343354 | 0.0621582 | 0.0671645 | 0 | 0.7567817 | 0.6889725 | 0.6420637 | 5 | E |

-1.950454 | 0.0160303 | -0.0134764 | -0.0345278 | 0.0778201 | 0.0333858 | 0.0738508 | 0.0650410 | 0 | 0.6311185 | 0.8552054 | 0.5955144 | 6 | F |

-1.970271 | -0.0026762 | 0.0750061 | -0.0063660 | 0.0645989 | 0.0341936 | 0.0615159 | 0.0668723 | 0 | 0.9376159 | 0.2227322 | 0.9241583 | 7 | G |

-1.964322 | 0.0141545 | -0.0610982 | -0.0578682 | 0.0809950 | 0.0336580 | 0.0800989 | 0.0656801 | 0 | 0.6740920 | 0.4455920 | 0.3782847 | 8 | H |

-2.074035 | -0.0178188 | -0.0436111 | 0.0259748 | 0.0597958 | 0.0362203 | 0.0587687 | 0.0707913 | 0 | 0.6227507 | 0.4580383 | 0.7136780 | 9 | I |

-2.046058 | 0.0307022 | 0.0227234 | -0.0246096 | 0.0714146 | 0.0354844 | 0.0702255 | 0.0691813 | 0 | 0.3869124 | 0.7462578 | 0.7220452 | 10 | J |

When testing subject-level variables, it should be kept in mind that
the actual sample size is the number of subjects, not the number of
cells in the data set. At least a moderate number of subjects (>30)
are required for testing a subject-level variable using
`nebula`

simply because a small number of subjects are not
enough to accurately estimate the subject-level overdispersion. As shown
in the original article (He et al. 2021),
even 30 subjects lead to mild inflated type I errors in most simulated
scenarios. If the number of subjects is very small, methods designed for
small sample size (e.g., DESeq2, edgeR) should be used for testing
subject-level variables.

In addition, when the ratio between the number of subjects and the
number of subject-level variables is small (<10), it is recommended
to instead use a restricted maximum likelihood (REML) estimate, which is
provided in `nebula`

through the argument `reml`

.
Please see (He et al. 2021) for more
details about the formula of REML. This is because the number of
subject-level fixed-effects parameters should be much smaller than the
number of subjects in order to make the maximum likelihood estimation
(MLE) work properly. For example, if the data set has 50 subjects, it is
a good practice to keep the number of subject-level variables below 5
based on our simulation study. Increasing the number of subject-level
parameters will gradually inflate the type I error rate due to an
underestimated overdispersion. When these two numbers are at the same
magnitude, the MLE for the overdispersion will break down and
consequently, the NBMM can degenerate to a negative binomial model. In
contrast, REML takes into account the uncertainty of the estimated fixed
effects and controls the false positive rate even if many subject-level
covariates are included in the model. As shown in the following example,
one could simply specify `reml=1`

to use REML, which is
supported only for `model='NBLMM'`

in the current
version.

`re = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,model='NBLMM',reml=1,ncore=1)`

In some situations, a user may want to test a combination (contrast)
of the log(FC) or perform a global test for multiple variables or
levels. For example, a user may want to test whether the log(FC) of two
variables are the same. Here, we show how `nebula`

can be
used for this kind of analysis.

The first step is to tell `nebula`

to output the
covariance matrix of the estimated log(FC). This can be done by
specifying `covariance=TRUE`

in `nebula`

. To save
storage, the covariance returned by `nebula`

only contains
the elements in the lower triangular part including the diagonal. Here
is an example to recover the covariance matrix from the output of
`nebula`

.

```
df = model.matrix(~X1+X2+cc, data=sample_data$pred)
re_ln = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,method='LN',covariance=TRUE,ncore=1)
#> Remove 0 genes having low expression.
#> Analyzing 10 genes with 30 subjects and 6176 cells.
cov= matrix(NA,4,4)
cov[lower.tri(cov,diag=T)] = as.numeric(re_ln$covariance[1,])
cov[upper.tri(cov)] = t(cov)[upper.tri(cov)]
cov
#> [,1] [,2] [,3] [,4]
#> [1,] 4.014261e-03 2.499051e-05 1.384999e-04 -5.197643e-05
#> [2,] 2.499051e-05 1.249382e-03 9.212341e-06 -1.167080e-05
#> [3,] 1.384999e-04 9.212341e-06 4.159507e-03 5.142249e-05
#> [4,] -5.197643e-05 -1.167080e-05 5.142249e-05 4.732936e-03
```

Note that if there are *K* variables, the covariance table in
the output will have *(K+1)K/2* columns. So, for a large
*K*, substantial increase of computational intensity should be
expected.

The second step is to build the contrast vector for your hypothesis.
In this example, we want to test whether the log(FC) of *X1* and
*X2* are equal for the first gene. This hypothesis leads to the
contrast vector `(0 1 -1 0)`

. Thus, the test can be performed
using the following code.

```
df = model.matrix(~X1+X2+cc, data=sample_data$pred)
## the gene to test
gene_i = 1
## output covariance
re_ln = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,method='LN',covariance=TRUE,ncore=1)
#> Remove 0 genes having low expression.
#> Analyzing 10 genes with 30 subjects and 6176 cells.
## recover the covariance matrix
cov= matrix(NA,4,4)
cov[lower.tri(cov,diag=T)] = as.numeric(re_ln$covariance[gene_i,])
cov[upper.tri(cov)] = t(cov)[upper.tri(cov)]
## build the contrast vector
contrast = c(0,1,-1,0)
## testing the hypothesis
eff = sum(contrast*re_ln$summary[gene_i,1:4])
p = pchisq(eff^2/(t(contrast)%*%cov%*%contrast),1,lower.tail=FALSE)
p
#> [,1]
#> [1,] 0.2692591
```

Pearson residuals are the distances between the raw count and its expected value standardized by its standard deviation. Pearson residuals obtained from fitting the NBMM can be used for normalization and downstream analyses. The marginal Pearson residuals are obtained by removing from the raw count the contribution from all fixed-effect variables included in the model. The conditional Pearson residuals further remove the subject-level random effects, which capture the contribution of all other potential subject-level variables that are not explicitly included in the model. Therefore, the conditional Pearson residuals are very useful in a situation where one needs to remove the subject-level batch effects from the normalized residuals for downstream analyses.

Both Pearson residuals can be easily extracted by using the
`nbresidual`

function after successfully running the
`nebula`

function. To extract the marginal Pearson residuals,
one provides in `nbresidual`

the object returned by
`nebula`

together with the same arguments including the count
matrix, `id`

, `pred`

and `offset`

used
in running the `nebula`

function. Here is an example.

```
re = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset)
pres = nbresidual(re,count=sample_data$count,id=sample_data$sid,pred=df,offset=sample_data$offset)
```

The parameters `count`

, `id`

, `pred`

and `offset`

should be the same in these two functions. Then,
the marginal Pearson residuals are available in the matrix
`pres$residuals`

. The rows in `pres$residuals`

correspond to the genes in the output of `nebula`

, and the
columns are the cells in `count`

.

To extract the conditional Pearson residuals, we need to first let
`nebula`

output subject-level random effects by setting
`output_re=TRUE`

when running `nebula`

as shown
below.

`re = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,output_re=TRUE)`

The returned object will include an *M* by *L* matrix
of the random effects, where *L* is the number of subjects. In
the current version, this option does NOT support
`model="PMM"`

. Then, the conditional Pearson residuals can be
extracted by running `nbresidual`

with
`conditional=TRUE`

.

`pres = nbresidual(re,count=sample_data$count,id=sample_data$sid,pred=df,offset=sample_data$offset,conditional=TRUE)`

Starting with version 1.4.0, *nebula* supports parallel
computing to accelerate tasks. To specify the number of logical cores
(threads), users can use the `ncore`

argument when running
`nebula`

. By default, `nebula`

uses two cores if
more than two cores are available. However, if the specified value for
`ncore`

exceeds the number of available cores,
*nebula* will use available cores detected and issue a warning.
It’s important to note that the maximum number of available processors
on a computing cluster may be restricted by the job scheduler’s
configuration.

While parallel computing significantly improves computational time, users should monitor memory usage when exploiting multiple cores. Insufficient memory allocation to a new thread may cause the R session or job to be terminated by the operating system. In addition, if the number of cells is small, using too many cores might not improve or even reduce the efficiency because the overhead for creating new threads exceeds the speed gain.

He, Liang, Jose Davila-Velderrain, Tomokazu S. Sumida, David A. Hafler,
Manolis Kellis, and Alexander M. Kulminski. 2021. “NEBULA Is a
Fast Negative Binomial Mixed Model for Differential or Co-Expression
Analysis of Large-Scale Multi-Subject Single-Cell Data.”
*Communications Biology*, no. 629 (May). https://doi.org/10.1038/s42003-021-02146-6.

Lab, Satija. 2019. *Panc8.SeuratData: Eight Pancreas Datasets Across
Five Technologies*.