• NEBULA v1.1.8
  • Overview
  • Installation
    • Most recent version
  • Functions
  • Basic usage
    • Example
  • Specifying scaling factors
    • Example
  • Selection between NEBULA-LN and NEBULA-HL
  • Filtering low-expressed genes
  • Checking convergence for the summary statistics
  • Using other mixed models
    • Example
  • Testing contrasts

NEBULA v1.1.8

Overview

The R package, nebula, 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). The package nebula can be used for e.g., identifying marker genes, testing treatment effects, detecting genes with differentail expression, and performing cell-level co-expression analysis.

More details can be found in the manuscript “NEBULA: a fast negative binomial mixed model for differential expression and co-expression analyses of large-scale multi-subject single-cell data” (https://www.nature.com/articles/s42003-021-02146-6).

Installation

Most recent version

To install the lastest version from github:

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

To install the lastest version from R-forge:

install.packages("nebula", repos="http://R-Forge.R-project.org")

Because the package nebula uses the R package Rfast, the installation process may first install Rfast, which requires that GSL is installed or available in the environment.

The installation has been tested on R-3.6 and R-3.5. Please contact liang.he@duke.edu for more information.

Functions

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.

Basic usage

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 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.

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 function nebula. The count matrix is an M by N matrix, where M is the number of genes, and N is the number of cells.

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

The function by default fitted the negative binomial gamma mixed model (NBGMM) for each of the genes, and return a list of summary statistics including the fold change, p-values, and both subject-level and cell-level overdispersions (σ² and ϕ^− 1). The cells 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 nebula function. If the cells are not grouped, the group_cell function can be used to first reorder the cells, as shwon 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.

Example

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 intecept term by default. This can be used when only the overdispersions are of interest.

Specifying scaling factors

The scaling factor for each cell is specified in nebula using the argument offset. The argument offset has to be a positive vector of length N. Note that log(offset) will be the offset term in the NBMM. If not specified, nebula will set offset as 1 by default, which means that each cell is treated equally. Common scaling factors include the library size of a cell or a normalizing factor adjusted using e.g., TMM.

Example

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

Selection between NEBULA-LN and NEBULA-HL

In nebula, a user can choose one of the two algorithms to fit an NBMM. 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 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$offset,method='LN')
#> Remove  0  genes having low expression.
#> Analyzing  10  genes with  30  subjects and  6176  cells.
re_hl = nebula(sample_data$count,sample_data$sid,pred=df,offset=sample_data$offset,method='HL')
#> 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  0.08432321 0.9284703 0.08125256 0.8840821
#> 2  0.07455464 0.9726512 0.07102681 0.9255032
#> 3  0.17403276 0.9817570 0.17159404 0.9266395
#> 4  0.05352148 0.8516682 0.05026165 0.8124118
#> 5  0.07480033 1.3254379 0.07075366 1.2674146
#> 6  0.12372426 1.1653128 0.12086392 1.1096065
#> 7  0.07724824 0.9578169 0.07360445 0.9112956
#> 8  0.13797646 0.7991954 0.13571262 0.7549629
#> 9  0.05879492 0.8568854 0.05541398 0.8139652
#> 10 0.09782335 0.9940223 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  0.6373037 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 0.9787882   0.5087304 0.6271261 0.9792875   0.5058082
#> 4  0.8786074 0.6278826   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  0.7017083 0.4443604   0.3955343 0.7009654 0.4409831   0.3949916
#> 9  0.6505414 0.4561469   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. In contrast, NEBULA-HL is slower, but its accuracy of estimating the overdispersions depends less on these factors.

When NEBULA-LN is used, the user can opt for better accuracy of estimating a smaller subject-level overdispersion through the argument κ. NEBULA first fits the data using NEBULA-LN. If the estimated κ 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 κ is 800, which can provide a good estimate of the subject-level overdispersion as low as ~0.005. Our simulation results suggest that κ = 200 is often sufficent for achieving a well controlled false positive rate of testing a cell-level predictor. We do not recommend using a smaller κ than 200. Specifying a larger κ 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 κ = 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.

Filtering low-expressed genes

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 substantailly 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.

Checking convergence for the summary statistics

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 <= -20 suggests that the algorithm does not converge well. If the convergence code is -30, which indicates a failure of convergence, their summary statistics should NOT be used. If the convergence code is -20 or -40, it indicates that the optimization algorithm stops at the maximum step limit before the complete convergence. The results should be interpreted with caution in this case. 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 which case the likelihood is flat 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.

Using other mixed models

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.

Example

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

Testing contrasts

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)
#> 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)
#> 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