Size factors

Introduction

I’m interested in the family of data transformations \[ Y_{ij} = \log \left( \frac{X_{ij}}{\lambda_{j}} + \alpha \right), \] where the \(X_{ij}\)s are raw counts, the \(\lambda_j\)s are cell-specific “size factors,” and \(\alpha\) is a “pseudocount” that must be added to avoid taking logarithms of zeros.

Note that, by subtracting \(\log \alpha\) from each side (and letting this constant be absorbed into the “mean factor” fitted by flashier), I can instead consider the family of sparity-preserving transformations \[ Y_{ij} = \left( \frac{X_{ij}}{\alpha \lambda_j} + 1 \right). \]

Without loss of generality, I constrain the size factors to have median one. In this analysis, I fix \(\alpha = 1\); in a subsequent analysis, I’ll consider the effect of varying the pseudocount.

source("./code/utils.R")
droplet <- readRDS("./data/droplet.rds")
droplet <- preprocess.droplet(droplet)
res <- readRDS("./output/size_factors/sizefactor_fits.rds")

Calculating size factors

I consider three methods. The method which I’ve been using in previous analyses normalizes cell counts so that all scaled cells have the same total count. That is, letting \(n_j\) be the cell-wise sum \(\sum_i X_{ij}\), it sets \[ \lambda_j = n_j / \text{median}_\ell(n_\ell) \] This method, commonly known as “library size normalization,” has been criticized because library size is strongly correlated with the expression levels of the most highly expressed genes, so that normalizing by library size tends to suppress relevant variation in those genes’ expression.

More sophisticated methods have been proposed. I particularly like an approach by Aaron Lun, Karsten Bach, and John Marioni which pools cells together and uses robust statistics to get more accurate estimates of size factors. The approach, detailed here, is implemented in the R package scran. It took about minute to calculate size factors for the Montoro et al. droplet dataset.

A second approach that has gained some traction is Rhonda Bacher’s method implemented in the R package SCnorm. I attempted to run it on the Montoro et al. dataset, but it simply set all scale factors to one (and it took a very long time to do so).

The third method I consider is to not use scale factors at all. This is not as dumb as it might seem! Write the EBMF model as \[ Y = \ell f' + LF' + E, \] where \(\ell f'\) is the first factor fitted by flashier. This factor, which I usually refer to as the “mean factor,” typically accounts for by far the largest proportion of variance explained. Then, \[ X_{ij} + 1 = e^{\ell_i} e^{f_j} \exp \left( (LF')_{ij} + E_{ij} \right), \] so that \(f\) essentially acts as a vector of size factors. The difference is that this scaling is done after the log1p transformation rather then before.

In this analysis, I’ll argue that it’s worth it to calculate scran size factors. Qualitatively, factors are very similar, but there are measurable differences in fits, it doesn’t take very long to calculate the size factors, and in many ways it’s the correct thing to do.

All fits were produced by adding 20 “greedy” factors to the Montoro et al. droplet dataset and backfitting. The code can be viewed here.

Results: scran size factors

The size factors obtained using scran are very different from the size factors obtained via library size normalization. In particular, scran yields a somewhat narrower range of size factors.

sf.df <- data.frame(libsize = res$libsize$size.factors, scran = res$scran$size.factors)
ggplot(sf.df, aes(x = libsize, y = scran)) +
  geom_point(size = 0.2) +
  geom_abline(slope = 1) + 
  labs(x = "library size normalization", y = "scran")

Past versions of sf-1.png

Results: ELBO

I use a change-of-variables formula to compare ELBOs. It’s probably not surprising that the most sophisticated method (scran) does best. The improvement in the ELBO is substantial: it’s approximately 600,000, which is the equivalent of 8-10 greedy factors. More surprisingly, library-size normalization does very poorly: worse, in fact, than not normalizing at all.

method.names <- c("No size factors", 
                  "Library size normalization",
                  "scran size factors")
elbo.df <- data.frame(method = method.names,
                      elbo = sapply(res, function(x) x$fl$elbo + x$elbo.adj))
ggplot(elbo.df, aes(x = method, y = elbo)) + 
  geom_point() +
  scale_x_discrete(limits = method.names) +
  labs(x = NULL, y = "ELBO")

Past versions of elbo-1.png

Results: Distribution of p-values

The distribution of \(p\)-values is very similar across methods, but the small differences confirm the above ranking: scran does best, while library-size normalization does worst. (I omit the individual plots, which are nearly indistinguishable to the eye.)

KL.df <- data.frame(method = method.names, 
                    KL = sapply(res, function(x) x$p.vals$KL.divergence))
ggplot(KL.df, aes(x = method, y = KL)) +
  geom_bar(stat = "identity") +
  scale_x_discrete(limits = method.names) +
  labs(x = NULL, y = "KL divergence relative to uniform")

Past versions of KL-1.png

Results: Log likelihood of implied distribution

Using the log likelihood of the implied discrete distribution — which, I’ve suggested, might be more reliable than the ELBO when different data transformations are in play — the relative rankings of the methods are more in line with what I’d expect. Once again, scran wins out by a large margin.

llik.df <- data.frame(method = method.names, 
                      llik = sapply(res, function(x) x$p.vals$llik))
ggplot(llik.df, aes(x = method, y = llik)) + 
  geom_point() +
  scale_x_discrete(limits = method.names) +
  labs(x = NULL, y = "Log likelihood of implied distribution")

Past versions of llik-1.png

Results: Factors and cell types

Qualitatively, factors are very similar across methods. Factor 20 looks a bit cleaner using scran size factors than using library size normalization. The fit that doesn’t use size factors yields an extra ciliated-specific factor for factor 20 and handles droplet cells somewhat differently.

plot.factors(res$scran, droplet$cell.type, 1:20, title = "scran")

Past versions of plot.factors-1.png

scran.v.libsize <- compare.factors(res$scran$fl, res$libsize$fl)
plot.factors(res$libsize, droplet$cell.type, scran.v.libsize$fl2.k,
             title = "Library size normalization")

Past versions of plot.factors-2.png

scran.v.noscale <- compare.factors(res$scran$fl, res$noscale$fl)
plot.factors(res$noscale, droplet$cell.type, scran.v.noscale$fl2.k,
             title = "No size factors")

Past versions of plot.factors-3.png

Session information

sessionInfo()

R version 3.5.3 (2019-03-11)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Mojave 10.14.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] flashier_0.1.15 ggplot2_3.2.0   Matrix_1.2-15  

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.1        plyr_1.8.4        compiler_3.5.3   
 [4] pillar_1.3.1      git2r_0.25.2      workflowr_1.2.0  
 [7] iterators_1.0.10  tools_3.5.3       digest_0.6.18    
[10] evaluate_0.13     tibble_2.1.1      gtable_0.3.0     
[13] lattice_0.20-38   pkgconfig_2.0.2   rlang_0.3.1      
[16] foreach_1.4.4     parallel_3.5.3    yaml_2.2.0       
[19] ebnm_0.1-24       xfun_0.6          withr_2.1.2      
[22] stringr_1.4.0     dplyr_0.8.0.1     knitr_1.22       
[25] fs_1.2.7          rprojroot_1.3-2   grid_3.5.3       
[28] tidyselect_0.2.5  glue_1.3.1        R6_2.4.0         
[31] rmarkdown_1.12    mixsqp_0.1-119    reshape2_1.4.3   
[34] ashr_2.2-38       purrr_0.3.2       magrittr_1.5     
[37] whisker_0.3-2     MASS_7.3-51.1     codetools_0.2-16 
[40] backports_1.1.3   scales_1.0.0      htmltools_0.3.6  
[43] assertthat_0.2.1  colorspace_1.4-1  labeling_0.3     
[46] stringi_1.4.3     pscl_1.5.2        doParallel_1.0.14
[49] lazyeval_0.2.2    munsell_0.5.0     truncnorm_1.0-8  
[52] SQUAREM_2017.10-1 crayon_1.3.4