A huge thanks to Jolanta Szkodon for her custom function to reproduce DESeq2 normalization.
The median of ratios method is a normalization method used to account for sequencing differences in library size and RNA composition of samples. This makes it a suitable method to use when comparing between samples. It does not account for differences in gene length, making it unsuitable for within sample comparisons.
The median of ratios method is applied in the DESeq2 package, used for differential expression testing. In the DESeq2 package, normalization is conducted easily with a function. Although this is convenient, it is useful to understand the step-by-step breakdown of the median of ratios method.
Sample expression count data was taken from the GSE49110 dataset and cut to create a manageable dataset for the tutorial.
In Section 1, steps of the median of ratios method are shown.
In Section 2, a function including all steps is shown.
In Section 3, the normalized counts are compared with the DESeq2 method to ensure the results are the same. References used are listed at the end.
Table of Contents
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Section 1: step-by-step procedure
- Step 1: Upload data
- Step 2: Take the log of all values
- Step 3: Take the average of each row
- Step 4: Filter out all of the genes with -Inf as their average
- Step 5: Subtract the gene pseudo-references from log counts
- Step 6: Find the median of the ratios for each sample
- Step 7: Convert medians to scaling factors
- Step 8: Divide the original counts (not log version) by the scaling factors
- Section two: a function to normalize the DESeq2 way
- Section 3: sanity check
- References
Section 1: step-by-step procedure
Step 1: Upload data
Rows (genes) and columns (samples) are seen below.
data = read.csv("sample_count_data.csv", row.names = 1)
head(data)
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
12SBSRNA4 0 2 3 3 2 4 6 2
A1BG 50 52 86 80 66 94 67 83
A1BG-AS1 7 9 12 10 11 18 7 8
A1CF 2 1 4 7 7 1 1 7
A2LD1 58 76 57 52 67 97 101 98
A2M 0 1 0 1 1 0 0 0
The following few steps are used to create a pseudo-reference sample for each gene. To do this, we need to calculate the geometric mean for each gene.
Step 2: Take the log of all values
log_data = log(data)
head(log_data)
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
12SBSRNA4 -Inf 0.6931472 1.098612 1.098612 0.6931472 1.386294 1.791759 0.6931472
A1BG 3.9120230 3.9512437 4.454347 4.382027 4.1896547 4.543295 4.204693 4.4188406
A1BG-AS1 1.9459101 2.1972246 2.484907 2.302585 2.3978953 2.890372 1.945910 2.0794415
A1CF 0.6931472 0.0000000 1.386294 1.945910 1.9459101 0.000000 0.000000 1.9459101
A2LD1 4.0604430 4.3307333 4.043051 3.951244 4.2046926 4.574711 4.615121 4.5849675
A2M -Inf 0.0000000 -Inf 0.000000 0.0000000 -Inf -Inf -Inf
Step 3: Take the average of each row
library(dplyr)
library(tibble)
log_data = log_data %>%
rownames_to_column('gene') %>%
mutate (pseudo_reference = rowMeans(log_data))
head(log_data)
gene siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32 pseudo_reference
1 12SBSRNA4 -Inf 0.6931472 1.098612 1.098612 0.6931472 1.386294 1.791759 0.6931472 -Inf
2 A1BG 3.9120230 3.9512437 4.454347 4.382027 4.1896547 4.543295 4.204693 4.4188406 4.2570154
3 A1BG-AS1 1.9459101 2.1972246 2.484907 2.302585 2.3978953 2.890372 1.945910 2.0794415 2.2805306
4 A1CF 0.6931472 0.0000000 1.386294 1.945910 1.9459101 0.000000 0.000000 1.9459101 0.9896465
5 A2LD1 4.0604430 4.3307333 4.043051 3.951244 4.2046926 4.574711 4.615121 4.5849675 4.2956204
6 A2M -Inf 0.0000000 -Inf 0.000000 0.0000000 -Inf -Inf -Inf -Inf
Step 4: Filter out all of the genes with -Inf as their average
They will not be used to calculate the median.
filtered_log_data = log_data %>% filter(pseudo_reference != "-Inf")
head(filtered_log_data)
gene siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32 pseudo_reference
1 A1BG 3.9120230 3.951244 4.454347 4.382027 4.189655 4.543295 4.2046926 4.418841 4.2570154
2 A1BG-AS1 1.9459101 2.197225 2.484907 2.302585 2.397895 2.890372 1.9459101 2.079442 2.2805306
3 A1CF 0.6931472 0.000000 1.386294 1.945910 1.945910 0.000000 0.0000000 1.945910 0.9896465
4 A2LD1 4.0604430 4.330733 4.043051 3.951244 4.204693 4.574711 4.6151205 4.584967 4.2956204
5 A4GALT 6.1506028 6.100319 6.270988 6.565265 6.335054 6.428105 6.3952616 6.171701 6.3021621
6 A4GNT 0.6931472 0.000000 0.000000 1.609438 1.945910 1.098612 0.6931472 0.000000 0.7550318
Step 5: Subtract the gene pseudo-references from log counts
In this step, you are subtracting the average of the logs from the log of the original data. Log(x/y) is a ratio.
ratio_data = sweep(filtered_log_data[,2:9], 1, filtered_log_data[,10], "-")
head(ratio_data)
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
1 -0.34499242 -0.30577171 0.19733187 0.12501121 -0.06736068 0.2862794 -0.05232281 0.1618252
2 -0.33462050 -0.08330607 0.20437600 0.02205444 0.11736462 0.6098411 -0.33462050 -0.2010891
3 -0.29649932 -0.98964650 0.39664786 0.95626365 0.95626365 -0.9896465 -0.98964650 0.9562637
4 -0.23517736 0.03511297 -0.25256910 -0.34437665 -0.09092775 0.2790906 0.31950015 0.2893471
5 -0.15155934 -0.20184315 -0.03117367 0.26310286 0.03289215 0.1259432 0.09309949 -0.1304615
6 -0.06188466 -0.75503184 -0.75503184 0.85440607 1.19087831 0.3435804 -0.06188466 -0.7550318
Step 6: Find the median of the ratios for each sample
sample_medians = apply(ratio_data, 2, median)
sample_medians
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
-0.02605145 -0.04549215 -0.07749397 0.12030642 0.06268842 0.05039392 -0.06025587 0.03220589
Step 7: Convert medians to scaling factors
# scaling factor = e^median
scaling_factors = exp(sample_medians)
scaling_factors
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
0.9742850 0.9555271 0.9254326 1.1278424 1.0646951 1.0516853 0.9415236 1.0327301
Step 8: Divide the original counts (not log version) by the scaling factors
manually_normalized = sweep(data, 2, scaling_factors, "/")
head(manually_normalized)
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
12SBSRNA4 0.000000 2.093086 3.241727 2.6599461 1.8784721 3.8034192 6.372650 1.936614
A1BG 51.319688 54.420225 92.929511 70.9318970 61.9895807 89.3803503 71.161255 80.369498
A1BG-AS1 7.184756 9.418885 12.966909 8.8664871 10.3315968 17.1153862 7.434758 7.746458
A1CF 2.052788 1.046543 4.322303 6.2065410 6.5746525 0.9508548 1.062108 6.778150
A2LD1 59.530838 79.537252 61.592816 46.1057330 62.9288167 92.2329146 107.272936 94.894106
A2M 0.000000 1.046543 0.000000 0.8866487 0.9392361 0.0000000 0.000000 0.000000
Section two: a function to normalize the DESeq2 way
Below is one function to “manually” normalize data with the median of ratios method. Import data before using the function.
mor stands for “median of ratios”.
mor_normalization = function(data){
library(dplyr)
library(tibble)
# take the log
log_data = log(data)
# find the psuedo-references per sample by taking the geometric mean
log_data = log_data %>%
rownames_to_column('gene') %>%
mutate (gene_averages = rowMeans(log_data)) %>%
filter(gene_averages != "-Inf")
# the last columns is the pseudo-reference column
pseudo_column = ncol(log_data)
# where to stop before the pseudo column
before_pseduo = pseudo_column - 1
# find the ratio of the log data to the pseudo-reference
ratios = sweep(log_data[,2:before_pseduo], 1, log_data[,pseudo_column], "-")
# find the median of the ratios
sample_medians = apply(ratios, 2, median)
# convert the median to a scaling factor
scaling_factors = exp(sample_medians)
# use scaling factors to scale the original data
manually_normalized = sweep(data, 2, scaling_factors, "/")
return(manually_normalized)
}
You can check that this function returns the same result as the step-by-step procedure above 
.
mor_normalization(data) == manually_normalized
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
12SBSRNA4 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A1BG TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A1BG-AS1 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A1CF TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A2LD1 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
... (more lines) ...
Section 3: sanity check
A sanity check to see if the normalization within DESeq2 is identical to our manually normalized data.
library(DESeq2)
# samples (columns names) of the data should be named
samples = as.data.frame(colnames(data))
# create a DESeqDataSet object. The design can be altered based on experimental design. A design of 1 means no design.
dds = DESeqDataSetFromMatrix(countData = data, colData = samples, design = ~1)
# this function generates the size factors
dds = estimateSizeFactors(dds)
We can compare the scaling factors that were found manually vs. the scaling factors (termed, “size factors”) from DESeq2.
# scaling_factors were manually computed using our mor_normalization function
# sizeFactors(dds) is used to find the scaling factors from DESeq2
scaling_factors == sizeFactors(dds)
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
Lastly, we can use counts() to normalize the original data in DESeq2. When counts() is used with the normalized = TRUE arguments, the counts will be normalized. When you do not use this argument, counts(dds) returns the original counts.
normalized_deseq2 = counts(dds, normalized = TRUE)
The normalized counts are identical between DESeq2 and the manual method.
normalized_deseq2 == manually_normalized
siC1 siC2 siE11 siE12 siE21 siE22 siE31 siE32
12SBSRNA4 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A1BG TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A1BG-AS1 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A1CF TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A2LD1 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A2M TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A2ML1 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
A2MP1 TRUE TRUE TRUE TRUE TRUE TRUE TRUE TRUE
... (more lines) ...
References
- Love et al. (2014) Moderated estimation of fold change and dispersion for RNA-seq data with DESeq2. Genome Biology, 15:550. doi:10.1186/s13059-014-0550-8.
- Statquest
- DGE count normalization