Quantile normalization
Quantile normalization is frequently used in microarray data analysis. It was introduced as quantile standardization and then renamed as quantile normalization.
Quantile, quartile, percentile ???
Quantiles are just the lines that divide data into equally sized groups.percentiles are just quantiles that divide the data into 100 equally sized groups
Example:
0 quartile = 0 quantile = 0th percentile
1 quartile = 0.25 quantile = 25th percentile
2 quartile = .5 quantile = 50th percentile (median)
3 quartile = .75 quantile = 75th percentile
4 quartile = 1 quantile = 100th percentile
Quantile normalization
Quantile normalization transform the statistical distributions across samples to be the same.
Assumptions
- The roughly same distribution of values across samples
- Most genes are not differentially expressed
Assume global differences in the distribution are induced by only technical variation!
How Q-normalization work
row: genes
column: samples/Arrays
Procedure:
- order values within each sample
- determine a rank from lowest to highest and record the order within each sample
- Average across rows and substitute value with average
- re-order averaged values in the original order recorded in 2.
Tied rank entries ?
Average the tied rank entries’ mean values and substitute.
When NOT to normalize
Consider a dilution experiment. In which distributions are supposed to decrease (left plot), Q-normalization does the totally wrong thing (right plot). When you expect a real difference in distributions, Q-normalization will create weird artifacts.
Smooth quantile ormalizaiton
Assumptions
the statistical distribution of each sample should be the same ( or have the same distributional shape) within biological groups or conditions, but allowing that they may differ between groups
How to
At each quantile, a weight is computed comparing the variability between groups relative to the total variability between and within groups
Let gene(g) denote the $g^{th}$ row after sorting each column in the data. For each row, gene(g), we compute the weight $w(g)$ ∈ [0,1], where a weight of 0 implies quantile normalization within groups is applied and a weight of 1 indicates quantile normalization is applied. The weight at each row depends on the between group sum of squares SSB(g) and total sum of squares SST(g), as follows:
$$ w_{(g)} = \operatorname{median} \bigg\lbrace 1- \frac{SSB_{(i)}}{SST_{(i)}} \bigg\rbrace \text{for } i = g -k, \cdots, g, \cdots, g+k $$
where $k$ = floor(Total number of genes * 0.05). The number 0.05 is a flexible parameter that can be altered to change the window of the number of genes considered.
StatQuest: Quantile Normalization
reference
- https://en.wikipedia.org/wiki/Quantile_normalization
- BIOMEDIN 245: Statistical and Machine Learning Methods for Genomics, Stanford
- Hicks SC, Okrah K, Paulson JN, Quackenbush J, Irizarry RA, Bravo HC. Smooth quantile normalization. Biostatistics. 2018;19(2):185‐198. doi:10.1093/biostatistics/kxx028