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# Fuzzy zeros in percentile normalization method to correct for batch effects

We recently published a method to normalize data that corrects for batch effects and allows for pooling of raw data across studies. In this post, I want to discuss more about the important (but potentially confusing) aspect of our method, where we add noise to the zeros before performing the normalization. I’ve gotten a few questions about this, and wanted to write this short blog post to illustrate why we did this and why it improves the method.

In the paper, we present a simple non-parametric normalization procedure where features (i.e. bacterial taxa) in case samples are converted to percentiles of the equivalent features in control samples within a study prior to pooling data across studies. In other words, for each feature in a study, we convert its abundances across control samples into a uniform distribution, and then convert the abundances in case samples to the percentile of that uniform that they fall on. Another way to think about this method is that the controls are a null distribution and cases are converted to values describing where they fall on the control distribution. We’re excited about this method because it allows us to combine “raw” data across disparate studies, as long as all studies have some subset of samples which are comparable. In the paper, we focus on case/control studies, but this could be applied to any meta-analysis with a common comparison group among all the studies.

In this post, I want to provide some discussion on how the percentile normalization actually works, and one of the big challenges we had to overcome in developing it. The main workhorse of this method is the conversion of control abundances to a uniform and subsequent mapping of case abundances onto that uniform. To do this, we use  SciPy’s percentileofscore function.

The percentile rank of a score relative to a list of scores.

A `percentileofscore` of, for example, 80% means that 80% of the
scores in `a` are below the given score. In the case of gaps or
ties, the exact definition depends on the optional keyword, `kind`.

For each feature in each case sample, we use this function to convert its abundance to a percentile rank of that feature’s abundances across all control samples. For example, if we start out with 10 control samples with the following abundances for a given feature: [0.1, 0.1, 0.2, 0.2, 0.3, 0.3] and we have a case sample with abundance of 0.2, then it gets converted to 0.5, because 0.2 is in the 50th percent rank of the control distribution (in fact, it is the median value):

import scipy as sp

sp.percentileofscore([0.1, 0.1, 0.2, 0.2, 0.3, 0.3], 0.2, ‘mean’)
#Out: 50.0

When we first started working on the method, we called this function straight on the relative abundances, without correcting for zeros. However, we noticed some weird behavior for more sparse OTUs: when we percentile normalized the data, we got a pile-up of values corresponding to zero abundance. The problem was that this rank pile-up wasn’t in the same place across studies, since it depends on what proportion of controls are zero, which is different in every study. So when we pooled the percentile normalized data, we noticed that these different rank pile-ups actually led to spurious associations driven by batch effects. In the figure below, you can see this clearly: the ranks pile up in different places across the different studies, and so pooling the data leads to data distributions that can mess up downstream non-parametric tests. Oh no! Percentile normalization data for Enterobacter genus across 4 colorectal cancer studies (colors) and pooled (gray), before adding fuzzy zeros. Controls are on the left side of each study (median indicated by X) and cases are on the right (median indicated by +).

The reason this is happening actually makes total sense, if you think about how the percentile normalization function is working. In our case, we’re calling it with the “mean” score: the value is converted to its rank in the provided array, and in cases of ties it is the mean rank of all those tied values. So if the first 50% of your data is zeros, then all of those zeros will be converted to 0.25, because 25% is the mean rank of the values that make up the first 50% of the data.

# Make a list with 5 zeros and 5 random numbers between 0 and 1
a = 5*[0.0] + list(np.random.uniform(0, 1, 5))

# Print a
print(‘, ‘.join([‘{:.2f}’.format(i) for i in a]))
#0.00, 0.00, 0.00, 0.00, 0.00, 0.36, 0.97, 0.47, 0.11, 0.66

# Percentile normalize 0.0 with respect to a
sp.percentileofscore(a, 0.0, ‘mean’)
#Out: 25.0

In other words, if zero is the first 20% of the cumulative distribution of your data, then the mean percentile rank of the value zero is 10%.

The way we got around this problem was by adding noise to the zeros, drawn randomly from a uniform going between 0 and something smaller than the smallest value in the data. Thus, we convert all the zeros in the data to values that are different from each other (and thus won’t have the same rank), but which are less than the smallest non-zero number (and thus will have a lower rank than the “real” data).

If we go back to our previous example, where 0.11 is the lowest value, this would look like:

# Add noise to the zeros
a_fuzzy = [np.random.uniform(0, 0.01) if i == 0.0 else i for i in a]

# Print a
print(‘, ‘.join([‘{:.3f}’.format(i) for i in a_fuzzy]))
#0.004, 0.006, 0.007, 0.006, 0.007, 0.360, 0.974, 0.471, 0.108, 0.659

# Check that this fixes the rank pile-up problem

## Percentile normalize 0.0 with respect to a
sp.percentileofscore(a_fuzzy, 0.0, ‘mean’)
#Out: 0.0

## Percentile normalize a non-zero value with respect to a
sp.percentileofscore(a_fuzzy, 0.8, ‘mean’)
#Out: 90.0

## Check that this non-zero values has the same percentile in the original data
sp.percentileofscore(a, 0.8, ‘mean’)
#Out: 90.0

If we do this to our real data, we see two things: (1) the control distributions are now actually uniform (unlike before, where we saw the pile-up at the mean “zero” rank) and (2) we’ve gotten rid of the rank pile-ups in each study and the pooled data: Same genus as above, but this time with the fuzzy zeros added before percentile normalization. Control values are now uniformly distributed across the entire 0-100% range, and rank pile-ups are no longer leading to spurious results after pooling!

Theoretically speaking, we’re basically removing any claim on the informativeness of the value of zero in our data. All we know is that it is a smaller value than the smallest observed value, but we don’t know whether some of those zeros are “true” zeros (i.e. the organism is not present) vs. “false” zeros (i.e. the organism is present but unobserved). Thus, adding this noise is actually a fairly conservative thing to do. But it allows us to correct for batch effects and still sleep at night knowing that we’re not doing something weird to the data – win win!

A modified version of this post was originally published on my personal website: https://cduvallet.github.io/posts/2018/06/fuzzy-zeros

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