In adw96/DivNet: Diversity Estimation in Networked Ecological Communities

The point of this vignette

It's fairly common to summarize beta diversity analyses with some appeal to "centroids" of various communities -- but what exactly do we mean by "centroid", and what role does our choice of beta diversity play here? This vignette provides some background on centroids and their links to beta diversities, as well as a brief discussion of how DivNet handles some of the intricacies of centroid and beta diversity estimation.

What is a centroid?

In the context we'll discuss here, a centroid is a way to summarize the "typical" microbial community in some group of microbial communities. For example, we might be interested in describing the typical microbial community present in the cheek mucosa of a healthy human adult. If we were able to (accurately) measure the cheek microbiome of every human adult (in terms of relative abundances, let's assume), we could calculate a centroid by taking an average in each taxon (e.g., strain, species, genus, etc.). That is, one kind of centroid we could calculate is just a vector giving the mean relative abundance of microbes in the healthy cheek microbiome by taxon.

By no means is this the only choice of centroid available to us -- we could, for instance, calculate median instead of mean relative abundances by taxon, and this would also give us a (different kind of) centroid.

Another complication we run into here is that it is basically impossible to measure the cheek microbiome of every healthy human adult. The centroids we've discussed so far are what statisticians refer to as "population" parameters -- idealized values we would calculate if we were able to observe the entire population (here, all healthy human cheek microbiomes) we are interested in. A standard statistical task, which DivNet provides tools for, is to reason about population parameters on the basis of a sample from that population (in our example, the people whose cheeks we were able to measure). Before we get into DivNet-specific details, however, let's first discuss beta diversities and their relation to centroids.

What is a beta diversity?

A beta diversity is a way of quantifying the degree of similarity or dissimilarity between two microbial communities. This is, so far, extremely general -- there are countless ways to quantify how (dis)similar two communities are. For the moment, however, let's consider just two: Euclidean distance and Bray-Curtis dissimilarity.

Suppose we have (relative abundance) measurements $\vec{x}A$ and $\vec{x}_B$ on communities $A$ and $B$ covering taxa $j = 1, \dots, J$ (so $\vec{x}_A = [x{A1}, \dots x_{AJ}]$, the vector of relative abundances of taxa $j = 1, \dots, J$ in community A, and similarly for community $\vec{x}_B$ with community $B$). Then the Euclidean distance between $\vec{x}_A$ and $\vec{x}_B$ is

$$d_{Euc}(\vec{x}A,\vec{x}_B) = \sqrt{\sum{j = 1}^J (x_{Aj} - x_{Bj})^2}$$ In words, the Euclidean distance is the square root of the sum of squared between-community differences of relative abundances in each taxon.

As the term "dissimilarity" suggests, the Bray-Curtis dissimilarity is not always a distance, but on relative abundances it is, with

$$d_{BC}(\vec{x}A,\vec{x}_B) = \frac{1}{2}\sum{j = 1}^J |x_{Aj} - x_{Bj}|$$ That is, the Bray-Curtis dissimilarity is one half the sum of the absolute value of the between-community differences of relative abundances in each taxon.

These two choices of beta diversity are not equivalent -- we will get different summaries of how different or similar communities are depending on which one we use. Additionally, as we'll see below, given a choice of beta diversity, we can summarize our analysis in terms of a centroid or a set of centroids.

What do centroids have to do with beta diversities?

It turns out that there is a correspondence between commonly used beta diversities and centroids. To see this, let's continue our cheek microbiome example. As we've already discussed, we can calculate one type of centroid for our (hypothetical) set of observations of all healthy human cheek microbiomes by taking a mean of the relative abundance in each taxon across all our observations. We can also characterize this centroid in a seemingly completely different (but in reality equivalent) way -- as the point that minimizes the sum of (squared) Euclidean distances to all of our observations.

If we let $(\vec{x}1, \dots, \vec{x}_N)$ be our set of (relative abundance) observations on all healthy cheek microbiomes (here we are imagining this set to consist of a large but finite number $N$ of cheek microbiomes) and we let $\vec{x}{mean}$ indicate the centroid defined by taxon-wise means over our population of cheek microbiomes, we can write out a characterization of the population centroid in terms of Euclidean distances to our observations:

$$\vec{x}{mean} = \text{arg min}{\vec{x}} \sum_{i = 1}^N d_{Euc}(\vec{x},\vec{x}_i)^2$$ In other words, the (population) mean minimizes the sum of squared Euclidean distances to observations in our population of healthy human cheek microbiomes. (We can write out this correspondence more generally to include the case of infinite populations by writing righthand side of the above equation in terms of an expectation, but for simplicity we'll stick to the finite case here and leave the general case as an exercise for the curious reader.)

What about Bray-Curtis dissimilarity? It turns out we have a similar equivalence in this case as well! The centroid $\vec{x}_{med}$ defined by taking taxon-wise medians over our population of cheek microbiomes is

$$\vec{x}{med} = \text{arg min}{\vec{x}} \sum_{i = 1}^N d_{BC}(\vec{x},\vec{x}_i)$$ Hence the (population) minimizer of Bray-Curtis dissimilarities is just the taxon-wise median relative abundance! (Note that -- again for simplicity -- we're stating, and not proving, these results.)

Why haven't you discussed Aitchison distance?

For clarity, we've stuck to two dissimilarities commonly used in microbial ecology -- Euclidean distance and Bray-Curtis dissimilarity. Another possible choice of beta diversity is Aitchison distance, which is equivalent to Euclidean distance calculated on centered-log-ratio-transformed relative abundances. As you might expect from this characterization of Aitchison distance, the group centroids this choice of beta diversity induces are the mean centered-log-ratio-transformed relative abundances in each group. (We can either keep these centroids on the centered-log-ratio scale or back-transform them to relative abundances.) DivNet accommodates Aitchison-distance-based beta diversity analyses -- see our beta diversity vignette for more details.

What about measurement?

In the discussion above, we've implicitly assumed (for simplicity) that our sample relative abundances are measured without error (or at least with negligible error). In reality, this doesn't appear to be the case! Whatever amplicon protocol we choose, we are likely to overdetect some taxa and underdetect others (McLaren et al., 2019). As a result, measured sample compositions are biased for the true compositions, and so our beta diversity analyses are in general estimating not true group centroids, but instead average measured group centroids, which are not typically equal to true group centroids.

Addressing measurement error in microbiome analyses is an area of ongoing methods development. While methods are still being developed, we suggest being explicit the limitations of existing technologies and analyses. In the beta diversity vignette for DivNet, we provide some examples of formal write ups of DivNet beta diversity analyses that acknowledge this issue.

References

McLaren, Michael R., Amy D. Willis, and Benjamin J. Callahan. "Consistent and correctable bias in metagenomic sequencing experiments." Elife 8 (2019): e46923.

adw96/DivNet documentation built on Oct. 2, 2023, 11:49 a.m.