Fitting qpAdm models with a 'rotation' strategy
In admixr: An Interface for Running 'ADMIXTOOLS' Analyses

evaluate <- .Platform$OS.type == "unix" && system("which qpDstat", ignore.stdout = TRUE) == 0 && Sys.getenv("TRAVIS_R_VERSION") == ""

knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  eval = evaluate
)

Introduction

qpAdm model fitting is a complex topic. To navigate it successfuly requires solid knowledge of the $f$-statistics theory first introduced by Nick Patterson and colleagues in 2012. As part of our tutorial, we have looked at a very basic overview of the qpAdm-related functionality implemented in admixr. We also talked about the most important resources for learning more about this powerful method pioneered by Iosif Lazaridis in 2015.

Recently, Harney et al. published an exciting new preprint called "Assessing the Performance of qpAdm: A Statistical Tool for Studying Population Admixture". Before we go any further, I encourage everyone to read it and the superb tutorial/guide available as its supplementary pdf on bioRxiv. There really isn't a better source of information on how to run and interpret qpAdm analyses.

Please, only attempt to run qpAdm if you have familiarized yoursef with all of the above-mentioned resources. I have had many people ask questions via email (not only about qpAdm but also other topics) to which the only sensible answer was - "you have to read the papers and understand the statistics first." I know it's frustrating but there really are no shortcuts here.

qpAdm "rotation"

If you have ever worked with qpAdm, you are well aware of the intricacies of finding the most suitable set of models that can explain the data. Among other things, we have to make a decision about the number of admixture sources and which populations are the most appropriate surrogates for those source populations (because only rarely we have sampled them directly). Furthermore, we need to carefully choose a number of so called 'outgroup' populations (also called 'references' or 'right' populations, depending on whom you talk to).

The preprint by Harney et al. described an interesting idea to find a set of the most appropriate models (i.e. combinations of source and outgroup populations) which has been sucessfully used in the past. They call the method a "rotating population" strategy.

This approach starts by defining a set of "candidate" populations from which we iteratively sample a defined number of "sources" of ancestry for our "target" population of interest (most commonly two or three sources). After removing the sources from the candidate list, we then define all the remaining populations as "outgroups". Finally, we iteratively fit qpAdm models for each combination of target, sources and outgroups, extracting $p$-values and other statistics of interest. After finishing the exhaustive fitting of source-outgroup combinations, we examine all explored models, selecting those that seem most appropriate.

Implementation in admixr

In admixr, I have implemented a function qpAdm_rotation() which does exactly what is described paragraph with one additional feature. Given the sensitivity of qpAdm to large numbers of potential outgroups (references), for each combination of sources and outgroups we also explore models for all possible subsets of outgroups. This is to find models which are as small as possible, possibly determining which outgroups are potentially redundant and not actually needed.

Let's say that we have a target population T and a set of candidates for potential sources and outgroups C = {a, b, c, d, e, f}. Then, if we imagine an iteration of the rotation scheme in which we fixed sources S = {a, b}, we have remaining candidates for outgroups C - S = {c, d, e, f}. The basic implementation of the rotation procedure would simply take C - S as the full set of outgroups and fitted the following model:

model #1: target T, sources S = {a, b} and outgroups = {c, d, e, f}

However, in admixr, we would evaluate the following models in addition to the model #1:

model #2: target T, sources S = {a, b} and outgroups = {c, d, e}
model #3: target T, sources S = {a, b} and outgroups = {c, d, f}
model #4: target T, sources S = {a, b} and outgroups = {c, e, f}
model #5: target T, sources S = {a, b} and outgroups = {d, e, f}.

Therefore, our implementation in qpAdm_rotation() explores all posible outgroup combinations, allowing us to look for the smallest model (in terms of outgroup size) that can explain our data.

Concrete example

Performing exhaustive search by rotating sources/outgroups

As an example, let's revisit the problem of estimating the level of Neandertal ancestry in a French person from the main tutorial. We use this as an illustration because:

It's the simplest possible analysis one could do with qpAdm.
It gives us a clear expectation of what the "truth" is.
It gives us a clear expectation of what models we should definitely reject.

First, let's download and install a development version of admixr to get access to the new features, and download a small example data set:

library(admixr)

snps <- eigenstrat(download_data(dirname = tempdir()))

These are the individuals for which we have genotype data:

read_ind(snps)

The qpAdm_rotation() function is very simple. It accepts:

a name of the target population,
a list of candidate populations,
a logical parameter minimize, determining whether to perform the "minimization" of the outgroup size described in the previous section,
the assumed number of sources of ancestry,
the number of CPU cores to use for analysis (be careful with this options as many ADMIXTOOLS analyses run in parallel can consume a lot of memory!),
parameter fulloutput specifying whether we want to have all the "ranks" and "subsets/patterns" statistics (see the main tutorial for more information) or if we just want the proportions of ancestry and significance values for individual models (this is the default, i.e. fulloutput = FALSE).

So, let's say we are interested in finding the proportions of archaic human ancestry in a French individual, and we also want to see what sorts of possible models we could find that match archaic introgression. We would run the following:

models <- qpAdm_rotation(
    data = snps,
    target = "French",
    candidates = c("Dinka", "Mbuti", "Yoruba", "Vindija", "Altai", "Denisova", "Chimp"),
    minimize = TRUE,
    nsources = 2,
    ncores = 2,
    fulloutput = TRUE
)

Here is what the full output looks like:

models

We can see a list with three components, as we would expect from any other qpAdm() run (again, see the manual page and the tutorial for description of all three elements and their meaning). The first column of each component is always named model - this contains a short identifier of each individual "rotation" run (i.e., a combination target & sources & outgroups). It's values don't have any particular meaning - the order is completely arbitrary!, This variable is useful for later filtering and examination of individual models in detail.

Let's ignore the $ranks and $subsets elements for now. We will focus only on the first element, $proportions which contains the main qpAdm summary.

Examining and filtering fitted models

The $proportions table shown above contains information about all models, regardless of their plausibility. We can see that by examining the distributions of p-values (column pvalue) and admixture proportions (columns prop1 and prop2) of each evaluated model in the figure below.

Notice two things (each dot represents one examined qpAdm model):

Many models have inferred admixture proportions way outside the [0, 1] interval - those are clearly nonsensical.
Many models have very low p-values - this means these are incompatible with the data and can be rejected.

library(tidyverse)

select(models$proportions, model, pvalue, prop1, prop2) %>%
    gather(parameter, value, -model) %>%
    ggplot(aes(parameter, value)) +
    geom_jitter() +
    facet_wrap(~ parameter, scales = "free")

To make it easier to narrow down the list of all models, admixr package contains a function qpAdm_filter(). This function accepts the result of the qpAdm_rotation() function (either the fulloutput = TRUE version or the simple data frame with admixture proportions, p-values etc. produced by usingfulloutput = FALSE) and filters out models with any of the proportions outside of the [0, 1] range and with p-values lower than a specified cutoff (0.05 by default):

# filter out models which can clearly be rejected
fits <- qpAdm_filter(models)

We can verify that the filtering worked by visualizing the filtered set of models again. Note that the p-values are distributed across the range of "insigificance" (i.e., "non-rejection") between [0.05, 1.0]. Furthermore - remember that we originally set out to find combinations of sources-outgroups that model archaic ancestry in a French individual? We can clearly see two tidy clusters of estimated ancestry proportions. One is very small (this corresponds to the Neandertal component in modern humans - we would expect about 2-3% based on many previous analyses) and one large ("modern human" component, non-Neandertal ancestry):

select(fits$proportions, model, pvalue, prop1, prop2) %>%
    gather(parameter, value, -model) %>%
    ggplot(aes(parameter, value)) +
    geom_jitter() +
    facet_wrap(~ parameter, scales = "free") +
    coord_cartesian(y = c(0, 1))

Let's now focus only on the proportions table. We will also ignore a couple of columns for brevity. Note that we are now also completely ignoring p-values because we cannot used those for model selection - they are not statistically meaningful at this stage! Higher p-value never implies higher likelihood of the model. Finally, we order the models based on the size of the outgroup set (smaller models first):

props <- fits$proportions %>%
    arrange(noutgroups) %>%
    select(-c(target, noutgroups, stderr1, stderr2, nsnps_used, nsnps_target))

print(props, n = Inf)

Fun fact: notice in the table below that there are many models in which the chimpanzee was fitted as a source of ancestry! Interestingly, qpAdm used Chimp to infer archaic human ancestry. This is because you could think of Neandertal ancestry as an "ancestral component" of a modern human genome and the qpAdm rotation procedure therefore concludes that Chimpanzee is not be an unreasonable surrogate for a source population. Of course, we know there are better sources in our candidates set - we have the archaic humans!

filter(props, source1 == "Chimp" | source2 == "Chimp")

Another interesting fact: notice that the rotating population procedure selected another plausible model characterizing the ancestry of the French individual. However, this of course doesn't represent Neandertal introgression. What it might possibly represent is left as an exercise for the reader... :)

filter(props, prop1 < 0.9, prop2 < 0.9)

Conclusions

At this stage of analysis, you would have to decide which of the models produced by qpAdm_filter() that cannot be immediately rejected are more reasonable than others and why. Possibly based on both some prior knowledge and additional statistics (such as the details information available in the full log output information shown by loginfo()). You could say that the qpAdm methodology, while rooted in strong statistics, is from a certain point as much art as it is science. Interpreting the results and finding the most appropriate models can be quite a challenge.

Happy modeling and please, do let me know if you discover bugs or missing features. My goal with this tool is to streamline qpAdm model fitting as much as possible and I can do it only with your input.

Final remarks

As a reminder, keep in mind that admixr gives you tools for filtering SNPs and also grouping samples into populations on the fly! You can easily process and group samples before plugging them into qpAdm_rotation()!
Also note that you can use the function loginfo() to examine the complete log output of any model by specifying the model identifier. This is helpful not only for debugging purposes but also for cases when you need a particular statistic in the full qpAdm log report which is not currently parsed by admixr: