Guidelines for standardising the application of discriminant analysis of principal components to genotype data

Abstract

Multivariate methods are incredibly beneficial for population genetic analyses where the number of measured variables (genetic loci) can easily exceed the number of sampled individuals. Discriminant analysis of principal components (DAPC) has become a popular method for visualising population structure in genotype datasets due to its simplicity, its computational speed, and its freedom from demographic assumptions. Despite the popularity of DAPC in population genetic studies, there has been little discussion on best practise and parameterisation. Unappreciated, perhaps, is the fact that unlike principal component analysis (PCA), which is a hypothesis free method, discriminant analysis (DA) is a hypothesis driven method. That is, when performing a DA, a researcher is making an explicit hypothesis about how variation in a set of predictor variables is organised among pre-defined groups in a sample set. Parameter choice is critical to ensure the results produced by a DA are biologically meaningful. In a DAPC, one of the most important parameter choices is the number of PC axes, paxes, to use as predictors in a DA of among-population differences. Yet there are no clear guidelines on how researchers should choose paxes. In this work, I propose that the value of paxes is a deterministic feature of a genotype dataset based on population genetic theory. For k discrete populations, only the first k - 1 PC axes are expected to be biologically informative and capture population structure. DAs fit using more than the first k - 1 PC axes are over-parametrised and may discriminate groups using biologically uninformative predictors. Using samples drawn from simulated metapopulations, I show that DAPCs parameterised with the appropriate k - 1 PC axes are: (1) more parsimonious; (2) capture the maximal amount of among-population variation using biologically relevant predictors; (3) are less sensitive to unintended interpretations of population structure; and (4) are more generally applicable to independent sample sets.

Guidelines for standardizing the application of discriminant analysis of principal components to genotype data

“…The data and scripts associated with this paper have been deposited into Dryad (Thia, 2022): https://doi.org/10.5061/dryad.b8gtht7f0.…”

A roadmap to robust discriminant analysis of principal components

“…The second is to select components that describe the between cluster variation and use those in a discriminant analysis (DA), which builds a model that can predict the population for each individual. The number of components selected from the PCA is critical as selecting too many axes will result in overfitting the model and create an inflated estimate of the amount of differentiation among clusters (Thia, 2022). While the developers of the DAPC method indicate the importance of this step, they do not provide a defined set of rules for selecting the components (Jombart & Collins, 2022).…”

“…The selection of the number of PC axes has primarily relied on three approaches (Miller et al, 2020): (1) selecting the number of PC axes which explain an arbitrary amount of variance in the data set (often ≥80%), (2) the xvalDapc function, which uses a training and testing set to find the optimal trade‐off between selecting too few, and too many PC axes, or (3) the optim.a.score function, which looks at the assignment success of individuals based on different numbers of PC axes retained, and selects the number of PCs that maximizes assignment success. The study by Thia (2022) highlights caveats to consider for each of these methods. The first method often leads to inclusion of components which explain individual differences, rather than just including those which capture between population differences.…”

“…When there is no migration ( F ST = 0.99 in simulated data), there is little impact on the conclusions made because the majority of variance is between populations. However, when migration is medium to high ( F ST = 0.09 and 0.0009, respectively) there is an inflation of how differentiated populations appear (Figure 8 in Thia, 2022). The second method suffered from similar issues to the first, where the xvalDapc function indicates more PC axes be selected than those that just explained population structure in the simulated data sets, again resulting in an over‐fit model.…”

“…As an alternative to these approaches, Thia (2022) suggests selecting the number of PC axes using the K − 1 rule from Patterson et al (2006), where K is the number of genetic clusters. This initially seems like a very conservative approach; however, Thia (2022) clearly demonstrates these components capture the between population differences, and when migration is moderate, provide the best model for predicting population membership. Including additional PC axes can inflate or obscure the true population structure.…”