# Inferring phenotypic trait evolution on large trees with many incomplete measurements

## TL;DR

This paper introduces a scalable inference method for analyzing phenotypic trait evolution across large phylogenetic trees with many missing measurements, improving computational efficiency significantly.

## Contribution

It presents an analytical integration technique for missing data under a multivariate Brownian diffusion model that scales linearly with the number of taxa.

## Key findings

- Achieves up to two orders-of-magnitude speedup over existing methods.
- Successfully applied to diverse biological datasets including mammals, prokaryotes, and HIV.
- Extends the model to incorporate sampling error and residual variance.

## Abstract

Comparative biologists are often interested in inferring covariation between multiple biological traits sampled across numerous related taxa. To properly study these relationships, we must control for the shared evolutionary history of the taxa to avoid spurious inference. Existing control techniques almost universally scale poorly as the number of taxa increases. An additional challenge arises as obtaining a full suite of measurements becomes increasingly difficult with increasing taxa. This typically necessitates data imputation or integration that further exacerbates scalability. We propose an inference technique that integrates out missing measurements analytically and scales linearly with the number of taxa by using a post-order traversal algorithm under a multivariate Brownian diffusion (MBD) model to characterize trait evolution. We further exploit this technique to extend the MBD model to account for sampling error or non-heritable residual variance. We test these methods to examine mammalian life history traits, prokaryotic genomic and phenotypic traits, and HIV infection traits. We find computational efficiency increases that top two orders-of-magnitude over current best practices. While we focus on the utility of this algorithm in phylogenetic comparative methods, our approach generalizes to solve long-standing challenges in computing the likelihood for matrix-normal and multivariate normal distributions with missing data at scale.