Motivation: High dimensional and diverse phenotypes

How to handle a large number of phenotypes?

More and more phenotypes are being generated across time and space

Challenges:

• high dimensional phenotypes
• diverse phenotypes
• how to make sense of these data and interpret
• multi-trait linear mixed model is computationally challenging

Objective:

• leverage Bayesian confirmatory factor analysis and Bayesian network to characterize a wide spectrum of rice phenotypes

Bayesian confirmatory factor analysis

Assume observed phenotypes are derived from underlying latent variables

$$\begin{align*} \mathbf{T} = \mathbf{\Lambda} \mathbf{F} + \mathbf{s} \end{align*}$$

• $\mathbf{T}$ is the $t \times n$ matrix of observed phenotypes (413 accessions)

• $\mathbf{\Lambda}$ is the $t \times q$ factor loading matrix

• $\mathbf{F}$ is the $q \times n$ latent variables matrix

• $\mathbf{s}$ is the $t \times n$ matrix of specific effects.

Variance-covariance model: $$\begin{align*} var\mathbf{(T)} &= \mathbf{\Lambda}\mathbf{\Phi}\mathbf{\Lambda}' + \mathbf{\Psi}, \end{align*}$$

• $\mathbf{\Phi}$ is the variance of latent variables

• $\mathbf{\Psi}$ is the variance of specific effects

Prior distributions

$$\begin{align*} var\mathbf{(T)} &= \mathbf{\Lambda}\mathbf{\Phi}\mathbf{\Lambda}' + \mathbf{\Psi}, \end{align*}$$

• Factor loading matrix: $$\begin{align*} & \mathbf{\Lambda} \sim \mathcal{N}(0, 0.01) \end{align*}$$

• Variance of latent variables: $$\begin{align*} \mathbf{\Phi} \sim \mathcal{W}^{-1}(\mathbf{I}_{66}, 7) \end{align*}$$

• Variance of specific effects: $$\begin{align*} \mathbf{\Psi} \sim \Gamma^{-1} (1, 0.5) \end{align*}$$

Define 6 latent variables from 48 phenotypes

1. Grain Morphology (Grm, 11)
   • Seed length (Sl), Seed width (Sw), Seed volume (Sv), etc
2. Morphology (Mrp, 14)
   • Flag leaf length (Fll), Flag leaf width (Flw), etc
3. Flowering Time (Flt, 7)
   • Flowering time in Arkansas (Fla), Flowering time in Aberdeen (Flb), etc
4. Ionic components of salt stress (Iss, 6)
   • Na shoot (Nas), K shoot salt (Kss), etc
5. Yield (Yid, 5)
   • Panicle number per plant (Pnu), Panicle length (Pal), etc
6. Morphological salt response (Msr, 5)
   • Shoot BM ratio (Sbr), Root BM ratio (Rbr), etc

Study the genetics of each latent variable

Multivariate analysis

Bayesian genomic best linear unbiased prediction

- separate genetic effects from noise (44K SNPs)

$$\begin{align*} \mathbf{F} = \boldsymbol{\mu} + \mathbf{Xb} + \mathbf{Zu} + \boldsymbol{\epsilon} \end{align*}$$

• $\mathbf{F}$ : Vector of factor scores
• $\mathbf{X}$ : Incidence matrix for fixed effects
• $\mathbf{Z}$ : Incidence matrix for additive genetic effects
• $\mathbf{b}$: Vector of fixed effects
• $\mathbf{u}$: Vector of additive genetic effects
• $\mathbf{e}$: Vector of residuals

Piror distributions

$$\begin{align*} \mathbf{F} = \boldsymbol{\mu} + \mathbf{Xb} + \mathbf{Zu} + \boldsymbol{\epsilon} \end{align*}$$

• $\mathbf{\mu}$, $\mathbf{b}$ were assigned a flat prior

• $\boldsymbol{\Sigma_{u}}$, $\boldsymbol{\Sigma_{e}}$ are variance-covariance matrix between latent variables

$$\begin{align*} \boldsymbol{\Sigma_{u}}, \boldsymbol{\Sigma_{e}} \sim \mathcal{W}^{-1}(\mathbf{I}_{66}, 6) \end{align*}$$

Bayesian Network

Probabilistic directed acyclic graphical model (DAG)

- interrelationships (Edges) among latent variables (Nodes)
- genetic selection for breeding requires causal assumptions

Constraint-based learning

Score-based learning

Examples of algorithms

• Score-based algorithm

  • Hill climbing
  • Tabu search

• Hybrid algorithm

  • Max-Min Hill Climbing algorithm
  • General 2-Phase Restricted Maximization algorithm

Standardized factor loadings

Standardized factor loadings

Genetic correlations among latent variable

Hill Climbing algorithm

Tabu algorithm

Max-Min Hill Climbing algorithm

General 2-Phase Restricted Maximization algorithm

Consensus Bayesian network

FA vs. PCA

• What is the main difference between principal component analysis (PCA) and factor analysis (FA)?

• Confirmatory factor analysis (CFA) vs. Explanatory factor analysis (EFA)

CFA vs. EFA

Summary

• Bayesian cofirmatory factor analysis allows to work at the level of latent variables

• Bayesian network can be applied to predict the potential influence of external interventions or selection associated with target traits

• Provide greater insights than pairwise-association measures of multiple phenotypes

• It is possible to dissect genetic signals from high-dimensional phenotypes if we focus on underlying patterns in big data

Useful links

• Genomic Bayesian confirmatory factor analysis and Bayesian network to characterize a wide spectrum of rice phenotypes (https://www.g3journal.org/content/9/6/1975)

• An illustrative example of Bayesian Network (https://haipengu.github.io/Rmd/GBN.html)