Kron2Sum

class glimix_core.lmm.Kron2Sum(Y, A, X, G, rank=1, restricted=False)

LMM for multi-traits fitted via maximum likelihood.

This implementation follows the work published in [CA05]. Let n, c, and p be the number of samples, covariates, and traits, respectively. The outcome variable Y is a n×p matrix distributed according to:

vec(Y) ~ N((A ⊗ X) vec(B), K = C₀ ⊗ GGᵀ + C₁ ⊗ I).

A and X are design matrices of dimensions p×p and n×c provided by the user, where X is the usual matrix of covariates commonly used in single-trait models. B is a c×p matrix of fixed-effect sizes per trait. G is a n×r matrix provided by the user and I is a n×n identity matrices. C₀ and C₁ are both symmetric matrices of dimensions p×p, for which C₁ is guaranteed by our implementation to be of full rank. The parameters of this model are the matrices B, C₀, and C₁.

For implementation purpose, we make use of the following definitions:

• 𝛃 = vec(B)

• M = A ⊗ X

• H = MᵀK⁻¹M

• Yₓ = LₓY

• Yₕ = YₓLₕᵀ

• Mₓ = LₓX

• Mₕ = (LₕA) ⊗ Mₓ

• mₕ = Mₕvec(B)

where Lₓ and Lₕ are defined in glimix_core.cov.Kron2SumCov.

References

CA05

Casale, F. P., Rakitsch, B., Lippert, C., & Stegle, O. (2015). Efficient set tests for the genetic analysis of correlated traits. Nature methods, 12(8), 755.

__init__(Y, A, X, G, rank=1, restricted=False)

Constructor.

Parameters

• Y ((n, p) array_like) – Outcome matrix.

• A ((n, n) array_like) – Trait-by-trait design matrix.

• X ((n, c) array_like) – Covariates design matrix.

• G ((n, r) array_like) – Matrix G from the GGᵀ term.

• rank (optional, int) – Maximum rank of matrix C₀. Defaults to 1.

Methods

__init__(Y, A, X, G[, rank, restricted])	Constructor.
covariance()	Covariance K = C₀ ⊗ GGᵀ + C₁ ⊗ I.
fit([verbose])	Maximise the marginal likelihood.
get_fast_scanner()	Return FastScanner for association scan.
gradient()	Gradient of the log of the marginal likelihood.
lml()	Log of the marginal likelihood.
mean()	Mean 𝐦 = (A ⊗ X) vec(B).
value()	Log of the marginal likelihood.

Attributes

A	A from the equation 𝐦 = (A ⊗ X) vec(B).
B	Fixed-effect sizes B from 𝐦 = (A ⊗ X) vec(B).
C0	C₀ from equation K = C₀ ⊗ GGᵀ + C₁ ⊗ I.
C1	C₁ from equation K = C₀ ⊗ GGᵀ + C₁ ⊗ I.
M	M = (A ⊗ X).
X	X from equation M = (A ⊗ X).
beta	Fixed-effect sizes 𝛃 = vec(B).
beta_covariance	Estimates the covariance-matrix of the optimal beta.
name	Name of this function.
ncovariates	Number of covariates, c.
nsamples	Number of samples, n.
ntraits	Number of traits, p.