lmm.aireml: Linear mixed model fitting with AIREML

In gaston: Genetic Data Handling (QC, GRM, LD, PCA) & Linear Mixed Models

Linear mixed model fitting with AIREML

Description

Estimate the parameters of a linear mixed model, using Average Information Restricted Maximum Likelihood (AIREML) algorithm.

Usage

lmm.aireml(Y, X = matrix(1, nrow = length(Y)), K,
           EMsteps = 0L, EMsteps_fail = 1L, EM_alpha = 1,
           min_tau, min_s2 = 1e-06, theta, constraint = TRUE, max_iter = 50L,
           eps = 1e-05, verbose = getOption("gaston.verbose", TRUE),
           contrast = FALSE, get.P = FALSE) 

Arguments

Y	Phenotype vector
X	Covariable matrix. By default, a column of ones to include an intercept in the model
K	A positive definite matrix or a list of such matrices
EMsteps	Number of EM steps ran prior the AIREML
EMsteps_fail	Number of EM steps performed when the AIREML algorithm fail to improve the likelihood value
EM_alpha	Tweaking parameter for the EM (see Details)
min_tau	Minimal value for model parameter tau (if missing, will be set to 1e-6)
min_s2	Minimal value for model parameter sigma^2
theta	(Optional) Optimization starting point theta = c(sigma^2, tau)
constraint	If TRUE, the model parameters respect the contraints given by min_tau and min_s2
max_iter	Maximum number of iterations
eps	The algorithm stops when the gradient norm is lower than this parameter
verbose	If TRUE, display information on the algorithm progress
contrast	If TRUE, use a contrast matrix to compute the Restricted Likelihood (usually slower)
get.P	If TRUE, the function sends back the last matrix P computed in the optimization process

Details

Estimate the parameters of the following linear mixed model, using AIREML algorithm:

Y =X beta + omega_1 + ... + omega_k + epsilon

with omega_i ~ N(0, tau_i K_i) for i \in 1, …,k and epsilon ~ N(0, sigma^2 I_n).

The variance matrices K_1, ..., K_k, are specified through the parameter K.

If EMsteps is positive, the function will use this number of EM steps to compute a better starting point for the AIREML algorithm. Setting EMsteps to a value higher than max_iter leads to an EM optimization. It can happen that after an AIREML step, the likelihood did not increase: if this happens, the functions falls back to EMsteps_fail EM steps. The parameter EM_alpha can be set to a value higher than 1 to attempt to accelerate EM convergence; this could also result in uncontrolled behaviour and should be used with care.

After convergence, the function also compute Best Linear Unbiased Predictors (BLUPs) for beta and omega, and an estimation of the participation of the fixed effects to the variance of Y.

Value

A named list with members:

sigma2	Estimate of the model parameter sigma^2
tau	Estimate(s) of the model parameter(s) tau_1, ..., tau_k
logL	Value of log-likelihood
logL0	Value of log-likelihood under the null model (without random effect)
niter	Number of iterations done
norm_grad	Last computed gradient's norm
P	Last computed value of matrix P (see reference)
Py	Last computed value of vector Py (see reference)
BLUP_omega	BLUPs of random effects
BLUP_beta	BLUPs of fixed effects beta
varbeta	Variance matrix for beta estimates
varXbeta	Participation of fixed effects to variance of Y

If get.P = TRUE, there is an additional member:

P	The last matrix P computed in the AIREML step

Author(s)

Hervé Perdry and Claire Dandine-Roulland

References

Gilmour, A. R., Thompson, R., & Cullis, B. R. (1995), Average information REML: an efficient algorithm for variance parameter estimation in linear mixed models, Biometrics, 1440-1450

See Also

lmm.diago, logistic.mm.aireml, lmm.simu

Examples

# Load data
data(AGT)
x <- as.bed.matrix(AGT.gen, AGT.fam, AGT.bim)

# Compute Genetic Relationship Matrix
standardize(x) <- "p"
K <- GRM(x)

# Simulate a quantitative genotype under the LMM
set.seed(1)
y <- 1 + x %*% rnorm(ncol(x), sd = 1)/sqrt(ncol(x)) + rnorm(nrow(x), sd = sqrt(2))

# Estimates
estimates <- lmm.aireml(y, K = K, verbose = FALSE)
str(estimates)

gaston documentation built on Dec. 28, 2022, 1:30 a.m.