lmm.aireml: Linear mixed model fitting with AIREML
In gaston: Genetic Data Handling (QC, GRM, LD, PCA) & Linear Mixed Models

lmm.aireml

R Documentation

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 `10^{-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 + \ldots + \omega_k + \varepsilon

with \omega_i \sim N(0,\tau_i K_i) for i \in 1, \dots,k and \varepsilon \sim 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, \dots, \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

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 May 29, 2024, 7:33 a.m.