AI3: Average Information Algorithm

Description

This function is used internally in the function mmer when MORE than 1 variance component needs to be estimated through the use of the average information (AI3) algorithm.

Usage

AI3(y, X=NULL, ZETA=NULL, R=NULL, draw=TRUE, REML=TRUE, silent=FALSE, init=NULL, iters=50, forced=NULL, sherman=FALSE)

Arguments

a numeric vector for the response variable

an incidence matrix for fixed effects.

ZETA

an incidence matrix for random effects. This can be for one or more random effects. This NEEDS TO BE PROVIDED AS A LIST STRUCTURE. For example Z=list(list(Z=Z1, K=K1), list(Z=Z2, K=K2), list(Z=Z3, K=K3)) makes a 2 level list for 3 random effects. The general idea is that each random effect with or without its variance-covariance structure is a list, i.e. list(Z=Z1, K=K1) where Z is the incidence matrix and K the var-cov matrix. When moving to more than one random effect we need to make several lists that need to be inside another list. What we call a 2-level list, i.e. list(Z=Z1, K=K1) and list(Z=Z2, K=K2) would need to be put in the form; list(list(Z=Z1, K=K1),list(Z=Z1, K=K1)), which as can be seen, is a list of lists (2-level list).

a matrix for variance-covariance structures for the residuals, i.e. for longitudinal data. if not passed is assumed an identity matrix.

draw

a TRUE/FALSE value indicating if a plot of updated values for the variance components and the likelihood should be drawn or not. The default is TRUE. COMPUTATION TIME IS SMALLER IF YOU DON'T PLOT SETTING draw=FALSE

REML

a TRUE/FALSE value indicating if restricted maximum likelihood should be used instead of ML. The default is TRUE.

silent

a TRUE/FALSE value indicating if the function should draw the progress bar or iterations performed while working or should not be displayed.

init

Initial values for the variance components to use during the optimization process.

iters

a scalar value indicating how many iterations have to be performed if the EM is performed. There is no rule of tumb for the number of iterations. The default value is 100 iterations or EM steps.

forced

indexes of the variance components to be forced during the boosting process performed when one or more variance components are close to the zero boundary.

sherman

a TRUE/FALSE value indicating if Sherman-Morrison-Woodbury formula (Seber, 2003, p. 467) should be used when estimating variance components in order to perform faster when a mixed model with no covariance structure using the average information algorithm is fitted. The default is FALSE since this software was designed for unreplicated data (altough can fit models with replicated data but slower than lme4).

Value

If all parameters are correctly indicated the program will return a list with the following information:

$Vu: a scalar value for the variance component estimated
$Ve: a scalar value for the error variance estimated
$V.inv: a matrix with the inverse of the phenotypic variance V = ZGZ+R, V^-1
$u.hat: a vector with BLUPs for random effects
$Var.u.hat: a vector with variances for BLUPs
$PEV.u.hat: a vector with predicted error variance for BLUPs
$beta.hat: a vector for BLUEs of fixed effects
$Var.beta.hat: a vector with variances for BLUEs
$X: incidence matrix for fixed effects, if not passed is assumed to only include the intercept
$Z: incidence matrix for random effects, if not passed is assumed to be a diagonal matrix
$K: the var-cov matrix for the random effect fitted in Z
$ll: the log-likelihood value for obtained when optimizing the likelihood function when using ML or REML

Details

This algorithm is based on Gilmour et al. (1995), it is based on REML. This handles models of the form:

y = Xb + Zu + e

b ~ N[b.hat, 0] ............zero variance because is a fixed term

u ~ N[0, K*sigma(u)] .......where: K*sigma(u) = G

e ~ N[0, I*sigma(e)] .......where: I*sigma(e) = R

y ~ N[Xb, var(Zu+e)] ......where;

var(y) = var(Zu+e) = ZGZ+R = V which is the phenotypic variance

The function allows the user to specify the incidence matrices with their respective variance-covariance matrix in a 2 level list structure. For example imagine a mixed model with the following design:

fixed = only intercept.....................b ~ N[b.hat, 0]

random = GCA1 + GCA2 + SCA.................u ~ N[0, G]

where G is:

|K*sigma(gca1).....................0..........................0.........| |.............0.............S*sigma(gca2).....................0.........| = G

|.............0....................0......................W*sigma(sca)..|

The likelihood function optimized in this algorithm is:

logL = -0.5 * (log( | V | ) + log( | X'VX | ) + y'Py

where: | | refers to the derminant of a matrix

The algorithm can be summarized in the next steps:

1) provide initial values for the variance components

2) estimate the phenotypic variance matrix V = ZGZ + R

3) obtain Vinv by inverting V

4) obtain the projection matrix P = Vinv - [Vinv X (X'V-X)- X Vinv]

5) evaluate the logLikelihood as shown above

6) fill the average information matrix (AI3) with equation provided in Gilmour et al. (1995)

7) obtain AI3.inv by inverting AI3 (the average information matrix)

8) calculate scores by first derivatives refer as "B" in Gilmour et al. (1995)

9) update the values of variance components by : k(i+1) = k(i) + [ B(i) * AI3.inv ]

10) steps are repeated in a while loop until convergence is reached, the likelihood doesn't increase anymore.

References

Covarrubias-Pazaran G (2016) Genome assisted prediction of quantitative traits using the R package sommer. PLoS ONE 11(6): doi:10.1371/journal.pone.0156744

Gilmour et al. 1995. Average Information REML: An efficient algorithm for variance parameter estimation in linear mixed models. Biometrics 51(4):1440-1450.