# lme.Lb.dist.theta: Calculate mean and variance for linear combination of the... In pass.lme: Power and Sample Size for Linear Mixed Effect Models

## Description

Consider the following model:

Y=XB+Zu+e, u~N(0,D), e~N(0,R)
Yi~N(XBi,Vi), Vi=Zi*D*Zi'+Ri,

for each independent observation i

estimate of fixed effect coefficient B, denoted by b:

b=inv(sum(Xi'*inv(Vi)*Xi))*(sum(Xi'*inv(Vi)*Yi))

variance of b:

var(b)=Vb/n=inv(sum(Xi'*inv(Vi)*Xi))

where Vb=inv(Xi'*inv(Vi)*Xi)

distribution of any linear combinations L of b is given by:

Lb~N(mu,Sigma/n)

where mu = LB, Sigma = L*Vb*L'

## Usage

 `1` ```lme.Lb.dist.theta(B, D, R, X, Z, m = NULL, L) ```

## Arguments

 `B` fixed effect beta in px1 matrix `D` list of qxq random effect variance matrix; where the first element corresponding to the highest-level effect, the last element corresponding to the level 1 effect `R` residual variance `X` nxp matrix representing the covariates for the fixed effects `Z` nxq matrix representing the covariates for each level of random effects `m` vector of repeated measures from the highest to lowest level (level 1 effects are addressed by Z and X and no need to be specified) `L` lxp matrix, representing l-linear-combinations of beta interested, if L is not defined, it will be auto-created to select the last coefficient

## Value

theta: parameters (mu and Sigma) of the normal distribution for Lb

## Author(s)

Marco Chak Yan YU
Maintainer: Marco Chak Yan YU <marcocyyu@gmail.com>

`pass.lme.CLb.test`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```#Example 1 # calc BLUE for 1-level LME model, # with covariates X, Z: (1,t), t=1,2,3 # for both fixed and random effects, # with fixed effect coefficients B: (100,-0.5), # random effect variance D: (2 1;1 2), # residual variance R: 0.2 B <- matrix(c(100,-0.5),2,1) D <- matrix(c(2,1,1,2),2,2) R <- 0.2 X <- cbind(rep(1,3),1:3) Z <- X lme.Lb.dist.theta(B,D,R,X,Z) #Example 2 # calc BLUE for 3-levels LME model, # with level 1 same as the above example # with 3 repeated-measures in level 2 # and 5 repeated-measures in the highest level, # assuming random effect variance for level 2 = (3 1;1 3), # and random effect variance for highest level = (5 1;1 5) D <- list(matrix(c(2,1,1,2),2,2),matrix(c(3,1,1,3),2,2), matrix(c(5,1,1,5),2,2)) lme.Lb.dist.theta(B,D,R,X,Z,m=c(5,3)) ```