Page 338 - Contributed Paper Session (CPS) - Volume 2

CPS1878 Zakir H. et al.
In this paper, the randomization approach of Brien and Bailey (2006) is
used for deriving the variance-covariance matrices of the random effects in a
GLMM. It is shown that the random effects are correlated. We then study the
impact of misspecification of both the correlation structure and the random
effects distribution on the estimated model parameters via simulation studies,
which has not been considered in the previous studies.

2. Methodology
Let Y be the bt × 1 vector of responses and X be the bt × (t + 1) design
matrix for the fixed treatment effects. We then write the GLMM for the RCBD
with random blocks in matrix notation as

((|, )) = + + (1)

where = (µ, , . . . , ) is the (t + 1) × 1 vector of fixed treatment effect
⊺
1

parameters, Z is the model matrix of order bt × b corresponding to random
block effects, B is the b × 1 vector of random block effects, is the bt × 1
vector of random errors, (|, ) is the conditional expectation, and g is the
link function.
For the RCBD with random blocks, we consider the randomization for
deriving moments of the random effects which is in line with the approach of
Brien and Bailey (2006). Let St and Sb be the two symmetric groups of
permutations of the sets {1,2, . . . , } and {1,2, . . . , } respectively. The
corresponding randomization of b blocks and t units within each block is
modelled by elements of the wreath product St ≀ Sb of two symmetric groups
St and Sb. In this setup, the symmetric group St represents the randomization
of units and Sb stands for the randomization of blocks.
We derive from the randomization () = 0 and the variance-covariance
2
matrix () = 0 diag(Q,...,Q) with Q = ( − 1 1 ) = , where =
2
Τ
−1
1
Τ
( − 1 1 ) is a matrix of order t×t. We also derive () = 0 and () =
−1
2 ( − 1 1 ) = with the b×b matrix = 1 ( − 1 1 ). It follows
⊺
2
Τ
−1 −1
that the random blocks and errors are correlated due to randomization.

3. Simulation study and results
The vector of linear predictors η, corresponding to a GLMM (M1) with
correlated random effects is de ned as
1 ∶ = + +
and for a standard GLMM (M2) with uncorrelated random effects as

2 ∶ = + + .
∗
∗

327 | I S I W S C 2 0 1 9