Page 21 - Contributed Paper Session (CPS) - Volume 1

CPS653 Chang-Yun L.
and choose more reasonable models which follow the effect heredity principle.
The Markov chain Monte Carlo and Gibbs sampling are applied and a general
WinBUGS code that can be used for any SPDS and BDS designs is provided.
Simulation studies are conducted and results show that the proposed SSVS
method well controls the false discovery rate and has higher detection
capability than the regression methods.

2. Models and estimations
For an m-factor SPDS (or BDS) design with w whole plots (or blocks), let
denote the response from the th run in the th whole plot (block), where

(1)
= 1, … , , = 1, … , , and is the size of the th whole plot (block). Then

the full second-order model for the SPDS (BDS) design can be written as
−1
= + ∑ + ∑ ∑ + ∑ 2 + + ,

0

,
,
, ,
=1 =1 =+1 =1
where is the intercept, , is the level of factor for the th run in the th
0

whole plot (block), and are the main effect and quadratic effect of factor

, respectively, is the interaction of factors and , is the random

effect for the th whole-plot (block), and is the random error for the th run

in the th whole plot (block). It is assumed that and have zero means with

variances and , respectively, and are mutually independent. Equation (1)
2
2

can be further expressed in the form of matrices as follows. Let =
′
11 12,…, ( −1) ,
( ) be the × 1 vector of responses, where =
′
(∑ , = ( , … , , , … , (−1) , , … , ) be the (1 + ) × 1
0, 1
11
12

=1
vector of the intercept and fixed effects, where = 2 + ( − 1)/
2, be the × (1 + ) model matrix corresponding to , = ( ,· · ·
1
, ) be the × 1 vector of the random whole-plot (block) effects, U be the
′

× indicator matrix corresponding to and =
′
( , , … , ( −1) , ) be the × 1 vector of the random errors. Then
12
11
Equation (1) can be expressed as

= + + .

If and are known, the GLS estimate for is

̂
′ −1
−1 ′ −1
= ( Σ Χ) Χ Σ (2)

and the covariance matrix of is
̂

−1
′ −1
= ( Σ Χ) , (3)

Where ∑ = UU + I (I is an x identity matrix) is the covariance
2
′
2

matrix of . In practice, and are usually unknown and can be estimated

using the REML method. By replacing Σ in (2) and (3) with Σ = ̂ UU + ̂ I ,
′
̂
2
2

we obtain = (X′∑ X) X ∑ y and V = (X ∑ X) . Although it is known
̂
−1 ′ ̂ −1
̂ −1
̂
−1
′ ̂ −1
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