Page 23 - Contributed Paper Session (CPS) - Volume 1

CPS653 Chang-Yun L.
The resulting values of the latent variables which have higher marginal or joint
posterior probability are then used to identify active factors and select
promising models for further consideration. This procedure is called the
stochastic search variable selection (SSVS) by George and McCulloch (1993).

4. Examples: the SSVS method for SPDS designs
We apply the proposed SSVS method to analyze the data for the eight-factor
SPSD design in Lin and Yang (2015). The SPDS design and the responses are
listed in Table 1. This design was constructed by arranging a 17×8 definitive
screening design into nine unbalanced whole plots ( = 17, = 8, and =
9), in which and are whole-plot factors and ,· · · , are subplot
2
1
8
3
factors. The responses were generated from the true model ( ) = 50 −
2
2
4 + 3.5 + + 3.5 − 4.5 and the covariance matrix = ′ +
3
1
3
1 3
1
. We choose the tuning parameters = 10 and = .5. To ensure that the

selected models follow the effect heredity principle, we set = = .01
00
0
and = = 11 = .75 To reduce the uncertainty, we use = = =
1
11
1
.75 (lower probability increases the model uncertainty). The Gibbs sampling
with 11000 iterations are conducted by using the WinBUGS code. The
algorithm starts with = 0 for the intercept, = 0 and = 0 for the
0

fixed effects, = 1 for the random effects, and = = 1 for the

variances of the whole-plot and random errors. A burn-in of the first 1000
iterations is taken for eliminating the non-stationary portion of the chain at
beginning and then only every 5th value from the Gibbs sampling

Table 1: The SPDS design and data in Lin and Yang (2015).
Whole plot X1 X2 X3 X4 X5 X6 X7 X8 Y
1 -1 -1 1 -1 1 0 -1 1 55.073
1 -1 -1 1 1 -1 1 0 -1 56.359
1 -1 -1 -1 1 1 -1 1 0 50.529
2 -1 0 -1 -1 -1 1 1 1 50.349
3 -1 1 0 1 -1 -1 -1 1 58.019
3 -1 1 -1 0 1 1 -1 -1 49.619
3 -1 1 1 -1 0 -1 1 -1 55.049
4 0 1 1 1 1 1 1 1 48.806
5 0 0 0 0 0 0 0 0 48.955
6 0 -1 -1 -1 -1 -1 -1 -1 42.708
7 1 -1 0 -1 1 1 1 -1 50.650
7 1 -1 1 0 -1 -1 1 1 51.752
7 1 -1 -1 1 0 1 -1 1 41.401
8 1 0 1 1 1 -1 -1 -1 48.219
9 1 1 -1 1 -1 0 1 -1 41.676
9 1 1 -1 -1 1 -1 0 1 42.943
9 1 1 1 -1 -1 1 -1 0 52.089

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