Biomedical Engineering Reference
In-Depth Information
tion, we used the pseudo-data set to compute a pseudo-posterior. The average
of the 1,000 pseudo-posterior means was used as the prior mean of : For the
third step, we calibrated the variances of the entries of to ensure a suit-
ably noninformative prior in terms of the prior effective sample sizes (ESSs)
of
T
(s;c;q;) and F
E
(s;c;q;); and also to obtain good simulated perfor-
mance of the design across a diverse set of scenarios. Denoting
T
(s;c;q;) or
F
E
(s;c;q;) for s = 0 or 1 by p(), the ESS of the prior on p() was approxi-
mated by matching its mean and variance with those of a beta(a;b) distribu-
tion and approximating the ESS as a+b Efp()g[1Efp(g]=varfp()g1:
This motivated setting
2
= varflog(
j
)g = 81 for each element
j
of ; with
ESS values of each p() ranging from 0:17 to 0:22:
Computations for each interim decision include obtaining the posterior
probabilities in the admissibility criteria and posterior mean utility for all (c;q)
combinations. This was done using MCMC with Gibbs sampling (Robert and
Cassella, 1999) to compute all posterior quantities, based on the full condi-
tionals. Each sample parameter series
(1)
; ;
(N)
distributed proportionally
to the posterior integrand was initialized at the mode using the two-level al-
gorithm given in Braun et al. (2007), which reliably identifies the region of
highest posterior probability density, so no burn-in was required and a sin-
gle chain was used for each computation. MCMC sample sizes of N =2,000
were used for choosing (c;q) during the trial, and N =16,000 for selecting
(c;q) at the end of the trial. For each sample
(i)
= (
(i)
;
(i)
), p
0
(c;q;
(i)
),
(Y
E
;c;q;
(i)
), F
E
(Y
E
;c;q;
(i)
); and
T
(Y
E
;c;q;
(i)
) were computed for all
interval endpoints and (c;q) pairs, with
E;T
(I
m
;y
T
jc;q;
(i)
) computed from
Equation (12:10). The elicited utilities were averaged over these distributions
to obtain a utility for (c;q) given
(i)
. Posterior mean utilities were obtained
from posterior sample averages. The Monte Carlo standard error (MCSE) was
computed using the batch-means method for F
E
(1;c;q),
T
(1;c;q) and u(c;q)
at the highest and lowest (c;q) pairs, with MCMC convergence concluded if
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