Technical Complements to the paper
Shared Uncertainty in Measurement Error Problems, With Application to
Nevada Test Site Fallout Data
by Yehua Li, Annamaria Guolo, F. Owen Hoffman and Raymond J. Carroll
Web Appendix A: Sketch of Technical Arguments
A.1
Computation of Σ−1
s (ρs , γC )
As is easily seen,
Σs (ρs , γC ) = (1 − γC ){(1 − ρs )Ds + ρs vs vsT };
2
Ds = diag(σ1s,total
, ..., σn2 s s,total );
vs = (σ1s,total , ..., σns s,total )T ;
Σ−1
s (ρs , γC )
(
Ds−1 vs vsT Ds−1
ρs
(1 − ρs )
−
= (1 − γC )
(1 − ρs )2 1 + ρs (1 − ρs )−1 vsT Ds−1 vs
)
(
ρs
−1
T −1
−1
−1
−1
D v s v s Ds ;
Ds −
= (1 − γC ) (1 − ρs )
1 + (ns − 1)ρs s
−1
−1
Ds−1
)
|Σs (ρs , γC )| = (1 − γC )ns |(1 − ρs )Ds | × |1 + ρs (1 − ρs )−1 vsT Ds−1 vs |
= (1 − γC )ns (1 − ρs )ns −1 {1 + (ns − 1)ρs }
ns
Y
2
σis,total
.
i=1
A.2
Complete Conditionals for Gibbs Sampler
We will use Gibbs sampler to draw samples from the posterior distributions of Xis , Lis , ρs ,
γC , β, µs , σs2 , θ. First we need to derive the complete conditional distributions for these
variables and decide the sampling schemes from these complete conditionals. For those
variables whose complete conditional distribution can not be easily generated, we will use a
Metropolis step to update them. This algorithm is also referred as Metropolis-within-Gibbs
algorithm.
• If we impose prior distribution Normal(0, σµ2 s ) to µs , then
ns
X
σs2 σµ2 s
σµ2 s
Lis , 2
π(µs |rest) = Normal
σs2 + ns σµ2 s i=1
σs + ns σµ2 s
!
.
• Let π(σs2 ) be an Inverse Gamma distribution, i.e., π(σs2 ) ∝ (σs2 )−(as +1) exp{−1/(bs σs2 )},
then

(
π(σs2 |rest) = IG as + ns /2, (1/2)
ns
X
(Lis − µs )2 + b−1
s
i=1
)−1 
.
• Let 1ns be a vector of ns 1’s, then the complete conditional for Ls = (L1s , · · · , Lns s )T
is
(
1
1
(Ls − µs 1ns )T (Ls − µs 1ns ) −
(Ls − Ws )T Ds−1 (Ls − Ws )
2
2σs
2γC
1
T −1
− (Ls − Xs ) Σs (ρs , γC )(Ls − Xs )
2
1 T −2
∝ exp − Ls {σs I + γC−1 Ds−1 + Σ−1
s (ρs , γC )}Ls
2
π(Ls |rest) ∝ exp −
−2
−1 −1
−1
+LT
s {µs σs 1ns + γC Ds Ws + Σs (ρs , γC )Xs }
−1
= Normal(Ω−1
s hs , Ωs ),
(A.1)
−1 −1
−2
−1
where Ωs = σs−2 I + γC−1 Ds−1 + Σ−1
s (ρs , γC ), hs = µs σs 1ns + γC Ds Ws + Σs (ρs , γC )Xs .
• We let the prior for β to be Normal(0, σβ2 I), then the complete conditional for β is
π(β|rest) ∝
T
Y (exp[Zis
β + log{1 + θ exp(Xis )}])Yis
i,s
(
)
1
exp − 2 β T β .
T
1 + exp[Zis β + log{1 + θ exp(Xis )}]
2σβ
We use a Metropolis step to update β. Denote the current value of β as βcurr , then
generate a candidate value of β by kernel q(βcurr , βnew ) = Normal(βcurr , 0.01I). The
new value is accepted with probability
(
)
π(β = βnew |rest)
α = min 1,
.
π(β = βcurr |rest)
• Complete conditional for γC :
−N/2
π(γC |rest) ∝ I(ac ≤ γC ≤ bc )γC


exp −
1 X
2γC
i,s


2
(Wis − Lis )2 /σis,total

×
Y
−1/2
|Σs (ρs , γC )|
s
(
)
1X
(Xs − Ls )T Σ−1
× exp −
s (ρs , γC )(Xs − Ls )
2 s




1 X
−N/2
2
(Wis − Lis )2 /σis,total
∝ I(ac ≤ γC ≤ bc )γC (1 − γC )−N/2 exp −

2γC i,s
"
#
X
1
(Xs − Ls )T {(1 − ρs )Ds + ρs vs vsT }−1 (Xs − Ls ) .
× exp −
2(1 − γC ) s
Again, we use a Metropolis step to update γC . A new value γC,new is generated from
Uniform[ac , bc ]. We accept the new value with probability
(
)
π(γC = γC,new |rest)
α = min 1,
.
π(γC = γC,curr |rest)
• Complete conditional for ρs :
1
π(ρs |rest) ∝ I(cs ≤ ρs ≤ ds )|Σs (ρs , γC )|−1/2 exp − (Xs − Ls )T Σ−1
s (ρs , γC )(Xs − Ls )
2
"
1
−(ns −1)/2
−1/2
∝ I(cs ≤ ρs ≤ ds )(1 − ρs )
{1 + (ns − 1)ρs }
exp − (1 − γC )−1
2
)
#
(
ρs
−1
T −1
−1
−1
T
D vs vs Ds (Xs − Ls ) .
×(1 − ρs ) (Xs − Ls ) Ds −
1 + (ns − 1)ρs s
We use a Metropolis step similar to that of γC to update ρs . The candidate value is
generated from Uniform[cs , ds ].
• Complete conditional for θ:
π(θ|rest) ∝ π(θ)
Y
(H[ZisT β + log{1 + θ exp(Xis )}])Yis
i,s
×(1 − H[ZisT β + log{1 + θ exp(Xis )}])1−Yis .
Again, θ is updated by a Metropolis step similar to above. We consider the following
two schemes.
(θ.1) For the truncated normal prior, we use the prior π(θ) as the proposal density. A
new value θnew is generated from π(·), and then it is determine to be accepted with
probability
(
)
Pθ (θnew )
α = min 1,
,
Pθ (θcurr )
where Pθ (t) =
Q
T
Yis
(1−H[ZisT β+log{1+t exp(Xis )}])1−Yis .
i,s (H[Zis β+log{1+t exp(Xis )}])
In the analysis, we used proper priors for θ, such as the Truncated Normal priors.
(θ.2) For the approximate Jeffreys prior, we used a transition kernel q(θcurr , θnew ) =
T N (θcurr , σq2 ), where T N (µ, σ 2 ) is the truncated normal distribution with density function I(x ≥ 0)φ{(x − µ)/σ}/Φ(µ/σ), φ and Φ are the density and distribution function
of a standard normal random variable. Then the new value θnew is accepted with
probability
(
)
π(θ = θnew |rest)/Φ(θnew /σq )
α = min 1,
.
π(θ = θcurr |rest)/Φ(θcurr /σq )
A.3
Method of Scoring for EM Algorithm
The method of scoring in Section 4.2 is
(k+1)
b
Θ
1
(k)
(k)
(k)
−1
b
=Θ
1 − A (Θ1 )B(Θ1 ),
T
b
b
where A(Θ1 ) = ∂ 2 Q/∂Θ
1 ∂Θ1 and B(Θ1 ) = ∂ Q/∂Θ1 , these terms given as
(b)
(b)
Pis (β, δ, Xis ) = H[ZisT β + log{1 + exp(δ + Xis )}],
b
XX
∂Q
(b)
= B −1
{Yis − Pis (β, δ, Xis )}Zis ,
∂β
b s,i
(b)
b
XX
∂Q
exp(δ + Xis )
(b)
,
= B −1
{Yis − Pis (β, δ, Xis )}
(b)
∂δ
1 + exp(δ + Xis )
b s,i
b
XX
∂2Q
(b)
(b)
−1
=
−B
Pis (β, δ, Xis ){1 − Pis (β, δ, Xis )}Zis ZisT ,
T
∂β∂β
b s,i
(b)
b
XX
exp(δ + Xis )
∂2Q
(b)
(b)
= −B −1
Pis (β, δ, Xis ){1 − Pis (β, δ, Xis )}
Zis ,
(b)
∂β∂δ
1 + exp(δ + Xis )
b s,i

(b)
2
b
 exp(δ + X ) 
XX
∂ 2Q
(b)
(b)
is
−1
=
−B
P
(β,
δ,
X
){1
−
P
(β,
δ,
X
)}
.
is
is
is
is
(b) 
2

∂δ
1
+
exp(δ
+
X
)
s,i
b
is
A.4
Formulae For Computing (8)
Let Pis (β, δ, x) = H[ZisT β + log{1 + exp(δ + x)}]. Then
X
∂ log f1
=
{Yis − Pis (β, δ, Xis )}Zis ,
∂β
s,i
X
∂ log f1
exp(δ + Xis )
=
,
{Yis − Pis (β, δ, Xis )}
∂δ
1 + exp(δ + Xis )
s,i
X
∂ log f1
= (σs2 )−1 (Lis − µs ),
∂µs
i
X
∂ log f1
= (2σs4 )−1 (Lis − µs )2 − ns /(2σs2 ),
2
∂σs
i
−
−
−
X
∂ 2 log f1
=
P (β, δ, Xis ){1 − P (β, δ, Xis )}Zis ZisT ,
T
∂β∂β
s,i
X
∂ 2 log f1
exp(δ + Xis )
=
Zis ,
P (β, δ, Xis ){1 − P (β, δ, Xis )}
∂β∂δ
1 + exp(δ + Xis )
s,i
X
∂ 2 log f1
exp(δ + Xis )
=
−
{Yis − Pis (β, δ, Xis )}
2
∂δ
{1 + exp(δ + Xis )}2
s,i
+
X
s,i
exp(δ + Xis )
P (β, δ, Xis ){1 − P (β, δ, Xis )}
1 + exp(δ + Xis )
∂ 2 log f1
= I(s = s′ )ns /σs2 ,
∂µs ∂µs′
(
)
X
∂ 2 log f1
′
−6
2
4
(Lis − µs ) − ns /(2σs ) ,
− 2 2 = I(s = s ) σs
∂σs ∂σs′
i
−
(
(
)
X
∂ 2 log f1
−
= I(s = s′ ) σs−4 (Lis − µs ) .
2
∂µs ∂σs′
i
)2
,
Web Figures W.1 and W.2
30
0
10
θ
20
0
e+00
2
e+04
4
e+04
6
e+04
8
e+04
1
e+05
e+04
8
e+04
1
e+05
1.0
1.5
β1
2.0
2.5
Iteration
0
e+00
2
e+04
4
e+04
6
Interation
Figure W.1: History for the excess relative risk parameter θ per gray and the gender effect
β1 for thyroiditis, in the case of shared Berkson uncertainties and using the approximate
Jeffreys prior.
300
250
200
0
50
100
θ
150
0e+00
2e+04
4e+04
6e+04
8e+04
1e+05
6e+04
8e+04
1e+05
2
0
1
β1
3
4
Iteration
0e+00
2e+04
4e+04
Iteration
Figure W.2: History for the excess relative risk parameter θ per gray for neoplasm, in the
case of shared Berkson uncertainties and using the approximate Jeffreys prior.