Review: A ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (PART 2)

A ROBUST BAYESIAN ESTIMATE OF THE
CONCORDANCE CORRELATION COEFFICIENT (2)
Dai Feng, Richard Baumgartner & Vladimir Svetnik
Jan 18, 2019
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 1 / 16

1 INTRODUCTION
2 A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
3 SIMULATION STUDY
4 REAL-LIFE EXAMPLES
5 CONCLUSION AND DISCUSSION

INTRODUCTION
INTRODUCTION

A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
A ROBUST PARAMETRIC MODEL BASED ON
MULTIVARIATE t-DISTRIBUTION

t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
Under the multivariate t-distribution, the CCC is deﬁned as
CCCt =
2 d−1
i=1
d
j=i+1
ν
ν−2 σij
(d − 1) d
i=1
ν
ν−2 σ2
i + d−1
i=1
d
j=i+1(µi − µj)2
(1)
where ν = the degrees of freedom,
µi s = components of the location vector,
σ2
i s = diagonal elements of the scale matrix,
σij s = oﬀ diagonal elements of the scale matrix.

t-DISTRIBUTION
(conti.)
Conjugate priors
µ ∼ MVN(µ0, Σ0)
Σ−1
∼ Wishart(ρ, V)
ν ∼ U(νmin, νmax)
Noninformative priors
µ0 = 0, Σ0 = very large,
ρ = d, V = diagonal matrix,
νmin = 4, νmax = 25 ← produces accurate estimates
for the CCC in the scenarios they studied

t-DISTRIBUTION
(conti.)
Posterior distributions under the conjugate priors
µ ∼ MVN(A−1
b, A−1
)
Σ−1
∼ Wishart

ρ + n, V−1
+
n
i=1
λi (Yi − µ)(Yi − µ)T
−1


λi ∼ Γ
ν + d
2
,
ν + (Yi − µ)T Σ−1
(Yi − µ)
2
where A =
n
i=1
λi Σ−1
+ Σ−1
0 , b = Σ−1
n
i=1
λi Yi + Σ−1
0 µ0.

t-DISTRIBUTION
(conti.)
The logarithm of the posterior distribution of ν is proportional to
n
i=1
ν
2
log
ν
2
− log Γ
ν
2
+
ν
2
− 1 log λi −
ν
2
λi
and within the interval of [νmin, νmax].
To update ν with a bounded support, they use the slice sampling (Neal,
2003), which is easy to choose the magnitude of changes adaptively.
To run multiple sequences of the MCMC, the MLE-based estimates could
be used to create a starting point as suggested by Liu (1994).

t-DISTRIBUTION
(conti.)
Slice sampling (Neal, 2003)
Neal, R. M. (2003). Slice Sampling. The Anals of Statistics 31:705-741.
To implement Gibbs sampling, one may need to devise methods for
sampling from nonstandard univariate distributions.
To use the Metropolis algorithm, one must ﬁnd an appropriate
“proposal” distribution that will lead to eﬃcient sampling.
Simple forms of univariate slice sampling are an alternative to Gibbs
sampling that avoids the need to sample from nonstandard
distributions.
It is easier to tune than Metropolis methods and also avoids problems
that arise when the appropriate scale of changes varies over the
distribution.

t-DISTRIBUTION
(conti.)
The single variable slice sampling methods.
x0 : current value , x1 : a new value
1 Draw a real value, y, uniformly from (0, f (x0)), thereby deﬁning a
horizontal “slice” : S = {x : y < f (x)}.
2 Find an interval, I = (L, R), around x0 that contains all, or much, of
the slice.
3 Draw the new point, x1, from the part of the slice within this interval.

t-DISTRIBUTION
(conti.)
Multivariate slice sampling methods.
x0 = (x0,1, . . . , x0,n) : current value
x1 = (x1,1, . . . , x1,n) : a new value
1 Draw a real value, y, uniformly from (0, f (x0)), thereby deﬁning the
slice S = {x : y < f (x)}.
2 Find a hyperrectangle, H = (L1, R1) × · · · × (Ln, Rn), around x0, which
preferably contains at least a big part of the slice.
3 Draw the new point, x1, from the part of the slice within this
hyperrectangle.

t-DISTRIBUTION
(conti.)
Various other things
Finding an appropriate hyperrectangle.
Sampling from the part of the slice within the hyperrectangle.
Overrelaxed slice sampling.
Reﬂective slice sampling.

t-DISTRIBUTION
2.2 Robust Bayesian Estimation of the CCC with
Accommodation of Covariates and Multiple Replications

SIMULATION STUDY
SIMULATION STUDY

REAL-LIFE EXAMPLES
REAL-LIFE EXAMPLES

CONCLUSION AND DISCUSSION
CONCLUSION AND DISCUSSION

Review: A ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (PART 2)

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