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

1. 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

2. 1 INTRODUCTION
2 A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
3 SIMULATION STUDY
4 REAL-LIFE EXAMPLES
5 CONCLUSION AND DISCUSSION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 2 / 16

3. INTRODUCTION
INTRODUCTION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 3 / 16

4. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
A ROBUST PARAMETRIC MODEL BASED ON
MULTIVARIATE t-DISTRIBUTION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 4 / 16

5. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
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.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 5 / 16

6. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(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
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 6 / 16

7. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(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.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 7 / 16

8. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(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).
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 8 / 16

9. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(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.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 9 / 16

10. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(conti.)
Slice sampling (Neal, 2003)
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.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 10 / 16

11. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(conti.)
Slice sampling (Neal, 2003)
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.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 11 / 16

12. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(conti.)
Slice sampling (Neal, 2003)
Various other things
Finding an appropriate hyperrectangle.
Sampling from the part of the slice within the hyperrectangle.
Overrelaxed slice sampling.
Reflective slice sampling.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 12 / 16

13. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.2 Robust Bayesian Estimation of the CCC with
Accommodation of Covariates and Multiple Replications
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 13 / 16

14. SIMULATION STUDY
SIMULATION STUDY
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 14 / 16

15. REAL-LIFE EXAMPLES
REAL-LIFE EXAMPLES
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 15 / 16

16. CONCLUSION AND DISCUSSION
CONCLUSION AND DISCUSSION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 18, 2019 16 / 16