Review: A ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (PART 1)

A ROBUST BAYESIAN ESTIMATE OF THE
CONCORDANCE CORRELATION COEFFICIENT (1)
Dai Feng, Richard Baumgartner & Vladimir Svetnik
Jan 11, 2019
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICJan 11, 2019 1 / 18

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

INTRODUCTION
The Concordance Correlation Coefficient (CCC)
The CCC was proposed in a paper, Lin (1989).
It quantifies the closeness of the measurements from two observers
(could be two measurement methods, instruments, assays, etc.)
It could also be generalized to multiple observers.
Assume measurements from d observers have multivariate distribution
with a mean vector µ and covariance matrix Σ, then the CCC is
defined as
CCC =
2 d−1
i=1
d
j=i+1 σij
(d − 1) d
i=1 σ2
i + d−1
i=1
d
j=i+1(µi − µj)2
where σ2
i and µi are the variance and mean of the measurements made
by observer i and σij is the covariance between the measurements from
observers i and j.

INTRODUCTION
The CCC (conti.)
The range of the value of CCC is [−1, 1]−The larger the value, the
better the agreement.
The cutoﬀ points of strength-of-agreement
McBride (2005)
Lin et al. (2007)
The deﬁnition of the original proposed CCC was later extended to
multiple observers for data with or without replications (pure replicates
or repeated measures).
To conduct inference, a normality assumption was adopted (e.g., Lin,
1989; Carrasco and Jover, 2003).
Besides this the semiparametric generalized estimating equations and
nonparametric approaches were proposed (Barnhart et al., 2007; Lin et
al., 2007; and references therein).

INTRODUCTION
Robust estimators of the CCC
Motivation :

INTRODUCTION
Robust estimators of the CCC (conti.)
King and Chinchilli (2001)
By using alternative distance functions
Limitations
The new metrics lack nice interpretation in terms of precision and
accuracy,
and subsequent diagnosis functions that the original CCC possesses.
There is no easy way to choose the cutoff points.
Different distance functions → different geometric interpretations → not
comparable.
There is no adjustment of confounding covariates and accommodation
of various versions of the CCC under replication.
Only two observers but not beyond.

INTRODUCTION
In this article
Proposing a robust procedure for the CCC estimation.
To overcome the methods proposed by King and Chinchilli (2001).
Based on multivariate t-distributions (which have been widely used in
robust statistics).
Inference : MCMC rather than MLE (multivariate t-distributions can
have many modes)

INTRODUCTION
In this article (conti.)
Section 2 : The introduction for Bayesian method for CCC estimation
based on the multivariate t-distribution.
Section 3 : Simulation study.
Section 4 : Real-life examples.
Section 5 : Discussion and conclusions.

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
Multivariate t-distributions
(Liu, 1994; Wakeﬁeld, 1996; Kotz and Nadarajah, 2004)
p(Yi |µ, Σ, λi ) ∼ MVN(µ, λ−1
i Σ) (1)
p(λi |ν) ∼ Γ(ν/2, ν/2) (2)
Then the marginal distribution of Yi is
f (Yi |µ, Σ, ν) ∼
Γ ν+d
2
Γ(ν
2 )(νπ)
d
2 |Σ|
1
2 1 + 1
ν (Yi − µ)T Σ−1
(Yi − µ)
ν+d
2
.
where d = the dimensions of vector Yi ,
ν = degrees of freedom, µ = location parameter, Σ = scale parameter.
Using EM-type algorithms and MCMC for the implementation.

t-DISTRIBUTION
(conti.)
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
(3)
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.)
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.)
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
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 1)

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Review: A ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (PART 1)