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

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

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 COEFFICIENT (1)Jan 11, 2019 1 / 18

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 COEFFICIENT (1)Jan 11, 2019 2 / 18

3. INTRODUCTION
INTRODUCTION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 3 / 18

4. INTRODUCTION
The Concordance Correlation Coecient (CCC)
The CCC was proposed in a paper, Lin (1989).
It quanties 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 fromdobservers have multivariate distribution
with a mean vectorand covariance matrix, then the CCC is
dened as
CCC=
2
P
d1
i=1
P
d
j=i+1
ij
(d1)
P
d
i=1

2
i
+
P
d1
i=1
P
d
j=i+1
(ij)
2
where
2
iandiare the variance and mean of the measurements made
by observeriandijis the covariance between the measurements from
observersiandj.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 4 / 18

5. INTRODUCTION
The CCC (conti.)
The range of the value of CCC is[1;1]The larger the value, the
better the agreement.
The cuto points of strength-of-agreement
McBride (2005)
Lin et al. (2007)
The denition 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).
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 5 / 18

6. INTRODUCTION
Robust estimators of the CCC
Motivation :
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 6 / 18

7. 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 cuto points.
Dierent distance functions!dierent 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.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 7 / 18

8. INTRODUCTION
In this article
Proposing a robust procedure for the CCC estimation.
To overcome the methods proposed by King and Chinchilli (2001).
Based on multivariatet-distributions (which have been widely used in
robust statistics).
Inference : MCMC rather than MLE (multivariatet-distributions can
have many modes)
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 8 / 18

9. INTRODUCTION
In this article (conti.)
Section 2 : The introduction for Bayesian method for CCC estimation
based on the multivariatet-distribution.
Section 3 : Simulation study.
Section 4 : Real-life examples.
Section 5 : Discussion and conclusions.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 9 / 18

10. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
A ROBUST PARAMETRIC MODEL BASED ON
MULTIVARIATEt-DISTRIBUTION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 10 / 18

11. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
Multivariatet-distributions
(Liu, 1994; Wakeeld, 1996; Kotz and Nadarajah, 2004)
p(Yij;; i)MVN(;
1
i
) (1)
p(ij)(=2; =2) (2)
Then the marginal distribution ofYiis
f(Yij;; )


+d
2

(

2
)()
d
2jj
1
2
h
1+
1

(Yi)
T

1
(Yi)
i+d
2
:
whered=the dimensions of vectorYi,
=degrees of freedom;=location parameter;=scale parameter.
Using EM-type algorithms and MCMC for the implementation.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 11 / 18

12. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(conti.)
Under the multivariatet-distribution, the CCC is dened as
CCCt=
2
P
d1
i=1
P
d
j=i+1

2
ij
(d1)
P
d
i=1

2

2
i
+
P
d1
i=1
P
d
j=i+1
(ij)
2
(3)
where=the degrees of freedom,
is=components of the location vector,

2
i
s=diagonal elements of the scale matrix,
ijs=o diagonal elements of the scale matrix.
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 12 / 18

13. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(conti.)
Priors
MVN(
0;0)

1
Wishart(;V)
U(min; max)
Noninformative priors

0=0;0=very large;
=d;V=diagonal matrix;
min=4; max=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 COEFFICIENT (1)Jan 11, 2019 13 / 18

14. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
2.1. Robust Bayesian Method for the CCC Estimation
(conti.)
Conjugate priors
MVN(A
1
b;A
1
)

1
Wishart
0
@+n;
"
V
1
+
n
X
i=1
i(Yi)(Yi)
T
#
1
1
A
i

+d
2
;
+ (Yi)
T

1
(Yi)
2
!
whereA=

n
X
i=1
i
!

1
+
1
0
;b=
1

n
X
i=1
iYi
!
+
1
0

0
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 14 / 18

15. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
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 COEFFICIENT (1)Jan 11, 2019 15 / 18

16. SIMULATION STUDY
SIMULATION STUDY
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 16 / 18

17. REAL-LIFE EXAMPLES
REAL-LIFE EXAMPLES
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 17 / 18

18. CONCLUSION AND DISCUSSION
CONCLUSION AND DISCUSSION
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (1)Jan 11, 2019 18 / 18