Review: A ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (PART 3)

A ROBUST BAYESIAN ESTIMATE OF THE
CONCORDANCE CORRELATION COEFFICIENT (3)
Dai Feng, Richard Baumgartner & Vladimir Svetnik
Feb 20, 2019
Dai Feng, Richard Baumgartner & Vladimir SvetnikA ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICFeb 20, 2019 1 / 34

1 ISSUES OF THE LAST
2 2. A ROBUST PARAMETRIC MODEL BASED ON MULTIVARIATE
t-DISTRIBUTION
3 3. SIMULATION STUDY
4 4. REAL-LIFE EXAMPLES
5 5. CONCLUSION AND DISCUSSION

ISSUES OF THE LAST
ISSUES OF THE LAST

ISSUES OF THE LAST
ISSUES OF THE LAST
1 To run multiple sequences of the MCMC, the MLE-based estimates
could be used to create a starting point as suggested by Liu
(1994).
Liu , C., Rubin, D. B. (1994). The ECME algorithm: A simple extension
of EM and ECM with faster monotone convergence. Biometrika
81:633-648.
ECM (the expectation/conditional maximization; Liu and Rubin, 1994)
ECM replaces each M step with a sequence of conditional maximization
(CM) steps in which each parameter θi is maximized individually,
conditionally on the other parameters remaining ﬁxed.
ECME (the expectation/conditional maximization either)
Each CM-step maximises either the expected complete-data
loglikelihood, as with ECM, or the actual likelihood function, subject to
the same constraints on θ.

ISSUES OF THE LAST
ISSUES OF THE LAST (cont.)
1 To run multiple sequences of the MCMC, the MLE-based estimates
could be used to create a starting point as suggested by Liu
(1994).
Van Ravenzwaaij, D., Cassey, P. & Brown, S.D. Psychon Bull Rev
(2018). A simple introduction to Markov Chain Monte–Carlo sampling.
Psychonomic Bulletin & Review. 25: 143.
https://doi.org/10.3758/s13423-016-1015-8
“Starting values that are closer to the mode of the posterior distribution
will ensure faster burn–in and fewer problems with convergence. It can
be diﬃcult in practice to ﬁnd starting points near the posterior mode,
but maximum–likelihood estimation (or other approximations to that)
can be useful in identifying good candidates.”
“Multiple sequences”는 무슨 의미에서 썼는지 잘 모르겠습니다. . .

ISSUES OF THE LAST
2 It is easier to tune than Metropolis methods and also avoids problems
that arise when the appropriate scale of changes varies over the
distribution.
Roberts, G. O., Rosenthal, J. S. (2001). Optimal Scaling for Various
Metropolis–Hastings Algorithms. Statistical Science 16:351-367.
Matropolis-Hastings : Given Xn, a “proposed value” Yn+1 is generated
from some prespeciﬁed density q (Xn, y) and is then accepted with
probability α (Xn, Yn+1), given by
α (x, y) = min
π(y)
π(x)
q(y, x)
q(x, y)
, 1
.
The scale of q(x, y) is extremely small ⇒ the algorithm will propose
small jumps ⇒ It will take a long time to converge to its stationary
distribution.
The scale of q(x, y) is extremely large ⇒ the algorithm will propose
large jumps ⇒ It will reject most of its proposed moves.

ISSUES OF THE LAST
3 Reﬂective slice sampling.
To improve sampling eﬃciency by suppressing random walks.

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

2.1 Robust Bayesian Method for the CCC Estimation.
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
(I) Yi |µ, Σ, λi ∼ MVN(µ, λ−1
i Σ)
(II) µ ∼ MVN(µ0, Σ0)
Σ−1
∼ Wishart(ρ, V)
λi ∼ Γ (ν/2, ν/2)
(III) ν ∼ U (νmin, νmax)

2.2 Robust Bayesian Estimation of the CCC with
Accommodation of Covariates and Multiple Replications
To accomodate covariates
They assume a multivariate linear model.
Yi = µi + i = Xi β + i
where i are iid MVT(0, Σ, ν) distributed.

(cont.)
As I understand it,
(I) Yi |β, Σ, λi ∼ MVN(Xi β, λ−1
i Σ)
(II) β ∼ MVN(β0, Σ0)
Σ−1
∼ Wishart(ρ, V)
λi ∼ Γ (ν/2, ν/2)
(III) ν ∼ U (νmin, νmax)

(cont.)
When there are multiple replications
yijk = µij + ijk
where yijk is the kth reading for subject i by observer j,
i = 1, . . . , n; j = 1, . . . , d; k = 1, . . . , K
Assumptions: given λi s,
1 µij and ijk are independent, E (µij ) = µj and E ( ijk ) = 0
2 between-subject variances Var(µij ) = λ−1
i σ2
Bj
within-subject variances Var( ijk ) = λ−1
i σ2
Wj
total variation of observer j is σ2
j = λ−1
i σ2
Bj + σ2
Wj
3 Corr (µij , µij ) = ρµjj , Corr (µij , ij k ) = 0, Corr ( ijk , ijk ) = 0
for all j, j , k, k .

(cont.)
As I understand it,
Yi |µ, Σ, λi ∼ MVN (Id ⊗ 1K ) µ, λ−1
i Σ
Var (µ) = λ−1
i






σ2
B1 ρµ12σB1σB2 · · · ρµ1d σB1σBd
ρµ12σB1σB2 σ2
B2 · · · ρµ2d σB2σBd
...
...
...
...
ρµ1d σB1σBd ρµ2d σB2σBd · · · σ2
Bd






Σ = diag(σ2
W 1, . . . , σ2
Wd ) ⊗ IK
여기에 MVT를 어떻게 걸며, Prior는 어떻게 주는지?

(cont.)
I guess..
(I) Yi |µ, Σ, λi ∼ MVN (Id ⊗ 1K ) µ, λ−1
i diag(σ2
W 1, . . . , σ2
Wd ) ⊗ IK
(II) µ ∼ MVN(µ0, Σ0)
σ2
Wj
−1
∼ Γ(α, β)
λi ∼ Γ (ν/2, ν/2)
(III) µ0 ∼ MVN (µ00, Σ00)
Σ0 ∼ Wishart(ρ, V)
ν ∼ U (νmin, νmax)

(cont.)
An overview on assessing agreement with continuous measurements
Huiman X. Barnhart; Michael J. Haber; Lawrence I. Lin.
CCCtotal = 1 −
d−1
j=1
d
j =j+1 E Yijk − Yij k
2
d−1
j=1
d
j =j+1 EI Yijk − Yij k
2
CCCinter = 1 −
d−1
j=1
d
j =j+1 E µij − µij
2
d−1
j=1
d
j =j+1 EI µij − µij
2
CCCj,intra = 1 −
K−1
k=1
K
k =k+1 E Yijk − Yijk
2
K−1
k=1
K
k =k+1 EI Yijk − Yij k
2
where EI is the conditional expectation given indep. of µij and µij .

(cont.)
An overview on assessing agreement with continuous measurements
Huiman X. Barnhart; Michael J. Haber; Lawrence I. Lin.
CCCtotal =
2 d−1
j=1
d
j =j+1 σBjσBj ρµjj
d−1
j=1
d
j =j+1 µj − µj
2
+ σ2
Bj + σ2
Bj + σ2
Wj + σ2
Wj
CCCinter =
2 d−1
j=1
d
j =j+1 σBjσBj ρµjj
d−1
j=1
d
j =j+1 µj − µj
2
+ σ2
Bj + σ2
Bj
CCCj,intra =
σ2
Bj
σ2
Bj + σ2
Wj

(cont.)
Under MVT instead of MVN distribution,
CCCtotal =
2 d−1
j=1
d
j =j+1
ν
ν−2 σBjσBj ρµjj
d−1
j=1
d
j =j+1 µj − µj
2
+ ν
ν−2 σ2
Bj + σ2
Bj + σ2
Wj + σ2
Wj
CCCinter =
2 d−1
j=1
d
j =j+1
ν
ν−2σBjσBj ρµjj
d−1
j=1
d
j =j+1 µj − µj
2
+ ν
ν−2 σ2
Bj + σ2
Bj
CCCj,intra =
σ2
Bj
σ2
Bj + σ2
Wj

3. SIMULATION STUDY
3. SIMULATION STUDY

3. SIMULATION STUDY
SIMULATION STUDY
Cases
16 cases as adopted by Carrasco and Jover (2003).
Carrasco, J. L., Jover, L. (2003) Estimating the generalized concordance
correlation coeﬃcient through variance components.. Biometrics
59:849-858.
Combination µ1 µ2 σ2
1 σ2
2 ρ12
1 100 100 100 100 0.99
2 0.9
3 0.7
4 0.5
5 100 100 100 125 0.99
6 0.9
7 0.7
8 0.5

3. SIMULATION STUDY
SIMULATION STUDY (cont.)
Cases (cont.)
Combination µ1 µ2 σ2
1 σ2
2 ρ12
9 100 105 100 100 0.99
10 0.9
11 0.7
12 0.5
13 100 105 100 125 0.99
14 0.9
15 0.7
16 0.5
For each combination, sample sizes of 30 and 100 were run with 1000
iterations.
The dists for each case they consider are multivariate t with df 5 and
normal dist.

3. SIMULATION STUDY
Statistics to compare
coverage of credible/conﬁdence intervals.
bias of point estimates.
root mean square error of point estimates.

3. SIMULATION STUDY
Methods
1 Bayes : Bayesian method.
(# of MCMC simulations) = 10,000 , (burn-in) = 1,000
?? For the Bayesian method, a single chain was used. For each
simulation scenario, from 1000 iterations, one was picked
randomly and multiple chains were run. ??
Convergence check
Cowles, M. K., Carlin, B. P. (1996). Markov Chain Monte Carlo
Convergence Diagnostics: A Comparative Review. Journal of the
American Statistical Association 91:883-904.
Gelman and Rubin (1992); Raftery and Lewis (1992); Geweke (1992);
Roberts (1992); Ritter and Tanner (1992); Zellner and Min (1995); Liu,
Liu, and Rubin (1992); Garren and Smith (1993); Johnson (1994);
Heidelberger and Welch and Schruben, Singh, and Tierney (1982-1983);
Mykland, Tierney, and Yu (1995); Yu (1994); Yu and Mykland (1994).
point estimate = posterior median
interval estimation = the highest posterior density credible interval

3. SIMULATION STUDY
Methods (cont.)
2 JZ : jackknife method with the Fisher’s Z-transformation
They use method of moment to estimate the CCC. (as in Carrasco et al.
(2007) and Lin (1989))
3 J : jackknife method without the Fisher’s Z-transformation
(2007) and Lin (1989))
4 Boot : bootstrap method
(2007) and Lin (1989))
(# of bootstrap replicates) = 999
Three CIs: basic, percentile, and bias-corrected accelerated (Efron and
Tibshirani, 1993; Carpenter and Bithell, 2000). Since the results from the
other two are inferior, only use the results from the percentile are shown.

3. SIMULATION STUDY
여기서 드는 의문 : 이전에 개발된 metric을 바꾸는 방법 (King and
Chinchilli, 2001)과는 비교할 수 없어서 안 한걸까?

3. SIMULATION STUDY
Table 2 Estimation of the coverage probability of nominal 95% credible/conﬁdence intervals for the data generated from
bivariate t-distributions
Combination n Bayes JZ J Boot
1 30 0.939 0.916 0.907 0.906
100 0.941 0.937 0.920 0.928
...
...
...
...
...
...
16 30 0.905 0.917 0.898 0.903
100 0.932 0.931 0.917 0.927
Table 3 Estimation of the coverage probability of nominal 95% credible/conﬁdence intervals for the data generated from
bivariate normal distributions
Combination n Bayes JZ J Boot
1 30 0.961 0.929 0.912 0.920
100 0.966 0.950 0.939 0.940
...
...
...
...
...
...
16 30 0.934 0.934 0.917 0.918
100 0.957 0.957 0.938 0.933

3. SIMULATION STUDY
In general, the Bayes method provides more accurate and slightly
conservative coverage probabilities.
For the bivariate t cases
All the methods exhibit under coverage.
The larger the sample size, the closer the coverage probability.
“Bayes” tends to provide more accurate coverage than the others.
“Boot” is uniformly worse than the “JZ”
The success of the “Boot” method is based on the assumption that the
bootstrap distribution is similar to the sample distribution. However, the
bootstrap replicates again tend to be skewed to the left (see Fig. 1(a)
and Fig. 1(b)).
For “JZ”, the non-normality is greatly corrected (see Fig. 1(c)) by the
transformation.
According to Efron and Tibshirani (1993), the jackknife is less suﬃcient
than “Boot”, since the former uses only limited information.
For the normal cases, the coverage prob. of the Bayes method tend to
be higher than that from the other resampling methods.

3. SIMULATION STUDY
(a) Histogram of the
CCC from 1000 samples
from the corresponding
bivariate t-dist.
(b) Histogram of the
CCC using "Boot" from
the corresponding
bivariate t-dist.
(c) Histogram of the
CCC using "JZ" from
the corresponding
bivariate t-dist.
Figure 1: The histograms. The arrow lines mark the true values.

4. REAL-LIFE EXAMPLES

REAL-LIFE EXAMPLES
Examples from biomarker study from EEG measurements in a
neuroscience study of insomnia.
LPS : the latency to persistent sleep
It is obtained typically for each PSG (polysomnographic) by a trained
polysomnographist. (“PSG scoring”)
The data from Two observers, WideMedAuto vs. manual, are plotted in
Fig. 2.
The point estimates and credible/confidence intervals of different
methods are shown in Table 1.
The point estimates and credible/confidence intervals of WideMedAuto,
WideMedPartial, and manual are shown in Table 2.
In general, the Bayes method provides a more robust estimate than the
others.

REAL-LIFE EXAMPLES (cont.)
Figure 2: Scatter plot of measurements of log in minutes from WideMedAuto and
Manual. The 45° line through origin (identity line) is given as a reference line.
Three suspicious "outliers" from subjects 1, 30, and 79 were highlighted.

Table 1: Point estimate and credible/conﬁdence intervals of the CCC between
WideMedAuto and Manual
With all subjects Without subjects 1, 30, 79
Methods Point estimate CI Point estimate CI
Bayes 0.870 (0.799, 0.927) 0.894 (0.835, 0.939)
J 0.672 (0.457, 0.888) 0.862 (0.777, 0.948)
JZ 0.656 (0.346, 0.837) 0.856 (0.729, 0.926)
Boot 0.675 (0.475, 0.861) 0.861 (0.711, 0.931)

Table 2: Point estimate and credible/conﬁdence intervals of the CCC between
WideMedAuto, WideMedPartial, and Manual
With all subjects Without subjects 1, 30, 79
Methods Point estimate CI Point estimate CI
Bayes 0.914 (0.870, 0.946) 0.931 (0.898, 0.958)
J 0.752 (0.591, 0.914) 0.931 (0.898, 0.958)
JZ 0.740 (0.502, 0.873) 0.890 (0.831, 0.949)
Boot 0.752 (0.602, 0.900) 0.888 (0.816, 0.936)

5. CONCLUSION AND DISCUSSION

CONCLUSION AND DISCUSSION
Bayesian method for robust estimation of the CCC under multivariate t
model allows for data with heavier tailed distribution with potential
outliers.
In their work, they used noninformative priors. However, They would
like to point out that the Bayesian paradigm allows for a
straightforward adoption of informative priors in cases use conjugate
priors for parameters µ and Σ and a uniform prior for ν.
The impact of informative prior needs further investigation especially
when the sample size is small.
As a topic for further research, they are currently developing statistical
approaches for robust CCC that would be able to handle categorical,
ordinal, or hightly skewed data.

Review: A ROBUST BAYESIAN ESTIMATE OF THE CONCORDANCE CORRELATION COEFFICIENT (PART 3)

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