Arnošt Komárek
Dept. of Probability and Mathematical Statistics
Regression modelling of misclassified
correlated interval-censored data
Workshop on Flexible Models for Longitudinal and Survival Data
with Applications in Biostatistics
Warwick, July 27 – 29, 2015
Joint work
with Marı́a José Garcı́a-Zattera
and Alejandro Jara
Pontificia Universidad Católica de Chile
Santiago de Chile
Outline
1
Misclassified interval-censored data.
2
Model for misclassified interval-censored data.
a
Misclassification model.
b
Event-time model.
3
Estimation and inference.
4
Simulation study.
5
Models comparison.
6
The Signal Tandmobielr study.
7
Summary and conclusions.
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.
Part I
Misclassified interval-censored data
Motivating dataset
The Signal Tandmobielr study
‘
Longitudinal dental study, Flanders (Belgium), 1996 – 2001.
‘
2 315 boys, 2 153 girls followed from 7 until 12 years old (primary
school time).
‘
Annual dental examinations.
‘
Sixteen trained dental examiners.
Each child examined in general by different examiner in each year.
w Clinical data.
’
‘
Data on oral hygiene and dietary habits.
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I. Misclassified interval-censored data
Main aim
Model the relationship between time to
caries experience (CE) and potential
risk factors.
‘
Gender (boys vs. girls).
‘
Presence of sealants.
‘
Frequency of brushing (daily / not daily).
‘
Geographical location.
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I. Misclassified interval-censored data
Caries experience (CE)
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I. Misclassified interval-censored data
Caries experience (CE)
?
Reversible
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I. Misclassified interval-censored data
Caries experience (CE)
What is oral health research and why of interest?
Cariology
6
Caries
(Irreversible)
Emmanuel Lesaffre (ERASMUS & KUL)
Statistical Modeling in OH Research
22 July 2009
7 / 94
?
Reversible
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I. Misclassified interval-censored data
Statistical modelling challenges
‘
CE is a progressive disease
we deal with a monotone 0/1 process.
‘
CE status checked only at discrete occasions
(visits/dental examinations)
interval censoring.
w
w
‘
Teeth in one mouth share common environment, genetical
dispositions, . . .
dependence among processes on different teeth
in one mouth.
w
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I. Misclassified interval-censored data
CE process & interval censoring
1
Y(i,j) (t)
T(i,j)
0
Y(i,j)
0
0
v(i,1)
pp
pp
pp
pp
p
-ppp
pp
pp
pp
pp
pp
pp
pp
p
v(i,2)
0
1
1
-
v(i,3)
v(i,4)
t
T(i,j) ∈ v(i,2) , v(i,3) ,
Y(i,j) = 0, 0, 1, 1
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>
.
I. Misclassified interval-censored data
Summary of notation
‘
T(i,j) : event (CE) time of tooth j on subject i,
i = 1, . . . , N, j = 1, . . . , J.
‘
Y(i,j) (t): 0/1 CE status of tooth (i, j) at time t.
‘
x(i,j) : potential risk factors, covariates to explain T(i,j) Y(i,j) (t).
‘
0 = v(i,0) < v(i,1) < v(i,2) < · · · < v(i,Ki ) < v(i,Ki +1) = ∞:
visit times (of dental examinations) for subject i.
‘
Y(i,j) = Y(i,j,1) , . . . , Y(i,j,Ki )
>
:
recorded 0/1 CE status of tooth (i, j) at performed visits.
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I. Misclassified interval-censored data
Interval-censored data
Interest in
Regression
T(i,j) ∼ x(i,j)
≡ Y(i,j) (t) ∼ x(i,j) .
Observed data
Monotone 0/1 sequence Y(i,j) = Y(i,j,1) , . . . , Y(i,j,Ki )
visit times v(i,1) , . . . , v(i,Ki ) .
>
together with the
≡ Intervals (L(i,j) , U(i,j) ]
such that T(i,j) ∈ (L(i,j) , U(i,j) ]
and L(i,j) , U(i,j) ∈ 0, v(i,1) , . . . , v(i,Ki ) , ∞ .
L(i,j) :
the last visit time when Y(i,j,∗) = 0,
U(i,j) :
the first visit time when Y(i,j,∗) = 1.
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I. Misclassified interval-censored data
Life is not so easy. . .
‘
Not easy and somehow subjective diagnosis of CE
misclassification in recorded values Y(i,j,1) , . . . , Y(i,j,Ki ) .
sensitivity/specificity of the diagnostic test towards caries are
not one.
w
w
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I. Misclassified interval-censored data
Life is not so easy. . .
‘
Not easy and somehow subjective diagnosis of CE
misclassification in recorded values Y(i,j,1) , . . . , Y(i,j,Ki ) .
sensitivity/specificity of the diagnostic test towards caries are
not one.
w
w
Misclassified correlated
interval-censored data.
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I. Misclassified interval-censored data
CE process & misclassified interval-censored data
1
Y(i,j) (t)
T(i,j)
0
Y(i,j)
0
0
v(i,1)
pp
pp
pp
pp
p
-ppp
pp
pp
pp
pp
pp
pp
pp
p
0
0
-
v(i,2)
T(i,j) ∈ v(i,3) , v(i,4)
1
v(i,3)
v(i,4)
t
really?,
>
Y(i,j) = 0, 0, 0, 1 .
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I. Misclassified interval-censored data
CE process & misclassified interval-censored data
1
Y(i,j) (t)
T(i,j)
0
Y(i,j)
0
1
v(i,1)
pp
pp
pp
pp
p
-ppp
pp
pp
pp
pp
pp
pp
pp
p
v(i,2)
0
0
1
-
v(i,3)
v(i,4)
t
T(i,j) ∈ ???,
Y(i,j) = 0, 1, 0, 1
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>
.
I. Misclassified interval-censored data
Misclassified interval-censored data
Interest in
Regression
T(i,j) ∼ x(i,j)
≡ Y(i,j) (t) ∼ x(i,j) .
Observed data
T(i,j) Y(i,j) observed only indirectly through
Y(i,j) = Y(i,j,1) , . . . , Y(i,j,Ki )
>
:
w not necessarily monotone sequence of 0/1 possibly misclassified
CE status indicators from visits performed at times v(i,1) , . . . , v(i,Ki ) .
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I. Misclassified interval-censored data
Study design that leads to misclassified
interval-censored data
‘
Longitudinal follow-up.
‘
Event status checked at pre-specified time points.
’
‘
‘
Assumption here: visit times independent of the event time.
Occurrence of event is determined by a diagnostic test (with possibly imperfect sensitivity and/or specificity).
’
Frequent for many non-death events.
’
Nevertheless, data are mostly analyzed as if both sensitivity and specificity are equal to one and hence there is no event status misclassification.
Follow-up is not scheduled to stop after the first positive result.
’
Frequent in longitudinal studies where the event is not the primary study
outcome.
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I. Misclassified interval-censored data
Principal questions
Using just the observed data – Y(i,j)
1
Can we do a valid statistical inference on the time to event T(i,j)
in presence of event misclassification even if no external information is available on the magnitude of the misclassification?
’
No external information on the sensitivity/specificity values.
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I. Misclassified interval-censored data
Principal questions
Using just the observed data – Y(i,j)
1
Can we do a valid statistical inference on the time to event T(i,j)
in presence of event misclassification even if no external information is available on the magnitude of the misclassification?
’
2
No external information on the sensitivity/specificity values.
Can we evaluate the magnitude of misclassification?
’ Can we estimate sensitivity/specificity of the event
classification?
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I. Misclassified interval-censored data
Principal questions
Using just the observed data – Y(i,j)
1
Can we do a valid statistical inference on the time to event T(i,j)
in presence of event misclassification even if no external information is available on the magnitude of the misclassification?
’
2
3
No external information on the sensitivity/specificity values.
Can we evaluate the magnitude of misclassification?
’ Can we estimate sensitivity/specificity of the event
classification?
Do we get a valid inference on the time to event T(i,j) if misclassification ignored and it is assumed that T(i,j) lies in the first
“possible” observed interval?
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I. Misclassified interval-censored data
Part II
Modelling approach
Hierarchical model
Hierarchically specified model (likelihood) for observed data
Yi = Y(i,1) , . . . , Y(i,J) .
Start with a joint likelihood of unobservable Ti and observed Yi :
p(Yi , Ti ) = p Yi Ti p(Ti ).
’
p Yi Ti : (mis)classification model
w visit times v
(i,1) ,
’
. . . , v(i,Ki ) act as covariates here.
p(Ti ): survival model for (correlated) event times
w risk factors x
(i,1) , . . . , x(i,J)
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act as covariates here.
II. Modelling approach
Hierarchical model
Hierarchically specified model (likelihood) for observed data
Yi = Y(i,1) , . . . , Y(i,J) .
Start with a joint likelihood of unobservable Ti and observed Yi :
p(Yi , Ti ) = p Yi Ti p(Ti ).
’
p Yi Ti : (mis)classification model
w visit times v
(i,1) ,
’
. . . , v(i,Ki ) act as covariates here.
p(Ti ): survival model for (correlated) event times
w risk factors x
(i,1) , . . . , x(i,J)
act as covariates here.
Likelihood of observed data on subject i:
Z
p(Yi ) =
p(Yi , Ti ) dTi .
RJ+
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II. Modelling approach
Overall likelihood
Overall likelihood
‘
Independence among subjects (children):
p(Y1 , . . . , YN ) =
N
Y
p(Yi ).
i=1
Z
p(Yi ) =
p(Yi , Ti )
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=
RJ+
p(Yi , Ti ) dTi ,
p Yi Ti
p(Ti )
| {z }
| {z }
event-time
model
misclassification model
II. Modelling approach
Part III
Misclassification model
(Mis)classification model p Y i T i
‘
For each i (each child), some conditional independence assumed.
‘
Event classification Y(i,j,k) for given unit (tooth j) at given time (k) is
(conditionally) independent of
(a) event classification Y(i,j ∗ ,l) for other units (other teeth, j ∗ 6= j) at
arbitrary times (arbitrary l);
(b) event classification Y(i,j,k ∗ ) for the same unit (the same tooth) at
other times (k ∗ 6= k );
(c) event times T(i,j ∗ ) of other units (other teeth, j ∗ 6= j).
www
p(Yi | Ti ) =
Ki
J Y
Y
p(Y(i,j,k) | T(i,j) ).
j=1 k=1
‘
In the rest: form of p(Y(i,j,k) | T(i,j) ) for given j (tooth) and k (visit time).
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III. Misclassification model
Simple (mis)classification model p Y i T i
Only one examiner
Model parameters:
’ α: examiner’s sensitivity
α = P Y(i,j,k) = 1 T(i,j) ≤ v(i,k) .
’
η: examiner’s specificity
η = P Y(i,j,k ) = 0 T(i,j) > v(i,k ) .
Likelihood contribution:
p(Y(i,j,k) | T(i,j) ) = p(Y(i,j,k ) | T(i,j) ; α, η, vi,k )
 Y
α (i,j,k ) (1 − α)1−Y(i,j,k ) , if T(i,j) ≤ v(i,k)





(correct Y(i,j,k) equals 1),
=
Y(i,j,k) 1−Y(i,j,k )


η
, if T(i,j) > v(i,k)
 (1 − η)


(correct Y(i,j,k) equals 0).
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III. Misclassification model
Slightly
complicated (mis)classification model
more
p Yi Ti
‘
More (Q > 1) examiners involved in a study
Signal Tandmobielr study: Q = 16.
‘
Different examiners have different ability to detect event (caries)
sensitivity/specificity should be allowed to depend on the
examiner.
w
w
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III. Misclassification model
Slightly
complicated (mis)classification model
more
p Yi Ti
‘
More (Q > 1) examiners involved in a study
Signal Tandmobielr study: Q = 16.
‘
Different examiners have different ability to detect event (caries)
sensitivity/specificity should be allowed to depend on the
examiner.
‘
It is not necessarily as easy to detect caries on all teeth
(j = 1, . . . , J) in the mouth
sensitivity/specificity should be allowed to depend on tooth (j).
w
w
w
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III. Misclassification model
Slightly
complicated (mis)classification model
more
p Yi Ti
Q examiners, dependence of sensitivity/specificity on tooth (j)
One more set of covariates in a model: ξ(i,k) ∈ {1, . . . , Q}
’
index (id) of examiner who scored (all) teeth of the ith child during his/her
kth visit at time v(i,k) .
Slightly more unknown parameters of a model (q = 1, . . . , Q):
>
αq = α(q,1) , . . . , α(q,J) ,
ηq =
η(q,1) , . . . , η(q,J)
>
.
α(q,j) , η(q,j) : sensitivity and specificity if the event classification of tooth j
is performed by examiner q, i.e.,
α(q,j) = P Y(i,j,k) = 1 T(i,j) ≤ v(i,k) ; ξ(i,k) = q ,
η(q,j) = P Y(i,j,k) = 0 T(i,j) > v(i,k) ; ξ(i,k) = q .
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III. Misclassification model
Slightly
complicated (mis)classification model
more
p Yi Ti
Q examiners, dependence of sensitivity/specificity on tooth (j)
Model parameters:
’
’
> >
α = α>
: sensitivites of all examiners for all teeth.
1 , . . . , αQ
>
>
η = η>
: specificities of all examiners for all teeth.
1 , . . . , ηQ
Likelihood contribution:
p(Y(i,j,k) | T(i,j) ) = p(Y(i,j,k ) | T(i,j) ; α, η,
 Y
)

α(ξ(i,j,k
(1 − α(ξ(i,k ) ,j) )1−Y(i,j,k ) ,

(i,k ) ,j)




=
1−Y

)

(1 − η(ξ(i,k ) ,j) )Y(i,j,k ) η(ξ(i,k(i,j,k
,


) ,j)


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vi,k , ξ(i,k ) )
if T(i,j) ≤ v(i,k)
(correct Y(i,j,k ) equals 1),
if T(i,j) > v(i,k)
(correct Y(i,j,k ) equals 0).
III. Misclassification model
Evenmore
complicated (mis)classification model
p Yi Ti
Sensitivies/specificities can further be modelled in a structural way as
functions of characteristics (covariates) of examiners/teeth:
’
age of examiner,
’
gender of examiner,
’
tooth position in the mouth,
..
.
’
“Only” one more hierarchical level of the model.
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III. Misclassification model
Hierarchical model
Reminder
Likelihood contribution of observed data of the ith child:
Z
p(Yi ) =
RJ+
Z
=
RJ+
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p(Yi , Ti ) dTi
p Yi Ti p(Ti ) dTi .
III. Misclassification model
Part IV
Event-time model
Event time model p(T i )
One more reminder
Ti = T(i,1) , . . . , T(i,J)
≡ possibly correlated times to CE for J teeth of the ith child.
xi = x(i,1) , . . . , x(i,J)
≡ covariates that may explain Ti .
Form of p(Ti ) can in principle be derived from any regression model
for correlated event time data (if we believe that a given model is suitable for data at hand):
’
frailty Cox model,
’
random intercept accelerated failure time (AFT) model,
..
.
’
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IV. Event-time model
Event time model p(T i )
Random intercept AFT model
log T(i,j) = x>
(i,j) β + bi + ε(i,j)
i = 1, . . . , N, j = 1, . . . , J,
‘
β: regression coefficients.
‘
ε(1,1) , . . . , ε(N,J) : i.i.d. with zero-mean density gε (·).
‘
b1 , . . . , bN : i.i.d. with density gb (·) .
’
bi (common for all j) induces dependence between T(i,1) , . . . , T(i,J) ,
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IV. Event-time model
Event time model p(T i )
Distributional assumptions
gε (·) ∼ N (0, σε2 ).
gb (·) ∼ penalized Gaussian mixture (PGM):
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
κ−9
κ−6
κ−3
κ0
κ3
κ6
κ9
κ−9
κ−6
κ−3
κ0
κ3
κ6
κ9
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
κ−9
κ−6
κ−3
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κ0
κ3
κ6
κ9
κ−9
κ−6
κ−3
κ0
κ3
κ6
κ9
IV. Event-time model
Event time model p(T i )
Distributional assumptions
gε (·) ∼ N (0, σε2 ),
XM
wl N (κl , ζ 2 )
{z
}
|
penalized Gaussian mixture (PGM)
gb (·) ∼ µ + τ
’
’
Model parameters: σε2 , w = w−M , . . . , wM , µ, τ 2 .
Penalized Gaussian mixture:
M ≈ 15, ζ ≈ 0.2,
w
’
l=−M
κ−M , . . . , κM : equidistant knots on interval approx. [−4.5, 4.5];
flexible model for distribution with approximately zero mean and
unit variance.
Regularization using penalized differences of (transformed) weights
w−M , . . . , wM .
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IV. Event-time model
Event time model p(T i )
log T(i,j) = x>
(i,j) β + bi + ε(i,j)
i = 1, . . . , N, j = 1, . . . , J,
Distribution of the event time T(i,j)
Up to the log-transformation:
’ convolution of a full parametric N ormal and a semi-parametric
PGM.
Also distribution of the event time is specified semi-parametrically.
More details:
’
Komárek, Lesaffre & Hilton (2005, J. of Computat. and Graphical Stat.),
’
Komárek, Lesaffre & Legrand (2007, Statistics in Medicine),
’
Komárek & Lesaffre (2008, J. of the American Statistical Association).
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IV. Event-time model
Part V
Estimation and inference
Likelihood
p(Y1 , . . . , YN ) =
N
Y
p(Yi ) =
i=1
N Z
Y
i=1
RJ+
p(Yi , Ti )dTi
=
N Z
Y
i=1
RJ+
p Yi Ti )p(Ti ) dTi .
p Yi Ti : (mis)classification model
’
>
>
unknown parameters: α = α(1,1) , . . . , α(Q,J) , η = η(1,1) , . . . , η(Q,J) :
sensitivities and specificities for examiners and teeth.
p(Ti ): event-time model
’
random intercept AFT model with a PGM distribution of random intercept;
’
unknown parameters: regression coefficients β, intercept µ, variances τ 2
and σε2 , mixture weights w.
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V. Estimation and inference
Likelihood
For each i = 1, . . . , N:Z
p(Yi ) =
p Yi Ti )
| {z }
RJ+
QJ
j=1
QKi
k =1
p
p(Ti ) dTi .
Y(i,j,k ) T(i,j)
Misclassification part
p Y(i,j,k) T(i,j)
1−Y(i,j,k ) I(T(i,j) ){T ∈(0,v ]}
(i,j)
(i,k )
= α(ξ(i,k ) ,j) Y(i,j,k ) 1 − α(ξ(i,k ) ,j)
×
=
I(T(i,j) )
Y(i,j,k)
{T(i,j) ∈(v(i,k ) ,+∞)}
1 − η(ξ(i,k ) ,j)
η(ξ(i,k) ,j) 1−Y(i,j,k )
k 1−Y(i,j,k ) I(T(i,j) ){T ∈(v
Y
(i,j)
(i,l−1) ,v(i,l) ]}
α(ξ(i,k ) ,j) Y(i,j,k ) 1 − α(ξ(i,k) ,j)
×
l=1
Ki +1
×
Y 1 − η(ξ(i,k ) ,j)
Y(i,j,k)
η(ξ(i,k) ,j) 1−Y(i,j,k )
I(T(i,j) )
{T(i,j) ∈(v(i,l−1) ,v(i,l) ]}
.
l=k +1
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V. Estimation and inference
Likelihood
For each i = 1, . . . , N:
Z
p(Yi ) =
RJ+
p Yi Ti ) p(Ti ) dTi .
Event-time part
Z
p(Ti ) =
p T(i,j) bi p(bi ) dbi ,
R
‘
p T(i,j) bi : log-normal following from the AFT model with a normal
error
’
‘
Unknown parameters: β, σε2 .
p(bi ): normal mixture following from the PGM model.
’
Unknown parameters: w, α, τ 2 .
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V. Estimation and inference
Estimation and inference
Maximum-likelihood clearly not tractable.
Bayesian specification of the model (with weakly informative priors)
and MCMC based inference
’ Possible.
’ All integrals in the likelihood disappear in calculations if Bayesian
data augmentation used
w unobserved event times T ;
w random effects (frailties) b .
(i,j)
i
package bayesSurv (≥2.3).
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V. Estimation and inference
Prior distributions
‘
AFT regression parameters:
β ∼ Normal (with large variances);
‘
(Inverted) variance of the AFT error terms:
σε−2 ∼ Gamma (with small rate and shape params.);
‘
Location of the random intercepts:
µ ∼ Normal (with large variance);
‘
(Inverted) squared scale of the random intercepts:
τ −2 ∼ Gamma (with small rate and shape params.).
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V. Estimation and inference
Random intercept distribution
Remember: bi ∼ penalized Gaussian mixture (PGM)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
κ−9
κ−6
κ−3
κ0
κ3
κ6
κ9
κ−9
κ−6
κ−3
κ0
κ3
κ6
κ9
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
κ−9
κ−6
κ−3
41/87 Arnošt Komárek
κ0
κ3
κ6
κ9
κ−9
κ−6
κ−3
κ0
κ3
κ6
κ9
V. Estimation and inference
Prior for PGM weights
Mixture weights (from the PGM model for the distribution of the
random intercept)
PM
2
‘ Remember: bi ∼ µ + τ
l=−M wl N (κl , ζ ), where M is relatively
large.
‘
Weights w should sum-up to one.
‘
It is primarily worked
with the transformed weights
a = a−M , . . . , aM :
wl = PM
exp(al )
m=−M
‘
exp(am )
,
l = −M, . . . , M,
a0 = 0.
Regularization prior for the (transformed) weights.
42/87 Arnošt Komárek
V. Estimation and inference
Prior for PGM weights
Mixture weights (from the PGM model for the distribution of the
random intercept)
Regularization prior for the (transformed) weights:

 λ
p a λ ∝ exp −
 2
’
M
X
j=−M+o


λ
2
∆o aj
= exp − a> P>
P
a
.
o
o

2
∆o : difference operator of order o
w P : corresponding difference operator matrix.
o
’
λ: smoothing hyperparameter
w prior: λ ∼ Gamma.
43/87 Arnošt Komárek
V. Estimation and inference
Prior for misclassification parameters
Sensitivities and specificities of the event-classification
‘
For each q (examiners) and j (unit – tooth)
0 < α(q,j) < 1: sensitivity of examiner q when scoring the jth unit (tooth);
0 < η(q,j) < 1: specificity of examiner q when scoring the jth unit (tooth).
Identification constraint: α(q,j) + η(q,j) > 1.
‘ Prior: α(q,j) , η(q,j) ∼ Beta × Beta truncated by the identification
constraint.
‘
44/87 Arnošt Komárek
V. Estimation and inference
Markov chain Monte Carlo
MCMC – Block Gibbs sampler
‘
Parameters of the event-time model (β, σε2 , PGM parameters
w/a, λ, µ, τ 2 and augmented random effects b1 , . . . , bN ):
’
Nothing new compared to the situation without misclassification, see
earlier papers Komárek, Lesaffre (& Legrand) (2007, 2008).
45/87 Arnošt Komárek
V. Estimation and inference
Markov chain Monte Carlo
MCMC – Block Gibbs sampler
‘
Parameters of the event-time model (β, σε2 , PGM parameters
w/a, λ, µ, τ 2 and augmented random effects b1 , . . . , bN ):
’
‘
Nothing new compared to the situation without misclassification, see
earlier papers Komárek, Lesaffre (& Legrand) (2007, 2008).
Augmented event times T(i,j) :
’
Sampling from a mixture of truncated log-normals.
’
Truncation: intervals between the visit times.
’
Mixture weights: binomial probabilities that depend on sensitivities/specificities and observed Y(i,j,l) values of 0/1 event classifications.
45/87 Arnošt Komárek
V. Estimation and inference
Markov chain Monte Carlo
MCMC – Block Gibbs sampler
‘
Parameters of the event-time model (β, σε2 , PGM parameters
w/a, λ, µ, τ 2 and augmented random effects b1 , . . . , bN ):
’
‘
Nothing new compared to the situation without misclassification, see
earlier papers Komárek, Lesaffre (& Legrand) (2007, 2008).
Augmented event times T(i,j) :
’
Sampling from a mixture of truncated log-normals.
’
Truncation: intervals between the visit times.
’
Mixture weights: binomial probabilities that depend on sensitivities/specificities and observed Y(i,j,l) values of 0/1 event classifications.
What would be changed if other than AFT model with normal errors
assumed for event times?
45/87 Arnošt Komárek
V. Estimation and inference
Markov chain Monte Carlo
MCMC – Block Gibbs sampler
‘
Parameters of the event-time model (β, σε2 , PGM parameters
w/a, λ, µ, τ 2 and augmented random effects b1 , . . . , bN ):
’
‘
Nothing new compared to the situation without misclassification, see
earlier papers Komárek, Lesaffre (& Legrand) (2007, 2008).
Augmented event times T(i,j) :
’
Sampling from a mixture of truncated log-normals.
’
Truncation: intervals between the visit times.
’
Mixture weights: binomial probabilities that depend on sensitivities/specificities and observed Y(i,j,l) values of 0/1 event classifications.
What would be changed if other than AFT model with normal errors
assumed for event times?
‘
Sensitivities (α’s) and specificities (η’s):
’
Sampling from truncated Beta distributions.
45/87 Arnošt Komárek
V. Estimation and inference
Part VI
Simulation study
Simulation study
J = 4 (teeth),
N = 500, 1 000, 2 000 (children).
log T(i,j) = 2.0 + 0.2 x(i,j),1 − 0.1 x(i,j),2 + bi + ε(i,j) .
x(i,j),1 ∼ Uniform(0, 1), x(i,j),2 ∼ Bernoulli(0.5).
var(bi ) + var(ε(i,j) ) = 0.1.
q
σb
var(bi )
var(ε(i,j) ) = σ = 0.5, 1, 2, 5.
ε
gb : (a) N ormal,
(b) clearly bimodal two-component N ormal mixture,
(c) Gumbel.
Ki = 10 visits (in random intervals).
Q = 5 examiners randomly assigned to the visits.
Sensitivities and specificities ranging 0.60 – 0.96.
47/87 Arnošt Komárek
VI. Simulation study
Simulation study
500 data sets for each scenario.
Each dataset also analyzed while ignoring misclassification. The first
Y(i,j,k) = 1 determined the “observed” interval where T(i,j) occurred
leading to “standard” interval-censored data.
48/87 Arnošt Komárek
VI. Simulation study
Sensitivity α(1,1) = 0.60
gb : bimodal two-component N mixture
σb σε:
0.5
1
2
5
●
●
●
0.65
●
●
●
●
●
●
●
●
●
●
●
●
0.60
●
●
●
α11
●
●
●
●
●
●
●
0.55
●
●
●
●
●
●
●
500
●
●
●
●
●
●
●
1000
2000
●
●
●
●
●
500
●
●
1000
2000
500
1000
2000
500
1000
2000
N
49/87 Arnošt Komárek
VI. Simulation study
Sensitivity α(1,1) = 0.60
gb : Gumbel
σb σε:
0.5
1
2
●
●
●
●
●
●
●
●
●
●
●
●
●
0.65
●
●
●
●
●
●
0.60
●
●
α11
5
●
●
0.55
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
500
1000
2000
500
●
●
●
1000
2000
500
●
1000
2000
500
1000
2000
N
50/87 Arnošt Komárek
VI. Simulation study
Sensitivity α(4,4) = 0.91
gb : bimodal two-component N mixture
σb σε:
0.5
1
2
5
●
0.94
●
●
●
●
●
●
●
0.90
α44
0.92
●
0.88
●
●
●
●
●
●
●
●
●
●
●
●
0.86
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
51/87 Arnošt Komárek
VI. Simulation study
Sensitivity α(4,4) = 0.91
gb : Gumbel
σb σε:
0.5
1
2
5
0.96
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.88
0.90
α44
0.92
0.94
●
●
●
●
●
0.86
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.84
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
52/87 Arnošt Komárek
VI. Simulation study
Regression parameter β1 = 0.20
gb : bimodal two-component N mixture
σb σε:
0.5
1
2
5
0.30
●
●
●
●
●
●
●
0.25
●
●
●
●
●
●
●
●
●
0.20
β1
●
●
●
●
●
●
0.15
●
●
●
●
●
●
●
0.10
●
●
●
●
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
53/87 Arnošt Komárek
VI. Simulation study
Regression parameter β1 = 0.20
0.30
gb : Gumbel
σb σε:
0.5
1
2
5
●
●
●
●
●
●
●
●
●
0.25
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.20
β1
●
●
●
●
●
0.15
●
●
●
●
●
●
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
54/87 Arnošt Komárek
VI. Simulation study
Regression parameter β1 = 0.20
gb : bimodal two-component N mixture
σb σε:
0.5
1
2
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.05
0.00
β1
5
●
0.10
0.15
0.20
IGNORED MISCLASSIFICATION
−0.05
●
●
●
●
●
●
●
●
●
●
−0.10
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
500
1000
55/87 Arnošt Komárek
2000
500
1000
2000
500
N
1000
2000
500
1000
2000
VI. Simulation study
Regression parameter β1 = 0.20
gb : Gumbel
IGNORED MISCLASSIFICATION
σb σε:
1
0.20
0.5
2
0.15
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.05
0.10
●
0.00
β1
5
●
●
●
−0.10
−0.05
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
500
1000
56/87 Arnošt Komárek
2000
500
1000
2000
500
N
1000
2000
500
1000
2000
VI. Simulation study
Survival function for a certain covariates combination
σb /σε = 5
gb : bimodal two-component N mixture
0
5
10
15
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
0.4
0.0
0.2
S(t)
N = 2000
1.0
N = 1000
1.0
N = 500
0
5
10
15
0
5
10
Time
Time
Time
N = 500
N = 1000
N = 2000
15
0
5
10
Time
57/87 Arnošt Komárek
15
1.0
S(t)
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.6
0.4
0.0
0.2
S(t)
0.8
1.0
gb : Gumbel
0
5
10
Time
15
0
5
10
15
Time
VI. Simulation study
Survival function for a certain covariates combination
σb /σε = 0.5
gb : bimodal two-component N mixture
0
5
10
15
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
0.4
0.0
0.2
S(t)
N = 2000
1.0
N = 1000
1.0
N = 500
0
5
10
15
0
5
10
Time
Time
Time
N = 500
N = 1000
N = 2000
15
0
5
10
Time
58/87 Arnošt Komárek
15
1.0
S(t)
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.6
0.4
0.0
0.2
S(t)
0.8
1.0
gb : Gumbel
0
5
10
Time
15
0
5
10
15
Time
VI. Simulation study
Survival function for a certain covariates combination
σb /σε = 5
IGNORED MISCLASSIFICATION
gb : bimodal two-component N mixture
0
5
10
15
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
0.4
0.0
0.2
S(t)
N = 2000
1.0
N = 1000
1.0
N = 500
0
5
10
15
0
5
10
Time
Time
Time
N = 500
N = 1000
N = 2000
15
0
5
10
59/87 Arnošt Komárek
Time
15
1.0
S(t)
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.6
0.4
0.0
0.2
S(t)
0.8
1.0
gb : Gumbel
0
5
10
Time
15
0
5
10
15
VI. Time
Simulation study
Survival function for a certain covariates combination
σb /σε = 0.5
IGNORED MISCLASSIFICATION
gb : bimodal two-component N mixture
0
5
10
15
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
0.4
0.0
0.2
S(t)
N = 2000
1.0
N = 1000
1.0
N = 500
0
5
10
15
0
5
10
Time
Time
Time
N = 500
N = 1000
N = 2000
15
0
5
10
60/87 Arnošt Komárek
Time
15
1.0
S(t)
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.6
0.4
0.0
0.2
S(t)
0.8
1.0
gb : Gumbel
0
5
10
Time
15
0
5
10
15
VI. Time
Simulation study
Simulation study 2
How about if no misclassification
present but we use the model that
accounts for possible misclassification?
Simulation study 2 where data generated without misclassification (all
sensitivities and specificities being equal to one).
61/87 Arnošt Komárek
VI. Simulation study
Sensitivity α(1,1) = 1.00
1.00
gb : bimodal two-component N mixture
σb σε:
0.5
●
●
1
●
●
●
2
●
●
●
5
●
●
●
●
●
●
●
●
●
●
●
0.99
●
●
0.98
●
●
●
●
●
●
●
0.97
α11
●
●
●
●
●
●
●
●
●
0.96
●
0.95
●
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
62/87 Arnošt Komárek
VI. Simulation study
Sensitivity α(1,1) = 1.00
gb : Gumbel
σb σε:
0.5
1
0.999
●
●
●
●
●
●
2
●
●
5
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.997
α11
0.998
●
●
0.996
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.995
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
63/87 Arnošt Komárek
VI. Simulation study
Regression parameter β1 = 0.20
gb : bimodal two-component N mixture
σb σε:
0.5
1
2
5
●
0.25
●
●
●
●
●
●
●
●
●
●
●
●
●
0.20
β1
●
●
●
●
●
●
●
●
●
●
●
0.15
●
●
●
●
●
●
●
●
●
500
1000
2000
500
1000
2000
500
1000
2000
500
1000
2000
N
64/87 Arnošt Komárek
VI. Simulation study
Regression parameter β1 = 0.20
gb : Gumbel
σb σε:
0.5
1
2
5
●
●
●
●
●
●
●
0.24
0.26
●
●
●
●
●
●
●
●
●
●
●
●
●
●
0.20
●
●
●
●
0.18
β1
0.22
●
●
●
●
2000
500
●
●
●
●
●
●
●
0.16
●
●
●
●
●
●
●
●
●
●
0.14
●
●
500
1000
2000
500
1000
2000
500
1000
1000
2000
N
65/87 Arnošt Komárek
VI. Simulation study
Survival function for a certain covariates combination
σb /σε = 5
gb : bimodal two-component N mixture
0
5
10
15
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
0.4
0.0
0.2
S(t)
N = 2000
1.0
N = 1000
1.0
N = 500
0
5
10
15
0
5
10
Time
Time
Time
N = 500
N = 1000
N = 2000
15
0
5
10
Time
66/87 Arnošt Komárek
15
1.0
S(t)
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.6
0.4
0.0
0.2
S(t)
0.8
1.0
gb : Gumbel
0
5
10
Time
15
0
5
10
15
Time
VI. Simulation study
Survival function for a certain covariates combination
σb /σε = 0.5
gb : bimodal two-component N mixture
0
5
10
15
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
S(t)
0.0
0.2
0.4
0.8
0.6
0.4
0.0
0.2
S(t)
N = 2000
1.0
N = 1000
1.0
N = 500
0
5
10
15
0
5
10
Time
Time
Time
N = 500
N = 1000
N = 2000
15
0
5
10
Time
67/87 Arnošt Komárek
15
1.0
S(t)
0.0
0.2
0.4
0.6
0.8
1.0
0.8
0.6
S(t)
0.0
0.2
0.4
0.6
0.4
0.0
0.2
S(t)
0.8
1.0
gb : Gumbel
0
5
10
Time
15
0
5
10
15
Time
VI. Simulation study
Part VII
Models comparison
Models comparison
Two competing models M1 and M2 .
’
May differ in specification of the event-time and/or the misclassification
model.
Pseudo Bayes factor (PsBF)
(Geisser and Eddy, 1979, JASA; Gelfand and Dey, 1994, JRSS, B):
PsMLM1
PsBF(M1 , M2 ) =
,
PsMLM2
PsMLM : pseudo marginal likelihood given model M:
PsMLM =
N Y
J
Y
pM Y(i,j,1) , . . . , Y(i,j,Ki ) Y[−(i,j)]
i=1 j=1
’
Y[−(i,j)] : data without observation of unit (tooth) j of subject (child) i;
’
pM (· | ·): posterior predictive distribution.
69/87 Arnošt Komárek
VII. Models comparison
Pseudo marginal likelihood
Approximation based on the proposal of Gelfand and Dey (1994,
JRSS, B):
pM Y(i,j,1) , . . . , Y(i,j,Ki ) Y[−(i,j)]
(
=
Eα, η, β, bi , σε2 | Y
(
≈
B
1X
B
b=1
!)−1
1
P Y(i,j,1) , . . . , Y(i,j,Ki ) α, η, β, bi , σε2
!)−1
1
(b)
2(b) P Y(i,j,1) , . . . , Y(i,j,Ki ) α(b) , η (b) , β (b) , bi , σε
.
,
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VII. Models comparison
Pseudo marginal likelihood
P Y(i,j,1) , . . . , Y(i,j,Ki ) α, η, β, bi , σε2
Ki +1
=
X
k =1
(Z
v(i,k )
v(i,k −1)
1
2
ϕ log t x>
(i,j) β + bi , σε dt
t
)
× W(i,j,k ) (Y(i,j) , α, η).
W(i,j,k) (Y(i,j) , α, η): quantity for which we have a closed-form expression and which is also needed in the MCMC procedure.
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VII. Models comparison
Part VIII
The Signal Tandmobielr study
Models
Event-time model
log T(i,j) = bi + x>
(i,j) β + ε(i,j)
‘
T(i,j) Age at getting caries on tooth j (∈ {1, 2, 3, 4}) of a child i.
‘
x(i,j) : gender, presence of sealants, frequency of brushing, x and y
geographical coordinate.
Misclassification models
‘
16 examiners.
‘
Model M1 : sensitivities/specificities both examiner and tooth specific
(64 + 64 sensitivities and specificities).
‘
Model M2 : sensitivities/specificities only examiner specific
(16 + 16 sensitivities and specificities).
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VIII. The Signal Tandmobiel study
Models comparison
Pseudo marginal log-likelihoods: M1 : −16 545,
M2 : −16 515.
PsBF(M1 , M2 ) = exp(−30) ≈ 10−13 .
From a predictive point of view, the simpler model M2 (sensitivities/specificities only examiner specific) is better.
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VIII. The Signal Tandmobiel study
Sensitivities
0.85
0.70
0.75
0.80
Sensitivity
0.90
0.95
1.00
(posterior means and 95% HPD credible intervals)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Examiner
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VIII. The Signal Tandmobiel study
Specificities
0.85
0.70
0.75
0.80
Specificity
0.90
0.95
1.00
(posterior means and 95% HPD credible intervals)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Examiner
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VIII. The Signal Tandmobiel study
Random intercept density
0.6
0.0
0.2
0.4
g(b)
0.8
1.0
1.2
(standardized, pointwise posterior means)
−3
−2
−1
0
1
2
3
b
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VIII. The Signal Tandmobiel study
Posterior summary for regression parameters
Model
Posterior
Mean
Median
95%HPD
Gender
(Girl)
1
2
−0.05971
−0.05984
−0.05964
−0.05993
Sealants
(Present)
1
2
0.19027
0.19067
0.19016
0.19054
( 0.16319 ;
( 0.16378 ;
0.21762)
0.21787)
Freq. of Brush.
(Daily)
1
2
0.16564
0.16538
0.16562
0.16542
( 0.12168 ;
( 0.12242 ;
0.21056)
0.20938)
x-ordinate
1
2
−0.00092
−0.00092
−0.00092
−0.00092
(−0.00122 ; −0.00062)
(−0.00122 ; −0.00062)
y -ordinate
1
2
−0.00002
−0.00007
−0.00004
−0.00007
(−0.00010 ;
(−0.00101 ;
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(−0.09115 ; −0.02854)
(−0.09098 ; −0.02814)
0.00087)
0.00080)
VIII. The Signal Tandmobiel study
Survival functions
0.8
1.0
(pointwise posterior means)
0.0
0.2
0.4
S(t)
0.6
Boy: Seal:More freq.
Girl: Seal:More freq.
Boy: Seal:Less freq.
Boy: No seal:More freq.
Girl: Seal:Less freq.
Girl: No seal:More freq.
Boy: No seal:Less freq.
Girl: No seal:Less freq.
0
5
10
15
Age
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VIII. The Signal Tandmobiel study
Hazard functions
0.4
(pointwise posterior means)
0.2
0.0
0.1
h(t)
0.3
Girl: No seal:Less freq.
Boy: No seal:Less freq.
Girl: No seal:More freq.
Girl: Seal:Less freq.
Boy: No seal:More freq.
Boy: Seal:Less freq.
Girl: Seal:More freq.
Boy: Seal:More freq.
0
5
10
15
Age
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VIII. The Signal Tandmobiel study
Part IX
Summary and conclusions
Misclassified interval-censored data
1
Y(i,j) (t)
T(i,j)
0
Y(i,j)
0
1
v(i,1)
pp
pp
pp
pp
p
-ppp
pp
pp
pp
pp
pp
pp
pp
p
v(i,2)
0
0
1
-
v(i,3)
v(i,4)
t
T(i,j) ∈ ???,
Y(i,j) = 0, 1, 0, 1
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>
.
IX. Summary and conclusions
Conclusions
Interval-censored event time data are encountered whenever a certain evaluation (examination/labo/. . . ) is needed to determine the
event status.
Event status evaluation is often subject to misclassification.
’
Not only human examiners but also labo procedures have usually sensitivity and/or specificity < 1.
Ignoring misclassification may lead to seriously biased results of the
event time analysis.
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IX. Summary and conclusions
Conclusions
Interval-censored event time data are encountered whenever a certain evaluation (examination/labo/. . . ) is needed to determine the
event status.
Event status evaluation is often subject to misclassification.
’
Not only human examiners but also labo procedures have usually sensitivity and/or specificity < 1.
Ignoring misclassification may lead to seriously biased results of the
event time analysis.
Joint modelling of the misclassification and event-time processes allows for unbiased/consistent estimation of parameters of:
’
the event-time process (survival functions, regression parameters, . . . );
’
the misclassification process (sensitivities, specificities).
No need for external (validation) data to get sensitivities/specificities
related to classification.
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IX. Summary and conclusions
Possible extensions/modifications
Other than AFT model with random intercept as the event-time model.
’
Only small parts of the MCMC scheme would have to be modified.
Examiner-specific covariates to model sensitivities/specificities in the
misclassification model.
’
Logit model.
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IX. Summary and conclusions
Possible extensions/modifications
Other than AFT model with random intercept as the event-time model.
’
Only small parts of the MCMC scheme would have to be modified.
Examiner-specific covariates to model sensitivities/specificities in the
misclassification model.
’
Logit model.
Time-dependent sensitivities/specificities.
’
Useful if a learning-by-doing can be expected in event-classification.
’
Likely not possible with our “joint” approach due to identifiability problems.
’
External (validation) data needed to estimate parameters of the misclassification process.
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IX. Summary and conclusions
Applicability
Designed longitudinal studies with visit times pre-specified (being independent of the event times).
Event status checked at each visit independently of previous examination results by imperfect diagnostic procedure.
At least three visits (for at least some subjects) needed to identify
parameters of the misclassification process (sensitivities and specificities).
Above conditions quite often satisfied in practice and misclassification
ignored. . .
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IX. Summary and conclusions
Applicability
Designed longitudinal studies with visit times pre-specified (being independent of the event times).
Event status checked at each visit independently of previous examination results by imperfect diagnostic procedure.
At least three visits (for at least some subjects) needed to identify
parameters of the misclassification process (sensitivities and specificities).
Above conditions quite often satisfied in practice and misclassification
ignored. . .
Practically nothing is lost if misclassification considered even if not
present.
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IX. Summary and conclusions
THANK YOU FOR YOUR
ATTENTION!
References
G ARC ÍA -Z ATTERA , J ARA , KOM ÁREK (2015+). A flexible AFT model for misclassified clustered interval-censored data. Under review.
KOM ÁREK , L ESAFFRE , H ILTON (2005). Accelerated failure time model for arbitrarily
censored data with smoothed error distribution. Journal of Computational and Graphical Statistics, 14(3), 726–745.
KOM ÁREK , L ESAFFRE , L EGRAND (2007). Baseline and treatment effect heterogeneity for survival times between centers using a random effects accelerated failure time
model with flexible error distribution. Statistics in Medicine, 26(30), 5457–5472.
KOM ÁREK , L ESAFFRE (2008). Bayesian accelerated failure time model with multivariate
doubly-interval-censored data and flexible distributional assumptions. Journal of the
American Statistical Association, 103(482), 523–533.
G ARC ÍA -Z ATTERA , M UTSVARI , J ARA , D ECLERCK , L ESAFFRE (2010). Correcting for
misclassification for a monotone disease process with an application in dental research.
Statistics in Medicine, 29(30), 3103–3117.
G ARC ÍA -Z ATTERA , J ARA , L ESAFFRE , M ARSHALL (2012). Modeling of multivariate
monotone disease processes in the presence of misclassification. Journal of the American Statistical Association, 107(499), 976–989.
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IX. Summary and conclusions