Flexible Accelerated Failure Time Frailty Models for ......Flexible Mixed-Effects Accelerated Failure Time Regression Model Emmanuel Lesaffre (ERASMUS and KUL) AFT models for multivariate

Flexible Accelerated Failure Time Frailty Modelsfor Multivariate Interval-Censored Datawith an Application in Caries Research

Emmanuel Lesaffrejoint work with Arnošt Komárek and Dominique Declerck

Department of BiostatisticsErasmus Medical Center, Rotterdam, the Netherlands

L-BiostatKU Leuven, Leuven, Belgium

Haceteppe University

December 2012

Emmanuel Lesaffre (ERASMUS and KUL) AFT models for multivariate IC data Ankara, December 2012 1 / 53

Outline

1 Signal Tandmobielr (STM) Study

2 Suggested statistical model

3 The Mixed-Effects AFT regression model

4 Mixed-Effects AFT model = linear mixed model

5 Analysis of STM data - Doubly Interval Censoring

6 Discussion


Signal Tandmobielr (STM) Study

Outline

1 Signal Tandmobielr (STM) StudyDescription studyResearch QuestionsInterval censoringClustering & dependence





6 Discussion


Signal Tandmobielr (STM) Study Description study

Signal Tandmobielr Study

Longitudinal dental study

Flanders (Dutch speaking part of Belgium) 1996 – 2001

4 468 children followed from 7 till 12 years of age

Annual (pre-scheduled) dental examinations

Caries times recorded for deciduous teeth

Emergence and caries times recorded for permanent teeth

Many other covariates collected, e.g.:Status of primary teeth (dmft score)Frequency of brushingAmount of plaquePresence of sealants


Signal Tandmobielr (STM) Study Research Questions

Research Questions

Primary Research Question:

Influence ofcaries status of deciduous second molars (teeth 55, 65, 75, 85)

on caries susceptibilityof the adjacent permanent first molars (teeth 16, 26, 36, 46)

Other Research Questions:Impact of other covariates (sealants, gender, etc) on caries statusof permanent molarsLeft-Right symmetry, Maxilla-Mandibular differenceIdentify periods of high risk for caries

taking into account time at “risk”.



Research Questions








Research Questions








Deciduous & Permanent Teeth









Transition from deciduous to permanent teeth

From 7 to 12 years of age children have mixed dentition.



Research Questions involve

Regression model for emergence times

Regression model for times to caries (from birth)

Better: Modelling time to caries given period at risk=⇒ jointly modelling emergence/caries times

In a multivariate sense (4 molars jointly)
























Research questions involve also:

1 Interval censoring (emergence time & time to caries)!!! doubly interval censoring (time to caries, given period at risk)!!!

2 Clustering & dependence



Research questions involve also:

1 Interval censoring (emergence time & time to caries)!!! doubly interval censoring (time to caries, given period at risk)!!!

2 Clustering & dependence


Signal Tandmobielr (STM) Study Interval censoring

Emergence Times (similar for time to caries)

Response of main interest: emergence time U

We only know uL < U ≤ uU : interval censoring

uL= last dental examination where tooth hadn’t still not emerged

uU= first dental examination where tooth emerged

For uL = 0 =⇒ left censoring

For uU →∞ =⇒ right censoring


Signal Tandmobielr (STM) Study Clustering & dependence

1 ClusteringTeeth of a single child

share the same environment

share the same dietary and brushing habits

⇒ Marginal dependence but possibly conditional independence

2 (Conditional) Dependence (especially for time to caries)

Spatial structure of the mouth

adjacent teeth (not applicable here)

vert opp teeth occluding temporarily(?) (ignored here)






















Suggested statistical model

Outline






6 Discussion



‘Possible’ statistical model

What characteristics should the statistical model have?

Applicable to interval-censored data

Allowing for extensions to multivariate outcomes

Allowing for correction wrt covariates

Allowing for clustering

Computationally feasible, while minimizing on parametricassumptions

Suggestion:Flexible Mixed-Effects Accelerated Failure Time Regression Model




















































The Mixed-Effects AFT regression model

Outline



3 The Mixed-Effects AFT regression modelNotationComparison with (frailty) PH model



6 Discussion


The Mixed-Effects AFT regression model Notation

Notation

Ui,l : event time of the l th unit of the i th cluster(i = 1, . . . ,N; l = 1, . . . ,ni)

xi,l : covariates to explain Ui,l

β: regression parameters (fixed effects)

bi : cluster-specific random effect, bii.i.d.∼ gb(b)

Survival function: S(u |x i,l , β, bi)

Hazard function: }(u |x i,l , β, bi)



Notation









Notation









Notation









Notation









Notation








The Mixed-Effects AFT regression model Comparison with (frailty) PH model

PH and AFT Model

Cox PH model

}(u |x i,l , β) = exp(β′x i,l) }0(u)

S(u |x i,l , β) = S0(u)exp(β′x i,l )

AFT model

}(u |x i,l , β) = exp(−β′x i,l) }0{

u exp(−β′x i,l)︸︷︷︸}S(u |x i,l , β) = S0

{u exp(−β′x i,l)︸︷︷︸}



PH and AFT Model

Cox PH model

}(u |x i,l , β) = exp(β′x i,l) }0(u)

S(u |x i,l , β) = S0(u)exp(β′x i,l )

AFT model

}(u |x i,l , β) = exp(−β′x i,l) }0{

u exp(−β′x i,l)︸︷︷︸}S(u |x i,l , β) = S0

{u exp(−β′x i,l)︸︷︷︸}



Graphical RepresentationGraphical Representation

0 1 2 3 4 5

12

34

t

haza

rd(t)

AFT model, β = 0.5

x = 1.2

x = 0.6

x = 0

5 0 1 2 3 4 5

12

34

t

haza

rd(t)

PH model, β = − 0.5

x = 1.2

x = 0.6

x = 0

16

Graphical Representation

0 1 2 3 4 5

12

34

t

haza

rd(t)

AFT model, β = 0.5

x = 1.2

x = 0.6

x = 0

5 0 1 2 3 4 5

12

34

t

haza

rd(t)


x = 1.2

x = 0.6

x = 0

16



Graphical RepresentationGraphical Representation

0 1 2 3 4 5

12

34

t

haza

rd(t)

AFT model, β = 0.5

x = 1.2

x = 0.6

x = 0

5 0 1 2 3 4 5

12

34

t

haza

rd(t)


x = 1.2

x = 0.6

x = 0

16

Graphical Representation

0 1 2 3 4 5

12

34

t

haza

rd(t)

AFT model, β = 0.5

x = 1.2

x = 0.6

x = 0

5 0 1 2 3 4 5

12

34

t

haza

rd(t)


x = 1.2

x = 0.6

x = 0

16Emmanuel Lesaffre (ERASMUS and KUL) AFT models for multivariate IC data Ankara, December 2012 19 / 53


Frailty PH and Mixed-Effects AFT Model

Frailty Cox PH model

}(u |x i,l , β, bi) = exp(β′x i,l + bi) }0(u)

S(u |x i,l , β, bi) = S0(u)exp(β′x i,l+bi )

Mixed-Effects AFT model

}(u |x i,l , β, bi) = exp(−β′x i,l − bi) }0{

u exp(−β′x i,l − bi)︸︷︷︸}S(u |x i,l , β, bi) = S0

{u exp(−β′x i,l − bi)︸︷︷︸}



Frailty PH and Mixed-Effects AFT Model

Frailty Cox PH model

}(u |x i,l , β, bi) = exp(β′x i,l + bi) }0(u)

S(u |x i,l , β, bi) = S0(u)exp(β′x i,l+bi )

Mixed-Effects AFT model

}(u |x i,l , β, bi) = exp(−β′x i,l − bi) }0{

u exp(−β′x i,l − bi)︸︷︷︸}S(u |x i,l , β, bi) = S0

{u exp(−β′x i,l − bi)︸︷︷︸}



Robustness properties of (Mixed-Effects) AFT Model

Robust against uncorrelated omitted covariates

Inference for the fixed effects β does not (heavily) depend on thechosen distribution gb for the random effects

Marginal survival distribution (after integrating bi out)is of the same form as theconditional survival distribution (given bi )

Above robustness properties (generally) not truefor (frailty) Cox PH model























Mixed-Effects AFT model = linear mixed model

Outline




4 Mixed-Effects AFT model = linear mixed modelLinear Mixed-Effects model with interval-censored responseDistributional assumptionsFlexible distributions


6 Discussion


Mixed-Effects AFT model = linear mixed model Linear Mixed-Effects model with interval-censored response

Mixed-effects AFT Model = Linear Mixed Model

}(u |x i,l , β, bi ) = exp(−β′x i,l − bi ) }0{

u exp(−β′x i,l − bi )}

S(u |x i,l , β, bi ) = S0{


Linear mixed model with interval-censored response

log(Ui,j) = Yi,l = β′x i,l + bi + εi,l(i = 1, . . . ,N; l = 1, . . . ,ni)

with

β = fixed effectbi = random effect bi ∼ gb(·)εi,l = error random variable εi,l ∼ gε(·)


Mixed-Effects AFT model = linear mixed model Linear Mixed-Effects model with interval-censored response

Mixed-effects AFT Model = Linear Mixed Model

}(u |x i,l , β, bi ) = exp(−β′x i,l − bi ) }0{


S(u |x i,l , β, bi ) = S0{


Linear mixed model with interval-censored response

log(Ui,j) = Yi,l = β′x i,l + bi + εi,l(i = 1, . . . ,N; l = 1, . . . ,ni)

with

β = fixed effectbi = random effect bi ∼ gb(·)εi,l = error random variable εi,l ∼ gε(·)


Mixed-Effects AFT model = linear mixed model Distributional assumptions

Distributional Assumptions

Distribution of the error terms εi,l have to be specified

=⇒Which parametric density gε(ε)?

Distribution of the random effects bi have to be specified=⇒Which parametric density gb(b)?

Parametric assumptions influence the shape of hazard andsurvivor curvesNeeded for prediction & detecting periods of high risk=⇒ flexible model for gε(ε) and gb(b) is needed
















Mixed-Effects AFT model = linear mixed model Flexible distributions

Generic model (error & random part)

Y = α + τY ∗

Y ∗ ∝∑

Kj=−K wjN (µj , σ

2)

Fixed:Number of components 2K + 1Mixture means(knots) µj on grid

Mixture variance σ2

To estimate:Weights w = (w−K , . . . ,wK )T and

∑Kj=−K wj = 1

Reparametrisation: wj =exp(aj )∑Kk=−K ak



Generic model (error & random part)

Y = α + τY ∗

Y ∗ ∝∑

Kj=−K wjN (µj , σ

2)

Fixed:Number of components 2K + 1Mixture means(knots) µj on grid

Mixture variance σ2

To estimate:Weights w = (w−K , . . . ,wK )T and

∑Kj=−K wj = 1

Reparametrisation: wj =exp(aj )∑Kk=−K ak



Grid of Normal ComponentsGrid of Normal Components

−3 −2 −1 0 1 2 3

0.0

0.5

1.0

1.5

2.0

y∗

g∗(y

∗)



Grid of Weighted Normal ComponentsGrid of Weighted Normal Components

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

y∗

g∗(y

∗)

Each curve multiplied by some wj ∈ (0, 1)



Mixture of Weighted Normal ComponentsMixture of Weighted Normal Components

−3 −2 −1 0 1 2 3

0.0

0.1

0.2

0.3

0.4

0.5

y∗

g∗(y

∗)

Weighted sum of the curves using weights wj



Normal Mixture - Properties

Properties:Only mixtures weights are assumed to be unknownNumber of mixture components K fixed and relatively high (≈ 30)Mixture means: fixed equidistant grid of knotsMixture variances: fixed and same for all mixture components

Advantages:FlexibilityStandard theories apply

Disadvantages:Relatively high number of parameters must be estimatedPotential overfitting and identifiability problems















Penalized Normal Mixture

To avoid overfitting & identifiability problems:

Put a penalty using the k th order difference ∆kaj :

∆1aj = aj − aj−1 ∆kaj = ∆k−1aj −∆k−1aj−1

Maximize penalized log-likelihood w.r.t. θ (= regression and modelparameters)




Penalized approach:

Penalized log-likelihood:Log-likelihood: log L(θ; Y )

Penalty term:∑J

j=k+1(∆k aj )2

Penalized log-lik: log LP(θ; Y ) = log L(θ; Y )− λ2

∑Jj=k+1(∆k aj )

2

Maximize log LP(θ; Y ) w.r.t. θ for a given λ

Choose optimal λ by maximizing (minimizing) AIC

In Bayesian approach⇔ difference prior on the a-coefficients(approach followed here)




Penalized approach:


Penalty term:∑J

j=k+1(∆k aj )2


∑Jj=k+1(∆k aj )

2







Penalized approach:


Penalty term:∑J

j=k+1(∆k aj )2


∑Jj=k+1(∆k aj )

2







Penalized approach:


Penalty term:∑J

j=k+1(∆k aj )2


∑Jj=k+1(∆k aj )

2





Analysis of STM data - Doubly Interval Censoring

Outline





5 Analysis of STM data - Doubly Interval CensoringSTM Study: Time to CariesReg model for clustered doubly IC dataApplication to STM Study

6 Discussion


Analysis of STM data - Doubly Interval Censoring STM Study: Time to Caries

Research Questions

Of interest to dental researchers is:

Effect of covariates (gender, caries on adjacent 2nd deciduousmolar , frequency of brushing, amount of plaque, presence ofsealants) on the time to caries of the permanent first molars (16,26, 36, 46)

Response of main interest:time at risk T = time to caries (V ) - emergence time (U)

Both U and V are interval-censored

⇒ Involves doubly-interval censoring



Doubly Interval CensoringDoubly Interval Censoring

AA AA AA AA��

uL

uU

vL

vU

Examinations

�� AA AA

U V

Emergence

time

Caries

time

Time to caries T

Warning

For modelling, distribution of both U and V need to be modelling.

It is not correct to assume that T is interval censored.

(De Gruttula & Lagakos, 1989).

37

For modelling, distribution of both U and V need to be modelled.It is not correct to model only the distribution of T as interval-censored

observation (De Gruttula & Lagakos, 1989)



Doubly Interval CensoringDoubly Interval Censoring

AA AA AA AA��

uL

uU

vL

vU

Examinations

�� AA AA

U V

Emergence

time

Caries

time

Time to caries T

Warning

For modelling, distribution of both U and V need to be modelling.

It is not correct to assume that T is interval censored.

(De Gruttula & Lagakos, 1989).

37

For modelling, distribution of both U and V need to be modelled.It is not correct to model only the distribution of T as interval-censored

observation (De Gruttula & Lagakos, 1989)


Analysis of STM data - Doubly Interval Censoring Reg model for clustered doubly IC data

Notation

Ui,l : emergence time of the l th unit of the i th clusterxu

i,l : covariates to explain Ui,l

AND

Ti,l : time from emergence to caries of the l th unit of the i th clusterx t

i,l : covariates to explain Ti,l



Notation

Ui,l : emergence time of the l th unit of the i th clusterxu

i,l : covariates to explain Ui,l

AND

Ti,l : time from emergence to caries of the l th unit of the i th clusterx t

i,l : covariates to explain Ti,l



Cluster-Specific AFT Model for D-I-C Data

Model for emergence time

log(Ui,l) = Yi,l = δ′xui,l + di + ζi,l

(i = 1, . . . ,N; l = 1, . . . ,ni)

Model for time to caries

log(Ti,l) = log(Vi,l − Ui,l) = β′x ti,l + bi + εi,l

(i = 1, . . . ,N; l = 1, . . . ,ni)

Distributions

Random effects distributions = gd (d) (emergence) and gb(b) (caries)Error distributions = gζ(ζ) (emergence) and gε(ε) (caries)

Model all distributions in a FLEXIBLE manner JOINTLY






(i = 1, . . . ,N; l = 1, . . . ,ni)



(i = 1, . . . ,N; l = 1, . . . ,ni)

Distributions








(i = 1, . . . ,N; l = 1, . . . ,ni)



(i = 1, . . . ,N; l = 1, . . . ,ni)

Distributions





Assumptions

Some simplifications were necessary (and reasonable).Given the covariates:

Independence of (bi , εi,1, . . . , εi,ni )′ and (di , ζi,1, . . . , ζi,ni )

′

=⇒ Time-at-risk Ti is independent of the emergence time Ui(given covariates)

Independence of bi and di=⇒Whether a child is an early emerger is independent ofwhether a child is more or less sensitive against caries

Independence of εi,l and ζi,l

=⇒Whether a specific tooth emerges early or late is independentof whether that tooth is more or less sensitive against caries



Assumptions



′







Assumptions



′







Likelihood Contribution of the i th Cluster

Li =

∫<

∫<

[ni∏

l=1

∫ uUi,l

uLi,l

{∫ vUi,l−ui,l

vLi,l−ui,l

p(ti,l |bi ) dti,l

}p(ui,l |di ) dui,l

]p(bi ) p(di ) dbi ddi

p(ti,l |bi ) = t−1i,l gε

{log(ti,l )− β′x t

i,l − bi}

p(ui,l |di ) = u−1i,l gζ

{log(ui,l )− δ′xu

i,l − di}

p(bi ) = gb(bi )p(di ) = gd (di )

=shifted and scalednormal mixtures

=⇒ Maximum-likelihood may be quite difficult



Parameter Estimation

In a Bayesian way

Natural solution to doubly interval censoringData-Augmentation

MCMC methodology

Does not maximize a complex likelihood

Sample latent event times together with remaining parameters

Base the inference on a sample from the posterior distribution ofthe model parameters



Parameter Estimation

In a Bayesian way

Natural solution to doubly interval censoringData-Augmentation

MCMC methodology

Does not maximize a complex likelihood

Sample latent event times together with remaining parameters

Base the inference on a sample from the posterior distribution ofthe model parameters



Prior Distributions

Vague normal priors for δ and β

Vague normal prior for α and vague gamma for τ−2

Markov random field prior for transformed weights aj

p(a|λ) ∝ exp

[−λ

2

K∑j=−K+s

(∆saj )2

]= exp

[−λ

2a′D′Da

]

∆s = sth order difference operator & D associated differenceoperator matrix

λ = smoothing parameter

p(λ) = vague gamma, acts as a precision parameter for a



Prior Distributions




p(a|λ) ∝ exp

[−λ

2

K∑j=−K+s

(∆saj )2

]= exp

[−λ

2a′D′Da

]






Prior Distributions




p(a|λ) ∝ exp

[−λ

2

K∑j=−K+s

(∆saj )2

]= exp

[−λ

2a′D′Da

]






Directed Acyclic GraphDirected Acyclic Graph

Emergence Caries

censoringi,l

uUi,luL

i,l vLi,l vU

i,l

vi,l

ui,l ti,l

δ di xui,l ζi,l εi,l xt

i,l bi β

rdi r

ζi,l

rεi,l rb

i

Gd Gζ Gε Gb

l=

1,.

..,n

i

i=

1,.

..,N

46



MCMC Sampling

Gibbs algorithm using full conditionals

When full conditional is not a standard distribution

Slice sampling (Neal, 2003, Ann. Stat)

Adaptive rejection sampling (Gilks, Wild, 1992, Appl. Stat)

R-package bayesSurv (A. Komárek)

Available from CRAN at http://www.R-project.org



MCMC Sampling









MCMC Sampling








Analysis of STM data - Doubly Interval Censoring Application to STM Study

Model (Komárek & Lesaffre, JASA, Applications & Case studies, 2008)

Emergence

xui,l = (genderi , tooth26i,l , tooth36i,l , tooth46i,l )

′

Caries

x ti,l = (genderi , statusi,l , brushingi , sealantsi,l , plaquei,l ,

tooth26i,l , tooth36i,l , tooth46i,l )′.

status status of the adjacent 2nd deciduous molar(0 = sound, 1 = decayed/filled/missing due to caries)

brushing frequency of brushing(0 = less than once a day, 1 = at least once a day )

sealants presence of sealants(0 = absent, 1 = present)

plaque presence of plaque on occlusal surfaces(0 = absent, 1 = present)




Emergence


′

Caries










Emergence


′

Caries









Posterior SummarySignal-Tandmobielr Study: Posterior Summary

Emergence Caries

Posterior Posterior

Parameter median 95% CR median 95% CR

Tooth p > 0.5 p > 0.5

tooth 26 −0.003 (−0.013, 0.007) −0.006 (−0.045, 0.031)

tooth 36 0.001 (−0.008, 0.011) −0.009 (−0.051, 0.034)

tooth 46 0.002 (−0.008, 0.012) −0.016 (−0.059, 0.026)

Gender p = 0.008 p = 0.085

girl −0.023 (−0.039, −0.007) −0.071 (−0.155, 0.009)

Status p < 0.001

dmf −0.140 (−0.193, −0.091)

Brushing p < 0.001

daily 0.337 (0.233, 0.436)

Sealants p < 0.001

present 0.119 (0.060, 0.178)

Plaque p < 0.001

present −0.114 (−0.171, −0.067)

55



Posterior Predictive Survival Function

S(t |data, x tnew ) =

∫S(t |θ, data, x t

new ) p(θ |data) dθ (θ = all parameters)

From our model

S(t |θ, x tnew ) = 1−

K∑j=−K

wεj Φ{

log(t)− β′x tnew − b

∣∣ αε + τεµεj , (σετε)2}

MCMC estimate of the predictive survivor function:


1M

M∑m=1

S(t |θ(m), x tnew )

θ(m), m = 1, . . . ,M . . . MCMC sample from PPD

All components of θ(m) directly available except b(m)

⇒ sample from G(m)b (normal mixture)







From our model

S(t |θ, x tnew ) = 1−

K∑j=−K

wεj Φ{





1M

M∑m=1











From our model

S(t |θ, x tnew ) = 1−

K∑j=−K

wεj Φ{





1M

M∑m=1







Caries Free (Survivor) Curves, Tooth 16, Boys

0 1 2 3 4 5 6

0.0

0.2

0.4

0.6

0.8

1.0

Time since emergence (years)

Car

ies

free

Daily brushing, sealed, no plaque

Not daily brushing, not sealed, present plaque

Sound primary predecessorDMF primary predecessor



Caries Hazard, Tooth 16, Boys

0 1 2 3 4 5 6

0.00

0.05

0.10

0.15

0.20

Time since emergence (years)

Haz

ard

of c

arie

s

Daily brushing, sealed, no plaque

Not daily brushing, not sealed, present plaque

DMF primary predecessorSound primary predecessor



Predictive Distributions

0.35 0.40 0.45 0.50 0.55

02

46

810

1214

Emergence: error

ζ

g(ζ)

−0.4 −0.2 0.0 0.2 0.4

0.0

0.5

1.0

1.5

Emergence: random

d

g(d)

−1 0 1 2 3 4

0.0

0.5

1.0

1.5

Caries: error

ε

g(ε)

−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

0.0

0.4

0.8

1.2

Caries: random

b

g(b)


Discussion

Outline






6 DiscussionCan it be simpler?Some criticismAnother criticism


Discussion Can it be simpler?

Earlier approach

Avoiding doubly interval censored outcome,by imputing emergence time and employing it as covariate

Problem: caries prior to emergence

Discussion

The advantages of survival analysis for the analysis oflongitudinal caries data have been emphasized in recentreports (18, 19). For each individual tooth (or toothsurface) the time at risk can be estimated, even in the caseof censoring (e.g. when a patient enters the study with adecayed tooth or leaves the study prematurely, etc.).Moreover, allowance is made for the changing numberof surfaces at risk, which is a result of the natural exfo-liation and emergence process in children.To obtain statistically valid estimates and CIs adjusted

for dependent censored observations, the GEE-type testfor bivariate right-censored data, proposed by Huster

et al. (16), was extended to the multivariate setting. Itpermits inferences to be made about the marginal dis-tributions, while treating the dependence between theteeth as a nuisance. Chuang et al. (20) compared pre-dicted dental implant survival estimates assuming theindependence or dependence of clustered observations.They found that the point estimates were similar, but thevariance estimates were drastically different. The 95%CIs for the naıve model were narrower, resulting in anincreased risk for type I error and erroneous rejection ofthe null hypothesis. In the present study the CIs were upto 10% wider when dependence was taken into account(data not shown).

Usually, tooth emergence is taken as the zero-pointof the analysis (6, 21–24). As it is impossible to assessthe exact time of emergence, it is often assumed thatemergence occurred in the middle of the intervalbetween two examinations (19, 21, 23). Parner et al.(25) noted that this approach is only sensible for properintervals between two examinations, and even then itmay give rise to bias. Other groups rely on publishedmean emergence ages (23) or fail to inform the readerof how exact emergence ages for individual teeth inindividual subjects were determined, in spite of annualor bi-annual examinations (6, 26). The same problemapplies for the outcome: in some studies it is assumedthat the event (e.g. cavity formation) occurred in themiddle of the interval between two examinations (19).In the present study, to avoid invalid inferences no suchassumptions were made. The date of birth was used asthe zero-point of the analysis, a date that is exactlyknown for each child. For the sake of completeness,the results were verified with additional analyseswhere tooth emergence was taken as the zero-point ofanalysis. These analyses revealed comparable results –occlusal plaque accumulation, reported brushing fre-quency, and caries experience in the deciduous dentitionwere highly significant covariates. As expected, genderbecame less significant (i.e. the overall P-value increasedfrom 0.062 in the original analysis to 0.106 in the

Age (yr)

Sur

viva

l pro

babi

lity

6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

0.8

1.0

Girls Tooth 16

ABC

ABC

α Age (yr)

Sur

viva

l pro

babi

lity

6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

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Girls Tooth 36

ABC

ABC

β

Age (yr)

Haz

ard

6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

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0.5

Girls Tooth 16

ABC

ABC

γ Age (yr)

Haz

ard

6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

0.4

0.5

Girls Tooth 36

ABC

ABC

δ

Fig. 1. a and b: survival curves for girls; c and d: hazard functions for girls. Unbroken line: frequent brushing, no cavities indeciduous dentition, and no visible plaque on the occlusal surfaces of permanent first molars. Broken line: infrequent brushing,cavities in deciduous dentition and visible plaque on the occlusal surfaces of permanent first molars. Line A represents the emergenceinterval […-6] yr; line B represents the emergence interval ]6–7] yr; and line C represents the emergence interval ]7-…] yr. Comparableresults were obtained for boys.

150 Leroy et al.

A = emergence < 6 yrs, B = emergence 6 < < 7 yrs, C = emergence > 7 years



Earlier approach

Avoiding doubly interval censored outcome,by imputing emergence time and employing it as covariateProblem: caries prior to emergence

Discussion





Age (yr)

Sur

viva

l pro

babi

lity

6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

0.8

1.0

Girls Tooth 16

ABC

ABC

α Age (yr)

Sur

viva

l pro

babi

lity

6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

0.8

1.0

Girls Tooth 36

ABC

ABC

β

Age (yr)

Haz

ard

6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

0.4

0.5

Girls Tooth 16

ABC

ABC

γ Age (yr)

Haz

ard

6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

0.4

0.5

Girls Tooth 36

ABC

ABC

δ


150 Leroy et al.




Earlier approach

Avoiding doubly interval censored outcome,by imputing emergence time and employing it as covariateProblem: caries prior to emergence

Discussion





Age (yr)

Sur

viva

l pro

babi

lity

6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

0.8

1.0

Girls Tooth 16

ABC

ABC

α Age (yr)

Sur

viva

l pro

babi

lity

6 7 8 9 10 11 12

0.0

0.2

0.4

0.6

0.8

1.0

Girls Tooth 36

ABC

ABC

β

Age (yr)

Haz

ard

6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

0.4

0.5

Girls Tooth 16

ABC

ABC

γ Age (yr)

Haz

ard

6 7 8 9 10 11 12

0.0

0.1

0.2

0.3

0.4

0.5

Girls Tooth 36

ABC

ABC

δ


150 Leroy et al.



Discussion Some criticism

Torturing of the data?

Perhaps, but

Shape of the distribution?

Simplifying methods (mid-point approach) are generally invalidAd-hoc approach gave (in retrospect) peculiar resultsSimulation study indicates good behavior

Other smoothing approaches are possible

Various extensions of current model are possible




Perhaps, but

Shape of the distribution?Simplifying methods (mid-point approach) are generally invalid

Ad-hoc approach gave (in retrospect) peculiar resultsSimulation study indicates good behavior






Perhaps, but

Shape of the distribution?Simplifying methods (mid-point approach) are generally invalidAd-hoc approach gave (in retrospect) peculiar results

Simulation study indicates good behavior






Perhaps, but

Shape of the distribution?Simplifying methods (mid-point approach) are generally invalidAd-hoc approach gave (in retrospect) peculiar resultsSimulation study indicates good behavior






Perhaps, but







Perhaps, but





Discussion Another criticism

Is approach general enough?

Possibly not, Bayesian non-parametric approaches offer a moreflexible and general approach


Discussion Another criticism

THANK YOU FOR YOUR ATTENTION

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