multivariate survival analysis luc duchateau, ghent university paul janssen, hasselt university 1
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Multivariate survival analysis
Luc Duchateau, Ghent UniversityPaul Janssen, Hasselt University
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Multivariate survival analysis
Overview of course material
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Lecture 1: Multivariate survival data examples
Univariate survival: independent event times
Multivariate survival data: clustered event times
Multivariate survival data Overview of course material
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Lecture 2: The different analysis approaches
Ignore dependence: basic survival analysis
The marginal model
The fixed effects model
Multivariate survival data Overview of course material
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The stratified model
The copula model
The frailty model
Multivariate survival data Overview of course material
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Lecture 3: Frailties: past, present and future
First introduction of longevity factor to better model population mortality= modeling overdispersion in univariate survival data
Populations consisting of subpopulations and the requirement for different frailties
Multivariate survival data Overview of course material
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Lecture 4: Basics of the parametric gamma frailty model
Analytical solution: marginal likelihood by integrating out frailties
Characteristics of this model through its Laplace transform
Multivariate survival data Overview of course material
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Parametric
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Lecture 5: Parametric frailty models with other frailty densities
Analytical solution exists also for positive stable No analytical solution exists for log normal, the
frailties are integrated out numerically
Multivariate survival data Overview of course material
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Parametric Positive stableLog normal
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Lecture 6: The semi-parametric gamma frailty model through EM and PPL
The marginal likelihood contains unknown baseline hazard
Solutions can be found through the EM-algorithm or through the penalised partial likelihood approach
Multivariate survival data Overview of course material
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UndefinedNuissance
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Lecture 7: Bayesian analysis of the semi-parametric gamma frailty model
Clayton developed an approach to fit this same model using MCMC algorithms
Multivariate survival data Overview of course material
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UndefinedNuissance
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Lecture 8: Multifrailty and multilevel models
Multifrailty: two frailties in one cluster
Solution based on Bayesian approach. The posterior densities are obtained by Laplacian integration
Multivariate survival data Overview of course material
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Multilevel: two or more cluster sizes, one clustering level nested in the other
Solution based on frequentist approach by integrating out one level analytically and the other level numerically
Multivariate survival data Overview of course material
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Multivariate survival data
Examples
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Multivariate survival data types Criteria to categorise
Cluster size: 1, 2, 3, 4, >4 Hierarchy: 1 or 2 nesting levels Event ordering: none or ordered in space/time
Multivariate survival data Examples
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Univariate survival data Clusters of fixed size 1 Example 1: East Coast fever transmission
dynamics
Multivariate survival data Examples
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Bivariate survival data Clusters of fixed size 2 Example 2: diagnosis of fracture healing
Multivariate survival data Examples
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Example 3: Udder quarter reconstitution
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Quadruples of correlated event times Cluster of fixed size 4 Example 4: Correlated infection times in 4 udder
quarters
Multivariate survival data Examples
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Event times in clusters of varying sizeFew clusters of large size One level of clustering, but varying cluster size Example 5: Peri-operative breast cancer
treatment
Multivariate survival data Examples
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Event times in clusters of varying sizeMany small clusters One level of clustering, but varying cluster size Example 6: Breast conserving therapy for Ductal
Carcinoma in Situ (DCIS)
Multivariate survival data Examples
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Event times in clusters of varying sizeMany large clusters One level of clustering, but varying cluster size Example 7: Time to culling of heifer cows as a
function of early somatic cell count
Multivariate survival data Examples
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Example 8: Time to first insemination of heifers as function of time-varying milk protein concentration
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Example 9: Time to first insemination of heifer cows as a function of initial milk protein concentration
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Example 10: Prognostic index evaluation in bladder cancer
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Clustered event times with ordering Event times in a cluster have a certain ordering Example 11: Recurrent asthma attacks in children
Multivariate survival data Examples
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Event times in 2 nested clustering levels Smaller clusters of event times are nested in a
larger cluster Example 12: Infant mortality in Ethiopia
Multivariate survival data Examples
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Modeling multivariate survival data. The different approaches.
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Overview
Basic quantities/likelihood in survival The different approaches
The marginal model The fixed effects model The stratified model The copula model The frailty model
Efficiency comparisons
Overview 28
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Basic quantities in survival
The probability density function of event time T
The cumulative distribution function
The survival function
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The hazard function
The cumulative hazard function
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Survival likelihoodWe consider the survival likelihood for right
censored data ( ) assuming uninformative censoring
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Survival likelihood maximisationLikelihood maximisation leads to ML
estimates, for instance, constant hazard
with likelihood and loglikelihood
Equating first derivative to zero gives
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Survival likelihood maximisation: Example Time to diagnosis using US
d=106,
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Survival likelihood maximisation: Example in R – explicit solutionTime to diagnosis using US
Basic quantities/likelihood in survival 34
library(survival)timetodiag <- read.table("c:\\docs\\bookfrailty\\data\\diag.csv", header = T,sep=";")t1<-timetodiag$t1/30;t2<-timetodiag$t2/30c1<-timetodiag$c1;c2<-timetodiag$c2
#Explicit solution for lambdasumy<-sum(t1);d<-sum(c1);lambda1<-d/sumysumy;d;lambda1
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Survival likelihood maximisation: Example in R – iterative solution#likelihood maximisationsurvlik<-function(p){d*log(p[1])+p[1]*sumy}nlm(survlik,0.5)
$minimum [1] 162.3838$estimate[1] 0.5874742$gradient[1] 4.447998e-05$code[1] 1$iterations[1] 4
Basic quantities/likelihood in survival 35
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Maximisation and Newton-RaphsonClosed form solutions exist only in exceptional casesIn most cases iterative procedures have to be used
for maximisationNewton Raphson (NR) procedure finds a point for
which Thus we can find the estimate that maximises the
likelihood by using the NR procedure on the first derivative of the (log) likelihood
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Newton-Raphson procedure Aim: find point of function with We start with point
and use Taylor series expansion
We only take first two terms and find
We take the right hand side as new estimate and proceed iteratively
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Newton-Raphson procedure: Illustration Iterative procedure: Example graphically:
Basic quantities/likelihood in survival 38
x
f(x)
0.0 0.5 1.0 1.5 2.0
02
46
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Newton-Raphson procedure Example Estimating though NR:
Basic quantities/likelihood in survival 39
0.45 0.50 0.55 0.60 0.65 0.70
-20
02
04
0
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Newton-Raphson procedure: Second iteration for the example
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Variance estimate from likelihood
Basic quantities/likelihood in survival 41
An estimate of the variance of is given by the inverse of minus the second derivative
evaluated at
Thus we have
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Variance from likelihood in R
Basic quantities/likelihood in survival 42
Minus the second derivative(s) of the log likelihood is obtained when using option ‘hessian=T’
survlik<-function(p){-d*log(p[1])+p[1]*sumy}results<-nlm(survlik,0.5,hessian=T)1/results$hessian
[,1][1,] 0.003257014
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Multidimensional NR procedure
Basic quantities/likelihood in survival 43
The NR procedure for one parameter
can be extended to multidimensional version
the observed information matrix and the score vector
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Overview
Basic quantities/likelihood in survival The different approaches
The marginal model The fixed effects model The stratified model The copula model The frailty model
Efficiency comparisons
Overview 44
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The marginal model The marginal model approach consists of
two stages Stage 1: Fit the model without taking into
account the clustering Stage 2: Adjust for the clustering in the data
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Consistency of marginal model parameter estimates
The ML estimate from the Independence Working Model (IWM)
is a consistent estimator for (Huster, 1989) More generally, the ML estimate ( and
possibly baseline parameters) from the IWM is also a consistent estimator for
Parameter refers to the wholewhole population
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Adjusting the variance of IWM estimates
The variance estimate based on the inverse of the information matrix of is an inconsistent estimator of Var( )
One possible solution: jackknife estimation General expression of jackknife estimator for iid
data (Wu, 1986)
with N the number of observations and a the number of parameters
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The grouped jackknife estimator
For clustered observations: grouped jackknife estimator
with s the number of clusters
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Sandwich estimator of variance of IWM estimates Implication: fit model s times, computer-
intensive! Lipsitz and Parzen (1996) propose the
following approximation Likelihood maximisation is based on
Newton-Raphson iteration with each step based on the Taylor series approximation
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A good starting value is , and the first Newton-Raphson iteration step is then
In this expression, however, we have
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… and can thus be rewritten as
Plugging this into the grouped jackknife:
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Further assume s>>a and always using the same information matrix, we obtain the robust variance estimator
with the information matrix and the score vector
This approximation corresponds to the sandwich estimator obtained by White (1982) starting from a different viewpoint.
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Example marginal model with jackknife estimator Example 3: Blood-milk barrier
reconstitution Estimates from IWM model with time-
constant hazard rate assumption are given by
Grouped jackknife = approximation
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The jackknife estimator in R Reading the datareconstitution<-read.table
("c://docs//presentationsfrailty//Rotterdam//data//reconstitution.csepv",header=T,sep=",")
#Create 5 column vectors, five different variablescowid<- reconstitution$cowidtimerec<-reconstitution$timerecstat<- reconstitution$stattrt<- reconstitution$trtheifer<- reconstitution$heifer
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Fitting the unadjusted model
res.unadjust<-survreg(Surv(timerec,stat)~trt,dist="exponential",data=reconstitution)
b.unadjust<- -res.unadjust$coef[2]stdb.unadjust<- sqrt(res.unadjust$var[2,2])l.unadjust<-exp(-res.unadjust$coef[1])
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Obtaining the grouped jackknife estimator
dat<-data.frame(timerec=timerec,trt=trt,stat=stat,cowid=cowid)
res<-survreg(Surv(timerec,stat)~trt,data=dat,dist="exp")
init<-c(res$coeff[1],res$coeff[2])
ncows<-length(levels(as.factor(cowid)))
bdel<-matrix(NA,nrow=ncows,ncol=2)
for (i in 1:ncows){
temp<-reconstitution[reconstitution$cowid!=i,]
coeff<-survreg(Surv(timerec,stat)~trt, data=temp,dist="exponential")$coeff
bdel[i,1]<- -coeff[2];bdel[i,2]<-exp(-coeff[1])}
sqrt(0.98*sum((bdel[,1]-b.unadjust)^2))
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Obtaining the grouped 1-step jackknife estimator
bdel<-matrix(NA,nrow=ncows,ncol=2)for (i in 1:ncows){temp<-reconstitution[reconstitution$cowid!=i,]coeff<-survreg(Surv(timerec,stat)~trt,data=temp,
init=init,maxiter=1,dist="exponential")$coeffbdel[i,1]<- -coeff[2]bdel[i,2]<-exp(-coeff[1])}sqrt(0.98*sum((bdel[,1]-b.unadjust)^2))
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The jackknife estimator in R Problems The grouped jackknife estimator can be
obtained right away by the ‘cluster’ command For Weibull, for instance, we have
res.adjust<-survreg(Surv(timerec,stat)~trt+cluster(cowid), data=reconstitution)
For exponential, this does not seem to work …
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Jackknife estimator-simulations In the example, the jackknife estimate of
the variance is SMALLER than the unadjusted variance!!
Is the jackknife estimate always smaller than estimate from unadjusted model, or does this depend on the data?
Generate data from the frailty model of time to reconstitution data with
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We generate 2000 datasets, each of 100 pairs of two subjects for the settings1. Matched clusters, no censoring2. 20% of clusters 2 treated or untreated
subjects, no censoring3. Matched clusters, 20% censoring
Matched clusters: the two animals sharing the clusters have different covariate levels
The different approaches - the marginal model 60
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Simulation results
The different approaches - the marginal model 61
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The fixed effects model The fixed effects model is given by
with the fixed effect for cluster i,
Assume for simplicity and
with
The different approaches - the fixed effects model 62
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The fixed effects model: ML solution General survival likelihood expression
For fixed effects model with constant hazard
The different approaches - the fixed effects model 63
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Fixed effects and loglinear model Software often uses loglinear model
with, for , Comparison can be based on S(t)
The different approaches - the fixed effects model 64
Gumbel
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Survival function loglinear model (1) We look for
General rule
Applied to
The different approaches - the fixed effects model 65
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Equivalence fixed effects model and loglinear model Therefore, we have for loglinear model
vs
The different approaches - the fixed effects model 66
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The delta method - general
Original parameters Interest in univariate cont. function Use one term Taylor expansion of
with
The different approaches - the fixed effects model 67
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The delta method - specific Interest in univariate cont. function
The one term Taylor expansion of
with
The different approaches - the fixed effects model 68
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Treatment effect for reconstitution data using R-function survreg (loglin. model):the effect of the drug
Example: within cluster covariate
The different approaches - the fixed effects model 69
res.fixed.trt<-survreg(Surv(timerec,stat)~ as.factor(cowid)+as.factor(trt), dist="exponential",data=reconstitution)summary(res.fixed.trt)
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The different approaches - the fixed effects model 70
Parameter estimates
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Parameter interpretation corresponds to constant hazard of
untreated udder quarter of cow 1 corresponds to constant
hazard of untreated udder quarter of cow i Cowid65 ≈ 0 Cowid100 exp(-21+18.8)=0.11
Treatment effect: HR=exp(0.185)=1.203 with 95% CI [0.83;1.75]
The different approaches - the fixed effects model 71
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Heifer effect for reconstitution dataintroducing heifer first in the model
Hazard ratio impossibly high
Example: between cluster covariate
The different approaches - the fixed effects model 72
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Heifer effect for reconstitution dataintroducing cowids first in the model
Hazard ratio equal to 1
The different approaches - the fixed effects model 73
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Heifer effect for reconstitution data
Two estimates for heifer effect, which one (if any) is correct? Write down incidence matrix for intercept, cowid and heifer for 6 cows, with first 3 cows being heifers
The different approaches - the fixed effects model 74
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The different approaches - the fixed effects model 75
Intercept ci2 ci3 ci4 ci5 ci6 heifer parity C1 C2 C3 C4 C5 C6 C7 C8
C7=C1-C4-C5-C6 COMPLETE CONFOUNDING
Incidence matrix heifer – cowid
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The different approaches - the fixed effects model 76
Intercept ci2 ci3 ci4 ci5 ci6 heifer parity C1 C2 C3 C4 C5 C6 C7 C8
C8=C1+C4+C5+3C6 COMPLETE CONFOUNDING
Incidence matrix heifer – cowid
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The stratified model The stratified model is given by
Maximisation of partial likelihood,
with the risk set for cluster i
The different approaches - the stratified model 77
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Example for bivariate data Consider the case of bivariate data, e.g. the reconstitution
data Write down a simplified version of the general expression
When does a cluster contribute to the likelihood?
The different approaches - the stratified model 78
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The partial likelihood for reconstitution data
Estimates
The different approaches - the stratified model 79
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The stratified model in R Use the coxph function for time to reconstution data
> res.strat.trt<-coxph(Surv(timerec,stat)~as.factor(trt) +strata(cowid),data=reconstitution)> summary(res.strat.trt)
coef exp(coef) se(coef) z pas.factor(trt)1 0.131 1.14 0.209 0.625 0.53 exp(coef) exp(-coef)
lower .95 upper .95as.factor(trt)1 1.14 0.878 0.757 1.72
Likelihood ratio test= 0.39 on 1 df, p=0.531Wald test = 0.39 on 1 df, p=0.532Score (logrank) test = 0.39 on 1 df, p=0.532
The different approaches - the stratified model 80
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The copula model If all clusters have the same size, another
useful approach is the copula model We consider bivariate data in which the
first (second) subject in each cluster has the same covariate information, and the covariate information differs between the two subjects in the same cluster
Time to diagnosis ofbeing healed
The different approaches - the copula model 81
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The two stage approach A copula model is often fitted using a two-
stage model: First specify the population (marginal) survival
functions for the first (second) subject in a cluster and obtain estimates for them.
We then generate the joint survival function by linking the population survival functions through the survival copula function (Frees et al., 1996).
The different approaches - the copula model 82
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Example of copula model
The different approaches - the copula model 83
Time to diagnosis of being healed
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Bivariate copula model likelihood Four different possible contributions of a
cluster
Estimated population survival functions are inserted, only copula parameters unknown
The different approaches - the copula model 84
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The Clayton copula The Clayton copula (Clayton, 1978) is
The Clayton copula corresponds to the family of Archimedean copulas, i.e.,
with in the Clayton copula case
The different approaches - the copula model 85
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Clayton copula likelihood for parametric marginals Two censored observations
Observation j censored
No obervations censored
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Example Clayton copula For diagnosis of being healed data, first fit
separate models for RX and US technique For instance, separate parametric models
The different approaches - the copula model 87
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Estimates for marginal models are
Based on these estimates we obtain
which can be inserted in the likelihood expression which is then maximised for
The different approaches - the copula model 88
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For parametric marginal models, the likelihood can also be maximised simul-taneously for all parameters leading to
Thus, for small sample sizes, the two-stage approach can differ substantially from the one-stage approach
The different approaches - the copula model 89
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Alternatives can be used for marginal models Nonparametric models Semiparametric models
leading to
The different approaches - the copula model 90
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Clayton copula in Rtimetodiag <- read.table("c:\\docs\\bookfrailty\\data\\diag.csv",
header = T,sep=";")
t1<-timetodiag$t1/30;t2<-timetodiag$t2/30c1<-timetodiag$c1;c2<-timetodiag$c2surv1<-survreg(Surv(t1,c1)~1); surv2<-survreg(Surv(t2,c2)~1)l1<- exp(-surv1$coeff/surv1$scale);r1<-(1/surv1$scale)l2<-exp(-surv2$coeff/surv2$scale);r2<-(1/surv2$scale)s1<-exp(-l1*t1^(r1));f1<-s1*r1*l1*t1^(r1-1)s2<-exp(-l2*t2^(r2));f2<-s2*r2*l2*t2^(r2-1)
The different approaches - the copula model 91
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R-function
loglikcon.gamma.twostage<-function(theta){P<-s1^(-theta)+ s2^(-theta)-1loglik<- -(1-c1)*(1-c2)*(1/theta)*log(P) +c1*(1-c2)*(-(1+1/theta)*log(P)-(theta+1)*log(s1)+log(f1)) +c2*(1-c1)*(-(1+1/theta)*log(P)-(theta+1)*log(s2)+log(f2)) +c1*c2*(log(1+theta)-(2+1/theta)*log(P)-
(theta+1)*log(s1)+log(f1)-(theta+1)*log(s2)+log(f2))-sum(loglik)}nlm(loglikcon.gamma.twostage,c(0.5))
The different approaches - the copula model 92
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Results of R-function
> nlm(loglikcon.gamma.twostage,c(0.5))$minimum[1] 233.0512
$estimate[1] 0.9659607
$gradient[1] 4.197886e-05
$code[1] 1
$iterations[1] 4
The different approaches - the copula model 93
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The positive stable copula The positive stable copula is given by
Is this an Archimedean copula?
The different approaches - the copula model 94
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The positive stable copula The positive stable copula is given by
Is this an Archimedean copula?
Yes, it is, take
The different approaches - the copula model 95
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The positive stable copula and likelihood contributions The positive stable copula is given by
Write down the likelihood assuming a semiparametric model for the population survival functions
The different approaches - the copula model 96
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The positive stable copula is given by
Write down the likelihood assuming a nonparametric model for the population survival functions No events:
The different approaches - the copula model 97
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Likelihood contributions No events
One event
Two events
The different approaches - the copula model 98
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The frailty model The ‘shared’ frailty model is given by
with the frailty An alternative formulation is given by
with
The different approaches - the frailty model 99
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The gamma frailty model Gamma frailty distribution is easiest choice
with and
The different approaches - the frailty model 100
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Marginal likelihood for the gamma frailty model Start from conditional (on frailty) likelihood
with containing the baseline hazard parameters, e.g., for Weibull
The different approaches - the frailty model 101
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Marginal likelihood: integrating out the frailties … Integrate out frailties using distribution
with
The different approaches - the frailty model 102
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Closed form expression for marginal likelihood Integration leads to
The different approaches - the frailty model 103
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Maximisation of marginal likelihood leads to estimates Marginal likelihood no longer contains frailties. By maximisation
estimates of are obtained Furthermore, the asymptotic variance-covariance matrix can be
obtained as the inverse of the observed information matrix with the Hessian matrix with entries
The different approaches - the frailty model 104
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Efficiency comparisons in the reconstitution data example Estimates (se) for reconstitution data
The different approaches – efficiency comparisons 105
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Efficiency comparisons-theory For parametric model with constant
hazard, we can obtain the expected information for (Wild, 1983)
Due to asymptotic independence of parameters, the asymptotic variance for is given by
The different approaches – efficiency comparisons 106
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Unadjusted model
Fixed effects model
Frailty model
The different approaches – efficiency comparisons 107
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Efficiency comparisons-simulations Generate data from the frailty model with
We generate 2000 datasets, each of 100 pairs of two subjects for the settings1. Matched clusters, no censoring2. 20% of clusters 2 treated or untreated
subjects, no censoring3. Matched clusters, 20% censoring
The different approaches – efficiency comparisons 108
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Simulation results
The different approaches – efficiency comparisons 109
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Semantics and History of the term
frailty
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Semantics of term frailty Medical field: gerontology
Frail people higher morbidity/mortality risk Determine frailty of a person (e.g. Get-up and Go test) Frailty: fixed effect, time varying, surrogate
Modelling: statistics Frailty often at higher aggregation level (e.g. hospital in
multicenter clinical trial) Frailty: random effect, time constant, estimable
Frailty semantics 111
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History of term frailty - Beard Introduced by Beard (1959) in univariate setting
to improve population mortality modelling by allowing heterogeneity
Beard (1959) starts from Makeham’s law (1868)
with the constant hazard and with the hazard increases with time
Longevity factor is added to model
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Beard’s model Population survival function
Population hazard function
Frailty semantics 113
Survival at time t forsubject with frailty u
Hazard at time t forsubject with frailty u
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With one parameter gamma distribution as frailty distribution, we obtain a Perk’s logistic curve
Frailty semantics 114
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With one parameter gamma distribution as frailty distribution, we obtain a Perk’s logistic curve
To what value does the hazard function goes asymtotically for time to infinity?
Frailty semantics 115
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The hazard function of Perk’s logistic curve starts at and goes asymptotically to
Example:
Frailty semantics 116
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Frailty semantics 117
Term frailty first introduced by Vaupel (1979) in univariate setting to obtain individual mortality curve from population mortality curve
For the case of no covariates
History of term frailty - Vaupel
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Frailty semantics 118
Vaupel assumed gamma frailties and thus
Conditional mean decreases with time, Individual hazard increases relatively with time
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Frailty semantics 119
From hazard to conditional probability
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Frailty semantics 120
Example Swedish population two centuries
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Frailty semantics 121
Example Swedish population two centuries
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Frailty semantics 122
Vaupel and Yashin (1985) studied heterogeneity due to two subpopulations Population 1: Population 2:
Frailty – two subpopulations
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Frailty semantics 123
Smokers:high and low recidivism rate
Is the population hazard constant, decreasing or increasing over time?
Calculate the population hazard at time 0 and at 20 years.
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Frailty semantics 124
Smokers:high and low recidivism rate
The population hazard At time 0:
At 20 years:
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Frailty semantics 125
Smokers:high and low recidivism rate: pictures
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Frailty semantics 126
Reliability engineering
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Frailty semantics 127
Two hazards increasing at different rates
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Frailty semantics 128
Two parallel hazards (at log scale)
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Basics of parametric gamma frailty model
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Overview
The gamma (frailty) density The marginal likelihood for the parametric
gamma frailty model An example: time to first insemination with
heifer as covariate Laplace transform From the joint survival function to the
marginal likelihood Functions at the population level An example: udder quarter infection data
Overview 130
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The gamma frailty density
Two-parameter gamma density
One-parameter gamma density:
The gamma frailty density 131
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The gamma frailty density 132
One-parameter gamma density: pictures
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Shared gamma frailty model (a conditional hazards model)
The conditional likelihood for cluster i is
The marginal likelihood for the parametric gamma frailty model
The marginal likelihood for the parametric gamma frailty model
133
with a random sample from one-parameter gamma density
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The marginal likelihood for cluster i is, with and ,
which has the following explicit solution
The marginal likelihood for the parametric gamma frailty model
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The marginal loglikelihood is
where
ML estimates for the components of can be found by maximising this loglikelihood.
The estimated variance-covariance matrix is obtained as the inverse of the observed information matrix evaluated at ,
The marginal likelihood for the parametric gamma frailty model
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See Example 9 The hazard for cow j from herd i is
Assume Weibull, then
i.e., given the value of the event times follow a Weibull distribution with parameters and
An example: time to first insem-ination with heifer as covariate
An example: time to first insemination with heifer as covariate
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with the heifer covariate ( for multiparous, for heifer), the heifer effect and the frailty term for herd i
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Maximising the marginal loglikelihood (with time in months – for convergence reasons) we obtain
For herd with the hazard ratio is exp(-0.153) = 0.858. The 95% CI is [0.820,0.897], i.e., the hazard of first insemination for heifers is significantly lower.
An example: time to first insemination with heifer as covariate
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Relation between months and days
(the cumulative hazard ratio at any time is required to be the same)
An example: time to first insemination with heifer as covariate
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Hazard functions for time to first insemination
An example: time to first insemination with heifer as covariate
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Median time to event for herd i in heifers, resp. multiparous cows ( ,resp. ): ( ,resp. ).
An example: time to first insemination with heifer as covariate
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Median time to first insemination
An example: time to first insemination with heifer as covariate
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First read in time to first insemination data
#Read datainsemfix<-read.table("c://docs//presentationsfrailty//
Rotterdam//data//insemination.datc", header=T,sep=",")#Create four column vectors, four different variablesherd<-insemfix$herdnr;timeto<-(insemfix$end*12/365.25)stat<-insemfix$score;heifer<-insemfix$par2#Derive some valuesn<-length(levels(as.factor(herd)))di<-aggregate(stat,by=list(herd),FUN=sum)[,2];r<-sum(di)
Maximising marginal likelihood in R
Maximising marginal likelihood in R 142
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Define loglikelihood function for transformed parameters
#Observable likelihood weibull with transformed variables#l=exp(p[1]), theta=exp(p[2]), beta=p[3], rho=exp(p[4])#r=No events,di=number of events by herdlikelihood.weibul<-function(p){cumhaz<-exp(heifer*p[3])*(timeto^(exp(p[4])))*exp(p[1])cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2]lnhaz<-stat*(heifer*p[3]+log((exp(p[4])*timeto^(exp(p[4])-
1))*exp(p[1])))lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2]lik<-r*log(exp(p[2]))-sum((di+1/
exp(p[2]))*log(1+cumhaz*exp(p[2])))+sum(lnhaz)+sum(sapply(di,function(x) ifelse(x==0,0,log(prod(x+1/exp(p[2])-seq(1,x))))))
-lik}
Maximising marginal likelihood in R143
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Maximise the function for the transformed parameters and return parameter estimates
res<-nlm(likelihood.weibul,c(log(0.174),log(0.39),-0.15,log(1.76)))
lambda<-exp(res$estimate[1])theta<-exp(res$estimate[2])beta<-res$estimate[3]rho<-exp(res$estimate[4])
Maximising marginal likelihood in R144
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Obtain standard errors for parameter estimates from Hessian matrix on original variables
#Observable likelihood weibull with original variables to obtain the Hessian
likelihood.weibul.natural<-function(p){
cumhaz<-exp(heifer*p[3])*(timeto^(p[4]))*p[1]
cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2]
lnhaz<-stat*(heifer*p[3]+log(p[4]*timeto^(p[4]-1)*p[1]))
lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2]
lik<-r*log(p[2])-sum((di+1/p[2])*log(1+cumhaz*p[2]))+sum(lnhaz)+
sum(sapply(di,function(x) log(prod(x+1/p[2]-seq(1,x)))))
-lik}
res.nat<-nlm(likelihood.weibul.natural, c(lambda,theta,beta,rho), hessian=T,iterlim=1)
stderrs<-sqrt(diag(solve(res.nat$hessian)))
Maximising marginal likelihood in R145
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Characteristic function
Moment generating function
Laplace transform for positive r.v.
Laplace transform
Laplace transform 146
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Generate nth moment Use nth derivative of Laplace transform
Evaluate at s=0
The Laplace transform for the one-parameter gamma density is
Laplace transform147
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Joint survival function in conditional model
Use as notation
For cluster of size n with covariates
From the joint survival and density functions to the marginal likelihood
From the joint survival and density functions to the marginal likelihood
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For cluster of size n with covariates the joint density is
From the joint survival and density functions to the marginal likelihood
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Applied to Laplace transform of gamma frailty we obtain
From the joint survival and density functions to the marginal likelihood
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=
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Note. Same as the , but now from more general Laplace point of view.
From the joint survival and density functions to the marginal likelihood
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Population survival function (integrate out the frailty from the conditional survival function)
Population density function
Population hazard function
Functions at the population level
Functions at the population level 152
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Ratio of population and conditional hazard
Applied to the gamma frailty we have
Functions at the population level 153
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Ratio of population and conditional hazard for the gamma frailty
Functions at the population level 154
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Population hazard ratio for a binary (0-1) covariate
Applied to the gamma frailty we have
Functions at the population level 155
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Population hazard ratio for the gamma frailty
Functions at the population level 156
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In general we have
Applied to the gamma frailty we have
Updating (no covariates, gamma frailty)
Updating (no covariates, gamma frailty)
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with conditionial mean
and conditional variance
Updating (no covariates, gamma frailty)
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Conditionial variance of U
Updating (no covariates, gamma frailty)
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See Example 4
Binary covariate: heifer/multiparous ( is the heifer effect)
Weibull baseline hazard
Gamma frailty density
Maximising the marginal loglikelihood we obtain
An example: udder quarter infection data
An example: udder quarter infection data 160
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Conditional and population hazard ratio functions
Updating (no covariates, gamma frailty)
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Hazard ratio multiparous cow versus heifer
Updating (no covariates, gamma frailty)
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0 1 2 3 4Time (year quarters)
1.0
1.1
1.2
1.3
1.4
1.5
1.6
Ha
zard
ra
tio
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Mean and variance of conditional frailty density
Updating (no covariates, gamma frailty)
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The parametric gamma frailty model:
variations on the theme
Overview A piecewise constant baseline hazard Recurrent events
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A piecewise constant baseline hazard
See Example 7 Event time: time to culling
Covariate: logarithm of somatic cell count (SCC)
The model
The MLE are given by
A piecewise constant baseline hazard 165
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LR test for constant baseline hazard versus piecewise constant baseline hazard
versus
The p-value (using a -distribution with 2 degrees of freedom) is smaller than 0.00001
A piecewise constant baseline hazard 166
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First read in time to culling data#Read dataculling<-read.table('c://culling.datc',header=T,sep=",")culltime<-culling$timetocul*12/365.25logSCC<-culling$logSCCherd<-as.factor(culling$herdnr)status<-culling$statusendp1<-70*12/365.25;endp2<-300*12/365.25n<-length(levels(as.factor(herd)))di<-aggregate(status,by=list(herd),FUN=sum)[,2]r<-sum(di))
Time to culling: piecewise constant hazard frailty model in R
A piecewise constant baseline hazard 167
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Fit model with linear effect of log(SCC), constant hazard
#parametric exponential covariate lnscc1likelihood.exp<-function(p){cumhaz<-exp(logSCC*p[1])*exp(p[3])*culltimecumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2]lnhaz<-status* (logSCC*p[1]+log(exp(p[3])))lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2]lik<-r*log(exp(p[2]))-sum((di+1/exp(p[2]))
*log(1+cumhaz*exp(p[2]))) +sum(lnhaz)-n*log(gamma(1/exp(p[2]))) +sum(log(gamma(di+1/exp(p[2]))))
-lik}initial<-c(0,log(0.2),log(0.01))t<-nlm(likelihood.exp,initial,print.level=2)beta<-t$estimate[1];theta<-exp(t$estimate[2])lambda<-exp(t$estimate[3]);likelihood.linear<-t$minimum
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Results model with linear effect of log(SCC), constant hazard
beta<-t$estimate[1]theta<-exp(t$estimate[2])lambda<-exp(t$estimate[3])likelihood.linear<-t$minimum > beta[1] 0.08813477> theta[1] 0.06230903> lambda[1] 0.01606552> likelihood.linear[1] 15079.36
A piecewise constant baseline hazard 169
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Fit model with linear effect of log(SCC); piecewise constant hazard
likelihood.piecexp<-function(p){cumhaz<-exp(logSCC*p[1])*exp(p[3])*pmin(culltime,endp1)cumhaz<-cumhaz+exp(logSCC*p[1]) *exp(p[4])*pmax(0,pmin(endp2-
endp1,culltime-endp1))cumhaz<-cumhaz+exp(logSCC*p[1])*exp(p[5]) *pmax(0,culltime-
endp2)cumhaz<-aggregate(cumhaz,by=list(herd),FUN=sum)[,2]lnhaz<-status*(logSCC*p[1]+log(as.numeric(culltime<endp1)*
exp(p[3])+as.numeric(endp1<=culltime) *as.numeric(culltime<endp2)*exp(p[4]) +as.numeric(culltime>=endp2)*exp(p[5])))
lnhaz<-aggregate(lnhaz,by=list(herd),FUN=sum)[,2]lik<-r*log(exp(p[2]))-sum((di+1/exp(p[2]))*log(1+cumhaz*exp(p[2])))
+ sum(lnhaz)-n*log(gamma(1/exp(p[2]))) +sum(log(gamma(di+1/exp(p[2]))))
-lik}
A piecewise constant baseline hazard 170
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initial<-c(0,log(0.2),log(0.01),log(0.01),log(0.01))t<-nlm(likelihood.piecexp,initial,print.level=2)beta<-t$estimate[1];theta<-exp(t$estimate[2])lambda1<-exp(t$estimate[3]);lambda2<-exp(t$estimate[4])lambda3<-exp(t$estimate[5])likelihood.piecewise<-t$minimum> beta[1] 0.08831314> theta[1] 0.1440544> lambda1[1] 0.006735632> lambda2[1] 0.01151625> lambda3[1] 0.09857648> likelihood.piecewise[1] 13562.13
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Comparing constant and piecewise constant hazard ffunction models
> LR<-2*(-likelihood.piecewise+likelihood.linear)> 1-pchisq(LR,2)[1] 0
A piecewise constant baseline hazard 172
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Recurrent events See Example 11 Time to asthma attack is the event time
Covariate: placebo versus drug Patient i has ni at risk periods
which can be represented in a graphical way (first two patients)
Recurrent events 173
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Different timescales can be considered
Calendar time
Gap time
Recurrent events 174
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We give four conditional hazard models and we use the marginal likelihood to estimate the model parameters
For all four models the marginal likelihood that needs to be maximised is
where
For each of the four models we need to specify the precise meaning of and
Recurrent events 175
drug effect
heterogeneity between patients
Weibull baseline hazard
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Calendar time model
Weibull baseline hazard
Recurrent events 176
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Gap time model. Information used is
Weibull baseline hazard
Recurrent events 177
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Gap time model with hazard for first event different and constant
Combining gap and calendar time
Recurrent events 178
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Overview of ML estimates for the four models
Recurrent events 179
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Graphical representation of the results
Recurrent events 180
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First read in asthma data#Read dataw <-
read.table("c:/docs/bookfrailty/data/asthma.dat",head=T)di<-aggregate(w$st.w,by=list(w$id.w),FUN=sum)[,2]r<-sum(di);n<-length(di)trt<-aggregate(w$trt.w,by=list(w$id.w),FUN=min)[,2]begin<-12*w$start.w/365.25end<-12*w$stop.w/365.25ptno<-w$id.w;status<-w$st.wgap<-end-begin;fevent<-w$fevent
Recurrent asthma attacks: Parametric frailty models in R
Recurrent events 181
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#Weibull-calendar-no specific hazard first eventlikelihood.weibull.cp.nof<-function(p){cumhaz<-p[3]*(end^p[4]-begin^p[4])cumhaz<-aggregate(cumhaz,by=list(ptno),FUN=sum)[,2]lnhaz<-status*(log(p[3]*p[4]*end^(p[4]-1)))lnhaz<-aggregate(lnhaz,by=list(ptno),FUN=sum)[,2]lik<-r*log(p[2])-n*log(gamma(1/p[2]))
+sum(log(gamma(di+1/p[2])))-sum((di+1/p[2])*log(1+p[2]*exp(p[1]*trt)*cumhaz))+sum(di*p[1]*trt)+sum(lnhaz)-lik}
Recurrent events 182
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#Weibull-gap-no specific hazardfirst eventlikelihood.weibull.gap.nof<-function(p){cumhaz<-p[3]*((end-begin)^p[4])cumhaz<-aggregate(cumhaz,by=list(ptno),FUN=sum)[,2]lnhaz<-status*(log(p[3]*p[4]*(end-begin)^(p[4]-1)))lnhaz<-aggregate(lnhaz,by=list(ptno),FUN=sum)[,2]lik<-r*log(p[2])-n*log(gamma(1/p[2]))
+sum(log(gamma(di+1/p[2])))-sum((di+1/p[2])*log(1+p[2]*exp(p[1]*trt)*cumhaz))+sum(di*p[1]*trt)+sum(lnhaz)-lik}
Recurrent events 183
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#Weibull-gap- first event exponentiallikelihood.weibull.gap.fev.exp<-function(p){cumhaz<- p[3]*(end-begin)*fevent+p[4]*((end-
begin)^p[5])*(1-fevent) cumhaz<-aggregate(cumhaz,by=list(ptno),FUN=sum)[,2]lnhaz<-status*(log(p[3])*fevent+ log(p[4]*p[5]*(end-
begin)^(p[5]-1))*(1-fevent))lnhaz<-aggregate(lnhaz,by=list(ptno),FUN=sum)[,2]lik<-r*log(p[2])-n*log(gamma(1/p[2]))
+sum(log(gamma(di+1/p[2])))-sum((di+1/p[2])*log(1+p[2]*exp(p[1]*trt)*cumhaz))+sum(di*p[1]*trt)+sum(lnhaz)-lik}
Recurrent events 184
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#Weibull-gap-calendar-nofirsteventlikelihood.weibull.all<-function(p){cumhaz<-p[3]*(end^p[4]-begin^p[4])+ (p[5]*((end-
begin)^p[6])*as.numeric(begin!=0))cumhaz<-aggregate(cumhaz,by=list(ptno),FUN=sum)[,2]lnhaz<-status*( log( (p[3]*p[4]*end^(p[4]-1)) +
as.numeric(begin!=0) * (p[5]*p[6]*(end-begin)^(p[6]-1))))lnhaz<-aggregate(lnhaz,by=list(ptno),FUN=sum)[,2]lik<-r*log(p[2])-n*log(gamma(1/p[2]))
+sum(log(gamma(di+1/p[2])))-sum((di+1/p[2])*log(1+p[2]*exp(p[1]*trt)*cumhaz))+sum(di*p[1]*trt)+sum(lnhaz)-lik}
Recurrent events 185
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Dependence measures
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Overview Setting, for i=1, …,s
binary data:covariate information:
Kendall’s : an overall measure of dependence The cross ratio function: a local measure of
dependence Other local dependence measures An example: the gamma frailty for
Overview 187
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Kendall’s : an overall measure of dependence
Kendall’s 188
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Express p in terms of joint survival and joint density function
using
Kendall’s 189
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Use
to obtain
Kendall’s 190
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Applied to the gamma frailty we obtain
Kendall’s 191
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The cross ratio function: a local measure of dependence
See Example 3: time to blood-milk barrier reconstitutionPositive experience: reconstitution at time t2
For positively correlated data, we assume that hazard in numerator > hazard in denominator
The cross ratio function 192
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Cross ratio as odds ratio Write
The cross ratio function 193
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Consider conditional 2x2 table
with
The cross ratio function 194
etc…
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Cross ratio as local Kendall’s
The cross ratio function 195
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Other local dependence measures
Other local dependence measures 196
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An example: the gamma frailty for =2/3
Other local dependence measures 197
see figures next slide for
and
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Dependence measures graphically for
Other local dependence measures 198
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Other choices for the frailty densities
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Overview The positive stable frailty
The positive stable (frailty) density The marginal likelihood for a positive stable frailty model The positive stable frailty density: functions at the population
level An example: udder quarter infection data Updating (no covariates, positive stable frailty) Bivariate data: positive stable Laplace transform
The lognormal frailty The lognormal (frailty) density Statistical inference for the parametric lognormal frailty model An example: udder quarter infection data
Overview 200
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The positive stable frailty density The positive stable density
with
Laplace transform
does not exist for and
This gives evidence for the fact that the mean of the positive stable density is infinite
The positive stable (frailty) density 201
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The positive stable density: pictures
The positive stable (frailty) density 202
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Joint survival function
This quantity is the key for obtaining the marginal likelihood
The positive stable (frailty) density 203
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The marginal likelihood for a positive stable frailty model We consider the special case of quadruple data
(clusters of size 4). See example 4 on udder quarter infection data. Five different types of contributions, according to ,the
number of events in cluster i ( ) Order subjects, put uncensored observations first (1,…, l) Contribution of cluster i to the marginal likelihood is
The marginal likelihood for a positive stable frailty model
204
=0
>0
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Note that the previous formula can (in terms of and ) be written as ( )
The marginal likelihood for a positive stable frailty model
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Derivatives of Laplace transforms
The marginal likelihood for a positive stable frailty model
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Marginal likelihood expression cluster i
The marginal likelihood for a positive stable frailty model
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The positive stable frailty density: functions at the population level Population survival function (integrate out the
frailty term from the conditional survival function).
Population density function
The positive stable frailty density: functions at the population level
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Population hazard function
Ratio of population and conditional hazard
The positive stable frailty density: functions at the population level
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Ratio of population and conditional hazard for the positive stable frailty
The positive stable frailty density: functions at the population level
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Population hazard ratio for a binary (0-1) covariate
Applied to the positive stable frailty we have
The positive stable frailty density: functions at the population level
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Population hazard ratio for the positive stable frailty:
The positive stable frailty density: functions at the population level
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An example: udder quarter infection data See Example 4 Binary covariate: heifer/multiparous ( is the heifer
effect)Weibull baseline hazardPositive stable frailty
Maximising the marginal likelihood we obtain (se=0.052) (se=0.114) (se=0.351) (se=0.039)
An example: udder quarter infection data
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The conditional hazard ratio is estimated as
The population hazard ratio is estimated as
An example: udder quarter infection data
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First read in udder quarter infection data#Read dataudder <- read.table("c:\\udderinfect.dat", header = T,skip=2)coword<-order(udder$cowid)cowid<-udder$cowid[coword]timeto<-4*(round(udder$timek[coword]*365.25/4))/365.25stat<-udder$censor[coword];laktnr<-udder$LAKTNR[coword]cluster<-as.numeric(levels(as.factor(udder$cowid)))G<-length(cluster)mklist<-function(clusternr){list(stat[cowid==clusternr],
timeto[cowid==clusternr],laktnr[cowid==clusternr])}clus<-lapply(cluster,mklist)
Udder quarter infections and the positive stable frailty model in R
An example: udder quarter infection data in R
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Function to fit positive stable frailty model#p[1]=lambda, p[2]=rho, p[3]=beta, p[4]=theta likelihood.posttab<-function(p){ll<-0for (clusternr in (1:G)){stat<-clus[[clusternr]][[1]];time<-clus[[clusternr]][[2]]trt<-clus[[clusternr]][[3]];Di<-sum(stat)SHij<-sum(exp(p[1])*time^(exp(p[2]))*exp(p[3]*trt))part1<-sum(stat*log(exp(p[1])*exp(p[2])*(time^(exp(p[2])-
1))*exp(p[3]*trt)))part2<-Di*log(exp(p[4]))+Di*(exp(p[4])-1)*log(SHij)-SHij^(exp(p[4]))part3<-0;if (Di==2)
{part3<-log(1+((1-exp(p[4]))*SHij^(-exp(p[4]))/exp(p[4])))}if (Di==3) {part3<-log(1+(3*(1-exp(p[4]))*SHij^(-exp(p[4]))/exp(p[4]))+
( (exp(p[4])^(-2))*(2-exp(p[4]))* (1-exp(p[4]))*SHij^(-2*exp(p[4]))))}if (Di==4) {part3<-log(1+(6*(1-exp(p[4]))*SHij^(-exp(p[4]))/exp(p[4]))+
( (exp(p[4])^(-2))*(1-exp(p[4]))*(11-7*exp(p[4]))*SHij^(-2*exp(p[4])))+( (exp(p[4])^(-3))*(3-exp(p[4]))*(2-exp(p[4]))*(1-exp(p[4]))*SHij^(-3*exp(p[4]))))}
ll<-ll+(part1+part2+part3)}-ll}
An example: udder quarter infection data in R
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Results fit positive stable frailty model
lambda<-exp(t$estimate[1])rho<-exp(t$estimate[2])beta<-t$estimate[3]theta<-exp(t$estimate[4]) > lambda[1] 0.1766017> rho[1] 2.129374> beta[1] 0.5374521> theta[1] 0.5285567
An example: udder quarter infection data in R
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Updating (no covariates, positive stable frailty) In general we have
Applied to the positive stable frailty we have
The updated density is not a positive stable density but is still a member of the power variance function family (see further)
Updating (no covariates, positive stable frailty)
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Bivariate data: positive stable Laplace transform
with
Bivariate data: positive stable Laplace transform
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Bivariate data: positive stable Laplace transform
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Bivariate data: positive stable Laplace transform
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Bivariate data: positive stable Laplace transform
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Bivariate data: positive stable Laplace transform
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The lognormal frailty density
Introduced by McGilchrist (1993) as
Therefore, for the frailty we have
The lognormal frailty density 224
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The lognormal frailty density: pictures
The lognormal frailty density 225
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Statistical inference for the parametric lognormal frailty model No explicit expression for Laplace transform …
difficult to compare Maximisation of the likelihood is based on
numerical integration of the normally distributed frailties
Statistical inference for the parametric lognormal frailty model
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An example: udder quarter infection data See Example 4 Binary covariate: heifer/multiparous ( is the heifer
effect)Weibull baseline hazardLognormal frailty density
Maximising the marginal loglikelihood (using numerical integration: Gaussian quadrature – nlmixed procedure) we obtain (se=0.094) (se=0.135) (se=0.378) (se=0.610)
Statistical inference for the parametric lognormal frailty model
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Difficult to compare with e.g. the gamma frailty model since, for the lognormal frailty model, the mean and variance both depend on . We therefore convert the results to the density function of the median event time
Statistical inference for the parametric lognormal frailty model
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Udder quarter infections and the lognormal frailty model in SAS
An example: udder quarter infection data in SAS
229
data mast;infile 'c:\mastitis.dat' delimiter=',' firstobs=2;input LAKTNR censor rightk leftk cowid timek;if leftk=0 then leftk=0.001;if censor=1 then midp=(leftk+rightk)/2;if censor=0 then midp=rightk;y=1;
proc nlmixed data=mast qpoints=10 cov;ebetaxb = exp(beta1*LAKTNR + b);G_t = exp(-lambda*ebetaxb*(midp**g));f_t = (lambda*ebetaxb*g*midp**(g-1))*exp(-lambda*ebetaxb*(midp**g));if (censor=1) then lik=f_t;else if (censor=0) then lik=G_t;llik=log(lik);model y~general(llik);random b~normal(0,theta) subject =cowid;run;
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The semiparametric frailty model
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Overview The semiparametric frailty model The marginal likelihood … a problem The EM-algorithm for the semiparametric frailty model The modified EM-algorithm An example: the DCIS (EM) The penalised partial likelihood approach for the
semiparametric frailty model An example: the DCIS (PPL: normal, resp. loggamma,
random effect) The performance of the PPL approach in estimating the
heterogeneity parameter
Overview 231
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The semiparametric frailty model A semiparametric frailty model is a model with
unspecified (nonparametric) baseline hazard
with and unspecified
For we will consider the one-parameter gamma and the lognormal frailty
The semiparametric frailty model 232
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The marginal likelihood … a problem We first discuss the one-parameter gamma frailty For parametric frailty models with gamma frailty
distribution we maximised the marginal log like-lihood obtained from cluster contributions of form
For the semiparametric frailty model, this is no longer possible since the baseline hazard is unspecified
The marginal likelihood … a problem 233
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Semiparametric survival models (Cox models) are typically fitted through partial likelihood maximisation
We will study two techniques to fit semiparametric frailty models, which combine the partial likelihood maximisation idea (classical approach for the Cox model) with the fact that, since the random frailty terms are unobserved, we have incomplete information (where the EM approach can help) The EM-algorithm The penalised partial likelihood approach
The marginal likelihood … a problem 234
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The EM algorithm for the semiparametric frailty model The general idea is as follows
The EM-algorithm iterates between an Expectation and Maximisation step
In the Expectation step, expected values for the frailties are obtained, conditional on the observed information and the current parameter estimates
In the Maximisation step, the expected values for the frailties are considered to be known (fixed offset terms), and partial likelihood is used to obtain new estimates
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The full likelihood
Consider the full loglikelihood for observed information and unobserved information
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Maximisation step using instead of Replace and in by expected
values and obtained in iteration k
Profile to a partial likelihood
with the risk set corresponding to
Maximise to obtain estimate
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Maximisation step for Replace in
by expected values and maximise wrt to obtain
With the ordered event times (r is the number of different events) and the number of events at time , obtain the Nelson-Aalen estimator for the baseline hazard with the frailties as fixed offset terms
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The expectation stepWe need expressions for and Obtain expectation
conditional on current parameter estimate
and
The conditional density of is (Bayes)
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Working out the previous expression we get
with
This corresponds to a gamma distribution with
parameters and
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Therefore we have
Furthermore, has a loggamma distribution and
thus, with the digamma function
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Asymptotic variances of estimates are obtained as entries of the inverse of the observed information matrix obtained from the marginal loglikelihood
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The EM algorithm for the semiparametric frailty model
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The modified EM-algorithm Convergence rate can be improved by using
modified profile likelihood (Nielsen et al., 1992)
The algorithm consists of two iteration levels Outer loop: maximisation for Inner loop: maximisation for all other parameters
conditional on chosen value
The modified EM-algorithm 244
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The modified EM-algorithm 245
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An example: the DICS (EM) See Example 6
We fit the semiparametric gamma frailty model
Ductal Carcinoma in Situ (DCIS) is a type of benign breast cancerBreast conserving therapy with/without radiotherapyTime to event: time to local recurrenceLarge number of clusters (46) with small cluster size ( )
An example: the DICS (EM) 246
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Statistical analysis (using the SAS macro of Klein and Moeschberger (1997))
Radiotherapy effect =-0.63 (se=0.17)
=0.53 [0.38;0.71]
Heterogeneity =0.086 (se=0.80)
An example: the DICS (EM) 247
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DCIS and the semiparametric gamma frailty model in SAS (Klein)
An example: DCIS in SAS 248
%macro gamfrail(mydata,factors,initbeta,convcrit,leftend,stepsize,options,data1,data2,data3);use &mydata;…log(gamma((1/thetacr)+di[,i]))….
Macro works for data with clusters with few events, such as DCIS
Does not work when clusters have many events, as gamma((1/thetacr)+di[,i])) gets very large
SAS macro requires IML
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The PPL approach for the semiparametric frailty model The PPL approach starts from model
with the frailty with variance and the random effect with variance
The two main ideas are Random effects are considered to be just another set of
parameters For values of far away from the mean (zero) the
penalty term has a large negative contribution to the full data (partial) loglikelihood
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The PPL is
with
with
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Maximisation step of the PPL An iterative procedure using an inner and an outer loop
Inner loop: Newton Raphson procedure to maximise for and
(BLUP’s) given provisional value for .
Outer loop: Obtain new value for using the current estimates for and in
the iteration process.
How this step works depends on the frailty density used in the model. We will
consider the outer loop for random effects with a normal distribution and random
effects with a loggamma distribution.
.
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The inner loop of the PPL approach Some notation
( ) denotes the outer (inner) loop index
is the estimate for at iteration
and are estimates at iteration given
At iteration step , the estimate is
with and
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and
the inverse of matrix
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The outer loop for the PPL approach: normal random effect For normal random effects
An estimate for is given by REML
where
Convergence obtained when
is sufficiently small
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The variance estimates: normal random effect
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The PPL approach for the semiparametric frailty model
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The outer loop for the PPL approach: loggamma random effect
The loggamma distribution is
Therefore, the penalty function is
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Outer loop: No REML estimator available!
Alternative: similar to modified EM-algorithm approach:
maximise marginal likelihood profiled for
For fixed value , obtain estimates by
maximising PPL
Plug in estimates in Nelson-Aalen estimator to obtain
estimates of the (cumulative) baseline hazard
Use estimates to obtain marginal likelihood
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The PPL approach for the semiparametric frailty model
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An example: the DCIS (PPL) See Example 6
We fit the PPL with normal random effect Radiotherapy effect =-0.63 (se=0.17)
= 0.53 [0.38;0.74]
Heterogeneity =0.087 (se=0.80)
We fit the PPL with loggamma random effect Radiotherapy effect =-0.63 (se=0.17)
= 0.53 [0.38;0.74] Heterogeneity =0.087 (se=0.80) The PPL can only be used to estimate , since the PPL
considered as a function of increases (see figure)
An example: the DICS (EM) 260
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Statistical analysis (using the SAS macro of Klein and Moeschberger (1997))
Radiotherapy effect =-0.63 (se=0.17)
=0.53 [0.38;0.71]
Heterogeneity =0.086 (se=0.80)
An example: the DICS (EM) 261
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An example: the DICS (EM) 262
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DCIS and the semiparametric gamma frailty model in R
An example: DCIS in SAS 263
dcis<-read.table("c://dcis.dat", header=T)dcis.gamfrail <-coxph(Surv(lrectime,lrecst)~trt+frailty(hospno,dist="gamma",eps=0.00000001),outer.max=100,data=dcis)summary(dcis.gamfrail)
coef se(coef) se2 Chisq DF p tr -0.628 0.167 0.167 14.1 1.00 0.00018frailty(hospno, dist = "g 11.5 7.82 0.16000 exp(coef) exp(-coef) lower .95 upper .95tr 0.534 1.87 0.385 0.741
Iterations: 10 outer, 29 Newton-Raphson Variance of random effect= 0.0865 I-likelihood = -1005 Degrees of freedom for terms= 1.0 7.8 Rsquare= 0.034 (max possible= 0.866 )Likelihood ratio test= 34.8 on 8.82 df, p=5.69e-05Wald test = 14.1 on 8.82 df, p=0.112 ….
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The performance of PPL in estimating heterogeneity Simulation setting
Number of clusters: 15 or 30 Number of subjects/cluster: 20,40 or 60 Constant hazard: 0.07 or 0.22 Accrual/follow-up period: 5/3 years resulting in 70% to 30% censoring Variance or : 0, 0.1, 0.2 HR: 1.3 6500 data sets generated
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Size for semiparametric gamma frailty model
=0
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Testing for heterogeneity
Hypothesis test: H0: =0 vs Ha: >0
-value in H0 is at boundary of parameter space
Use mixture of and
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Power For GammaFrailtymodel
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Power For NormalFrailtymodel
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Bayesian analysis of the semiparametric
frailty model
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Overview Introduction Frailty model for grouped data and gamma
process prior for the cumulative hazard Frailty model for observed event times and
gamma process prior for the cumulative hazard Examples
Overview 270
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Introduction Bayesian techniques can be used to fit frailty
modelsImportant references areKalbfleisch (1978): Bayesian analysis for the
semiparametric Cox model
Clayton (1991): Bayesian analysis for frailty models
Ibrahim et al. (2001): Bayesian survival analysis (book)
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Metropolis algorithm: sampling from the joint (multivariate) posterior density of the parameters of interest
For semiparametric frailty model: a problematic approach since hazard rate at each event time is considered as a parameter
high dimensional sampling problem (Metropolis no longer efficient)
Gibbs sampling: sampling from posterior density of each parameter conditional on the other parameters (iterative procedure)
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Frailty model for grouped data and gamma process prior for H(t) Consider the model
with actual value of
We need the conditional posterior density of each parameter (conditional on all other parameters) We therefore need the list of all parameters involved the prior densities of the parameters the (conditional) data likelihood
Grouped data: time axis is partitioned in z disjoint intervals
with Frailty model for grouped data 273
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For a particular interval and a particular subject three situations are possible(i) the subject experienced the event in (ii) the subject did not experience the event in and is still at risk
at (iii) the subject is no longer in the study at time and is lost to follow-
up in the interval
The increase of the cumulative (unspecified) baseline hazard in is
For these increments (considered as parameters) we will assume an independent increments gamma process prior (see further)
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The ‘parameters’ of interest are
the vector with the regression parameters
an hyperparameter
To see how the conditional data likelihood for grouped data is constructed we look at a concrete example
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Observed and grouped data: pictures
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The likelihood contributions of the six subjects are
Frailty model for grouped data 277
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General expression for the conditional data likelihood is in terms of the contributions of the subjects
with
in terms of the intervals
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This is different from previous survival expressions since we only specify cumulative hazard increments (and not the baseline hazard itself).
The expression resembles the likelihood used for interval-censored data.
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Specification of the prior densities (distributions) Independent increments gamma process for the cumulative baseline hazard
with an increasing continuous function such that
and independence between the increments
contained in
For the cumulative baseline hazard we have
with and
Note. Often a time-constant hazard and thus ;c large means variance small and therefore strong prior belief in
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( )
improper uniform prior
Clayton (1991)
Gelman (2003)
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Derivation of posterior densities: the general principle
with a -vector
and is the vector with component i
deleted
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Since (we assume inde-
pendence of prior densities) and since
does not depend on
Notes(i) Normalising factor often difficult to obtain (ii) If obtained the resulting posterior density does not take a form from which it is easy to sample (iii) An approximation for the derived likelihood expression will lead to more simple conditional posterior densities from which
sampling is easy
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unnormalised conditionalposterior density
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Frailty model for event times and gamma process prior for H(t) To obtain an appropriate data likelihood in case of
(censored) event times, we use ideas developed for grouped data with intervals with the ordered event times
Frailty model for observed event times 284
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Approximation for the derived likelihood. For instance, for the event time ,
For censored subjects the contribution is based on the cumulative hazard corresponding to the sum of the cumulative hazard increments of all event times before the actual censoring time (like in grouped data)
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This leads to the following likelihood expression
with
or
with if and
This expression resembles the likelihood of Poisson distributed data
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Prior densities: as before Posterior densities
regression parameters
this density is logconcave adaptive rejection
sampling can be used to generate a sample
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cumulative hazard increments
After a tedious derivation we obtain
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This is a gamma density
with
with the risk set at time
and the number of events at event time
sample from a gamma density
Frailty model for observed event times 289
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frailties
sample from a gamma density
the hyperparameter with
difficult density, slice sampling is needed
Frailty model for observed event times 290
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the hyperparameter with uniform on
difficult density, slice sampling is needed
A similar discussion can be given for the normal frailty model based on Poisson likelihood
Frailty model for observed event times 291
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Examples DCIS example (Example 6)
We focus on and (the parameters of interest)Four chains: burn-in 1000 iterations, followed by
5000 iterations
Gamma frailties
Examples 292
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Posterior densities – gamma frailties: pictures
Examples 293
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The trace – gamma frailties: pictures
Examples 294
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Normal random effects
Examples 295
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Posterior densities – normal random effects: pictures
Examples 296
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The trace – normal random effects: pictures
Examples 297
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Conclusion: Problem with convergence
(for , resp. )
Udder infection data (Example 4) with normal random effects
Examples 298
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Posterior densities – normal random effects: pictures
Examples 299
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The trace – normal random effects: pictures
Examples 300
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Conclusion: No problem with convergence
(for )
Examples 301
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Multilevel and multifrailty models
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Overview Multifrailty versus multilevel
Only one cluster, two frailties in cluster e.g., prognostic index (PI) analysis, with random center effect and ramdom PI
effect
More than one clustering level e.g., child mortality in Ethiopia with children clustered in village and village in
district Modelling techniques
Bayesian analysis through Laplacian integration Bayesian analysis throgh MCMC Frequentist approach through numerical integration
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Bayesian analysis through Laplacian integration Consider the following model with one clustering level
with the random center effect covariate information of first covariate, e.g. PIfixed effect of first covariaterandom first covariate by cluster interaction other covariate information other fixed effects
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Aim of Bayesian analysis: obtain posterior distributions for parameters of interest
Ducrocq and Casella (1996) proposed a method based on Laplacian integration rather than Gibbs sampling
Laplacian integration is much faster than Gibbs sampling, which makes it more suitable to their type of data, i.e., huge data sets in animal breeding, looking at survival traits
Emphasis is placed on heritability and thus estimation of variance components
Posterior distributions are only provided for variance components, fixed effects are only considered for adjustment
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The joint posterior density is given by
with
the variance of the random centre effect the variance of random covariate by cluster interaction
likelihood
furthermore, we have the prior distributions
random effects are independent from each other
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and for the other prior parameters flat priors are assumed
We leave baseline hazard unspecified, and therefore use partial likelihood (Sinha et al., 2003)
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The logarithm of the joint posterior density is then proportional to
The posterior distribution of the parameters of interest can be obtained by integrating out other parameters
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No analytical solution available for
Approximate integral for fixed value by Laplacian integration
For fixed value we use notation
First we obtain the mode of the joint posterior density function at
by maximising the logarithm of the joint posterior density wrt and (limited memory quasi-Newton method)
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We first rewrite integral as
and replace by the first terms of its Taylor expansion around the mode
The second term of the Taylor expansion cancels
For the third term we need the Hessian matrix
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The integral is then approximately
As has asymptotically a multivariate normal distribution with variance covariance matrix , we have
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The integral can therefore be simplified to
or on the logarithmic scale
Estimates for and are provided by the mode of this approximated bivariate posterior density
To depict the bivariate posterior density and determine other summary statistics by evaluating the integral on a grid of equidistant points
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The values can be standardised by computing
with the distance between two adjacent points
Univariate posterior densities for each of the two variance components can be obtained as
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This discretised version of the posterior densities can also be used to approximate the moments The first moment for is approximated by
In general, the cth moment is given by
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From these non-central moments, we obtain The mean
The variance
The skewness
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Bayesian analysis through Laplacian integration: Example Prognostic index heterogeneity for a bladder cancer multicentre trial Traditional method to validate prognostic index
Split database into a training dataset (typically 60% of the data) and validation dataset (remaining 40%)
Develop the prognostic index based on the training dataset and evaluate it based on the validation dataset
Flaws in the traditional method How to split the data, at random or picking complete centers? This will always work if sample sizes are sufficiently large It ignores the heterogeneity of the PI completely
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We fit the following frailty model
with the overall prognostic index effect
the random hospital effect
the random hospital* PI interaction
and consider disease free survival (i.e., time to relapse or death, whatever comes first).
Parameter estimates
= 0.737 (se=0.0964)
= 0.095
= 0.016
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Interpreting the parameter estimates Good prognosis in center i,
Poor prognosis in center i,
Hazard ratio in center i,
If for cluster i, , the mean of the distribution
with 95 % CI
This CI refers to the precision of the estimated conditional hazard ratio
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Bivariate posterior density for
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Univariate posterior densities for
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Plotting predicted random center and random interactions effects
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Interpretation of variance component
Heterogeneity of median event time quite meaningless, less than 50% of the patients had event, therefore rather use heterogeneity of five-year disease free percentage
Fit same model with Weibull baseline hazard
leading to the following estimate
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The density of the five-year disease-free percentage usingis then given by
where and FN standard normal
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The density of the five-year disease-free percentage
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Interpretation of variance component The hazard ratio for center i is given by
Derive the density function for this expression.
What are the 5th and 95th quantiles for this density assuming the parameter estimates
are the population parameters
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= 0.737 (se=0.0964)
= 0.016
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We look for
General rule
Applied to
=lognormal with parameters and
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The density function for
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For lognormal distribution we have
For the x% quantile of HR, , we have
Therefore, the 5th and 95th quantiles are given by
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Multilevel frailty models using a frequentist approach Consider the following model with two clustering levels
with the frailty term for cluster i the frailty term for subcluster j nested in cluster i
We further assume that frailty terms are independent and gamma distributed
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The conditional likelihood is then given by
The marginal likelihood for cluster i is
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Integrating out the frailties analytically, we obtain
with the number of events in subcluster j nested in cluster i number of events in cluster i
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Taking the logarithm we obtain the marginal log-likelihood
The marginal loglikelihood still contains the baseline hazard and cumulative baseline hazard Hazard functions are modeled using splines (Rondeau et al., 2006)
Cubic M-splines for baseline hazard function Integrated M-splines for baseline cumulative hazard
Penalty term added to marginal likelihood to cont-rol the degree of smoothness of hazard function
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The smoothing parameter can be chosen by visual inspection or by maximising a likelihood cross-validation criterion.
The penalised marginal loglikelihood is then maximised using the Marquardt algorithm. For evaluation of the penalised marginal loglikelihood, the frailties corresponding to the large clusters are integrated out
numerically using Gaussian quadrature.
This technique is implemented in the software package ‘frailtypack’, also available in R
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Frequentist approach for multilevel frailty model: Example Multilevel child mortality data (Example 12)
Gender effect: =-0.137 (se=0.071) HR (female versus male)= 0.87 [0.76;1.00]
Variance of village frailties: 0.0450 (se=0.028)
Variance of district frailties: 0.0073 (se=0.00007)
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