NonPH MV frailty models

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Non-PH MV frailty models
h - likelihood inference   by  
G. MacKenzie, I. D. Ha , Y. Lee M. Blagojevic
 
The Centre for Medical Statistics Keele
University England
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Overview   Ø Motivation MV Survival Data   Ø
Definition non-PH GTDL   Ø GTDL Frailty
Random Effect Models   Ø h Likelihood
Estimation   Ø Applications
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Introduction     Ø Coxs PH frailty model is
widely used for analysing multivariate
survival data.   Ø However the PH model makes
strong assumptions multiplicative hazard
structure and covariate effects linear on the
log-hazard scale.   Ø A possible
alternative is to have a non-PH hazard
structure with the effects linear on another
scale   Ø We consider non-PH frailty models
based on the GTDL
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Definition - GTDL Regression Model   Ø Hazard

    Ø Density   where  
 
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Definition - TDL Regression Model   Ø
Hazard
  Ø Density where
are 2 real scalars.   NB


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Key Properties Relative Risk Odds
Ratio   GTDL TDL
Ø RR
TDL
Ø OR
So RR OR multiplied by a factor exp(?) has an
OR interpretation.
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Hazard Plots (see next slide)

• Plots show characteristic logistic shape
• for heavy dependence on ?
• Plots are more linear when ? is near zero
• All plots are monotonic
• Wide range of shapes possible

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Frailty Models based on the GTDL   Ø
Multiplicative Frailty
        Ø Random Effect Model  
    where typically u g(u
?2) with E(U) 1 and V(U) ?2 and g (.) is
Gamma or Log-Normal.
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GTDL Scale of the Random Effect  
Ø                Multiplicative Frailty
    Ø Random Effect Model    

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Hierarchical Likelihood(s)   Thus, the
contribution over all n individuals leads
to But the h - Likelihood of Lee
Nelder is ie, now we consider the r.e.
v loge(u) rather than u !!
 
 
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The Survival Analysis Form  
Clearly the kernel for the first member is
where,
Is the conditional cumulative hazard
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h - likelihood Estimation   We use
inter-dependent up-dating algorithm   Step 1
Solve for ? (?,?,u) Location r. effects  
Step 2 Solve for ? Dispersion using
APHL.
 
Where
Penalty
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h - likelihood Estimation

• Ø   Think of hp (?) as a nuisance parameter
• elimination device (eg, when ? ? )
• When ? is a fixed effect, using the penalty
• is equivalent to conditioning (Cox-Reid,
• 1987)
• Ø When ? is a random effect it is equivalent
• to integrating the random effect out (Ist
• order Laplace) !

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Marginal Inference (by comparison).  
Ø Marginal Method

 
  Ø PH GTDL Gamma Frailty -gt closed-form
marginals. Ø MHLEs of ?, ? are
same as marginal estimators Ø m approach gt
? for other g(u) - eg Log Normal!
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• Examples
•  
• Ø Data Sets
•     Kidney Infection Data (McGilchrist
  Aisbett,
• 1991), 38 patients, gaps to 1st, 2nd
  recurrence
• only.
•         CGD data (Fleming Harrington,
  1991), 128
• Patients, times to 0 to 8 re-infections.
  
•  
• Ø Models Fitted
• M1 - Cox Model
• M2 - Cox LN Frailty
• M3 GTDL
• M4 GTDL LN Frailty
• M5 GTDL Random Effect

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• Kidney Infection Data
• All models give the same interpretation for the
  Sex effect
• 2. All non-Frailty models underestimate the
• size of the effect and its standard error
• 3. The non-PH frailty models suggest that
• the magnitude of the effect is larger than
• suggested by the PH models
• 4. However, here ? is non-significant - so we may
  prefer a PH interpretation
• 5. Non-PH frailty variances are 30 higher
  suggesting more heterogeneity

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• CGD Data
• All models give the same interpretation for the ?
  Interferon effect most other xs.
• 2. All non-Frailty models underestimate the
• standard errors throughout.
• 3. Here ? is definitely significant so model is
  non-PH, but the magnitude is so small as to be
  irrelevant. May prefer Non-PH in this case
• 4. Frailty variances are similar

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Remarks on Data Analysis 1. Frailty components
required by formal testing in both data
sets 2. No integration involved in these
analyses 3. CGD data analysed by full Bayesian
methods at IWSM (2004) same treatment
effect and se. 4. Open question on comparison of
semi- parametric frailty model with
parametric using h-likelihood.
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Conclusions   Ø NON PH models required for
MV Survival Data   Ø GTDL CTDL (non-PH)
flexible family   Ø Log-Normal Frailty
Random Effect Extensions   Ø Closed marginal
forms for Gamma Frailty   Ø Marginal
Likelihood requires ? for other g(u)   Ø
Thus h - likelihood can replace MCMC.
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\item Blagojevic, M., MacKenzie, G. and Ha,
I.D. (2003) A comparison of non-PH and PH gamma
frailty models. \textitProceedings of the 18th
International Workshop on Statistical Modelling.
Leuven, Belgium, 39-44. \item Chernoff, H.
(1954) On the distribution of the likelihood
ratio. \textitAnn. Math. Statist., \textbf25,
573-578. \vspace-3mm \item Cox, D. R.
(1970) \it The Analysis of Binary Data. London
Methuen. \vspace-3mm \item Cox, D. R.
(1972) Regression models and life-tables (with
discussion). \textitJ. R. Statist. Soc.
\textbfB, \textbf34, 187-220. \vspace-3mm
\item Fleming, T. R. and Harrington, D. P.
(1991) \textitCounting processes and survival
analysis. New York Wiley. \vspace-3mm \item
Ha, I. D. and Lee, Y. (2003) Estimating
frailty models via Poisson hierarchical
generalized linear models. \it Journal of
Computational and Graphical Statistics,
\textbf12, 663-681. \vspace-3mm \item Ha,
I. D. and Lee, Y. (2004) Multilevel mixed linear
models for survival data. \it Lifetime Data
Analysis, in press. \vspace-3mm \item Ha,
I. D., Lee, Y. and Song, J.-K. (2001)
Hierarchical likelihood approach for frailty
models. \textitBiometrika, \textbf88, 233-243.
\vspace-3mm \item Hougaard, P. (2000)
\textitAnalysis of multivariate survival
Data. New York Springer-Verlag. \vspace
-3mm \item Lee, Y. and Nelder, J. A. (1996)
Hierarchical generalized linear models (with
discussion). \textitJ. R. Statist. Soc.
\textbfB, \textbf58, 619-678.
\vspace-3mm \item Lee, Y and Nelder, J. A.
(2001) Hierarchical generalised linear models a
synthesis of generalised linear models,
random-effect models and structured dispersions.
\textitBiometrika, \bf 88, 987-1006.
\vspace-3mm \item MacKenzie, G. (1996)
Regression models for survival data the
generalised time dependent logistic family.
\textitJ. R. Statist. Soc. \textbfD,
\textbf45, 21-34. \vspace-3mm \item
MacKenzie, G. (1997) On a non-proportional
hazards regression model for repeated medical
random counts. \textitStatist. Med.,
\textbf16, 1831-1843. \vspace-3mm \item
McGilchrist, C. A. and Aisbett, C. W. (1991)
Regression with frailty in survival
analysis. \textitBiometrics, \textbf47,
461-466. \vspace-3mm \item Self, S. G. and
Liang, K. Y. (1987) Asymptotic properties of
maximum likelihood estimators and likelihood
ratio tests under nonstandard conditions.
\textitJournal of the American
Statistical Association, \textbf82, 605-610.
\vspace-3mm \item Stram, D. O. and Lee, J.
W. (1994) Variance components testing in
the longitudinal mixed effects model.
\textitBiometrics, \textbf50, 1171-1177,
1994. \vspace-3mm \item Vu, H. T. V. and
Knuiman, M. W. (2002) A hybrid ML-EM algorithm
for calculation of maximum likelihood estimates
in semiparametric shared frailty models.
\textitComputational Statistics and
Data Analysis \textbf40, 173-187.
\vspace-3mm \item Yau, K. K. W. and
McGilchrist, C. A. (1998) ML and REML estimation
in survival analysis with time dependent
correlated frailty. \textitStatist.
Med., \textbf17, 1201-1213. \vspace-3mm
Key References
\item Blagojevic, M., MacKenzie, G. and Ha,
I.D. (2003) A, comparison of non-PH and PH gamma
frailty models.\textitProceedings of the 18th
International Workshop on Statistical Modelling.
Leuven, Belgium, 39-44. \item Ha, I. D. and
Lee, Y. (2003) Estimating frailty models, via
Poisson hierarchical generalized linear models.
\it Journal of Computational and Graphical
Statistics, \textbf12, 663-681, \item Ha,
I. D. and Lee, Y. (2004) Multilevel mixed linear
models for survival data. \it Lifetime Data
Analysis, in press. \item Ha, I. D., Lee, Y.
and Song, J.-K. (2001) Hierarchical likelihood
Approach for frailty models. \textitBiometrika,
\textbf88, 233-243. \item Lee, Y. and
Nelder, J. A. (1996) Hierarchical generalized
linear, models (with discussion). \textitJ. R.
Statist. Soc. \textbfB,\textbf58, 619-678.
\item Lee, Y and Nelder, J. A. (2001)
Hierarchical generalised linear models a
synthesis of generalised linear models,
random-effect models and structured dispersions.
\textitBiometrika, \bf 88, 987-1006. \item
MacKenzie, G. (1996) Regression models for
survival data the generalised time dependent
logistic family. \textitJ. R. Statist. Soc.
\textbfD, \textbf45, 21-34. \item
MacKenzie, G. (1997) On a non-proportional
hazards regression model for repeated medical
random counts.\textitStatist. Med.,
\textbf16, 1831-1843. \item MacKenzie, G.
(1986) A Proportional hazards model for accident
data .\textitJRSS A, \textbf149,4 , 366-375.