empirical likelihood with arbitrary censored/truncated data by constrained em algorithm
DESCRIPTION
Empirical Likelihood with Arbitrary Censored/Truncated Data by Constrained EM Algorithm. Min Chen, Jingyu Luan, Mai Zhou Department of Statistics University of Kentucky 817 Patterson Office Tower Lexington, KY 40506 [email protected]. Outline. Introduction - PowerPoint PPT PresentationTRANSCRIPT
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Empirical Likelihood with Arbitrary Censored/Truncated Data by
Constrained EM Algorithm
Min Chen, Jingyu Luan, Mai ZhouDepartment of StatisticsDepartment of StatisticsUniversity of KentuckyUniversity of Kentucky
817 Patterson Office Tower817 Patterson Office TowerLexington, KY 40506Lexington, KY [email protected]@ms.uky.edu
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Outline
• Introduction• Maximize Empirical Likelihood under parameters
of weighted hazard• Maximize Empirical Likelihood under parameters
of weighted hazards for data with covariates• Examples and Numerical results • Conclusion
we can compute empirical likelihood ratio for arbitrary censored/truncated data with/without covariates under weighted hazard parameter.
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (1)
• Empirical Likelihood Ratio Method– Empirical Distribution for n iid observations
– Empirical Likelihood function
– Empirical Likelihood Ratio
[ ]1
1( )
i
n
n X xi
F x In
1
( ) ( ) ( )n
i ii
L F F X F X
( )( )
( )n
L FR F
L F
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (1)
– Empirical Likelihood ratio Function by Owen(1988)
( ) sup ( ) | ( ) ( ) , nF
R R F g x dF x F F
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (2)
• Arbitrary Censored/Truncated Data– According to Turnbull (1976) and Frydman (1994),
any observation in an arbitrary censored/Truncated Data can be described by , with ,
where
And we associate each observation with , the covariate, if any.
( , )i iA B i iA B
1
[ , ]ik
i ij ijj
A L R
1
( , )il
i ij ijj
B V U
iZ
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (2)
For example• Any exact observation without covariate could be
described as
• Any right censored observation with one covariate could be described as
[ , ], ( , )i i i iA x x B
[ , ), ( , ),i i i i iA x B Z z
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (2)
The likelihood for the arbitrary censored and/or
truncated data is proportion to
1
1
1
{ ( ) ( )}
( ) ( )
{ ( ) ( )}
i
i
k
ij ijNj
li
ij ijj
F R F L
L F
F U F V
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (3)
Based on Turnbull (1976) and Alioum (1996), for
arbitrary censored/truncated data, we could
construct a set such that
• Any c.d.f. jumps outside of C could not be MLE of the unknown distribution function
• The likelihood is independent of the behavior of the distribution inside each interval .
1
[ , ]m
j jj
C q p
[ , ]j jq p
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Introduction (3)
Now the problem of maximizing emprical likelihood
reduces to maximize
The MLE of can be obtained by self-
Consistency/EM algorithm.
*1
1 11
( ,..., ) ( / )N m m
m ij j ij jj ji
L s s s s
1,..., ms s
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Maximum of Empirical Likelihood under parameters of weighted hazard
• The empirical log likelihood for , i=1,…,N is proportional to
• and could be written in terms of hazards as
*
1 1 1
log ( ) [log log ]N m m
ij j ij ji j i
L s s s
( , )i iA B
: :*1
1 1 1
log ( ,..., ) [log log ]k kk t t k t tk j k j
N m mp p
m ij j ij ji j j
L p p p e p e
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Maximum Empirical Likelihood under parameters of weighted hazard
• Under hazard constraint , we may think of using the Lagrange Multiplier method to find the maximum log likelihood, but it turns out to be intractable.
( ) ( )g t dH t
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Maximum Empirical Likelihood under parameters of weighted hazard
• Modified self-consistency/EM algorithm
E-Step: Given current Estimate of H(.), we can compute a weight on each pseudo jump point of H(.).
M-Step: For the pseudo jump points associated with weights in E-Step, we could compute a new hazard jump.
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Maximum Empirical Likelihood under parameters of weighted hazard
Theorem 1
Under hazard type constraint ,
the NPMLE of hazard jumps obtained by the modified EM algorithm for the arbitrary censored and/or truncated data is equivalent to the solution by Lagrange Multiplier Method.
( ) ( )g t dH t
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Empirical ML under parameters of weighted hazards for data with covariates
• For arbitrary censored/truncated data with covariates, by Cox proportional hazards regression model, the log likelihood could be written as
: :*1
1 1 1
log ( ,..., , ) (log log )z zi i
k kk t t k t tk j k j
N m me p e p
m ij j ij ji j j
L p p p e p e
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Empirical ML under parameters of weighted hazards for data with covariates
• Under hazard constraint , and hypothesis about , obtain the maximum log likelihood by using Lagrange Multiplier method is even more complicate.
( ) ( )g t dH t
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Empirical ML under parameters of weighted hazards for data with covariates
Theorem 2
Under hazard type constraint, the NPMLE of hazard jumps obtained by the modified EM algorithm for the arbitrary censored and/or truncated data with covariates are equivalent to the solution by Lagrange Multiplier Method.
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Examples and Numerical results (1)
Left Truncated Right
Censored Data without
covariates under hazard
constraint
The maximum log
likelihood is achieved at
=0.33
[ 960] ( )tI dH t
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Examples and Numerical results (1)
Left Truncated Right
Censored Data without
covariates under hazard
constraint
The maximum log
likelihood is achieved at
=0.71
[ 1020] ( )tI dH t
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Examples and Numerical results (2)
Right Censored Data
with one covariate
under hazard constraint
The maximum log
likelihood is achieved at
=-0.01
[ 95] ( ) 2.05tI dH t
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Examples and Numerical results (3)
Interval Censored Data
with one covariate and
no hazard constraint
The maximum log
likelihood is achieved at
=0.1
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
Conclusion
we can compute empirical likelihood ratio for arbitrary censored/truncated data with/without covariates under weighted hazard parameter.
2004 ENAR Spring Meeting, Pittsburgh, PA2004 ENAR Spring Meeting, Pittsburgh, PA
References
• Alioum A. and Commenges D. (1996) A proportional Hazards Model for Arbitrarily Censored and Truncated Data. Biometrics, 52, 512-524.
• Cox, D.R. (1972) Regression models and life tables (with discussion). J. of the Royal Statistical Society, Series B, 34, 187-220.
• Frydman, H. (1994) A note on nonparametric estimation of the distribution function from interval-censored and truncated observations. Journals of the Royal Statistical Society, Series B, 56, 71-74.
• Gentleman, R. and Ihaka, R. (1996) R: A Language for data analysis and graphics. J. of Computational and Graphical Statistics, 5, 299-314.
• Luan J.Y., Chen M. and Zhou M. (2003) Empirical Likelihood Ratio with Right Censoring and Left Truncation Data. Technical Report.
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References
• Klein and Moeschberger (1997) Survival Analysis: Techniques for Censored and Truncated Data. Springer, New York.
• Owen, A. (2001) Empirical Likelihood. Chapman \& Hall. London.• Pan, X.R. and Zhou, M. (1999). Using one parameter sub-family of
distributions in empirical likelihood with censored data. J.Statist. Planning and Infer. 75, 379-392.
• Thomas, D. R. and Grunkemeier, G.L. (1975). Confidence Interval estimation of survival probabilities for censored data. Amer. Statist. Assoc. 70, 865-871.
• Turnbull B, The empirical distribution function with arbitrary grouped, censored and truncated data. JRSS B, 290-295.
• Zhou M. (2003). Empirical likelihood ratio with arbitrary censored/truncated data by EM algorithm. Technical Report.