Empirical%20Likelihood%20for%20Right%20Censored%20and%20Left%20Truncated%20data

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Empirical Likelihood for Right Censored and Left
Truncated data

• Jingyu (Julia) Luan
• University of Kentucky, Johns Hopkins University
• March 30, 2004

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Outline of the Presentation

• Part I Introduction and Background
• Part II Empirical Likelihood Theorem for
  Right-Censored and Left-Truncated Data
• Part III Future Research

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Part I Introduction and Background

• 1.1 Empirical Likelihood Ratio Test
• 1.2 Censoring and Truncation
• 1.3 Literature Review
• 1.4 Counting Process and Survival Analysis

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1.1 Empirical Likelihood Ratio Test
Likelihood Ratio Test

• Two cases
• Parametric
• Non-parametric

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1.1 Empirical Likelihood Ratio Test

• Parametric situation
• Wilks (1938) -2logR has an asymptotic ?p2
  distribution under null hypothesis..

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1.1 Empirical Likelihood Ratio Test

• Nonparametric situation
• Empirical Likelihood is defined as (Owen 1988)

Where X1, X2, , Xn are independent random
variables from an unknown distribution F. And
it is well known that L(F) is maximized by the
empirical distribution function Fn over all
possible CDFs, where
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1.1 Empirical Likelihood Ratio Test

• Owen focused on studying the properties of the
  likelihood ratio function when F0 satisfies
  certain constraint, (the null hypothesis)

for a given g(.).
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1.1 Empirical Likelihood Ratio Test

• Owen defined the empirical likelihood ratio
  function as

where L(F?) is the maximized empirical
likelihood function among all CDFs satisfying the
above constraint (null hypothesis). Owen (1988)
demonstrated when ??0, -2logR(F?) has an
asymptotic ?2 distribution with df1.
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1.2 Censoring and Truncation

• Survival analysis is the analysis of
  time-to-event data.
• Two important features of time-to-event data
• A. Censoring
• B. Truncation

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1.2 Censoring and Truncation

• Censoring occurs when an individuals life length
  is known to happen only in a certain period of
  time.
• A. Right Censoring
• B. Left Censoring
• C. Interval Censoring

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1.2 Censoring and Truncation

• Truncation
• A. Left Truncation it occurs when subjects
  enter a study at a particular time and are
  followed from this delayed entry time until the
  event happens or until the subject is censored
• B. Right Truncation it occurs when only
  individuals who have experienced the event of
  interest are included in the sample.

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1.2 Censoring and Truncation

• Example of Right-Censored and Left-Truncated
  data Klein, Moeschberger, p65
• In a survival study of the Channing House
  retirement center located in California, ages at
  which individuals entered the retirement
  community (truncation event) and ages when
  members of the community died or still alive at
  the end of the study (censoring event) were
  recorded.

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1.3 Literature Review

• Product limit estimator (based on right censored
  and left truncated data) of survival function an
  analogue of the Kaplan-Meier estimator of
  survival function under censoring
• For solely truncation data, Wang and Jewell
  (1985) and Woodroofe (1985) independently proved
  consistency results for the product limit
  estimator and showed weak convergence to a
  Brownian Motion process

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1.3 Literature Review

• Wang, Jewell, and Tsai (1987) described a
  description of the asymptotic behavior of the
  product limit estimator for right censoring and
  left truncation data
• For pure truncated data, Li (1995) studied the
  empirical likelihood theorem.

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1.3 Literature Review

• For solely censoring data
• Pan and Zhou (1999) showed that the empirical
  likelihood ratio with continuous mean and hazard
  constraint also have a chi-square limit.
• Fang (2000) proved the empirical likelihood ratio
  with discrete hazard constraint follows a
  chi-square distribution under one sample case and
  two sample case.

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1.4 Counting Process and Survival Analysis

• Counting process provides an elegant martingale
  based approach to study time-to-event data.
• Martingale methods can be used to obtain simple
  expressions for the moments of complicated
  statistics and to calculate and verify
  asymptotic distributions for test statistics and
  estimators.

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Part II Empirical Likelihood Theorem for
Right-Censored and Left-Truncated Data

• 2.1 One Sample Case
• 2.2 Two Sample Case

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Part III One Sample Case

• First, introduce some notations

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Part III One Sample Case

• Then, the likelihood function is

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Part III One Sample Case
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Part III One Sample Case
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Part III One Sample Case
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Part III Two Sample Case
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Part III Future research

• The same setting as one sample case. Under k
  constraints

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Part III Future research

• Let

be k observed samples.
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Part III Future research
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Part III Future Research

• Owen (1991) has demonstrated that empirical
  likelihood ratio can be applied to regression
  models.
• However, for censored/truncated data the
  empirical likelihood results are rare. Exception
  Li (2002).
• Notice we are talking about ordinary regression
  model, not the Cox proportional hazards
  regression model.

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Part III Regression models

• We propose a redistribution algorithm of
  estimating the parameters in the regression
  models with censored/truncated data.
• We think the Empirical likelihood method can be
  used there to do inference.

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Reference
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Reference
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Acknowledgement

• Dr. Mai Zhou
• Dr. Arne Bathke
• Dr. William Griffith
• Dr. William Rayens
• Dr. Rencang Li