Empirical Likelihood

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Empirical Likelihood

• Mai Zhou
• Dept. of Statistics, University of Kentucky

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• Any first year Statistical Inference course will
  talk about (parametric) likelihood function.
• Three inference methods (tests) based on
  likelihood
• 1. Wald test
• 2. Score test (Raos Score test)
• 3. Likelihood ratio test (Wilks)

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• Empirical likelihood is a nonparametric version
  of 3.

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• Empirical Likelihood allows the statistician to
  employ likelihood methods, without having to pick
  a parametric family of distributions for the
  data. --- Owen
• Empirical Likelihood allows for hypothesis
  testing and confidence region construction
  without an information/variance estimator.-- me
• Plus many additional nice properties.

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• First book on this subject (2001) by A. Owen
  Empirical Likelihood .
• But in Cox model the (partial) likelihood ratio
  exists for a long time (over 20 years). SAS proc
  phreg, Splus function coxph( ) all have it
  computed.
• Claim The (partial) likelihood ratio statistic
  for the regression coefficients in the Cox model
  can be interpreted as a case of Empirical
  Likelihood Ratio.

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• For n observations,
  
• independent, from the
  empirical likelihood is
• EL(F)
• Where

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• EL(F) is maximized by the empirical distribution
  function

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• An additional parameter of interest, when
  maximizing the EL(F)
• F(t) or can be considered as
  nuisance parameters

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Censored Observations

• For a right censored observation
• The likelihood contribution is
• For a left censored observation the contribution
  is
• Interval censored

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Truncated observations

• For a left truncated observation (often
  referred to as delayed entry)
• (entry time, survival time)
• The likelihood contribution is
• If the survival time is also right censored, then
  the likelihood contribution is

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• Empirical Likelihood Theorem
• If the null hypothesis is true then

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R Gnu S/Splus http//cran.us.r-project.org
many add-on packages A
Package for empirical likelihood with
censored/truncated data

• Contributed package emplik (maintained by
  Mai Zhou)
• It Does Empirical likelihood ratio tests for
  means or weighted hazard, based on
  left-truncated, right censored or left, right,
  doubly censored data.

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Tests hypothesis of the form
with right, left, doubly censored data. Or with
left-truncated, right censored data.
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• Example Data taken from Klein Moeschberger
  (1997) book as reported in their table 1.7
• y left truncation time
• (51, 58, 55, 28, 25, 48, 47, 25, 31, 30, 33,
  43, 45, 35, 36)
• x survival times of female psychiatric
  inpatients
• (52, 59, 57, 50, 57, 59, 61, 61, 62, 67, 68,
  69, 69, 65, 76)
• d censoring status
• ( 1, 1, 1, 1, 1, 1, 1, 1, 0,
  0, 0, 1, 1, 0, 1 )

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• gt library(emplik)
• gt el.ltrc.EM( y, x, d, mu62)
• The mean of the NPMLE is 63.18557.
• (if fun is left out, then funt, by default).
  Two of the outputs are
• -2LLR 0.2740571
• Pval 0.6006231

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• Repeat the test for many different values of the
  mean. (mu59, etc. )
• If the hypothesized mean is inside the interval
• 58.78936, 67.81304, the p-value of the test is
  larger then 0.05. ----- the 95 confidence
  interval for the mean is
• 58.78936, 67.81304

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• For doubly censored data, the standard deviation
  of the NPMLE is hard to compute.
• The Wald test/confidence interval is hard to do.
• No problem with empirical likelihood ratio!
• No need to estimate the standard deviation,
  instead, we need to maximize EL under some
  constraint.
• The maximization can be achieved with the help of
  modern computer. (E-M algorithm etc.)