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Empirical Likelihood

Empirical Likelihood. Mai Zhou Dept. of Statistics, University of Kentucky. Any first year Statistical Inference course will talk about (parametric) “likelihood function”. Three inference methods (tests) based on likelihood: 1. Wald test

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Empirical Likelihood

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  1. Empirical Likelihood Mai Zhou Dept. of Statistics, University of Kentucky

  2. 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 (Rao’s Score test) 3. Likelihood ratio test (Wilks)

  3. Empirical likelihood is a nonparametric version of 3.

  4. 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.

  5. 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.

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

  7. EL(F) is maximized by the empirical distribution function:

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

  9. Censored Observations • For a right censored observation • The likelihood contribution is • For a left censored observation the contribution is • Interval censored:

  10. Truncated observations For a left truncated observation (often referred toas delayed entry) : (entry time, survival time) = • The likelihood contribution is • If the survival time is also right censored, then the likelihood contribution is

  11. Empirical Likelihood Theorem: • If the null hypothesis is true then

  12. 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.

  13. Tests hypothesis of the form: with right, left, doubly censored data. Or with left-truncated, right censored data.

  14. 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 )

  15. > library(emplik) > el.ltrc.EM( y, x, d, mu=62) The mean of the NPMLE is 63.18557. • (if ‘fun’ is left out, then fun=t, by default). Two of the outputs are -2LLR = 0.2740571 Pval = 0.6006231

  16. Repeat the test for many different values of the mean. (mu=59, 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]

  17. 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.)

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