Empirical Likelihood

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.

15. 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 )

16. > 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

17. 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]

18. 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.)