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

2. Any first year Statistical Inference course will talk about “likelihood”. For independent observations the likelihood is Notice f(a)dx = P( X=a )

3. Three inference methods (tests) based on likelihood: 1. Wald test 2. Score test (Rao’s Score test) 3. Likelihood ratio test (Wilks) 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. A new (2001) book by A. Owen “Empirical Likelihood”. But in the 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 the Empirical Likelihood Ratio. (Pan 1997)

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

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

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

9. 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 right censored, then the likelihood contribution is

10. Empirical Likelihood Theorem: If the null hypothesis is true then if the hypothesis is not true, the quantity above will be larger.

11. R = Gnu S/Splus http://cran.us.r-project.org http://cran.us.r-project.org + many add-on packages + 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 mean or weighted hazard, based on left-truncated, right censored or left, right, doubly censored data.

12. 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) Survival Analysis 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 [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 computing. (E-M algorithm etc.)