intELtest: The integrated likelihood ratio test

In EL2Surv: Empirical Likelihood (EL) for Comparing Two Survival Functions

Description

intELtest gives a class of the weighted likelihood ratio statistics:

∑_{t\in U}w(t)\{-2\log R(t)\},

where w(t) is an objective weight function, and R(t) is an empirical likelihood (EL) ratio that compares two survival functions at each time point t in the set of observed uncensored lifetimes, U.

Usage

intELtest(data, g1 = 1, t1 = 0, t2 = Inf, sided = 2,
  nboot = 1000, wt = "p.event", alpha = 0.05, compo = FALSE,
  seed = 1011, nlimit = 200)

Arguments

data	a data frame/matrix with 3 columns. The first column is the survival time. The second is the censoring indicator. The last is the grouping variable. An example as the input to data provided is hepatitis.
g1	the group with longer survival in one-sided testing with the default value of 1.
t1	pre-specified t_1 based on domain knowledge with the default value of 0
t2	pre-specified t_2 based on domain knowledge with the default value of ∞
sided	2 if two-sided test, and 1 if one-sided test. It assumes the default value of 2.
nboot	number of bootstrap replications in calculating critical values with the defualt value of 1000.
wt	a string for the integral statistic with a specific weight function. There are four types of integral statistics provided: "p.event", "dF", "dt", and "db". It assumes the default value of "p.event". See 'Details' for more about the integral statistics.
alpha	pre-specified significance level of the test with the default value of 0.05
compo	FALSE if taking the standardized square of the difference as the local statisic for two-sided testing, and TRUE if constructing for one-sided testing, but only the positive part of the difference included. It assumes the default value of FALSE.
seed	the parameter with the default value of 1011 to set.seed for generating bootstrap-based critical values in R. The set.seed is used implicitly in intELtest.
nlimit	the splitting unit with the default value of 200. To deal with large data problems, the bootstrap algorithm is to split the number of bootstrap replicates into nsplit parts. The number nsplit is the smallest integer not less than ≤ft\| U\right\|/nlimit.

Details

intELtest calculates the weighted likelihood ratio statistics:

∑_{i=1}^{h}w_i\cdot \{-2\log R(t_i)\},

where w_1,...,w_h are the values of the weight function evaluated at the distinct ordered uncensored times t_1,...,t_h in U. There are four types of weight functions considered.

• (wt = "p.event")
  This default option is an objective weight,

  w_i=\frac{d_i}{n}

  In other words, this w_i assigns weight proportional to the number of events at each observed uncensored time t_i.

• (wt = "dF")
  Based on the integral statistic built by Barmi and McKeague (2013), another weigth function is

  w_i= \hat{F}(t_i)-\hat{F}(t_{i-1})

  for i=1,…,m,where \hat{F}(t)=1-\hat{S}(t), \hat{S}(t) is the pooled KM estimator, and t_0 \equiv 0. This reduces to the objective weight when there is no censoring.The resulting I_n can be seen as an empirical version of E(-2\log\mathcal{R}(T)), where T denotes the lifetime random variable of interest distributed as the common distribution under H_0.

• (wt = "dt")
  By means of an extension of the integral statistic derived by Pepe and Fleming (1989), another weight function is

  w_i= t_{i+1}-t_i

  for i=1,…,m, where t_{m+1} \equiv t_{m}. This gives more weight to the time intervals where there are fewer observed uncensored times, but may be affected by extreme observations.

• (wt = "db")
  According to a weigthing method mentioned in Chang and McKeague (2016), the other weight function is

  w_i= \hat{b}(t_i)-\hat{b}(t_{i-1})

  where \hat{b}(t)=\hat{σ}^2(t)/(1+\hat{σ}^2(t)), and \hat{σ}^2(t) is given. The \hat{b}(t) is chosen so that the limiting distribution is the same as the asymptotic null distribution in EL Barmi and McKeague (2013).

Value

intELtest returns a list with three elements:

• teststat the resulting integrated test statistic

• critval the critical value

• pvalue the p-value based on the integrated statistic

References

• H.-w. Chang and I. W. McKeague, "Empirical likelihood based tests for stochastic ordering under right censorship," Electronic Journal of Statistics, Vol. 10, No. 2, pp. 2511-2536 (2016).

• M. S. Pepe and T. R. Fleming, "Weighted Kaplan-Meier Statistics: A Class of Distance Tests for Censored Survival Data," Biometrics, Vol. 45, No. 2, pp. 497-507 (1989). https://www.jstor.org/stable/2531492?seq=1#page_scan_tab_contents

• H. Uno, L. Tian, B. Claggett, and L. J. Wei, "A versatile test for equality of two survival functions based on weighted differences of Kaplan-Meier curves," Statistics in Medicine, Vol. 34, No. 28, pp. 3680-3695 (2015). http://onlinelibrary.wiley.com/doi/10.1002/sim.6591/abstract

• H. E. Barmi and I. W. McKeague, "Empirical likelihood-based tests for stochastic ordering," Bernoulli, Vol. 19, No. 1, pp. 295-307 (2013). https://projecteuclid.org/euclid.bj/1358531751

See Also

hepatitis, supELtest, ptwiseELtest

Examples

library(EL2Surv)
intELtest(hepatitis)

## OUTPUT:
## $teststat
## [1] 1.406016
## 
## $critval
## [1] 0.8993514
## 
## $pvalue
## [1] 0.012

EL2Surv documentation built on May 1, 2019, 11:01 p.m.