intELtest gives a class of the weighted likelihood ratio statistics:
$$\sum_{t\in U}w(t)\{-2\log R(t)\},$$
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, \(U\).
intELtest(data, g1 = 1, t1 = 0, t2 = Inf, sided = 2,
nboot = 1000, wt = "p.event", alpha = 0.05, compo = FALSE,
seed = 1011, nlimit = 200)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.
the group with longer survival in one-sided testing with the default value of \(1\).
pre-specified \(t_1\) based on domain knowledge with the default value of \(0\)
pre-specified \(t_2\) based on domain knowledge with the default value of \(\infty\)
2 if two-sided test, and 1 if one-sided test. It assumes the default value of \(2\).
number of bootstrap replications in calculating critical values with the defualt value of \(1000\).
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.
pre-specified significance level of the test with the default value of \(0.05\)
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 \(FALSE\).
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.
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 \(\left\| U\right\|/\)nlimit.
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
intELtest calculates the weighted likelihood ratio statistics:
$$\sum_{i=1}^{h}w_i\cdot \{-2\log R(t_i)\},$$
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 \(i=1,\ldots,m\),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 \(i=1,\ldots,m\), 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 \(\hat{b}(t)=\hat{\sigma}^2(t)/(1+\hat{\sigma}^2(t))\), and \(\hat{\sigma}^2(t)\) 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).
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
# NOT RUN {
library(EL2Surv)
intELtest(hepatitis)
## OUTPUT:
## $teststat
## [1] 1.406016
##
## $critval
## [1] 0.8993514
##
## $pvalue
## [1] 0.012
# }
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