intELtest gives a class of integrated EL statistics:
$$\sum_{i=1}^{m}w_i\cdot \{-2\log R(t_i)\},$$
where \(R(t)\) is the EL ratio that compares the survival functions at each given time \(t\),
\(w_i\) is the weight at each \(t_i\), and \( 0<t_1<\ldots<t_m<\infty\) are the (ordered)
observed uncensored times at which the Kaplan--Meier estimate is positive and less than \(1\) for
each sample.
intELtest(
formula,
data = NULL,
group_order = NULL,
t1 = 0,
t2 = Inf,
sided = 2,
nboot = 1000,
wt = "p.event",
alpha = 0.05,
seed = 1011,
nlimit = 200
)a formula object with a Surv object as the response on the left of the ~ operator and
the grouping variable as the term on the right. The Surv object involves two variables: the observed survival
and censoring times, and the censoring indicator, which takes a value of \(1\) if the observed time is uncensored
and \(0\) otherwise. The grouping variable takes different values for different groups.
an optional data frame containing the variables in the formula: the observed survival and censoring times,
the censoring indicator, and the grouping variable. If not found in data, the variables in the formula should
be already defined by the user or in attached R objects. The default is the data frame with three columns of
variables taken from the formula: column 1 contains the observed survival and censoring times, column 2 the censoring
indicator, and column 3 the grouping variable.
a \(k\)-vector containing the values of the grouping variable, with the \(j\)-th element being the group hypothesized to have the \(j\)-th highest survival rates, \(j=1,\ldots,k\). The default is the vector of sorted grouping variables.
the first endpoint of a prespecified time interval, if any, to which the comparison of the survival functions is restricted. The default value is \(0\).
the second endpoint of a prespecified time interval, if any, to which the comparison of the survival functions is restricted. The default value is \(\infty\).
\(2\) if two-sided test, and \(1\) if one-sided test. The default value is \(2\).
the number of bootstrap replications in calculating critical values for the tests. The default value is \(1000\).
the name of the weight for the integrated EL statistics:
"p.event", "dF", or "dt". The default is "p.event".
the pre-specified significance level of the tests. The default value is \(0.05\).
the seed for the random number generator in R, for generating bootstrap samples needed
to calculate the critical values for the tests. The default value is \(1011\).
a number used to calculate nsplit= \(m\)/nlimit, the number of
parts into which the calculation of the nboot bootstrap replications is split.
The use of this variable can make computation faster when the number of time points \(m\) is large.
The default value for nlimit is 200.
intELtest returns a intELtest object, a list with 15 elements:
call the function call
teststat the resulting integrated EL statistics
critval the critical value based on bootstrap
pvalue the p-value of the test
formula the value of the input argument of intELtest
data the value of the input argument of intELtest
group_order the value of the input argument of intELtest
t1 the value of the input argument of intELtest
t2 the value of the input argument of intELtest
sided the value of the input argument of intELtest
nboot the value of the input argument of intELtest
wt the value of the input argument of intELtest
alpha the value of the input argument of intELtest
seed the value of the input argument of intELtest
nlimit the value of the input argument of intELtest
Methods defined for intELtest objects are provided for print and summary.
There are three options for the weight \(w_i\):
(wt = "p.event")
This default option is an objective weight,
$$w_i=\frac{d_i}{n},$$
which assigns weight proportional to the number of events \(d_i\)
at each observed uncensored time \(t_i\). Here \(n\) is the total sample size.
(wt = "dF")
Inspired by the integral-type statistics considered in 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 the expected negative two times log EL ratio under \(H_0\).
(wt = "dt")
Inspired by the integral-type statistics considered in 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 can be affected by extreme observations.
H. Chang, I.W. McKeague, "Nonparametric testing for multiple survival functions with non-inferiority margins," Annals of Statistics, Vol. 47, No. 1, pp. 205-232, (2019).
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. 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
hepatitis, supELtest, ptwiseELtest, nocrossings, print.intELtest, summary.intELtest
# NOT RUN {
library(survELtest)
intELtest(survival::Surv(hepatitis$time, hepatitis$censor) ~ hepatitis$group)
## OUTPUT:
## Call:
## intELtest(formula = survival::Surv(hepatitis$time, hepatitis$censor) ~
## hepatitis$group)
##
## Two-sided integrated EL test statistic = 1.42, p = 0.007
# }
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