supELtest {survELtest}

The maximally selected EL test

Description

supELtest provides the maximally selected EL statistics that is better adapted at detecting local differences:

\sup_{i=1,\ldots,m}\{-2\log R(t_i)\},

where R(t) is the EL ratio that compares the survival functions at each given time t, 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.

Usage

supELtest(
  formula,
  data = NULL,
  group_order = NULL,
  t1 = 0,
  t2 = Inf,
  sided = 2,
  nboot = 1000,
  alpha = 0.05,
  seed = 1011,
  nlimit = 200
)

Arguments

formula

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.

data

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.

group_order

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.

t1

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.

t2

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.

sided

2 if two-sided test, and 1 if one-sided test. The default value is 2.

nboot

the number of bootstrap replications in calculating critical values for the tests. The default value is 1000.

alpha

the pre-specified significance level of the tests. The default value is 0.05.

seed

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.

nlimit

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.

Value

supELtest returns a supELtest object, a list with 14 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 supELtest

• data the value of the input argument of supELtest

• group_order the value of the input argument of supELtest

• t1 the value of the input argument of supELtest

• t2 the value of the input argument of supELtest

• sided the value of the input argument of supELtest

• nboot the value of the input argument of supELtest

• alpha the value of the input argument of supELtest

• seed the value of the input argument of supELtest

• nlimit the value of the input argument of supELtest

Methods defined for supELtest objects are provided for print and summary.

References

• H. Chang, 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).

• 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).

See Also

hepatitis, intELtest, ptwiseELtest, nocrossings, print.supELtest, summary.supELtest

Examples

library(survELtest)
nocrossings(survival::Surv(hepatitis$time, hepatitis$censor)~
    hepatitis$group, sided = 1)

## OUTPUT:
## Call:
## nocrossings(formula = survival::Surv(hepatitis$time, hepatitis$censor) ~ 
##     hepatitis$group, sided = 1)
## 
## Decision = 1

## A decision value of 1 means the case of crossings or alternative orderings among the 
## survival functions is excluded. Thus, we can proceed to the one-sided test.

supELtest(survival::Surv(hepatitis$time, hepatitis$censor)~
    hepatitis$group, sided = 1)

## OUTPUT:
## Call:
## supELtest(formula = survival::Surv(hepatitis$time, hepatitis$censor) ~   
##     hepatitis$group, sided = 1)
## 
## One-sided maximally selected EL test statistic = 10.36, p = 0.006