R: Two- and K-Sample Tests for Censored Data

SurvivalTests {coin} | R Documentation

## Two- and K-Sample Tests for Censored Data

### Description

Testing the equality of the survival distributions in two or more independent groups.

### Usage

    ## S3 method for class 'formula'
    logrank_test(formula, data, subset = NULL, weights = NULL, ...)
    ## S3 method for class 'IndependenceProblem'
    logrank_test(object, ties.method = c("mid-ranks", "Hothorn-Lausen",
                                         "average-scores"),
                 type = c("logrank", "Gehan-Breslow", "Tarone-Ware", "Prentice",
                          "Prentice-Marek", "Andersen-Borgan-Gill-Keiding",
                          "Fleming-Harrington", "Self"),
                 rho = NULL, gamma = NULL, ...)

### Arguments

`formula` | a formula of the form `y ~ x | block` where `y` is a survival object (see `Surv` in package survival), `x` is a factor and `block` is an optional factor for stratification.
`data` | an optional data frame containing the variables in the model formula.
`subset` | an optional vector specifying a subset of observations to be used. Defaults to `NULL`.
`weights` | an optional formula of the form `~ w` defining integer valued case weights for each observation. Defaults to `NULL`, implying equal weight for all observations.
`object` | an object inheriting from class `"IndependenceProblem"`.
`ties.method` | a character, the method used to handle ties: the score generating function either uses mid-ranks (`"mid-ranks"`, default), the Hothorn-Lausen method (`"Hothorn-Lausen"`) or averages the scores of randomly broken ties (`"average-scores"`); see ‘Details’.
`type` | a character, the type of test: either `"logrank"` (default), `"Gehan-Breslow"`, `"Tarone-Ware"`, `"Prentice"`, `"Prentice-Marek"`, `"Andersen-Borgan-Gill-Keiding"`, `"Fleming-Harrington"` or `"Self"`; see ‘Details’.
`rho` | a numeric, the ρ constant when `type` is `"Tarone-Ware"`, `"Fleming-Harrington"` or `"Self"`; see ‘Details’. Defaults to `NULL`, implying `0.5` for `type = "Tarone-Ware"` and `0` otherwise.
`gamma` | a numeric, the γ constant when `type` is `"Fleming-Harrington"` or `"Self"`; see ‘Details’. Defaults to `NULL`, implying `0`.
`...` | further arguments to be passed to `independence_test`.

### Details

`logrank_test` provides the weighted logrank test reformulated as a linear rank test. The family of weighted logrank tests encompasses a large collection of tests commonly used in the analysis of survival data including, but not limited to, the standard (unweighted) logrank test, the Gehan-Breslow test, the Tarone-Ware class of tests, the Prentice test, the Prentice-Marek test, the Andersen-Borgan-Gill-Keiding test, the Fleming-Harrington class of tests and the Self class of tests. A general description of these methods is given by Klein and Moeschberger (2003, Ch. 7). See Letón and Zuluaga (2001) for the linear rank test formulation.

The null hypothesis of equality, or conditional equality given `block`, of the survival distribution of `y` in the groups defined by `x` is tested. In the two-sample case, the two-sided null hypothesis is H_0: theta = 1, where θ = λ_2 / λ_1 and λ_s is the hazard rate in the sth sample. In case `alternative = "less"`, the null hypothesis is H_0: theta >= 1, i.e., the survival is lower in sample 1 than in sample 2. When `alternative = "greater"`, the null hypothesis is H_0: theta <= 1, i.e., the survival is higher in sample 1 than in sample 2.

If `x` is an ordered factor, the default scores, `1:nlevels(x)`, can be altered using the `scores` argument (see `independence_test`); this argument can also be used to coerce nominal factors to class `"ordered"`. In this case, a linear-by-linear association test is computed and the direction of the alternative hypothesis can be specified using the `alternative` argument. This type of extension of the standard logrank test was given by Tarone (1975) and later generalized to general weights by Tarone and Ware (1977).

Let (t_i, δ_i), i = 1, 2, …, n, represent a censored random sample of size n, where t_i is the observed survival time and δ_i is the status indicator (δ_i is 0 for censored observations and 1 otherwise). To allow for ties in the data, let t_(1) < t_(2) < … < t_(m) represent the m, m ≤ n, ordered distinct event times. At time t_(k), k = 1, 2, …, m, the number of events and the number of subjects at risk are given by d_k = sum(i = 1, …, n) I(t_i = t_(k) | delta_i = 1) and n_k = n - r_k, respectively, where r_k depends on the ties handling method.

Three different methods of handling ties are available using `ties.method`: mid-ranks (`"mid-ranks"`, default), the Hothorn-Lausen method (`"Hothorn-Lausen"`) and average-scores (`"average-scores"`). The first and last method are discussed and contrasted by Callaert (2003), whereas the second method is defined in Hothorn and Lausen (2003). The mid-ranks method leads to

r_k = sum(i = 1, …, n) I(t_i < t_(k))

whereas the Hothorn-Lausen method uses

r_k = sum(i = 1, …, n) I(t_i <= t_(k)) - 1.

The scores assigned to censored and uncensored observations at the kth event time are given by

C_k = sum(j=1,…,k) w_j * (d_j / n_j) and c_k = C_k - w_k,

respectively, where w is the logrank weight. For the average-scores method, used by, e.g., the software package StatXact, the d_k events observed at the kth event time are arbitrarily ordered by assigning them distinct values t_(k_l), l = 1, 2, …, d_k, infinitesimally to the left of t_(k). Then scores C_k_l and c_k_l are computed as indicated above, effectively assuming that no event times are tied. The scores C_k and c_k are assigned the average of the scores C_k_l and c_k_l respectively. It then follows that the score for the ith subject is

C_k' if delta_i = 0 a_i = c_k' otherwise

where k' = max{k : t_i >= t_(k)}.

The `type` argument allows for a choice between some of the most well-known members of the family of weighted logrank tests, each corresponding to a particular weight function. The standard logrank test (`"logrank"`, default) was suggested by Mantel (1966), Peto and Peto (1972) and Cox (1972) and has w_k = 1. The Gehan-Breslow test (`"Gehan-Breslow"`) proposed by Gehan (1965) and later extended to K samples by Breslow (1970) is a generalization of the Wilcoxon rank-sum test, where w_k = n_k. The Tarone-Ware class of tests (`"Tarone-Ware"`) discussed by Tarone and Ware (1977) has w_k = n_k^ρ, where ρ is a constant; ρ = 0.5 (default) was suggested by Tarone and Ware (1977), but note that ρ = 0 and ρ = 1 lead to the the standard logrank test and Gehan-Breslow test respectively. The Prentice test (`"Prentice"`) is another generalization of the Wilcoxon rank-sum test suggested by Prentice (1978), where

w_k = prod(j = 1, …, k) n_j / n_j + d_j).

The Prentice-Marek test (`"Prentice-Marek"`) is yet another generalization of the Wilcoxon rank-sum test discussed by Prentice and Marek (1979), with

w_k = prod(j = 1, …, k) (n_j + 1 - d_j) / (n_j + 1).

The Andersen-Borgan-Gill-Keiding test (`"Andersen-Borgan-Gill-Keiding"`) proposed by Andersen et al. (1982) is a modified version of the Prentice-Marek test using

w_k = (n_k / (n_k + 1)) prod(j = 0, …, k - 1) (n_j + 1 - d_j) / (n_j+1)

where n_0 := n and d_0 := 0. The Fleming-Harrington class of tests (`"Fleming-Harrington"`) suggested by Fleming and Harrington (1991) uses w_k = Shat_k^rho * (1 - Shat_k)^gamma, where ρ and γ are constants and

Shat_k = prod(j = 0, …, k - 1) (n_j - d_j) / n_j, Shat_0 := 1

is the left-continuous Kaplan-Meier estimator of the survival function; ρ = 0 and γ = 0 lead to the standard logrank test. The Self class of tests (`"Self"`) proposed by Self (1991) has w_k = v_k^rho * (1 - v_k)^gamma, where

v_k = 1 / 2 * (t_(k - 1) + t_(k)) / t_(m), t_(0) := 1

is the standardized mid-point between the (k - 1)th and the kth event time. (This is a slight generalization of Self's original proposal in order to allow for non-integer follow-up times.) Again, ρ and γ are constants and ρ = 0 and γ = 0 lead to the standard logrank test.

The conditional null distribution of the test statistic is used to obtain p-values and an asymptotic approximation of the exact distribution is used by default (`distribution = "asymptotic"`). Alternatively, the distribution can be approximated via Monte Carlo resampling or computed exactly for univariate two-sample problems by setting `distribution` to `"approximate"` or `"exact"` respectively. See `asymptotic`, `approximate` and `exact` for details.

### Value

An object inheriting from class `"IndependenceTest"`.

### Note

The test statistic differs from the one used by `survdiff` (in package survival), since the conditional variance used by `logrank_test` is not identical to the variance estimate used by the classical logrank test.

Combining `independence_test` or `symmetry_test` with `logrank_trafo` offers more flexibility than `logrank_test` and allows for, among other things, maximum-type versatile test procedures (e.g., Lee, 1996; see ‘Examples’) and user-supplied logrank weights (see `GTSG` for tests against Weibull-type or crossing-curve alternatives).

Starting with version 1.1-0, `logrank_test` replaces `surv_test`, which is now deprecated and will be removed in a future release. Furthermore, `logrank_trafo` is now an increasing function for all choices of `ties.method`, implying that the test statistic has the same sign irrespective of the ties handling method. Consequently, the sign of the test statistic will now be the opposite of what it was in earlier versions unless `ties.method = "average-scores"`. (In versions of coin prior to 1.1-0, `logrank_trafo` was a decreasing function when `ties.method` was other than `"average-scores"`.)

### References

Andersen, P. K., Borgan, Ø., Gill, R. and Keiding, N. (1982). Linear nonparametric tests for comparison of counting processes, with applications to censored survival data (with discussion). International Statistical Review 50(3), 219–258.

Breslow, N. (1970). A generalized Kruskal-Wallis test for comparing K samples subject to unequal patterns of censorship. Biometrika 57(3), 579–594.

Callaert, H. (2003). Comparing statistical software packages: The case of the logrank test in StatXact. The American Statistician 57(3), 214–217.

Cox, D. R. (1972). Regression models and life-tables (with discussion). Journal of the Royal Statistical Society B 34(2), 187–220.

Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. New York: John Wiley & Sons.

Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily single-censored samples. Biometrika 52(1–2), 203–223.

Hothorn, T. and Lausen, B. (2003). On the exact distribution of maximally selected rank statistics. Computational Statistics & Data Analysis 43(2), 121–137.

Klein, J. P. and Moeschberger, M. L. (2003). Survival Analysis: Techniques for Censored and Truncated Data, Second Edition. New York: Springer.

Lee, J. W. (1996). Some versatile tests based on the simultaneous use of weighted log-rank statistics. Biometrics 52(2), 721–725.

Letón, E. and Zuluaga, P. (2001). Equivalence between scores and weighted tests for survival curves. Communications in Statistics – Theory and Methods 30(4), 591–608.

Mantel, N. (1966). Evaluation of survival data and two new rank order statistics arising in its consideration. Cancer Chemotherapy Reports 50(3), 163–170.

Peto, R. and Peto, J. (1972). Asymptotic efficient rank invariant test procedures (with discussion). Journal of the Royal Statistical Society A 135(2), 185–207.

Prentice, R. L. (1978). Linear rank tests with right censored data. Biometrika 65(1), 167–179.

Prentice, R. L. and Marek, P. (1979). A qualitative discrepancy between censored data rank tests. Biometrics 35(4), 861–867.

Self, S. G. (1991). An adaptive weighted log-rank test with application to cancer prevention and screening trials. Biometrics 47(3), 975–986.

Tarone, R. E. (1975). Tests for trend in life table analysis. Biometrika 62(3), 679–682.

Tarone, R. E. and Ware, J. (1977). On distribution-free tests for equality of survival distributions. Biometrika 64(1), 156–160.

### Examples

    ## Example data (Callaert, 2003, Tab.1)
    callaert <- data.frame(
        time = c(1, 1, 5, 6, 6, 6, 6, 2, 2, 2, 3, 4, 4, 5, 5),
        group = factor(rep(0:1, c(7, 8)))
    )

    ## Logrank scores using mid-ranks (Callaert, 2003, Tab.2)
    with(callaert,
         logrank_trafo(Surv(time)))

    ## Classical asymptotic logrank test (p = 0.0523)
    survdiff(Surv(time) ~ group, data = callaert)

    ## Exact logrank test using mid-ranks (p = 0.0505)
    logrank_test(Surv(time) ~ group, data = callaert, distribution = "exact")

    ## Exact logrank test using average-scores (p = 0.0468)
    logrank_test(Surv(time) ~ group, data = callaert, distribution = "exact",
                 ties.method = "average-scores")

    ## Lung cancer data (StatXact 9 manual, p. 213, Tab. 7.19)
    lungcancer <- data.frame(
        time = c(257, 476, 355, 1779, 355,
                 191, 563, 242, 285, 16, 16, 16, 257, 16),
        event = c(0, 0, 1, 1, 0,
                  1, 1, 1, 1, 1, 1, 1, 1, 1),
        group = factor(rep(1:2, c(5, 9)),
                       labels = c("newdrug", "control"))
    )

    ## Logrank scores using average-scores (StatXact 9 manual, p. 214)
    with(lungcancer,
         logrank_trafo(Surv(time, event), ties.method = "average-scores"))

    ## Exact logrank test using average-scores (StatXact 9 manual, p. 215)
    logrank_test(Surv(time, event) ~ group, data = lungcancer,
                 distribution = "exact", ties.method = "average-scores")

    ## Exact Prentice test using average-scores (StatXact 9 manual, p. 222)
    logrank_test(Surv(time, event) ~ group, data = lungcancer,
                 distribution = "exact", ties.method = "average-scores",
                 type = "Prentice")

    ## Approximative (Monte Carlo) versatile test (Lee, 1996)
    rho.gamma <- expand.grid(rho = seq(0, 2, 1), gamma = seq(0, 2, 1))
    lee_trafo <- function(y)
        logrank_trafo(y, ties.method = "average-scores",
                      type = "Fleming-Harrington",
                      rho = rho.gamma["rho"], gamma = rho.gamma["gamma"])

    it <- independence_test(Surv(time, event) ~ group, data = lungcancer,
                            distribution = approximate(B = 10000),
                            ytrafo = function(data)
                                trafo(data, surv_trafo = lee_trafo))
    pvalue(it, method = "step-down")

[Package coin version 1.1-3 Index]