coin (version 1.4-3)

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

Value

An object inheriting from class "IndependenceTest".

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", "Peto-Peto", "Prentice", "Prentice-Marek", "Andersen-Borgan-Gill-Keiding", "Fleming-Harrington", "Gaugler-Kim-Liao" or "Self"; see ‘Details’.

rho

a numeric, the \(\rho\) constant when type is "Tarone-Ware", "Fleming-Harrington", "Gaugler-Kim-Liao" or "Self"; see ‘Details’. Defaults to NULL, implying 0.5 for type = "Tarone-Ware" and 0 otherwise.

gamma

a numeric, the \(\gamma\) constant when type is "Fleming-Harrington", "Gaugler-Kim-Liao" 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 Peto-Peto test, the Prentice test, the Prentice-Marek test, the Andersen-Borgan-Gill-Keiding test, the Fleming-Harrington class of tests, the Gaugler-Kim-Liao 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 \(\theta = \lambda_2 / \lambda_1\) and \(\lambda_s\) is the hazard rate in the \(s\)th sample. In case alternative = "less", the null hypothesis is \(H_0\!: \theta \ge 1\), i.e., the survival is lower in sample 1 than in sample 2. When alternative = "greater", the null hypothesis is \(H_0\!: \theta \le 1\), 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 \((t_i, \delta_i)\), \(i = 1, 2, \ldots, n\), represent a right-censored random sample of size \(n\), where \(t_i\) is the observed survival time and \(\delta_i\) is the status indicator (\(\delta_i\) is 0 for right-censored observations and 1 otherwise). To allow for ties in the data, let \(t_{(1)} < t_{(2)} < \cdots < t_{(m)}\) represent the \(m\), \(m \le n\), ordered distinct event times. At time \(t_{(k)}\), \(k = 1, 2, \ldots, m\), the number of events and the number of subjects at risk are given by \(d_k = \sum_{i = 1}^n I\!\left(t_i = t_{(k)}\,|\, \delta_i = 1\right)\) 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 $$ r_k = \sum_{i = 1}^n I\!\left(t_i < t_{(k)}\right) $$ whereas the Hothorn-Lausen method uses $$ r_k = \sum_{i = 1}^n I\!\left(t_i \le t_{(k)}\right) - 1. $$ The scores assigned to right-censored and uncensored observations at the \(k\)th event time are given by $$ C_k = \sum_{j = 1}^k w_j \frac{d_j}{n_j} \quad \mathrm{and} \quad c_k = C_k - w_k, $$ 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 \(k\)th event time are arbitrarily ordered by assigning them distinct values \(t_{(k_l)}\), \(l = 1, 2, \ldots, d_k\), 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 \(i\)th subject is $$ a_i = \left\{ \begin{array}{ll} C_{k'} & \mathrm{if}~\delta_i = 0 \\ c_{k'} & \mathrm{otherwise} \end{array} \right. $$ where \(k' = \max \{k: t_i \ge t_{(k)}\}\).

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 \(w_k = n_k^\rho\), where \(\rho\) is a constant; \(\rho = 0.5\) (default) was suggested by Tarone and Ware (1977), but note that \(\rho = 0\) and \(\rho = 1\) lead to the standard logrank test and Gehan-Breslow test, respectively. The Peto-Peto test ("Peto-Peto") suggested by Peto and Peto (1972) is another generalization of the Wilcoxon rank-sum test, where $$ w_k = \hat{S}_k = \prod_{j = 0}^{k - 1} \frac{n_j - d_j}{n_j} $$ is the left-continuous Kaplan-Meier estimator of the survival function, \(n_0 \equiv n\) and \(d_0 \equiv 0\). The Prentice test ("Prentice") is also a generalization of the Wilcoxon rank-sum test proposed by Prentice (1978), where $$ w_k = \prod_{j = 1}^k \frac{n_j}{n_j + d_j}. $$ The Prentice-Marek test ("Prentice-Marek") is yet another generalization of the Wilcoxon rank-sum test discussed by Prentice and Marek (1979), with $$ w_k = \tilde{S}_k = \prod_{j = 1}^k \frac{n_j + 1 - d_j}{n_j + 1}. $$ The Andersen-Borgan-Gill-Keiding test ("Andersen-Borgan-Gill-Keiding") suggested by Andersen et al. (1982) is a modified version of the Prentice-Marek test using $$ w_k = \frac{n_k}{n_k + 1} \prod_{j = 0}^{k - 1} \frac{n_j + 1 - d_j}{n_j + 1}, $$ where, again, \(n_0 \equiv n\) and \(d_0 \equiv 0\). The Fleming-Harrington class of tests ("Fleming-Harrington") proposed by Fleming and Harrington (1991) uses \(w_k = \hat{S}_k^\rho (1 - \hat{S}_k)^\gamma\), where \(\rho\) and \(\gamma\) are constants; \(\rho = 0\) and \(\gamma = 0\) lead to the standard logrank test, while \(\rho = 1\) and \(\gamma = 0\) result in the Peto-Peto test. The Gaugler-Kim-Liao class of tests ("Gaugler-Kim-Liao") discussed by Gaugler et al. (2007) is a modified version of the Fleming-Harrington class of tests, replacing \(\hat{S}_k\) with \(\tilde{S}_k\) so that \(w_k = \tilde{S}_k^\rho (1 - \tilde{S}_k)^\gamma\), where \(\rho\) and \(\gamma\) are constants; \(\rho = 0\) and \(\gamma = 0\) lead to the standard logrank test, whereas \(\rho = 1\) and \(\gamma = 0\) result in the Prentice-Marek test. The Self class of tests ("Self") suggested by Self (1991) has \(w_k = v_k^\rho (1 - v_k)^\gamma\), where $$ v_k = \frac{1}{2} \frac{t_{(k-1)} + t_{(k)}}{t_{(m)}}, \quad t_{(0)} \equiv 0 $$ is the standardized mid-point between the \((k - 1)\)th and the \(k\)th event time. (This is a slight generalization of Self's original proposal in order to allow for non-integer follow-up times.) Again, \(\rho\) and \(\gamma\) are constants and \(\rho = 0\) and \(\gamma = 0\) 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.

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. tools:::Rd_expr_doi("10.2307/1402489")

Breslow, N. (1970). A generalized Kruskal-Wallis test for comparing \(K\) samples subject to unequal patterns of censorship. Biometrika 57(3), 579--594. tools:::Rd_expr_doi("10.1093/biomet/57.3.579")

Callaert, H. (2003). Comparing statistical software packages: The case of the logrank test in StatXact. The American Statistician 57(3), 214--217. tools:::Rd_expr_doi("10.1198/0003130031900")

Cox, D. R. (1972). Regression models and life-tables (with discussion). Journal of the Royal Statistical Society B 34(2), 187--220. tools:::Rd_expr_doi("10.1111/j.2517-6161.1972.tb00899.x")

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

Gaugler, T., Kim, D. and Liao, S. (2007). Comparing two survival time distributions: An investigation of several weight functions for the weighted logrank statistic. Communications in Statistics -- Simulation and Computation 36(2), 423--435. tools:::Rd_expr_doi("10.1080/03610910601161272")

Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily single-censored samples. Biometrika 52(1--2), 203--223. tools:::Rd_expr_doi("10.1093/biomet/52.1-2.203")

Hothorn, T. and Lausen, B. (2003). On the exact distribution of maximally selected rank statistics. Computational Statistics & Data Analysis 43(2), 121--137. tools:::Rd_expr_doi("10.1016/S0167-9473(02)00225-6")

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. tools:::Rd_expr_doi("10.2307/2532911")

Letón, E. and Zuluaga, P. (2001). Equivalence between score and weighted tests for survival curves. Communications in Statistics -- Theory and Methods 30(4), 591--608. tools:::Rd_expr_doi("10.1081/STA-100002138")

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. tools:::Rd_expr_doi("10.2307/2344317")

Prentice, R. L. (1978). Linear rank tests with right censored data. Biometrika 65(1), 167--179. tools:::Rd_expr_doi("10.1093/biomet/65.1.167")

Prentice, R. L. and Marek, P. (1979). A qualitative discrepancy between censored data rank tests. Biometrics 35(4), 861--867. tools:::Rd_expr_doi("10.2307/2530120")

Self, S. G. (1991). An adaptive weighted log-rank test with application to cancer prevention and screening trials. Biometrics 47(3), 975--986. tools:::Rd_expr_doi("10.2307/2532653")

Tarone, R. E. (1975). Tests for trend in life table analysis. Biometrika 62(3), 679--682. tools:::Rd_expr_doi("10.1093/biomet/62.3.679")

Tarone, R. E. and Ware, J. (1977). On distribution-free tests for equality of survival distributions. Biometrika 64(1), 156--160. tools:::Rd_expr_doi("10.1093/biomet/64.1.156")

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)))

## Asymptotic Mantel-Cox 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(nresample = 10000),
                        ytrafo = function(data)
                            trafo(data, surv_trafo = lee_trafo))
pvalue(it, method = "step-down")