SurvivalTests

Two- and \(K\)-Sample Tests for Censored Data

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", "Prentice",
                      "Prentice-Marek", "Andersen-Borgan-Gill-Keiding",
                      "Fleming-Harrington", "Gaugler-Kim-Liao", "Self"),
             rho = NULL, gamma = NULL, …)

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 Let<U+00F3>n 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 $$ C_k = \sum_{j = 1}^k w_j \frac{d_j}{n_j} \quad \mbox{and} \quad c_k = C_k - w_k, $$ 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 $$ a_i = \left\{ \begin{array}{ll} C_{k'} & \mbox{if } \delta_i = 0 \\ c_{k'} & \mbox{otherwise} \end{array} \right. $$ 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 the standard logrank test and Gehan-Breslow test respectively. The Prentice test ("Prentice") is another 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 \(n_0 \equiv n\) 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 and $$ \hat{S}_k = \prod_{j = 0}^{k - 1} \frac{n_j - d_j}{n_j}, \quad \hat{S}_0 \equiv 1 $$ is the left-continuous Kaplan-Meier estimator of the survival function; \(\rho = 0\) and \(\gamma = 0\) lead to the standard logrank 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. 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.

Note

Peto and Peto (1972) proposed the test statistic implemented in logrank_test and named it the logrank test. However, the Mantel-Cox test (Mantel, 1966; Cox, 1972), as implemented in survdiff (in package survival), is also known as the logrank test. These tests are similar, but differ in the choice of probability model: the (Peto-Peto) logrank test uses the permutational variance, whereas the Mantel-Cox test is based on the hypergeometric variance.

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 replaced surv_test which was made defunct in version 1.2-0. 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".)

Starting with version 1.2-0, mid-ranks and the Hothorn-Lausen method can no longer be specified with ties.method = "logrank" and ties-method = "HL" respectively.

References

Andersen, P. K., Borgan, <U+00D8>., 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. 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. 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. 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.

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. 10.1080/03610910601161272

Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily single-censored samples. Biometrika 52(1--2), 203--223. 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. 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. 10.2307/2532911

Let<U+00F3>n, E. and Zuluaga, P. (2001). Equivalence between score and weighted tests for survival curves. Communications in Statistics -- Theory and Methods 30(4), 591--608. 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. 10.2307/2344317

Prentice, R. L. (1978). Linear rank tests with right censored data. Biometrika 65(1), 167--179. 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. 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. 10.2307/2532653

Tarone, R. E. (1975). Tests for trend in life table analysis. Biometrika 62(3), 679--682. 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. 10.1093/biomet/64.1.156

Aliases

• surv_test
• logrank_test
• logrank_test.formula
• logrank_test.IndependenceProblem

Examples

# NOT RUN {
## 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")
# }