| 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",
"Peto-Peto", "Prentice", "Prentice-Marek",
"Andersen-Borgan-Gill-Keiding",
"Fleming-Harrington", "Gaugler-Kim-Liao", "Self"),
rho = NULL, gamma = NULL, ...)
Arguments
formula |
a formula of the form |
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 |
weights |
an optional formula of the form |
object |
an object inheriting from class |
ties.method |
a character, the method used to handle ties: the score generating function
either uses mid-ranks ( |
type |
a character, the type of test: either |
rho |
a numeric, the |
gamma |
a numeric, the |
... |
further arguments to be passed to |
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 sth 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
kth 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 kth 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
ith 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 kth
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.
Value
An object inheriting from class "IndependenceTest".
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 coin 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 coin 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, 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. 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. 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. 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. 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. doi:10.1080/03610910601161272
Gehan, E. A. (1965). A generalized Wilcoxon test for comparing arbitrarily single-censored samples. Biometrika 52(1–2), 203–223. 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. 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. 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. 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. doi:10.2307/2344317
Prentice, R. L. (1978). Linear rank tests with right censored data. Biometrika 65(1), 167–179. 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. 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. doi:10.2307/2532653
Tarone, R. E. (1975). Tests for trend in life table analysis. Biometrika 62(3), 679–682. 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. 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")