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, …)
Arguments
- formula
a formula of the form
y ~ x | block
wherey
is a survival object (seeSurv
in package survival),x
is a factor andblock
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 toNULL
, implying equal weight for all observations.- object
- 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"
,"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 toNULL
, implying0.5
fortype = "Tarone-Ware"
and0
otherwise.- gamma
a numeric, the \(\gamma\) constant when
type
is"Fleming-Harrington"
,"Gaugler-Kim-Liao"
or"Self"
; see ‘Details’. Defaults toNULL
, implying0
.- …
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 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.
Value
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
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")
# }