# Two- and $K$-Sample Tests for Censored Data Testing the equality of the survival distributions in two or more independent groups. ```r ## 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 `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 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 $H_0: theta >= 1$, 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 $(t_i, \delta_i)$, $i = 1, 2, \ldots, n$, 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 c("$t_(1) < t_(2) < ... <\n$", "$ 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 c("$d_k = sum(i = 1, ..., n)\n$", "$ 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 $$ r_k = \sum_{i = 1}^n I\!\left(t_i < t_{(k)}\right)r_k = sum(i = 1, ..., n) I(t_i < t_(k)) $$ whereas the Hothorn-Lausen method uses $$ r_k = \sum_{i = 1}^n I\!\left(t_i \le t_{(k)}\right) - 1.r_k = sum(i = 1, ..., n) I(t_i <= t_(k)) - 1. $$ The scores assigned to right-censored and uncensored observations at the $k$th event time are given by $$ C_k = \sum_{j = 1}^k w_j \frac{d_j}{n_j}\quad \mathrm{and} \quadc_k = C_k - w_k,C_k = sum(j = 1, ..., k) w_j * (d_j / n_j) and c_k = C_k - w_k, $$ 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 $k$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 $$ a_i = \left\{\begin{array}{ll}C_{k'} & \mathrm{if}~\delta_i = 0 \\c_{k'} & \mathrm{otherwise}\end{array}\right.C_k' if delta_i = 0a_i =c_k' otherwise $$ 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 c("$w_k =\n$", "$ 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 $$ w_k = \hat{S}_k = \prod_{j = 0}^{k - 1} \frac{n_j - d_j}{n_j}w_k = Shat_k = prod(j = 0, ..., k - 1) (n_j - d_j) / n_j $$ is the **left-continuous** Kaplan-Meier estimator of the survival function, $n_0 := n$ and $d_0 := 0$. The Prentice test (`"Prentice"`) is also a generalization of the Wilcoxon rank-sum test proposed by Prentice (1978), where $$ w_k = \prod_{j = 1}^k \frac{n_j}{n_j + d_j}.w_k = prod(j = 1, ..., k) n_j / (n_j + d_j). $$ The Prentice-Marek test (`"Prentice-Marek"`) is yet another generalization of the Wilcoxon rank-sum test discussed by Prentice and Marek (1979), with $$ w_k = \tilde{S}_k = \prod_{j = 1}^k \frac{n_j + 1 - d_j}{n_j + 1}.w_k = Stilde_k = prod(j = 1, ..., k) (n_j + 1 - d_j) / (n_j + 1). $$ 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 $$ w_k = \frac{n_k}{n_k + 1} \prod_{j = 0}^{k - 1} \frac{n_j + 1 - d_j}{n_j + 1},w_k = (n_k / (n_k + 1)) prod(j = 0, ..., k - 1) (n_j + 1 - d_j) / (n_j + 1), $$ where, again, $n_0 := n$ and $d_0 := 0$. The Fleming-Harrington class of tests (`"Fleming-Harrington"`) proposed by Fleming and Harrington (1991) uses $w_k = Shat_k^rho * (1 - Shat_k)^gamma$, 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 $Shat_k$ with $Stilde_k$ so that c("$w_k = Stilde_k^rho * (1 -\n$", "$ Stilde_k)^gamma$"), where $\rho$ and $\gamma$ are constants; c("$\\rho\n$", "$ = 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 $$ v_k = \frac{1}{2} \frac{t_{(k-1)} + t_{(k)}}{t_{(m)}},\quadt_{(0)} \equiv 0v_k = 1 / 2 * (t_(k - 1) + t_(k)) / t_(m), t_(0) := 0 $$ 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. ## Returns 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, ., 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") , 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 ```r ## 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") ```

SurvivalTests function