# compare survival curves ```r comp(x, ...) ## S3 method for class 'ten' comp(x, ..., p = 1, q = 1, scores = seq.int(attr(x, "ncg")), reCalc = FALSE) ``` ## Arguments - `x`: A `tne` object - `...`: Additional arguments (not implemented). - `p`: $p$ for Fleming-Harrington test - `q`: $q$ for Fleming-Harrington test - `scores`: scores for tests for trend - `reCalc`: Recalcuate the values? If `reCalc=FALSE` (the default) and the `ten` object already has the calculated values stored as an `attribute`, the value of the `attribute` is returned directly. ## Returns The `tne` object is given additional `attributes`. The following are always added: - **lrt**: The l og-r ank family of t ests - **lrw**: The l og-r ank w eights (used in calculating the tests). An additional item depends on the number of covariate groups. If this is $=2$: - **sup**: The sup remum or Renyi family of tests and if this is $>2$: - **tft**: Tests for trend. This is given as a `list`, with the statistics and the scores used. ## Details The log-rank tests are formed from the following elements, with values for each time where there is at least one event: * $W[i]$, the weights, given below. * $e[i]$, the number of events (per time). * $P[i]$, the number of **predicted** events, given by `predict`. * $COV[, , i]$, the covariance matrix for time $i$, given by `COV`. It is calculated as: $$ Q_i = \sum{W_i (e_i - \hat{e}_i)}^T\sum{W_i \hat{COV_i} W_i^{-1}}\sum{W_i (e_i - \hat{e}_i)}Q[i] = sum(W[i] * (e[i] - P[i]))^T *sum(W[i] * COV[, , i] * W[i])^-1 *sum(W[i] * (e[i] - P[i])) $$ If there are $K$ groups, then $K-1$ are selected (arbitrary). Likewise the corresponding variance-covariance matrix is reduced to the appropriate $K-1 * K-1$ dimensions. $Q$ is distributed as chi-square with $K-1$ degrees of freedom. For $2$ covariate groups, we can use: * $e[i]$ the number of events (per time). * $e[i]$ the number at risk overall. * $e1[i]$ the number of events in group $1$. * $n1[i]$ the number at risk in group $1$. Then: $$ Q = \frac{\sum{W_i [e1_i - n1_i (\frac{e_i}{n_i})]} }{\sqrt{\sum{W_i^2 \frac{n1_i}{n_i}(1 - \frac{n1_i}{n_i})(\frac{n_i - e_i}{n_i - 1}) e_i }}}Q = sum(W[i] * (e1[i] - n1[i] * e[i] / n[i])) /sqrt(sum(W[i]^2 * e1[i] / e[i] * (1 - n1[i] / n[i]) * (n[i] - e[i] / (n[i] - 1)) *e[i])) $$ Below, for the Fleming-Harrington weights, $S(t)$ is the Kaplan-Meier (product-limit) estimator. Note that both $p$ and $q$ need to be $>=0$. The weights are given as follows: |||| |:-:|:--|:--| |$1$|log-rank|| |$n[i]$|Gehan-Breslow generalized Wilcoxon|| |$sqrt(n[i])$|Tarone-Ware|| |$S1[i]$|Peto-Peto's modified survival estimate|c("$\n$", "$ S1(t) = cumprod(1 - e / (n + 1))$")| |$S2[i]$|modified Peto-Peto (by Andersen)|c("$\n$", "$ S2(t) = S1[i] * n[i] / (n[i] + 1) $")| |$FH[i]$|Fleming-Harrington|The weight at $t_0 = 1$ and thereafter is: c("$\n$", "$ S(t[i - 1])^p * (1 - S(t)[i - 1]^q)$")| The supremum (Renyi) family of tests are designed to detect differences in survival curves which **cross**. That is, an early difference in survival in favor of one group is balanced by a later reversal. The same weights as above are used. They are calculated by finding $$ Z(t_i) = \sum_{t_k \leq t_i} W(t_k)[e1_k - n1_k\frac{e_k}{n_k}], \quad i=1,2,...,kZ(t[i]) = SUM W(t[k]) [ e1[k] - n1[k]e[k]/n[k] ] $$ (which is similar to the numerator used to find $Q$ in the log-rank test for 2 groups above). and it's variance: $$ \sigma^2(\tau) = \sum_{t_k \leq \tau} W(t_k)^2 \frac{n1_k n2_k (n_k-e_k) e_k}{n_k^2 (n_k-1)}simga^2(tau) = sum(k=1, 2, ..., tau) W(t[k]) (n1[k] * n2[k] * (n[k] - e[k]) *e[k] / n[k]^2 * (n[k] - 1) ] $$ where $tau$ is the largest $t$ where both groups have at least one subject at risk. Then calculate: $$ Q = \frac{ \sup{|Z(t)|}}{\sigma(\tau)}, \quad t<\tauQ = sup( |Z(t)| ) / sigma(tau), t < tau $$ When the null hypothesis is true, the distribution of $Q$ is approximately $$ Q \sim \sup{|B(x)|, \quad 0 \leq x \leq 1}Q ~ sup( |B(x)|, 0 <= x <= 1) $$ And for a standard Brownian motion (Wiener) process: $$ Pr[\sup|B(t)|>x] = 1 - \frac{4}{\pi}\sum_{k=0}^{\infty}\frac{(- 1)^k}{2k + 1} \exp{\frac{-\pi^2(2k + 1)^2}{8x^2}}Pr[sup|B(t)| > x] = 1 - 4 / pi sum((-1)^k / (2 * k + 1) * exp(-pi^2 (2k + 1)^2 / x^2)) $$ Tests for trend are designed to detect ordered differences in survival curves. That is, for at least one group: $$ S_1(t) \geq S_2(t) \geq ... \geq S_K(t) \quad t \leq \tauS1(t) >= S2(t) >= ... >= SK(t) for t <= tau $$ where $tau$ is the largest $t$ where all groups have at least one subject at risk. The null hypothesis is that $$ S_1(t) = S_2(t) = ... = S_K(t) \quad t \leq \tauS1(t) = S2(t) = ... = SK(t) for t <= tau $$ Scores used to construct the test are typically $s = 1,2,...,K$, but may be given as a vector representing a numeric characteristic of the group. They are calculated by finding: $$ Z_j(t_i) = \sum_{t_i \leq \tau} W(t_i)[e_{ji} - n_{ji} \frac{e_i}{n_i}],\quad j=1,2,...,KZ[t(i)] = sum(W[t(i)] * (e[j](i) - n[j](i) * e(i) / n(i))) $$ The test statistic is: $$ Z = \frac{ \sum_{j=1}^K s_jZ_j(\tau)}{\sqrt{\sum_{j=1}^K \sum_{g=1}^K s_js_g \sigma_{jg}}}Z = sum(j=1, ..., K) s[j] * Z[j] / sum(j=1, ..., K) sum(g=1, ..., K)s[j] * s[g] * sigma[jg] $$ where $sigma$ is the the appropriate element in the variance-covariance matrix (see `COV`). If ordering is present, the statistic $Z$ will be greater than the upper $alpha$-th percentile of a standard normal distribution. ## Note Regarding the Fleming-Harrington weights: * $p = q = 0$ gives the log-rank test, i.e. $W=1$ * $p=1, q=0$ gives a version of the Mann-Whitney-Wilcoxon test (tests if populations distributions are identical) * $p=0, q\>0$ gives more weight to differences later on * $p\>0, q=0$ gives more weight to differences early on The example using `alloauto` data illustrates this. Here the log-rank statistic has a p-value of around 0.5 as the late advantage of allogenic transplants is offset by the high early mortality. However using Fleming-Harrington weights of $p=0, q=0.5$, emphasising differences later in time, gives a p-value of 0.04. Stratified models (`stratTen`) are **not** yet supported. ## Examples ```r ## Two covariate groups data("leukemia", package="survival") f1 <- survfit(Surv(time, status) ~ x, data=leukemia) comp(ten(f1)) ## K&M 2nd ed. Example 7.2, Table 7.2, pp 209--210. data("kidney", package="KMsurv") t1 <- ten(Surv(time=time, event=delta) ~ type, data=kidney) comp(t1, p=c(0, 1, 1, 0.5, 0.5), q=c(1, 0, 1, 0.5, 2)) ## see the weights used attributes(t1)$lrw ## supremum (Renyi) test; two-sided; two covariate groups ## K&M 2nd ed. Example 7.9, pp 223--226. data("gastric", package="survMisc") g1 <- ten(Surv(time, event) ~ group, data=gastric) comp(g1) ## Three covariate groups ## K&M 2nd ed. Example 7.4, pp 212-214. data("bmt", package="KMsurv") b1 <- ten(Surv(time=t2, event=d3) ~ group, data=bmt) comp(b1, p=c(1, 0, 1), q=c(0, 1, 1)) ## Tests for trend ## K&M 2nd ed. Example 7.6, pp 217-218. data("larynx", package="KMsurv") l1 <- ten(Surv(time, delta) ~ stage, data=larynx) comp(l1) attr(l1, "tft") ### see effect of F-H test data("alloauto", package="KMsurv") a1 <- ten(Surv(time, delta) ~ type, data=alloauto) comp(a1, p=c(0, 1), q=c(1, 1)) ``` ## References Gehan A. A Generalized Wilcoxon Test for Comparing Arbitrarily Singly-Censored Samples. Biometrika 1965 Jun. 52(1/2):203--23. http://www.jstor.org/stable/2333825 JSTOR Tarone RE, Ware J 1977 On Distribution-Free Tests for Equality of Survival Distributions. **Biometrika**;64 (1):156--60. http://www.jstor.org/stable/2335790 JSTOR Peto R, Peto J 1972 Asymptotically Efficient Rank Invariant Test Procedures. **J Royal Statistical Society** 135 (2):186--207. http://www.jstor.org/stable/2344317 JSTOR Fleming TR, Harrington DP, O'Sullivan M 1987 Supremum Versions of the Log-Rank and Generalized Wilcoxon Statistics. **J American Statistical Association** 82 (397):312--20. http://www.jstor.org/stable/2289169 JSTOR Billingsly P 1999 **Convergence of Probability Measures.** New York: John Wiley & Sons. http://dx.doi.org/10.1002/9780470316962 Wiley (paywall)

comp function