comp(x, ...)## S3 method for class 'survfit':
comp(x, ..., FHp = 1, FHq = 1, lim = 10000,
scores = NULL)
## S3 method for class 'coxph':
comp(x, ..., FHp = 1, FHq = 1, scores = NULL,
lim = 10000)
survfit or coxph objectlist with two elements, tne and tests.
The first is a data.table with one row for each time at
which an event occurred.
Columns show time, no. at risk and no. events
(by stratum and overall).
The second contains the tests, as a list.
The first element is the log-rank family of tests.
The following additional tests depend on the no. of strata:
For a survfit or a coxph object with 2 strata,
these are the Supremum (Renyi) family of tests.
For a survfit or coxph object with at least 3 strata,
there are tests for trend.covMatSurv.The sum is taken to the largest observed survival time
(i.e. censored observations are excluded).
If there are $K$ groups, then $K-1$ are selected (arbitrary).
Likewise the corresponding variance-covariance matrix is reduced to the
appropriate $K-1 \times K-1$ dimensions.
$Q$ is distributed as chi-square with $K-1$ degrees of freedom.
For 2 strata this simplifies to:
$$Q = \frac{ \sum{ W_i [e1_i - n1_i (\frac{e_i}{n_i})]} }{ \sqrt{ \sum{ W_i^2 \frac{n1_i}{n_i} ( 1- \frac{n1_i}{n_i} ) ( \frac{n_i - e_i}{n_i-1}) e_i }}}$$
Here $e$ and $n$ refer to
the no. events and no. at risk overall and
$e1$ and $n1$ refer to the no. events and no. at risk in group $1$.
The weights are given as follows: [object Object],[object Object],[object Object],[object Object],[object Object],[object Object]
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
$$Z(t_i) = \sum_{t_k \leq t_i} W(t_k)[e1_k - n1_k\frac{e_k}{n_k}], \quad i=1,2,...,k$$
(which is similar to the numerator used to find $Q$
in the log-rank test for 2 groups above).
and it's variance:
$$\sigma^2(\tau) = \sum_{t_k \leq \tau} W(t_k)^2 \frac{n1_k n2_k (n_k-e_k) e_k}{n_k^2 (n_k-1)}$$
where $\tau$ is the largest $t$
where both groups have at least one subject at risk.
Then calculate:
$$Q = \frac{ \sup{|Z(t)|}}{\sigma(\tau)}, \quad t<\tau$$ 0="" when="" the="" null="" hypothesis="" is="" true,="" distribution="" of="" $q$="" approximately="" $$q="" \sim="" \sup{|b(x)|,="" \quad="" \leq="" x="" 1}$$="" and="" for="" a="" standard="" brownian="" motion="" (wiener)="" process:="" $$pr[\sup|b(t)|="">x] = 1-\frac{4}{\pi} \sum_{k=0}^{\infty} \frac{(-1)^k}{2k+1} \exp{ \frac{-\pi^2(2k+1)^2}{8x^2}}$$
Tests for trend are designed to detect ordered differences in survival curves.
That is, for at least one group:
$$S_1(t) \geq S_2(t) \geq ... \geq S_K(t) \quad t \leq \tau$$
where $\tau$ is the largest $t$ where all groups have at least one subject at risk.
The null hypothesis is that
$$S_1(t) = S_2(t) = ... = S_K(t) \quad t \leq \tau$$
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
$$Z_j(t_i) = \sum_{t_i \leq \tau} W(t_i)[e_{ji} - n_{ji} \frac{e_i}{n_i}], \quad j=1,2,...,K$$
The test statistic is
$$Z = \frac{ \sum_{j=1}^K s_jZ_j(\tau)}{\sqrt{\sum_{j=1}^K \sum_{g=1}^K s_js_g \sigma_{jg}}}$$
where $\sigma$ is the the appropriate element in the
variance-covariance matrix (as in covMatSurv).
If ordering is present, the statistic $Z$ will be greater than the upper $\alpha$th
percentile of a standard normal distribution.
Tarone RE, Ware J 1977
On Distribution-Free Tests for Equality of Survival Distributions.
Biometrika;64(1):156--60.
Peto R, Peto J 1972
Asymptotically Efficient Rank Invariant Test Procedures.
J Royal Statistical Society 135(2):186--207.
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.
Billingsly P 1999
Convergence of Probability Measures.
New York: John Wiley & Sons.
Examples are from Klein J, Moeschberger M 2003 Survival Analysis, 2nd edition. New York: Springer. Examples 7.2, 7.4, 7.5, 7.6, 7.9, pp 210-225.
### 2 curves
data(kidney,package="KMsurv")
s1 <- survfit(Surv(time=time, event=delta) ~ type, data=kidney)
comp(s1)
### 3 curves
data(bmt, package="KMsurv")
comp(survfit(Surv(time=t2, event=d3) ~ group, data=bmt))
### see effect of F-H test
data(alloauto, package="KMsurv")
s3 <- survfit(Surv(time, delta) ~ type, data=alloauto)
comp(s3, FHp=0, FHq=1)
### see trend tests
data(larynx, package="KMsurv")
s4 <- survfit(Surv(time, delta) ~ stage, data=larynx)
comp(s4)
### Renyi tests
data("gastric", package="survMisc")
s5 <- survfit(Surv(time, event) ~ group, data=gastric)
comp(s5)
c1 <- coxph(Surv(time=time, event=delta) ~ type, data=kidney )
comp(c1)Run the code above in your browser using DataLab