Description Usage Arguments Details Value Note References Examples
compare survival curves
1 2 3 4 5 
x 
A 
... 
Additional arguments (not implemented). 
p 
p for FlemingHarrington test 
q 
q for FlemingHarrington test 
scores 
scores for tests for trend 
reCalc 
Recalcuate the values?

The logrank 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]  P[i]))^T * sum(W[i] * COV[, , i] * W[i])^1 * sum(W[i] * (e[i]  P[i]))
If there are K groups, then K1 are selected (arbitrary).
Likewise the corresponding variancecovariance matrix is reduced to the
appropriate K1 * K1 dimensions.
Q is distributed as chisquare with K1 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 = 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 FlemingHarrington weights,
S(t) is the KaplanMeier (productlimit) estimator.
Note that both p and q need to be >=0.
The weights are given as follows:
1  logrank  
n[i]  GehanBreslow generalized Wilcoxon  
sqrt(n[i])  TaroneWare  
S1[i]  PetoPeto's modified survival estimate  S1(t) = cumprod(1  e / (n + 1)) 
S2[i]  modified PetoPeto (by Andersen)  S2(t) = S1[i] * n[i] / (n[i] + 1) 
FH[i]  FlemingHarrington  The weight at t_0 = 1 and thereafter is: 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 W(t[k]) [ e1[k]  n1[k]e[k]/n[k] ]
(which is similar to the numerator used to find Q
in the logrank test for 2 groups above).
and it's variance:
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 = sup( Z(t) ) / sigma(tau), t < tau
When the null hypothesis is true, the distribution of Q is approximately
Q ~ sup( B(x), 0 <= x <= 1)
And for a standard Brownian motion (Wiener) process:
Pr[supB(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:
S1(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
S1(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[t(i)] = sum(W[t(i)] * (e[j](i)  n[j](i) * e(i) / n(i)))
The test statistic is:
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
variancecovariance matrix (see COV
).
If ordering is present, the statistic Z will be greater than the
upper alphath
percentile of a standard normal distribution.
The tne
object is given
additional attributes
.
The following are always added:
lrt 
The logrank family of tests 
lrw 
The logrank weights (used in calculating the tests). 
An additional item depends on the number of covariate groups.
If this is =2:
sup 
The supremum or Renyi family of tests 
and if this is >2:
tft 
Tests for trend. This is given as a 
Regarding the FlemingHarrington weights:
p = q = 0 gives the logrank test, i.e. W=1
p=1, q=0 gives a version of the MannWhitneyWilcoxon 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 logrank statistic
has a pvalue of around 0.5
as the late advantage of allogenic transplants
is offset by the high early mortality. However using
FlemingHarrington weights of p=0, q=0.5,
emphasising differences later in time, gives a pvalue of 0.04.
Stratified models (stratTen
) are not yet supported.
Gehan A. A Generalized Wilcoxon Test for Comparing Arbitrarily SinglyCensored Samples. Biometrika 1965 Jun. 52(1/2):203–23. http://www.jstor.org/stable/2333825 JSTOR
Tarone RE, Ware J 1977 On DistributionFree 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 LogRank 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)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30  ## 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 209210.
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; twosided; two covariate groups
## K&M 2nd ed. Example 7.9, pp 223226.
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 212214.
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 217218.
data("larynx", package="KMsurv")
l1 < ten(Surv(time, delta) ~ stage, data=larynx)
comp(l1)
attr(l1, "tft")
### see effect of FH test
data("alloauto", package="KMsurv")
a1 < ten(Surv(time, delta) ~ type, data=alloauto)
comp(a1, p=c(0, 1), q=c(1, 1))

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