comp: compare survival curves
In survMisc: Miscellaneous Functions for Survival Data
Description
Usage
Arguments
Details
Value
Note
References
Examples
Description
compare survival curves
Usage
1
2
3
4
5
Arguments
x
A tne object
p
p for Fleming-Harrington test
q
q for Fleming-Harrington test
scores
scores for tests for trend
...
Additional arguments (not implemented).
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.
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] - 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 = 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
S1(t) = cumprod(1 - e / (n + 1))
S2[i] modified Peto-Peto (by Andersen)
S2(t) = S1[i] * n[i] / (n[i] + 1)
FH[i] Fleming-Harrington
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 log-rank 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[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:
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
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.
Value
The tne object is given
additional attributes.
The following are always added:
lrt
The log-rank family of tests
lrw
The log-rank 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 list,
with the statistics and the scores used.
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.
References
Gehan A.
A Generalized Wilcoxon Test for Comparing Arbitrarily
Singly-Censored Samples.
Biometrika 1965 Jun. 52(1/2):203–23.
JSTOR
Tarone RE, Ware J 1977
On Distribution-Free Tests for Equality of Survival Distributions.
Biometrika;64(1):156–60.
JSTOR
Peto R, Peto J 1972
Asymptotically Efficient Rank Invariant Test Procedures.
J Royal Statistical Society 135(2):186–207.
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.
JSTOR
Billingsly P 1999
Convergence of Probability Measures.
New York: John Wiley & Sons.
Wiley (paywall)
Examples
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
## Two covariate groups
## 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))
Example output
Loading required package: survival
Q Var Z pNorm
1 -3.9636e+00 6.2368e+00 -1.587104 0.1124892
n 9.0000e+00 3.8919e+04 0.045621 0.9636127
sqrtN -1.3203e+01 4.3384e+02 -0.633875 0.5261626
S1 -2.4692e+00 4.3725e+00 -1.180837 0.2376674
S2 -2.3134e+00 4.2087e+00 -1.127672 0.2594583
FH_p=0_q=1 -1.4134e+00 2.0778e-01 -3.100747 0.0019303 **
FH_p=1_q=0 -2.5501e+00 4.7072e+00 -1.175397 0.2398361
FH_p=1_q=1 -1.0206e+00 1.0661e-01 -3.125846 0.0017729 **
FH_p=0.5_q=0.5 -2.4695e+00 6.6091e-01 -3.037697 0.0023839 **
FH_p=0.5_q=2 -3.2350e-01 1.2842e-02 -2.854717 0.0043075 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
maxAbsZ Var Q pSupBr
1 3.9636e+00 6.2368e+00 1.5871 0.2249746
n 2.8200e+02 3.8919e+04 1.4294 0.3057172
sqrtN 2.6224e+01 4.3384e+02 1.2590 0.4157260
S1 2.4692e+00 4.3725e+00 1.1808 0.4745421
S2 2.3134e+00 4.2087e+00 1.1277 0.5174828
FH_p=0_q=1 1.4134e+00 2.0778e-01 3.1007 0.0038607 **
FH_p=1_q=0 2.5501e+00 4.7072e+00 1.1754 0.4788290
FH_p=1_q=1 1.0206e+00 1.0661e-01 3.1258 0.0035459 **
FH_p=0.5_q=0.5 2.4695e+00 6.6091e-01 3.0377 0.0047679 **
FH_p=0.5_q=2 3.2350e-01 1.2842e-02 2.8547 0.0086150 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
1 n sqrtN S1 S2 FH_p=0_q=1 FH_p=1_q=0 FH_p=1_q=1
1: 1 119 10.908712 0.9500000 0.9420833 0.00000000 1.0000000 0.00000000
2: 1 103 10.148892 0.9408654 0.9318186 0.05042017 0.9495798 0.04787797
3: 1 98 9.899495 0.9218580 0.9125463 0.05963939 0.9403606 0.05608253
4: 1 89 9.433981 0.9013723 0.8913570 0.07883042 0.9211696 0.07261619
5: 1 79 8.888194 0.8788380 0.8678525 0.09953086 0.9004691 0.08962447
6: 1 73 8.544004 0.8669618 0.8552461 0.12232755 0.8776725 0.10736352
7: 1 66 8.124038 0.8540220 0.8412754 0.13435046 0.8656495 0.11630041
8: 1 55 7.416198 0.8235213 0.8088155 0.14746636 0.8525336 0.12572003
9: 1 49 7.000000 0.8070508 0.7909098 0.17846759 0.8215324 0.14661691
10: 1 45 6.708204 0.7895063 0.7723431 0.19523355 0.8047664 0.15711741
11: 1 40 6.324555 0.7702500 0.7514634 0.21311725 0.7868827 0.16769829
12: 1 25 5.000000 0.7110000 0.6836538 0.23278932 0.7672107 0.17859845
13: 1 23 4.795832 0.6813750 0.6529844 0.29416617 0.7058338 0.20763244
14: 1 20 4.472136 0.6489286 0.6180272 0.32485460 0.6751454 0.21932409
15: 1 9 3.000000 0.5840357 0.5256321 0.35861187 0.6413881 0.23000940
16: 1 5 2.236068 0.4866964 0.4055804 0.42987722 0.5701228 0.24508280
FH_p=0.5_q=0.5 FH_p=0.5_q=2 t
1: 0.0000000 0.000000000 0.5
2: 0.2188104 0.002477276 1.5
3: 0.2368175 0.003449162 2.5
4: 0.2694739 0.005964273 3.5
5: 0.2993735 0.009400478 4.5
6: 0.3276637 0.014018927 5.5
7: 0.3410285 0.016793815 6.5
8: 0.3545702 0.020078982 8.5
9: 0.3829059 0.028868954 9.5
10: 0.3963804 0.034193523 10.5
11: 0.4095098 0.040289532 11.5
12: 0.4226091 0.047466086 15.5
13: 0.4556670 0.072700383 16.5
14: 0.4683205 0.086711501 18.5
15: 0.4795929 0.102993492 23.5
16: 0.4950584 0.139531756 26.5
Q Var Z pNorm
1 2.1463e+00 1.9862e+01 0.48159 0.630098
n 4.9100e+02 6.0322e+04 1.99913 0.045594 *
sqrtN 4.3629e+01 9.8798e+02 1.38803 0.165129
S1 5.4126e+00 7.2723e+00 2.00710 0.044739 *
S2 5.3864e+00 7.0411e+00 2.02993 0.042364 *
FH_p=1_q=1 -8.9383e-02 7.1790e-01 -0.10549 0.915985
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
maxAbsZ Var Q pSupBr
1 9.8049 19.8617 2.2001 0.0556044 .
n 725.0000 60322.4412 2.9519 0.0063169 **
sqrtN 84.1532 987.9774 2.6773 0.0148437 *
S1 7.9752 7.2723 2.9574 0.0062054 **
S2 7.8688 7.0411 2.9654 0.0060450 **
FH_p=1_q=1 1.3396 0.7179 1.5811 0.2277168
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
chiSq df pChisq
1 13.8037 2 0.00100591 **
n 16.2407 2 0.00029743 ***
sqrtN 15.6529 2 0.00039904 ***
S1 15.7260 2 0.00038472 ***
S2 15.7781 2 0.00037483 ***
FH_p=1_q=0 15.6725 2 0.00039515 ***
FH_p=0_q=1 6.1097 2 0.04713019 *
FH_p=1_q=1 9.9331 2 0.00696710 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$tft
Q Var Z pNorm
1 -10.6695 42.7801 -1.63127 0.102834
n -1294.0000 439987.8847 -1.95081 0.051080 .
sqrtN -118.1769 4202.2583 -1.82302 0.068301 .
S1 -9.2667 23.2023 -1.92379 0.054380 .
S2 -9.1996 22.7588 -1.92839 0.053806 .
FH_p=1_q=0 -9.3529 23.6462 -1.92339 0.054432 .
FH_p=0_q=1 -1.3166 4.5757 -0.61551 0.538219
FH_p=1_q=1 -1.0948 1.4957 -0.89516 0.370703
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$scores
[1] 1 2 3
chiSq df pChisq
1 22.763 3 4.5252e-05 ***
n 23.177 3 3.7093e-05 ***
sqrtN 23.141 3 3.7746e-05 ***
S1 23.171 3 3.7199e-05 ***
S2 23.170 3 3.7213e-05 ***
FH_p=1_q=1 16.661 3 0.00082967 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$tft
Q Var Z pNorm
1 -25.8061 48.1505 -3.7190 0.00020005 ***
n -1939.0000 210644.6590 -4.2248 2.3919e-05 ***
sqrtN -221.9185 2990.6169 -4.0580 4.9493e-05 ***
S1 -21.3895 26.8311 -4.1293 3.6380e-05 ***
S2 -21.0942 26.0075 -4.1363 3.5292e-05 ***
FH_p=1_q=1 -2.9412 1.5056 -2.3970 0.01653027 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$scores
[1] 1 2 3 4
$tft
Q Var Z pNorm
1 -25.8061 48.1505 -3.7190 0.00020005 ***
n -1939.0000 210644.6590 -4.2248 2.3919e-05 ***
sqrtN -221.9185 2990.6169 -4.0580 4.9493e-05 ***
S1 -21.3895 26.8311 -4.1293 3.6380e-05 ***
S2 -21.0942 26.0075 -4.1363 3.5292e-05 ***
FH_p=1_q=1 -2.9412 1.5056 -2.3970 0.01653027 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$scores
[1] 1 2 3 4
Q Var Z pNorm
1 2.1698e+00 1.2353e+01 0.6173547 0.537001
n -8.1000e+01 6.7778e+04 -0.3111290 0.755703
sqrtN 1.8488e+00 8.7383e+02 0.0625435 0.950130
S1 1.4514e-02 7.0860e+00 0.0054524 0.995650
S2 -6.8838e-02 6.8889e+00 -0.0262274 0.979076
FH_p=0_q=1 2.0925e+00 1.0430e+00 2.0488836 0.040473 *
FH_p=1_q=1 1.0900e+00 4.0186e-01 1.7193838 0.085545 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
maxAbsZ Var Q pSupBr
1 4.5493e+00 1.2353e+01 1.2944 0.390844
n 4.1500e+02 6.7778e+04 1.5941 0.221844
sqrtN 4.3433e+01 8.7383e+02 1.4693 0.283499
S1 4.0680e+00 7.0860e+00 1.5282 0.252920
S2 4.0239e+00 6.8889e+00 1.5331 0.250497
FH_p=0_q=1 2.0925e+00 1.0430e+00 2.0489 0.080947 .
FH_p=1_q=1 1.0900e+00 4.0186e-01 1.7194 0.171089
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
survMisc documentation built on May 2, 2019, 5:14 p.m.
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Description Usage Arguments Details Value Note References Examples
Description
compare survival curves
Usage
1 2 3 4 5 |
Arguments
x |
A |
p |
p for Fleming-Harrington test |
q |
q for Fleming-Harrington test |
scores |
scores for tests for trend |
... |
Additional arguments (not implemented). |
reCalc |
Recalcuate the values?
|
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] - 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 = 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 | S1(t) = cumprod(1 - e / (n + 1)) |
| S2[i] | modified Peto-Peto (by Andersen) | S2(t) = S1[i] * n[i] / (n[i] + 1) |
| FH[i] | Fleming-Harrington | 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 log-rank 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[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:
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
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.
Value
The tne object is given
additional attributes.
The following are always added:
lrt |
The log-rank family of tests |
lrw |
The log-rank 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 |
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.
References
Gehan A. A Generalized Wilcoxon Test for Comparing Arbitrarily Singly-Censored Samples. Biometrika 1965 Jun. 52(1/2):203–23. JSTOR
Tarone RE, Ware J 1977 On Distribution-Free Tests for Equality of Survival Distributions. Biometrika;64(1):156–60. JSTOR
Peto R, Peto J 1972 Asymptotically Efficient Rank Invariant Test Procedures. J Royal Statistical Society 135(2):186–207. 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. JSTOR
Billingsly P 1999 Convergence of Probability Measures. New York: John Wiley & Sons. Wiley (paywall)
Examples
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 | ## Two covariate groups
## 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))
|
Example output
Loading required package: survival
Q Var Z pNorm
1 -3.9636e+00 6.2368e+00 -1.587104 0.1124892
n 9.0000e+00 3.8919e+04 0.045621 0.9636127
sqrtN -1.3203e+01 4.3384e+02 -0.633875 0.5261626
S1 -2.4692e+00 4.3725e+00 -1.180837 0.2376674
S2 -2.3134e+00 4.2087e+00 -1.127672 0.2594583
FH_p=0_q=1 -1.4134e+00 2.0778e-01 -3.100747 0.0019303 **
FH_p=1_q=0 -2.5501e+00 4.7072e+00 -1.175397 0.2398361
FH_p=1_q=1 -1.0206e+00 1.0661e-01 -3.125846 0.0017729 **
FH_p=0.5_q=0.5 -2.4695e+00 6.6091e-01 -3.037697 0.0023839 **
FH_p=0.5_q=2 -3.2350e-01 1.2842e-02 -2.854717 0.0043075 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
maxAbsZ Var Q pSupBr
1 3.9636e+00 6.2368e+00 1.5871 0.2249746
n 2.8200e+02 3.8919e+04 1.4294 0.3057172
sqrtN 2.6224e+01 4.3384e+02 1.2590 0.4157260
S1 2.4692e+00 4.3725e+00 1.1808 0.4745421
S2 2.3134e+00 4.2087e+00 1.1277 0.5174828
FH_p=0_q=1 1.4134e+00 2.0778e-01 3.1007 0.0038607 **
FH_p=1_q=0 2.5501e+00 4.7072e+00 1.1754 0.4788290
FH_p=1_q=1 1.0206e+00 1.0661e-01 3.1258 0.0035459 **
FH_p=0.5_q=0.5 2.4695e+00 6.6091e-01 3.0377 0.0047679 **
FH_p=0.5_q=2 3.2350e-01 1.2842e-02 2.8547 0.0086150 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
1 n sqrtN S1 S2 FH_p=0_q=1 FH_p=1_q=0 FH_p=1_q=1
1: 1 119 10.908712 0.9500000 0.9420833 0.00000000 1.0000000 0.00000000
2: 1 103 10.148892 0.9408654 0.9318186 0.05042017 0.9495798 0.04787797
3: 1 98 9.899495 0.9218580 0.9125463 0.05963939 0.9403606 0.05608253
4: 1 89 9.433981 0.9013723 0.8913570 0.07883042 0.9211696 0.07261619
5: 1 79 8.888194 0.8788380 0.8678525 0.09953086 0.9004691 0.08962447
6: 1 73 8.544004 0.8669618 0.8552461 0.12232755 0.8776725 0.10736352
7: 1 66 8.124038 0.8540220 0.8412754 0.13435046 0.8656495 0.11630041
8: 1 55 7.416198 0.8235213 0.8088155 0.14746636 0.8525336 0.12572003
9: 1 49 7.000000 0.8070508 0.7909098 0.17846759 0.8215324 0.14661691
10: 1 45 6.708204 0.7895063 0.7723431 0.19523355 0.8047664 0.15711741
11: 1 40 6.324555 0.7702500 0.7514634 0.21311725 0.7868827 0.16769829
12: 1 25 5.000000 0.7110000 0.6836538 0.23278932 0.7672107 0.17859845
13: 1 23 4.795832 0.6813750 0.6529844 0.29416617 0.7058338 0.20763244
14: 1 20 4.472136 0.6489286 0.6180272 0.32485460 0.6751454 0.21932409
15: 1 9 3.000000 0.5840357 0.5256321 0.35861187 0.6413881 0.23000940
16: 1 5 2.236068 0.4866964 0.4055804 0.42987722 0.5701228 0.24508280
FH_p=0.5_q=0.5 FH_p=0.5_q=2 t
1: 0.0000000 0.000000000 0.5
2: 0.2188104 0.002477276 1.5
3: 0.2368175 0.003449162 2.5
4: 0.2694739 0.005964273 3.5
5: 0.2993735 0.009400478 4.5
6: 0.3276637 0.014018927 5.5
7: 0.3410285 0.016793815 6.5
8: 0.3545702 0.020078982 8.5
9: 0.3829059 0.028868954 9.5
10: 0.3963804 0.034193523 10.5
11: 0.4095098 0.040289532 11.5
12: 0.4226091 0.047466086 15.5
13: 0.4556670 0.072700383 16.5
14: 0.4683205 0.086711501 18.5
15: 0.4795929 0.102993492 23.5
16: 0.4950584 0.139531756 26.5
Q Var Z pNorm
1 2.1463e+00 1.9862e+01 0.48159 0.630098
n 4.9100e+02 6.0322e+04 1.99913 0.045594 *
sqrtN 4.3629e+01 9.8798e+02 1.38803 0.165129
S1 5.4126e+00 7.2723e+00 2.00710 0.044739 *
S2 5.3864e+00 7.0411e+00 2.02993 0.042364 *
FH_p=1_q=1 -8.9383e-02 7.1790e-01 -0.10549 0.915985
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
maxAbsZ Var Q pSupBr
1 9.8049 19.8617 2.2001 0.0556044 .
n 725.0000 60322.4412 2.9519 0.0063169 **
sqrtN 84.1532 987.9774 2.6773 0.0148437 *
S1 7.9752 7.2723 2.9574 0.0062054 **
S2 7.8688 7.0411 2.9654 0.0060450 **
FH_p=1_q=1 1.3396 0.7179 1.5811 0.2277168
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
chiSq df pChisq
1 13.8037 2 0.00100591 **
n 16.2407 2 0.00029743 ***
sqrtN 15.6529 2 0.00039904 ***
S1 15.7260 2 0.00038472 ***
S2 15.7781 2 0.00037483 ***
FH_p=1_q=0 15.6725 2 0.00039515 ***
FH_p=0_q=1 6.1097 2 0.04713019 *
FH_p=1_q=1 9.9331 2 0.00696710 **
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$tft
Q Var Z pNorm
1 -10.6695 42.7801 -1.63127 0.102834
n -1294.0000 439987.8847 -1.95081 0.051080 .
sqrtN -118.1769 4202.2583 -1.82302 0.068301 .
S1 -9.2667 23.2023 -1.92379 0.054380 .
S2 -9.1996 22.7588 -1.92839 0.053806 .
FH_p=1_q=0 -9.3529 23.6462 -1.92339 0.054432 .
FH_p=0_q=1 -1.3166 4.5757 -0.61551 0.538219
FH_p=1_q=1 -1.0948 1.4957 -0.89516 0.370703
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$scores
[1] 1 2 3
chiSq df pChisq
1 22.763 3 4.5252e-05 ***
n 23.177 3 3.7093e-05 ***
sqrtN 23.141 3 3.7746e-05 ***
S1 23.171 3 3.7199e-05 ***
S2 23.170 3 3.7213e-05 ***
FH_p=1_q=1 16.661 3 0.00082967 ***
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$tft
Q Var Z pNorm
1 -25.8061 48.1505 -3.7190 0.00020005 ***
n -1939.0000 210644.6590 -4.2248 2.3919e-05 ***
sqrtN -221.9185 2990.6169 -4.0580 4.9493e-05 ***
S1 -21.3895 26.8311 -4.1293 3.6380e-05 ***
S2 -21.0942 26.0075 -4.1363 3.5292e-05 ***
FH_p=1_q=1 -2.9412 1.5056 -2.3970 0.01653027 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$scores
[1] 1 2 3 4
$tft
Q Var Z pNorm
1 -25.8061 48.1505 -3.7190 0.00020005 ***
n -1939.0000 210644.6590 -4.2248 2.3919e-05 ***
sqrtN -221.9185 2990.6169 -4.0580 4.9493e-05 ***
S1 -21.3895 26.8311 -4.1293 3.6380e-05 ***
S2 -21.0942 26.0075 -4.1363 3.5292e-05 ***
FH_p=1_q=1 -2.9412 1.5056 -2.3970 0.01653027 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
$scores
[1] 1 2 3 4
Q Var Z pNorm
1 2.1698e+00 1.2353e+01 0.6173547 0.537001
n -8.1000e+01 6.7778e+04 -0.3111290 0.755703
sqrtN 1.8488e+00 8.7383e+02 0.0625435 0.950130
S1 1.4514e-02 7.0860e+00 0.0054524 0.995650
S2 -6.8838e-02 6.8889e+00 -0.0262274 0.979076
FH_p=0_q=1 2.0925e+00 1.0430e+00 2.0488836 0.040473 *
FH_p=1_q=1 1.0900e+00 4.0186e-01 1.7193838 0.085545 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
maxAbsZ Var Q pSupBr
1 4.5493e+00 1.2353e+01 1.2944 0.390844
n 4.1500e+02 6.7778e+04 1.5941 0.221844
sqrtN 4.3433e+01 8.7383e+02 1.4693 0.283499
S1 4.0680e+00 7.0860e+00 1.5282 0.252920
S2 4.0239e+00 6.8889e+00 1.5331 0.250497
FH_p=0_q=1 2.0925e+00 1.0430e+00 2.0489 0.080947 .
FH_p=1_q=1 1.0900e+00 4.0186e-01 1.7194 0.171089
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
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