ci: *c*onfidence *i*ntervals for survival curves.
In survMisc: Miscellaneous Functions for Survival Data
Description
Usage
Arguments
Details
Value
Note
Source
References
See Also
Examples
Description
confidence intervals for survival curves.
Usage
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ci(x, ...)
## S3 method for class 'ten'
ci(x, ..., CI = c("0.95", "0.9", "0.99"), how = c("point",
"nair", "hall"), trans = c("log", "lin", "asi"), tL = NULL, tU = NULL,
reCalc = FALSE)
## S3 method for class 'stratTen'
ci(x, ..., CI = c("0.95", "0.9", "0.99"),
how = c("point", "nair", "hall"), trans = c("log", "lin", "asi"),
tL = NULL, tU = NULL)
Arguments
x
An object of class ten.
CI
Confidence intervals. As the function currently relies on lookup
tables, currently only 90%, 95% (the default) and 99% are supported.
how
Method to use for confidence interval.
point (the default) uses pointwise confirence intervals.
The alternatives use confidence bands (see details).
trans
Transformation to use.
The default is trans="log".
Also supported are linear and arcsine-square root transformations.
tL
Lower time point. Used in construction of confidence bands.
tU
Upper time point. Used in construction of confidence bands.
...
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
In the equations below
sigma^2(t) = V[S(t)]/[S(t)]^2
Where S(t) is the Kaplan-Meier survival estimate and
V[S(t)] is Greenwood's estimate of its
variance.
The pointwise confidence intervals are valid for individual
times, e.g. median and quantile values.
When plotted and joined for multiple points they tend to
be narrower than the bands described below.
Thus they tend to exaggerate the impression of certainty
when used to plot confidence intervals for a time range.
They should not be interpreted as giving the intervals
within which the entire survival function lies.
For a given significance level alpha,
they are calculated using the standard normal distribution Z
as follows:
linear
S(t)+- Z(1-alpha) sigma(t) S(t)
log transform
[S(t)^(1/theta), S(t)^theta]
where
theta = exp ( Z(1-alpha)sigma(t) / log(S(t)) )
arcsine-square root transform
upper:
sin^2(max[0, arcsin S(t)^0.5 - Z(1-alpha)sigma(t)/2 (S(t)/1-S(t))^0.5])
lower:
sin^2(min[pi/2, arcsin S(t)^0.5 + Z(1-alpha)sigma(t)/2 (S(t)/1-S(t))^0.5])
Confidence bands give the values within which the survival function
falls within a range of timepoints.
The time range under consideration is given so that
tL >= min(t), the minimum or lowest event time and
tU <= max(t), the maximum or largest event time.
For a sample size n and 0 < a_l < a_u <1:
a_l = n*sigma^2(t_l) / [1+n*sigma^2(t_l)]
a_u = n*sigma^2(t_u) / [1+n*sigma^2(t_u)]
For the Nair or equal precision (EP) confidence bands,
we begin by obtaining the relevant
confidence coefficient c[alpha]. This is obtained from
the upper a-th fractile of the random variable
U = sup{ |W(x)[x(1-x)]^0.5|, a_l <= x <= a_u}
Where W is a standard Brownian bridge.
The intervals are:
linear
S(t)+- c[alpha] sigma(t) S(t)
log transform (the default)
This uses theta as below:
theta = exp (c[alpha] * sigma(t) / log(S(t)))
And is given by:
[S(t)^(1/theta), S(t)^theta]
arcsine-square root transform
upper:
sin^2(max[0, arcsin S(t)^0.5 - c[alpha]*sigma(t)/2 (S(t)/1-S(t))^0.5])
lower:
sin^2(min[pi/2, arcsin S(t)^0.5 - c[alpha]*sigma(t)/2 (S(t)/1-S(t))^0.5])
For the Hall-Wellner bands the confidence coefficient
k[alpha]
is obtained from the upper a-th fractile of a
Brownian bridge.
In this case t_l can be =0.
The intervals are:
linear
S(t)+- k[alpha] [1+n*sigma^2(t)]*S(t) / n^0.5
log transform
[S(t)^(1/theta), S(t)^theta]
where
theta = exp(k[alpha] * [1 + n * sigma^2(t)] / n^0.5 * log(S(t)))
arcsine-square root transform
upper:
sin^2( max[0, arcsin S(t)^0.5 - k[alpha]*[1+n*sigma^2(t)]/(2*n^0.5) (S(t)/1-S(t))^0.5])
lower:
sin^2( min[pi/2, arcsin S(t)^0.5 - k[alpha]*[1+n*sigma^2(t)]/(2*n^0.5) (S(t)/1-S(t))^0.5])
Value
The ten object is modified in place by the additional of a
data.table as an attribute.
attr(x, "ci") is printed.
This A survfit object. The upper and lower
elements in the list (representing confidence intervals)
are modified from the original.
Other elements will also be shortened if the time range under consideration has been
reduced from the original.
Note
For the Nair and Hall-Wellner bands, the function currently
relies on the lookup tables in package:km.ci.
Generally, the arcsin-square root transform has the best coverage properties.
All bands have good coverage properties for samples as small as n=20,
except for the Nair / EP bands with a linear transformation,
which perform poorly when n < 200.
Source
The function is loosely based on km.ci::km.ci.
References
Nair V, 1984.
Confidence bands for survival functions with censored data: a comparative study.
Technometrics. 26(3):265-75.
JSTOR.
Hall WJ, Wellner JA, 1980.
Confidence bands for a survival curve from censored data.
Biometrika. 67(1):133-43.
JSTOR.
See Also
Examples
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## K&M 2nd ed. Section 4.3. Example 4.2, pg 105.
data("bmt", package="KMsurv")
b1 <- bmt[bmt$group==1, ] # ALL patients
## K&M 2nd ed. Section 4.4. Example 4.2 (cont.), pg 111.
## patients with ALL
t1 <- ten(Surv(t2, d3) ~ 1, data=bmt[bmt$group==1, ])
ci(t1, how="nair", trans="lin", tL=100, tU=600)
## Table 4.5, pg. 111.
lapply(list("lin", "log", "asi"),
function(x) ci(t1, how="nair", trans=x, tL=100, tU=600))
## Table 4.6, pg. 111.
lapply(list("lin", "log", "asi"),
function(x) ci(t1, how="hall", trans=x, tL=100, tU=600))
t1 <- ten(Surv(t2, d3) ~ group, data=bmt)
ci(t1, CI="0.95", how="nair", trans="lin", tL=100, tU=600)
## stratified model
data("pbc", package="survival")
t1 <- ten(coxph(Surv(time, status==2) ~ log(bili) + age + strata(edema), data=pbc))
ci(t1)
Example output
Loading required package: survival
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[1]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[2]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[3]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[1]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[2]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[3]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
20: 2 100 0.87 0.0021 0.0028 0.75 0.99
21: 2 210 0.85 0.0023 0.0032 0.72 0.98
22: 2 220 0.83 0.0026 0.0037 0.7 0.97
23: 2 250 0.81 0.0028 0.0042 0.67 0.96
24: 2 270 0.8 0.003 0.0047 0.65 0.94
25: 2 290 0.78 0.0032 0.0053 0.63 0.93
26: 2 380 0.76 0.0034 0.0059 0.6 0.92
27: 2 390 0.74 0.0036 0.0065 0.58 0.9
28: 2 410 0.72 0.0037 0.0071 0.56 0.89
29: 2 420 0.7 0.0039 0.0078 0.54 0.87
30: 2 480 0.69 0.004 0.0085 0.52 0.85
31: 2 490 0.67 0.0041 0.0093 0.49 0.84
32: 3 100 0.69 0.0048 0.01 0.5 0.88
33: 3 100 0.67 0.0049 0.011 0.47 0.86
34: 3 110 0.64 0.0051 0.012 0.45 0.84
35: 3 120 0.62 0.0052 0.013 0.42 0.82
36: 3 120 0.6 0.0053 0.015 0.4 0.8
37: 3 160 0.58 0.0054 0.016 0.38 0.78
38: 3 160 0.56 0.0055 0.018 0.35 0.76
39: 3 160 0.53 0.0055 0.019 0.33 0.74
40: 3 170 0.51 0.0056 0.021 0.31 0.72
41: 3 180 0.49 0.0056 0.023 0.28 0.69
42: 3 240 0.47 0.0055 0.025 0.26 0.67
43: 3 270 0.44 0.0055 0.028 0.24 0.65
44: 3 270 0.42 0.0054 0.03 0.22 0.62
45: 3 320 0.4 0.0053 0.033 0.2 0.6
46: 3 360 0.38 0.0052 0.037 0.18 0.58
47: 3 390 0.36 0.0051 0.04 0.16 0.55
48: 3 420 0.33 0.0049 0.044 0.14 0.53
49: 3 460 0.31 0.0048 0.049 0.12 0.5
50: 3 470 0.29 0.0046 0.055 0.1 0.47
cg t S Sv SCV lower upper
$`1`
cg t S Sv SCV lower upper
1: 1 1800 1 0 0 1 1
2: 2 1100 1 0 0 1 1
3: 3 1600 1 0 0 1 1
4: 4 1900 1 0 0 1 1
5: 5 3600 1 0 0 1 1
---
350: 350 1000 0 NaN NaN NaN NaN
351: 350 1400 0 NaN NaN NaN NaN
352: 350 2300 1 0 0 1 1
353: 350 220 0 NaN NaN NaN NaN
354: 350 850 0 NaN NaN NaN NaN
$`2`
cg t S Sv SCV lower upper
1: 1 1200 0 NaN NaN NaN NaN
2: 2 1400 0 NaN NaN NaN NaN
3: 3 190 0 NaN NaN NaN NaN
4: 4 390 0 NaN NaN NaN NaN
5: 5 140 0 NaN NaN NaN NaN
6: 6 110 0 NaN NaN NaN NaN
7: 7 3400 0 NaN NaN NaN NaN
8: 8 970 0 NaN NaN NaN NaN
9: 9 180 0 NaN NaN NaN NaN
10: 10 220 0 NaN NaN NaN NaN
11: 11 1300 1 0 0 1 1
12: 12 130 0 NaN NaN NaN NaN
13: 13 51 0 NaN NaN NaN NaN
14: 14 330 0 NaN NaN NaN NaN
15: 15 400 0 NaN NaN NaN NaN
16: 16 550 0 NaN NaN NaN NaN
17: 17 130 0 NaN NaN NaN NaN
18: 18 260 0 NaN NaN NaN NaN
19: 19 41 0 NaN NaN NaN NaN
20: 20 860 0 NaN NaN NaN NaN
cg t S Sv SCV lower upper
$`3`
cg t S Sv SCV lower upper
1: 1 1000 1 0 0 1 1
2: 2 4200 1 0 0 1 1
3: 3 190 0 NaN NaN NaN NaN
4: 4 2100 0 NaN NaN NaN NaN
5: 5 2500 1 0 0 1 1
6: 6 2700 1 0 0 1 1
7: 7 1900 1 0 0 1 1
8: 8 560 0 NaN NaN NaN NaN
9: 9 2200 1 0 0 1 1
10: 10 2400 1 0 0 1 1
11: 11 1400 1 0 0 1 1
12: 12 1300 1 0 0 1 1
13: 13 2600 1 0 0 1 1
14: 14 3000 1 0 0 1 1
15: 15 4000 1 0 0 1 1
16: 16 1200 1 0 0 1 1
17: 17 3600 0 NaN NaN NaN NaN
18: 18 1000 0 NaN NaN NaN NaN
19: 19 1900 0 NaN NaN NaN NaN
20: 20 1200 0 NaN NaN NaN NaN
21: 21 1100 1 0 0 1 1
22: 22 1400 1 0 0 1 1
23: 23 1600 0 NaN NaN NaN NaN
24: 24 610 0 NaN NaN NaN NaN
25: 25 1600 0 NaN NaN NaN NaN
26: 26 3300 0 NaN NaN NaN NaN
27: 27 3300 1 0 0 1 1
28: 28 760 0 NaN NaN NaN NaN
29: 29 94 0 NaN NaN NaN NaN
30: 30 600 0 NaN NaN NaN NaN
31: 31 800 0 NaN NaN NaN NaN
32: 32 350 0 NaN NaN NaN NaN
33: 33 460 0 NaN NaN NaN NaN
34: 34 1200 0 NaN NaN NaN NaN
35: 35 330 0 NaN NaN NaN NaN
36: 36 490 0 NaN NaN NaN NaN
37: 37 260 0 NaN NaN NaN NaN
38: 38 1100 1 0 0 1 1
39: 39 71 0 NaN NaN NaN NaN
40: 40 1700 1 0 0 1 1
41: 41 1200 0 NaN NaN NaN NaN
42: 42 1200 0 NaN NaN NaN NaN
43: 43 77 0 NaN NaN NaN NaN
44: 44 940 0 NaN NaN NaN NaN
cg t S Sv SCV lower upper
survMisc documentation built on May 2, 2019, 5:14 p.m.
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Description Usage Arguments Details Value Note Source References See Also Examples
Description
confidence intervals for survival curves.
Usage
1 2 3 4 5 6 7 8 9 10 11 | ci(x, ...)
## S3 method for class 'ten'
ci(x, ..., CI = c("0.95", "0.9", "0.99"), how = c("point",
"nair", "hall"), trans = c("log", "lin", "asi"), tL = NULL, tU = NULL,
reCalc = FALSE)
## S3 method for class 'stratTen'
ci(x, ..., CI = c("0.95", "0.9", "0.99"),
how = c("point", "nair", "hall"), trans = c("log", "lin", "asi"),
tL = NULL, tU = NULL)
|
Arguments
x |
An object of class |
CI |
Confidence intervals. As the function currently relies on lookup tables, currently only 90%, 95% (the default) and 99% are supported. |
how |
Method to use for confidence interval.
|
trans |
Transformation to use.
|
tL |
Lower time point. Used in construction of confidence bands. |
tU |
Upper time point. Used in construction of confidence bands. |
... |
Additional arguments (not implemented). |
reCalc |
Recalcuate the values?
|
Details
In the equations below
sigma^2(t) = V[S(t)]/[S(t)]^2
Where S(t) is the Kaplan-Meier survival estimate and
V[S(t)] is Greenwood's estimate of its
variance.
The pointwise confidence intervals are valid for individual
times, e.g. median and quantile values.
When plotted and joined for multiple points they tend to
be narrower than the bands described below.
Thus they tend to exaggerate the impression of certainty
when used to plot confidence intervals for a time range.
They should not be interpreted as giving the intervals
within which the entire survival function lies.
For a given significance level alpha,
they are calculated using the standard normal distribution Z
as follows:
linear
S(t)+- Z(1-alpha) sigma(t) S(t)
log transform
[S(t)^(1/theta), S(t)^theta]
where
theta = exp ( Z(1-alpha)sigma(t) / log(S(t)) )
arcsine-square root transform
upper:
sin^2(max[0, arcsin S(t)^0.5 - Z(1-alpha)sigma(t)/2 (S(t)/1-S(t))^0.5])
lower:
sin^2(min[pi/2, arcsin S(t)^0.5 + Z(1-alpha)sigma(t)/2 (S(t)/1-S(t))^0.5])
Confidence bands give the values within which the survival function
falls within a range of timepoints.
The time range under consideration is given so that
tL >= min(t), the minimum or lowest event time and
tU <= max(t), the maximum or largest event time.
For a sample size n and 0 < a_l < a_u <1:
a_l = n*sigma^2(t_l) / [1+n*sigma^2(t_l)]
a_u = n*sigma^2(t_u) / [1+n*sigma^2(t_u)]
For the Nair or equal precision (EP) confidence bands, we begin by obtaining the relevant confidence coefficient c[alpha]. This is obtained from the upper a-th fractile of the random variable
U = sup{ |W(x)[x(1-x)]^0.5|, a_l <= x <= a_u}
Where W is a standard Brownian bridge.
The intervals are:
linear
S(t)+- c[alpha] sigma(t) S(t)
log transform (the default)
This uses theta as below:theta = exp (c[alpha] * sigma(t) / log(S(t)))
And is given by:
[S(t)^(1/theta), S(t)^theta]
arcsine-square root transform
upper:sin^2(max[0, arcsin S(t)^0.5 - c[alpha]*sigma(t)/2 (S(t)/1-S(t))^0.5])
lower:
sin^2(min[pi/2, arcsin S(t)^0.5 - c[alpha]*sigma(t)/2 (S(t)/1-S(t))^0.5])
For the Hall-Wellner bands the confidence coefficient
k[alpha]
is obtained from the upper a-th fractile of a
Brownian bridge.
In this case t_l can be =0.
The intervals are:
linear
S(t)+- k[alpha] [1+n*sigma^2(t)]*S(t) / n^0.5
log transform
[S(t)^(1/theta), S(t)^theta]
where
theta = exp(k[alpha] * [1 + n * sigma^2(t)] / n^0.5 * log(S(t)))
arcsine-square root transform
upper:sin^2( max[0, arcsin S(t)^0.5 - k[alpha]*[1+n*sigma^2(t)]/(2*n^0.5) (S(t)/1-S(t))^0.5])
lower:
sin^2( min[pi/2, arcsin S(t)^0.5 - k[alpha]*[1+n*sigma^2(t)]/(2*n^0.5) (S(t)/1-S(t))^0.5])
Value
The ten object is modified in place by the additional of a
data.table as an attribute.
attr(x, "ci") is printed.
This A survfit object. The upper and lower
elements in the list (representing confidence intervals)
are modified from the original.
Other elements will also be shortened if the time range under consideration has been
reduced from the original.
Note
For the Nair and Hall-Wellner bands, the function currently relies on the lookup tables in
package:km.ci.Generally, the arcsin-square root transform has the best coverage properties.
All bands have good coverage properties for samples as small as n=20, except for the Nair / EP bands with a linear transformation, which perform poorly when n < 200.
Source
The function is loosely based on km.ci::km.ci.
References
Nair V, 1984. Confidence bands for survival functions with censored data: a comparative study. Technometrics. 26(3):265-75. JSTOR.
Hall WJ, Wellner JA, 1980. Confidence bands for a survival curve from censored data. Biometrika. 67(1):133-43. JSTOR.
See Also
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 | ## K&M 2nd ed. Section 4.3. Example 4.2, pg 105.
data("bmt", package="KMsurv")
b1 <- bmt[bmt$group==1, ] # ALL patients
## K&M 2nd ed. Section 4.4. Example 4.2 (cont.), pg 111.
## patients with ALL
t1 <- ten(Surv(t2, d3) ~ 1, data=bmt[bmt$group==1, ])
ci(t1, how="nair", trans="lin", tL=100, tU=600)
## Table 4.5, pg. 111.
lapply(list("lin", "log", "asi"),
function(x) ci(t1, how="nair", trans=x, tL=100, tU=600))
## Table 4.6, pg. 111.
lapply(list("lin", "log", "asi"),
function(x) ci(t1, how="hall", trans=x, tL=100, tU=600))
t1 <- ten(Surv(t2, d3) ~ group, data=bmt)
ci(t1, CI="0.95", how="nair", trans="lin", tL=100, tU=600)
## stratified model
data("pbc", package="survival")
t1 <- ten(coxph(Surv(time, status==2) ~ log(bili) + age + strata(edema), data=pbc))
ci(t1)
|
Example output
Loading required package: survival
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[1]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[2]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[3]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[1]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[2]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
[[3]]
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
cg t S Sv SCV lower upper
cg t S Sv SCV lower upper
1: 1 100 0.87 0.003 0.004 0.71 1
2: 1 110 0.84 0.0035 0.0049 0.67 1
3: 1 110 0.82 0.004 0.0059 0.63 1
4: 1 110 0.79 0.0044 0.007 0.6 0.98
5: 1 120 0.74 0.0051 0.0094 0.53 0.94
6: 1 130 0.71 0.0054 0.011 0.5 0.92
7: 1 170 0.68 0.0057 0.012 0.47 0.9
8: 1 190 0.66 0.0059 0.014 0.44 0.88
9: 1 190 0.63 0.0061 0.015 0.41 0.86
10: 1 230 0.63 0.0061 0.015 0.41 0.86
11: 1 230 0.6 0.0063 0.017 0.37 0.83
12: 1 280 0.58 0.0065 0.019 0.34 0.81
13: 1 330 0.55 0.0066 0.022 0.32 0.78
14: 1 380 0.52 0.0067 0.025 0.29 0.76
15: 1 420 0.49 0.0067 0.027 0.26 0.73
16: 1 470 0.47 0.0067 0.031 0.23 0.7
17: 1 490 0.44 0.0066 0.034 0.2 0.67
18: 1 530 0.41 0.0065 0.039 0.18 0.64
19: 1 530 0.41 0.0065 0.039 0.18 0.64
20: 2 100 0.87 0.0021 0.0028 0.75 0.99
21: 2 210 0.85 0.0023 0.0032 0.72 0.98
22: 2 220 0.83 0.0026 0.0037 0.7 0.97
23: 2 250 0.81 0.0028 0.0042 0.67 0.96
24: 2 270 0.8 0.003 0.0047 0.65 0.94
25: 2 290 0.78 0.0032 0.0053 0.63 0.93
26: 2 380 0.76 0.0034 0.0059 0.6 0.92
27: 2 390 0.74 0.0036 0.0065 0.58 0.9
28: 2 410 0.72 0.0037 0.0071 0.56 0.89
29: 2 420 0.7 0.0039 0.0078 0.54 0.87
30: 2 480 0.69 0.004 0.0085 0.52 0.85
31: 2 490 0.67 0.0041 0.0093 0.49 0.84
32: 3 100 0.69 0.0048 0.01 0.5 0.88
33: 3 100 0.67 0.0049 0.011 0.47 0.86
34: 3 110 0.64 0.0051 0.012 0.45 0.84
35: 3 120 0.62 0.0052 0.013 0.42 0.82
36: 3 120 0.6 0.0053 0.015 0.4 0.8
37: 3 160 0.58 0.0054 0.016 0.38 0.78
38: 3 160 0.56 0.0055 0.018 0.35 0.76
39: 3 160 0.53 0.0055 0.019 0.33 0.74
40: 3 170 0.51 0.0056 0.021 0.31 0.72
41: 3 180 0.49 0.0056 0.023 0.28 0.69
42: 3 240 0.47 0.0055 0.025 0.26 0.67
43: 3 270 0.44 0.0055 0.028 0.24 0.65
44: 3 270 0.42 0.0054 0.03 0.22 0.62
45: 3 320 0.4 0.0053 0.033 0.2 0.6
46: 3 360 0.38 0.0052 0.037 0.18 0.58
47: 3 390 0.36 0.0051 0.04 0.16 0.55
48: 3 420 0.33 0.0049 0.044 0.14 0.53
49: 3 460 0.31 0.0048 0.049 0.12 0.5
50: 3 470 0.29 0.0046 0.055 0.1 0.47
cg t S Sv SCV lower upper
$`1`
cg t S Sv SCV lower upper
1: 1 1800 1 0 0 1 1
2: 2 1100 1 0 0 1 1
3: 3 1600 1 0 0 1 1
4: 4 1900 1 0 0 1 1
5: 5 3600 1 0 0 1 1
---
350: 350 1000 0 NaN NaN NaN NaN
351: 350 1400 0 NaN NaN NaN NaN
352: 350 2300 1 0 0 1 1
353: 350 220 0 NaN NaN NaN NaN
354: 350 850 0 NaN NaN NaN NaN
$`2`
cg t S Sv SCV lower upper
1: 1 1200 0 NaN NaN NaN NaN
2: 2 1400 0 NaN NaN NaN NaN
3: 3 190 0 NaN NaN NaN NaN
4: 4 390 0 NaN NaN NaN NaN
5: 5 140 0 NaN NaN NaN NaN
6: 6 110 0 NaN NaN NaN NaN
7: 7 3400 0 NaN NaN NaN NaN
8: 8 970 0 NaN NaN NaN NaN
9: 9 180 0 NaN NaN NaN NaN
10: 10 220 0 NaN NaN NaN NaN
11: 11 1300 1 0 0 1 1
12: 12 130 0 NaN NaN NaN NaN
13: 13 51 0 NaN NaN NaN NaN
14: 14 330 0 NaN NaN NaN NaN
15: 15 400 0 NaN NaN NaN NaN
16: 16 550 0 NaN NaN NaN NaN
17: 17 130 0 NaN NaN NaN NaN
18: 18 260 0 NaN NaN NaN NaN
19: 19 41 0 NaN NaN NaN NaN
20: 20 860 0 NaN NaN NaN NaN
cg t S Sv SCV lower upper
$`3`
cg t S Sv SCV lower upper
1: 1 1000 1 0 0 1 1
2: 2 4200 1 0 0 1 1
3: 3 190 0 NaN NaN NaN NaN
4: 4 2100 0 NaN NaN NaN NaN
5: 5 2500 1 0 0 1 1
6: 6 2700 1 0 0 1 1
7: 7 1900 1 0 0 1 1
8: 8 560 0 NaN NaN NaN NaN
9: 9 2200 1 0 0 1 1
10: 10 2400 1 0 0 1 1
11: 11 1400 1 0 0 1 1
12: 12 1300 1 0 0 1 1
13: 13 2600 1 0 0 1 1
14: 14 3000 1 0 0 1 1
15: 15 4000 1 0 0 1 1
16: 16 1200 1 0 0 1 1
17: 17 3600 0 NaN NaN NaN NaN
18: 18 1000 0 NaN NaN NaN NaN
19: 19 1900 0 NaN NaN NaN NaN
20: 20 1200 0 NaN NaN NaN NaN
21: 21 1100 1 0 0 1 1
22: 22 1400 1 0 0 1 1
23: 23 1600 0 NaN NaN NaN NaN
24: 24 610 0 NaN NaN NaN NaN
25: 25 1600 0 NaN NaN NaN NaN
26: 26 3300 0 NaN NaN NaN NaN
27: 27 3300 1 0 0 1 1
28: 28 760 0 NaN NaN NaN NaN
29: 29 94 0 NaN NaN NaN NaN
30: 30 600 0 NaN NaN NaN NaN
31: 31 800 0 NaN NaN NaN NaN
32: 32 350 0 NaN NaN NaN NaN
33: 33 460 0 NaN NaN NaN NaN
34: 34 1200 0 NaN NaN NaN NaN
35: 35 330 0 NaN NaN NaN NaN
36: 36 490 0 NaN NaN NaN NaN
37: 37 260 0 NaN NaN NaN NaN
38: 38 1100 1 0 0 1 1
39: 39 71 0 NaN NaN NaN NaN
40: 40 1700 1 0 0 1 1
41: 41 1200 0 NaN NaN NaN NaN
42: 42 1200 0 NaN NaN NaN NaN
43: 43 77 0 NaN NaN NaN NaN
44: 44 940 0 NaN NaN NaN NaN
cg t S Sv SCV lower upper
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