Description Usage Arguments Details Value Note Source References See Also Examples
confidence intervals for survival curves.
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)

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?

In the equations below
sigma^2(t) = V[S(t)]/[S(t)]^2
Where S(t) is the KaplanMeier 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(1alpha) sigma(t) S(t)
log transform
[S(t)^(1/theta), S(t)^theta]
where
theta = exp ( Z(1alpha)sigma(t) / log(S(t)) )
arcsinesquare root transform
upper:
sin^2(max[0, arcsin S(t)^0.5  Z(1alpha)sigma(t)/2 (S(t)/1S(t))^0.5])
lower:
sin^2(min[pi/2, arcsin S(t)^0.5 + Z(1alpha)sigma(t)/2 (S(t)/1S(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 ath fractile of the random variable
U = sup{ W(x)[x(1x)]^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]
arcsinesquare root transform
upper:
sin^2(max[0, arcsin S(t)^0.5  c[alpha]*sigma(t)/2 (S(t)/1S(t))^0.5])
lower:
sin^2(min[pi/2, arcsin S(t)^0.5  c[alpha]*sigma(t)/2 (S(t)/1S(t))^0.5])
For the HallWellner bands the confidence coefficient
k[alpha]
is obtained from the upper ath 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)))
arcsinesquare 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)/1S(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)/1S(t))^0.5])
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.
For the Nair and HallWellner bands, the function currently
relies on the lookup tables in package:km.ci
.
Generally, the arcsinsquare 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.
The function is loosely based on km.ci::km.ci
.
Nair V, 1984. Confidence bands for survival functions with censored data: a comparative study. Technometrics. 26(3):26575. JSTOR.
Hall WJ, Wellner JA, 1980. Confidence bands for a survival curve from censored data. Biometrika. 67(1):13343. JSTOR.
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)

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