survMisc (version 0.4.6)

ci: Confidence intervals for survival curves

Description

Confidence intervals for survival curves

Usage

ci(x, ...)
## S3 method for class 'survfit':
ci(x, ..., CI = c("0.95", "0.9", "0.99"), how = c("point",
  "nair", "hall"), trans = c("log", "lin", "asin"), tL = NULL, tU = NULL)

Arguments

x

An object of class survfit

...

Additional arguments (not implemented)

CI

Confidence intervals. As the function currently relies on lookup tables, currently only 95% (the default), 90% 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.

Value

• 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.

source

The function is loosely based on km.ci::km.ci.

Details

In the equations below $$\sigma^2_s(t) = \frac{\hat{V}[\hat{S}(t)]}{\hat{S}^2(t)}$$ Where $\hat{S}(t)$ is the Kaplan-Meier survival estimate and $\hat{V}[\hat{S}(t)]$ 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$$\hat{S}(t) \pm Z_{1- \alpha} \sigma (t) \hat{S}(t)$$
• log transform$$[ \hat{S}(t)^{\frac{1}{\theta}}, \hat{S}(t)^{\theta} ]$$where$$\theta = \exp{ \frac{Z_{1- \alpha} \sigma (t)}{ \log{\hat{S}(t)}}}$$
• arcsine-square root transform upper: $$\sin^2(\max[0, \arcsin{\sqrt{\hat{S}(t)}} - \frac{Z_{1- \alpha}\sigma(t)}{2} \sqrt{ \frac{\hat{S}(t)}{1-\hat{S}(t)}}])$$lower:$$\sin^2(\min[\frac{\pi}{2}, \arcsin{\sqrt{\hat{S}(t)}} + \frac{Z_{1- \alpha}\sigma(t)}{2} \sqrt{ \frac{\hat{S}(t)}{1-\hat{S}(t)}}])$$

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 $t_l \geq t_{min}$, the minimum or lowest event time and $t_u \leq t_{max}$, the maximum or largest event time. For a sample size $n$ and $0 < a_l < a_u <1$: $$a_l="\frac{n\sigma^2_s(t_l)}{1+n\sigma^2_s(t_l)}$$" $$a_u="\frac{n\sigma^2_s(t_u)}{1+n\sigma^2_s(t_u)}$$

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 $\alpha$-th fractile of the random variable $$U = \sup{|W^o(x)\sqrt{[x(1-x)]}|, \quad a_l \leq x \leq a_u}$$ Where $W^o$ is a standard Brownian bridge. The intervals are:

• linear$$\hat{S}(t) \pm c_{\alpha} \sigma_s(t) \hat{S}(t)$$
• log transform (the default)$$[ \hat{S}(t)^{\frac{1}{\theta}}, \hat{S}(t)^{\theta} ]$$where$$\theta = \exp{ \frac{c_{\alpha} \sigma_s(t)}{ \log{\hat{S}(t)}}}$$
• arcsine-square root transform upper:$$\sin^2(\max[0, \arcsin{\sqrt{\hat{S}(t)}} - \frac{c_{\alpha}\sigma_s(t)}{2} \sqrt{ \frac{\hat{S}(t)}{1-\hat{S}(t)}}])$$lower:$$\sin^2(\min[\frac{\pi}{2}, \arcsin{\sqrt{\hat{S}(t)}} + \frac{c_{\alpha}\sigma_s(t)}{2} \sqrt{ \frac{\hat{S}(t)}{1-\hat{S}(t)}}])$$

For the Hall-Wellner bands the confidence coefficient $k_{\alpha}$ is obtained from the upper $\alpha$-th fractile of a Brownian bridge. In this case $t_l$ can be $=0$. The intervals are:

• linear$$\hat{S}(t) \pm k_{\alpha} \frac{1+n\sigma^2_s(t)}{\sqrt{n}} \hat{S}(t)$$
• log transform$$[ \hat{S}(t)^{\frac{1}{\theta}}, \hat{S}(t)^{\theta} ]$$where$$\theta = \exp{ \frac{k_{\alpha}[1+n\sigma^2_s(t)]}{ \sqrt{n}\log{\hat{S}(t)}}}$$
• arcsine-square root transform upper:$$\sin^2(\max[0, \arcsin{\sqrt{\hat{S}(t)}} - \frac{k_{\alpha}[1+n\sigma_s(t)]}{2\sqrt{n}} \sqrt{ \frac{\hat{S}(t)}{1-\hat{S}(t)}}])$$lower:$$\sin^2(\min[\frac{\pi}{2}, \arcsin{\sqrt{\hat{S}(t)}} + \frac{k_{\alpha}[1+n\sigma^2_s(t)]}{2\sqrt{n}} \sqrt{ \frac{\hat{S}(t)}{1-\hat{S}(t)}}])$$

References

Nair V, 1984. Confidence bands for survival functions with censored data: a comparative study. Technometrics. 26(3):265-75. http://www.jstor.org/stable/1267553{JSTOR}.

Hall WJ, Wellner JA, 1980. Confidence bands for a survival curve from censored data. Biometrika. 67(1):133-43. http://www.jstor.org/stable/2335326{JSTOR}.

Examples are from: K&M. Section 4.4, pg 111.

See Also

sf

quantile

Examples

data(bmt, package="KMsurv")
b1 <- bmt[bmt$group==1, ] # ALL patients
s1 <- survfit(Surv(t2, d3) ~ 1, data=bmt[bmt$group==1, ])
ci(s1, how="nair", trans="lin", tL=100, tU=600)
s2 <- survfit(Surv(t2, d3) ~ group, data=bmt)
ci(s2, CI="0.99", how="point", trans="asin", tL=100, tU=600)