The LIFETEST Procedure

Simultaneous Confidence Intervals for Kaplan-Meier Curves

The pointwise confidence interval for the survivor function $S(t)$ is valid for a single fixed time at which the inference is to be made. In some applications, it is of interest to find the upper and lower confidence bands that guarantee, with a given confidence level, that the survivor function falls within the band for all t in some interval. Hall and Wellner (1980) and Nair (1984) provide two different approaches for deriving the confidence bands. An excellent review can be found in Klein and Moeschberger (1997). You can use the CONFBAND= option in the PROC LIFETEST statement to select the confidence bands. The EP confidence band provides confidence bounds that are proportional to the pointwise confidence interval, while those of the HW band are not proportional to the pointwise confidence bounds. The maximum time, $t_ U$, for the bands can be specified by the BANDMAX= option; the minimum time, $t_ L$, can be specified by the BANDMIN= option. Transformations that are used to improve the pointwise confidence intervals can be applied to improve the confidence bands. It might turn out that the upper and lower bounds of the confidence bands are not decreasing in $t_ L < t < t_ U$, which is contrary to the nonincreasing characteristic of survivor function. Meeker and escobar (1998) suggest making an adjustment so that the bounds do not increase: if the upper bound is increasing on the right, it is made flat from the minimum to $t_ U$; if the lower bound is increasing from the right, it is made flat from $t_ L$ to the maximum. PROC LIFETEST does not make any adjustment for the nondecreasing behavior of the confidence bands in the OUTSURV= data set. However, the adjustment was made in the display of the confidence bands by using ODS Graphics.

For Kaplan-Meier estimation, let $t_1 < t_2 < \ldots < t_ D$ be the D distinct events times, and at time $t_ i$, there are $d_ i$ events. Let $Y_ i$ be the number of individuals who are at risk at time $t_ i$. The variance of $\hat{S}(t)$, given by the Greenwood formula, is $\hat{\sigma }^2[\hat{S}(t)] = \sigma _ S^2(t)\hat{S}^2(t)$, where

\[ \sigma _ S^2(t) = \sum _{t_ i \le t} \frac{d_ i}{Y_ i(Y_ i-d_ i)} \]

Let $t_ L < t_ U$ be the time range for the confidence band so that $t_ U$ is less than or equal to the largest event time. For the Hall-Wellner band, $t_ L$ can be zero, but for the equal-precision band, $t_ L$ is greater than or equal to the smallest event time. Let

\[ a_ L = \frac{n\sigma _ S^2(t_ L)}{1+n\sigma _ S^2(t_ L)} ~ ~ \mr{and}~ ~ a_ U = \frac{n\sigma _ S^2(t_ U)}{1+n\sigma _ S^2(t_ U)} \]

Let $\{ W^0(u), 0 \le u \le 1\} $ be a Brownian bridge.

Hall-Wellner Band

The 100(1–$\alpha $)% HW band of Hall and Wellner (1980) is

\[ \hat{S}(t) - h_\alpha (a_ L,a_ U) n^{-\frac{1}{2}} [1+n\sigma _ S^2(t)]\hat{S}(t) \le S(t) \le \hat{S}(t) + h_\alpha (a_ L,a_ U) n^{-\frac{1}{2}} [1+n\sigma _ S^2(t)]\hat{S}(t) \]

for all $t_ L\le t \le t_ U$, where the critical value $h_\alpha (a_ L,a_ U)$ is given by

\[ \alpha = \mr{Pr}\{ \sup _{a_ L \le u \le a_ U}|W^0(u)| > h_\alpha (a_ L,a_ U)\} \]

The critical values are computed from the results in Chung (1986).

Note that the given confidence band has a formula similar to that of the (linear) pointwise confidence interval, where $h_{\alpha }(a_ L,a_ U)$ and $n^{-\frac{1}{2}} [1+n\sigma _ S^2(t)] \hat{S}(t)$ in the former correspond to $z_{\frac{\alpha }{2}}$ and $\hat{\sigma }(\hat{S}(t))$ in the latter, respectively. You can obtain the other transformations (arcsine-square root, log-log, log, and logit) for the confidence bands by replacing $z_{\frac{\alpha }{2}}$ and $\hat{\tau }(t)$ in the corresponding pointwise confidence interval formula by $h_\alpha (a_ L,a_ U)$ and the following $\hat{\tau }(t)$, respectively:

  • arcsine-square root transformation:

    \[ \hat{\tau }(t)= \frac{1+n\sigma _ S^2(t)}{2} \sqrt {\frac{S(t)}{n[1-S(t)]}} \]

  • log transformation:

    \[ \hat{\tau }(t)= \frac{1+n\sigma _ S^2(t)}{\sqrt {n}} \]

  • log-log transformation:

    \[ \hat{\tau }(t)= \frac{1+n\sigma _ S^2(t)}{\sqrt {n}|\log [\hat{S}(t)]|} \]

  • logit transformation:

    \[ \hat{\tau }(t)= \frac{1+n\sigma _ S^2(t)}{\sqrt {n}[1-\hat{S}(t)]} \]
Equal-Precision Band

The 100(1–$\alpha $)% EP band of Nair (1984) is

\[ \hat{S}(t) - e_\alpha (a_ L,a_ U) \hat{S}(t) \sigma _ S(t) \le S(t) \le \hat{S}(t) +e_\alpha (a_ L,a_ U) \hat{S}(t) \sigma _ S(t) \]

for all $t_ L \le t \le t_ U$, where $e_\alpha (a_ L,a_ U)$ is given by

\[ \alpha = \mr{Pr}\{ \sup _{a_ L \le u \le a_ U} \frac{|W^0(u)|}{[u(1-u)]^{\frac{1}{2}}} > e_\alpha (a_ L,a_ U)\} \]

PROC LIFETEST uses the approximation of Miller and Siegmund (1982, Equation 8) to approximate the tail probability in which $e_\alpha (a_ L,a_ U)$ is obtained by solving x in

\[ \frac{4x\phi (x)}{x} + \phi (x)\left(x-\frac{1}{x}\right)\log \left[ \frac{a_ U(1-a_ L)}{a_ L(1-a_ U)}\right] = \alpha \]

where $\phi (x)$ is the standard normal density function evaluated at x. Note that the confidence bounds given are proportional to the pointwise confidence intervals. As a matter of fact, this confidence band and the (linear) pointwise confidence interval have the same formula except for the critical values ($z_{\frac{\alpha }{2}}$ for the pointwise confidence interval and $e_\alpha (a_ L,a_ U)$ for the band). You can obtain the other transformations (arcsine-square root, log-log, log, and logit) for the confidence bands by replacing $z_{\frac{\alpha }{2}}$ by $e_\alpha (a_ L,a_ U)$ in the formula of the pointwise confidence intervals.