The LIFETEST Procedure

Comparison of Two or More Groups of Survival Data

Let K be the number of groups. Let $S_ k(t)$ be the underlying survivor function of the kth group, $k=1,\ldots ,K$. The null and alternative hypotheses to be tested are

$H_0: S_1(t)= S_2(t) = \ldots = S_ K(t)$ for all $t\le \tau $

versus

$H_1:$ at least one of the $S_ k(t)$’s is different for some $t\le \tau $

respectively, where $\tau $ is the largest observed time.

Likelihood Ratio Test

The likelihood ratio test statistic (Lawless 1982) for test $H_0$ versus $H_1$ assumes that the data in the various samples are exponentially distributed and tests that the scale parameters are equal. The test statistic is computed as

\[ \chi ^2 = 2N \log \left( \frac{T}{N} \right) - 2 \sum _{k=1}^ K N_ k \log \left( \frac{T_ k}{N_ k} \right) \]

where $N_ k$ is the total number of events in the kth group, $N = \sum _{k=1}^ k N_ k$, $T_ k$ is the total time on test in the kth stratum, and $T = \sum _{k=1}^ K T_ k$. The approximate probability value is computed by treating $\chi ^2$ as having a chi-square distribution with K – 1 degrees of freedom.

Nonparametric Tests

Let ($T_ i,\delta _ i,X_ i), i=1,\ldots ,n,$ denote an independent sample of right-censored survival data, where $T_ i$ is the possibly right-censored time, $\delta _ i$ is the censoring indicator ($\delta _ i$=0 if $T_ i$ is censored and $\delta _ i$=1 if $T_ i$ is an event time), and $X_ i=1,\ldots ,K$ for K different groups. Let $t_1<t_2<\ldots <t_ D$ be the distinct event times in the sample. At time $t_ j, j=1,\ldots ,D,$ let $W(t_ j)$ be a positive weight function, and let $Y_{jk}=\sum _{i:T_ i\geq t_ j}I(Xi=k)$ and $d_{jk}=\sum _{i:T_ i=t_ j}\delta _ iI(X_ i=k)$ be the size of the risk set and the number of events in the kth group, respectively. Let $Y_ j = \sum _{k=1}^ K Y_{jk}$, $ d_ j=\sum _{k=1}^ K d_{jk}$.

The choices of the weight function $W(t_ j)$ are given in Table 70.3.

Table 70.3: Weight Functions for Various Tests

Test

$W(t_ i)$

Log-rank

1.0

Wilcoxon

$Y_ j$

Tarone-Ware

$\sqrt {Y_ j}$

Peto-Peto

$\tilde{S}(t_ j)$

Modified Peto-Peto

$\tilde{S}(t_ j) \frac{Y_ j}{Y_ j+1}$

Harrington-Fleming (p,q)

$[\hat{S}(t_{j-1})]^ p[1-\hat{S}(t_{j-1})]^ q, p\ge 0, q\ge 0$


In Table 70.3, $\hat{S}(t)$ is the product-limit estimate at t for the pooled sample, and $\tilde{S}(t)$ is a survivor function estimate close to $\hat{S}(t)$ given by

\[ \tilde{S}(t) = \prod _{t_ j\le t} \biggl (1 - \frac{d_ j}{Y_ j+1} \biggr ) \]
Unstratified Tests

The rank statistics (Klein and Moeschberger 1997, Section 7.3) for testing $H_0$ versus $H_1$ have the form of a K-vector $\mb{v}=(v_1,v_2,\ldots ,v_ K)^{\prime }$ with

\[ v_ k = \sum _{j=1}^ D \left[W(t_ j) \left( d_{jk} - Y_{jk}\frac{d_ j}{Y_ j} \right) \right] \]

and the variance of $v_ k$ and the covariance of $v_ k$ and $v_ h$ are, respectively,

\begin{eqnarray*} V_{kk} & =& \sum _{j=1}^ D \left[W^2(t_ j) \frac{ d_ j (Y_ j-d_ j) Y_{jk} (Y_ j - Y_{jk})}{ Y_ j^2 (Y_ j - 1) } \right], \mbox{~ ~ ~ } 1\leq k \leq K \\ V_{kh} & =& -\sum _{j=1}^ D \left[W^2(t_ j) \frac{ d_ j (Y_ j-d_ j) Y_{jk} Y_{jh} }{ Y_ j^2 (Y_ j - 1) }\right], \mbox{~ ~ ~ } 1 \leq k \neq h \leq K \end{eqnarray*}

The statistic $v_ k$ can be interpreted as a weighted sum of observed minus expected numbers of failure for the kth group under the null hypothesis of identical survival curves. Let $\bV =(V_{kh})$. The overall test statistic for homogeneity is $\mb{v}^{\prime }\mb{V^{-}v}$, where $\mb{V^{-}}$ denotes a generalized inverse of $\mb{V}$. This statistic is treated as having a chi-square distribution with degrees of freedom equal to the rank of $\mb{V}$ for the purposes of computing an approximate probability level.

Adjusted Log-Rank Test

PROC LIFETEST computes the weighted log-rank test (Xie and Liu 2005, 2011) if you specify the WEIGHT statement. Let ($T_ i,\delta _ i,X_ i,w_ i), i=1,\ldots ,n,$ denote an independent sample of right-censored survival data, where $T_ i$ is the possibly right-censored time, $\delta _ i$ is the censoring indicator ($\delta _ i$=0 if $T_ i$ is censored and $\delta _ i$=1 if $T_ i$ is an event time), $X_ i=1,\ldots ,K$ for K different groups, and $w_ i$ is the weight from the WEIGHT statement. Let $t_1<t_2<\ldots <t_ D$ be the distinct event times in the sample. At each $t_ j, j=1,\ldots ,D$, and for each $1\leq k \leq K$, let

\begin{eqnarray*} d_{jk}= \sum _{i:T_ i=t_ j} I(X_ i=k) & & d_{jk}^ w=\sum _{i:T_ i=t_ j} w_ iI(X_ i=k) \\ Y_{jk}=\sum _{i:T_ i\geq t_ j} I(X_ i=k) & & Y_{jk}^ w=\sum _{i:T_ i\geq t_ j} w_ iI(X_ i=k) \end{eqnarray*}

Let $d_ j = \sum _{k=1}^ K d_{jk}$ and $Y_ j= \sum _{k=1}^ K Y_{jk}$ denote the number of events and the number at risk, respectively, in the combined sample at time $t_ j$. Similarly, let $d^ w_ j = \sum _{k=1}^ K d^ w_{jk}$ and $Y^ w_ j= \sum _{k=1}^ K Y^ w_{jk}$ denote the weighted number of events and the weighted number at risk, respectively, in the combined sample at time $t_ j$. The test statistic is

\[ v_ k= \sum _{j=1}^ D \left( d^ w_{jk} - Y^ w_{jk} \frac{d^ w_ j}{Y^ w_ j} \right) \mbox{~ ~ }k= 1, \ldots , K \]

and the variance of $v_ k$ and the covariance of $v_ k$ and $v_ h$ are, respectively,

\begin{eqnarray*} {V}_{kk} & = & \sum _{j=1}^ D \left\{ \frac{d_ j(Y_ j-d_ j)}{Y_ j(Y_ j-1)} \sum _{i=1}^{Y_ j} \left[ \left( \frac{Y^ w_{jk}}{Y^ w_ j }\right)^2 w^2_ iI\{ X_ i\neq k\} + \left( \frac{Y^ w_ j - Y^ w_{jk}}{Y^ w_ j }\right)^2 w^2_ i I\{ X_ i=k\} \right] \right\} , \mbox{~ ~ ~ } 1\leq k \leq K \\ {V}_{kh} & = & \sum _{j=1}^ D \left\{ \frac{d_ j(Y_ j-d_ j)}{Y_ j(Y_ j-1)} \sum _{i=1}^{Y_ j} \left[ \frac{Y^ w_{jk} Y^ w_{jh}}{(Y^ w_ j)^2} w^2_ iI\{ X_ i\neq k, h\} - \frac{(Y^ w_ j - Y^ w_{jk})Y^ w_{jh}}{(Y^ w_ j)^2} w^2_ i I\{ X_ i=k\} \right. \right. \\ & & \left. \left. - \frac{(Y^ w_ j - Y^ w_{jh})Y^ w_{jk}}{(Y^ w_ j)^2} w^2_ i I\{ X_ i=h\} \right] \right\} , \mbox{~ ~ ~ } 1 \leq k \neq h \leq K \end{eqnarray*}

Let ${V}= ({V}_{kh})$. Under $H_0$, the weighted K-sample test has a $\chi ^2$ statistic given by

\[ \chi ^2= (v_1,\ldots ,v_ K) \bV ^{-} (v_1,\ldots ,v_ K)’ \]

with K – 1 degrees of freedom.

Stratified Tests

Suppose the test is to be stratified on M levels of a set of STRATA variables. Based only on the data of the sth stratum ($s=1 \dots M$), let $\mb{v}_ s$ be the test statistic (Klein and Moeschberger 1997, Section 7.5) for the sth stratum, and let $\bV _ s$ be its covariance matrix. Let

\begin{eqnarray*} \mb{v} & =& \sum _{s=1}^ M \mb{v}_ s \\ \bV & =& \sum _{s=1}^ M \bV _ s \end{eqnarray*}

A global test statistic is constructed as

\[ \chi ^2 = \mb{v}’ \bV ^{-} \mb{v} \]

Under the null hypothesis, the test statistic has a $\chi ^2$ distribution with the same degrees of freedom as the individual test for each stratum.

Multiple-Comparison Adjustments

Let $\chi ^2_ r$ denote a chi-square random variable with r degrees of freedom. Denote $\phi $ and $\Phi $ as the density function and the cumulative distribution function of a standard normal distribution, respectively. Let m be the number of comparisons; that is,

\begin{eqnarray*} m = \left\{ \begin{array}{ll} \frac{k(k-1)}{2} & \mr{DIFF=ALL} \\ k-1 & \mr{DIFF=CONTROL} \end{array} \right. \end{eqnarray*}

For a two-sided test that compares the survival of the jth group with that of lth group, $1\leq j \neq l \leq r$, the test statistic is

\[ z^2_{jl}= \frac{(v_ j - v_ l)^2}{V_{jj} + V_{ll} - 2V_{jl}} \]

and the raw p-value is

\[ p = \mr{Pr}(\chi ^2_1 > z^2_{jl}) \]

Adjusted p-values for various multiple-comparison adjustments are computed as follows:

  • Bonferroni adjustment:

    \[ p = \mr{min}\{ 1, m \mr{Pr}(\chi ^2_1 > z^2_{jl})\} \]

  • Dunnett-Hsu adjustment: With the first group being the control, let $\bC =(c_{ij})$ be the $(r-1)\times r$ matrix of contrasts; that is,

    \begin{eqnarray*} c_{ij} = \left\{ \begin{array}{ll} 1 & i=1,\ldots ,r-1,j=2,\ldots ,r\\ -1 & j=i+1, i=2,\ldots ,r \\ 0 & \mr{otherwise} \end{array} \right. \end{eqnarray*}

    Let $\bSigma \equiv (\sigma _{ij})$ and $\bR \equiv (r_{ij})$ be covariance and correlation matrices of $\bC \mb{v}$, respectively; that is,

    \[ \bSigma = \bC \bV \bC ’ \]

    and

    \[ r_{ij}= \frac{\sigma _{ij}}{\sqrt {\sigma _{ii}\sigma _{jj}}} \]

    The factor-analytic covariance approximation of Hsu (1992) is to find $\lambda _1,\ldots ,\lambda _{r-1}$ such that

    \[ \bR = \bD + \blambda \blambda ’ \]

    where $\bD $ is a diagonal matrix with the jth diagonal element being $1-\lambda _ j$ and $\blambda =(\lambda _1,\ldots ,\lambda _{r-1})’$. The adjusted p-value is

    \[ p= 1 - \int _{-\infty }^{\infty } \phi (y) \prod _{i=1}^{r-1} \biggl [ \Phi \biggl (\frac{\lambda _ i y + z_{jl}}{\sqrt {1-\lambda _ i^2}}\biggr ) - \Phi \biggl (\frac{\lambda _ i y - z_{jl}}{\sqrt {1-\lambda _ i^2}} \biggr ) \biggr ]dy \]

    which can be obtained in a DATA step as

    \[ p=\mr{PROBMC}(\mr{"DUNNETT2"}, z_{ij},.,.,r-1,\lambda _1,\ldots ,\lambda _{r-1}). \]

  • Scheffé adjustment:

    \[ p = \mr{Pr}(\chi ^2_{r-1} > z^2_{jl}) \]

  • Šidák adjustment:

    \[ p = 1-\{ 1- \mr{Pr}(\chi ^2_1 > z^2_{jl})\} ^ m \]

  • SMM adjustment:

    \[ p = 1 - [2\Phi (z_{jl}) -1]^ m \]

    which can also be evaluated in a DATA step as

    \[ p = 1 - \mr{PROBMC}(\mr{"MAXMOD"},z_{jl},.,.,m). \]

  • Tukey adjustment:

    \[ p = 1 - \int _{-\infty }^{\infty } r \phi (y)[\Phi (y) - \Phi (y-\sqrt {2}z_{jl})]^{r-1}dy \]

    which can also be evaluated in a DATA step as

    \[ p = 1 - \mr{PROBMC}(\mr{"RANGE"},\sqrt {2}z_{jl},.,.,r). \]
Trend Tests

Trend tests (Klein and Moeschberger 1997, Section 7.4) have more power to detect ordered alternatives as

$H_2: S_1(t) \ge S_2(t) \ge \ldots \ge S_ k(t), t\le \tau ,$ with at least one inequality

or

$H_2: S_1(t) \le S_2(t) \le \ldots \le S_ k(t), t\le \tau ,$ with at least one inequality

Let $a_1 < a_2 < \ldots < a_ k$ be a sequence of scores associated with the k samples. The test statistic and its standard error are given by $ \sum _{j=1}^ k a_ j v_ j $ and $ \sum _{j=1}^ k \sum _{l=1}^ k a_ j a_ l V_{jl} $, respectively. Under $H_0$, the z-score

\[ Z = \frac{ \sum _{j=1}^ k a_ j v_ j}{\sqrt \{ \sum _{j=1}^ k \sum _{l=1}^ k a_ j a_ l V_{jl}\} } \]

has, asymptotically, a standard normal distribution. PROC LIFETEST provides both one-tail and two-tail p-values for the test.