Let represent the distinct event times. For each , let be the number of surviving units (the size of the risk set) just prior to and let be the number of units that fail at . If the NOTRUNCATE option is specified in the FREQ statement, and can be nonintegers.
The Breslow estimate of the survivor function is
Note that the Breslow estimate is the exponentiation of the negative NelsonAalen estimate of the cumulative hazard function.
The FlemingHarrington estimate (Fleming and Harrington, 1984) of the survivor function is
If the frequency values are not integers, the FlemingHarrington estimate cannot be computed.
The KaplanMeier (productlimit) estimate of the survivor function at is the cumulative product
Notice that all the estimators are defined to be right continuous; that is, the events at are included in the estimate of . The corresponding estimate of the standard error is computed using Greenwood’s formula (Kalbfleisch and Prentice, 1980) as
The first quartile (or the 25th percentile) of the survival time is the time beyond which 75% of the subjects in the population under study are expected to survive. It is estimated by
If is exactly equal to 0.75 from to , the first quartile is taken to be . If it happens that is greater than 0.75 for all values of t, the first quartile cannot be estimated and is represented by a missing value in the printed output.
The general formula for estimating the 100p percentile point is
The second quartile (the median) and the third quartile of survival times correspond to p = 0.5 and p = 0.75, respectively.
Brookmeyer and Crowley (1982) have constructed the confidence interval for the median survival time based on the confidence interval for the . The methodology is generalized to construct the confidence interval for the 100p percentile based on a gtransformed confidence interval for (Klein and Moeschberger, 1997). You can use the CONFTYPE= option to specify the gtransformation. The % confidence interval for the first quantile survival time is the set of all points t that satisfy
where is the first derivative of and is the percentile of the standard normal distribution.
Consider the bone marrow transplant data described in Example 56.2. The following table illustrates the construction of the confidence limits for the first quartile in the ALL group. Values of that lie between = 1.965 are highlighted.
Constructing 95% Confidence Limits for the 25th Percentile 




t 


LINEAR 
LOGLOG 
LOG 
ASINSQRT 
LOGIT 
1 
0.97368 
0.025967 
8.6141 
2.37831 
9.7871 
4.44648 
2.47903 
55 
0.94737 
0.036224 
5.4486 
2.36375 
6.1098 
3.60151 
2.46635 
74 
0.92105 
0.043744 
3.9103 
2.16833 
4.3257 
2.94398 
2.25757 
86 
0.89474 
0.049784 
2.9073 
1.89961 
3.1713 
2.38164 
1.97023 
104 
0.86842 
0.054836 
2.1595 
1.59196 
2.3217 
1.87884 
1.64297 
107 
0.84211 
0.059153 
1.5571 
1.26050 
1.6490 
1.41733 
1.29331 
109 
0.81579 
0.062886 
1.0462 
0.91307 
1.0908 
0.98624 
0.93069 
110 
0.78947 
0.066135 
0.5969 
0.55415 
0.6123 
0.57846 
0.56079 
122 
0.73684 
0.071434 
–0.1842 
–0.18808 
–0.1826 
–0.18573 
–0.18728 
129 
0.71053 
0.073570 
–0.5365 
–0.56842 
–0.5222 
–0.54859 
–0.56101 
172 
0.68421 
0.075405 
–0.8725 
–0.95372 
–0.8330 
–0.90178 
–0.93247 
192 
0.65789 
0.076960 
–1.1968 
–1.34341 
–1.1201 
–1.24712 
–1.30048 
194 
0.63158 
0.078252 
–1.5133 
–1.73709 
–1.3870 
–1.58613 
–1.66406 
230 
0.60412 
0.079522 
–1.8345 
–2.14672 
–1.6432 
–1.92995 
–2.03291 
276 
0.57666 
0.080509 
–2.1531 
–2.55898 
–1.8825 
–2.26871 
–2.39408 
332 
0.54920 
0.081223 
–2.4722 
–2.97389 
–2.1070 
–2.60380 
–2.74691 
383 
0.52174 
0.081672 
–2.7948 
–3.39146 
–2.3183 
–2.93646 
–3.09068 
418 
0.49428 
0.081860 
–3.1239 
–3.81166 
–2.5177 
–3.26782 
–3.42460 
466 
0.46682 
0.081788 
–3.4624 
–4.23445 
–2.7062 
–3.59898 
–3.74781 
487 
0.43936 
0.081457 
–3.8136 
–4.65971 
–2.8844 
–3.93103 
–4.05931 
526 
0.41190 
0.080862 
–4.1812 
–5.08726 
–3.0527 
–4.26507 
–4.35795 
609 
0.38248 
0.080260 
–4.5791 
–5.52446 
–3.2091 
–4.60719 
–4.64271 
662 
0.35306 
0.079296 
–5.0059 
–5.96222 
–3.3546 
–4.95358 
–4.90900 
Consider the LINEAR transformation where . The event times that satisfy include 107, 109, 110, 122, 129, 172, 192, 194, and 230. The confidence of the interval [107, 230] is less than 95%. Brookmeyer and Crowley (1982) suggest extending the confidence interval to but not including the next event time. As such the 95% confidence interval for the first quartile based on the linear transform is [107, 276). The following table lists the confidence intervals for the various transforms.
95% CI’s for the 25th Percentile 

CONFTYPE 
[Lower 
Upper) 
LINEAR 
107 
276 
LOGLOG 
86 
230 
LOG 
107 
332 
ASINSQRT 
104 
276 
LOGIT 
104 
230 
Sometimes, the confidence limits for the quartiles cannot be estimated. For convenience of explanation, consider the linear transform . If the curve that represents the upper confidence limits for the survivor function lies above 0.75, the upper confidence limit for first quartile cannot be estimated. On the other hand, if the curve that represents the lower confidence limits for the survivor function lies above 0.75, the lower confidence limit for the quartile cannot be estimated.
The estimated mean survival time is
where is defined to be zero. When the largest observed time is censored, this sum underestimates the mean. The standard error of is estimated as
where
If the largest observed time is not an event, you can use the TIMELIM= option to specify a time limit L and estimate the mean survival time limited to the time L and its standard error by replacing k by k + 1 with .
The NelsonAalen cumulative hazard estimator, defined up to the largest observed time on study, is
and its estimated variance is
PROC LIFETEST computes the adjusted KaplanMeier estimate (AKME) of the survivor function if you specify both METHOD=KM and the WEIGHT statement. Let ( denote an independent sample of rightcensored survival data, where is the possibly rightcensored time, is the censoring indicator ( if is censored and if is an event time), and is the weight (from the WEIGHT statement). Let be the D distinct event times in the sample. At time , there are events out of subjects. The weighted number of events and the weighted number at risk are and , respectively. The AKME (Xie and Liu, 2005) is
The estimated variance of is
where
The lifetable estimates are computed by counting the numbers of censored and uncensored observations that fall into each of the time intervals , , where and . Let be the number of units that enter the interval , and let be the number of events that occur in the interval. Let , and let , where is the number of units censored in the interval. The effective sample size of the interval is denoted by . Let denote the midpoint of .
The conditional probability of an event in is estimated by
and its estimated standard error is
where .
The estimate of the survival function at is
and its estimated standard error is
The density function at is estimated by
and its estimated standard error is
The estimated hazard function at is
and its estimated standard error is
Let be the interval in which . The median residual lifetime at is estimated by
and the corresponding standard error is estimated by
If you want to determine the intervals exactly, use the INTERVALS= option in the PROC LIFETEST statement to specify the interval endpoints. Use the WIDTH= option to specify the width of the intervals, thus indirectly determining the number of intervals. If neither the INTERVALS= option nor the WIDTH= option is specified in the lifetable estimation, the number of intervals is determined by the NINTERVAL= option. The width of the time intervals is 2, 5, or 10 times an integer (possibly a negative integer) power of 10. Let (maximum observed time/number of intervals), and let b be the largest integer not exceeding c. Let and let
with I being the indicator function. The width is then given by
By default, NINTERVAL=10.
Pointwise confidence limits are computed for the survivor function, and for the density function and hazard function when the lifetable method is used. Let be specified by the ALPHA= option. Let be the critical value for the standard normal distribution. That is, , where is the cumulative distribution function of the standard normal random variable.
When the computation of confidence limits for the survivor function is based on the asymptotic normality of the survival estimator , the approximate confidence interval might include impossible values outside the range [0,1] at extreme values of t. This problem can be avoided by applying the asymptotic normality to a transformation of for which the range is unrestricted. In addition, certain transformed confidence intervals for perform better than the usual linear confidence intervals (Borgan and Liestøl, 1990). The CONFTYPE= option enables you to pick one of the following transformations: the loglog function (Kalbfleisch and Prentice, 1980), the arcsinesquare root function (Nair, 1984), the logit function (Meeker and Escobar, 1998), the log function, and the linear function.
Let g be the transformation that is being applied to the survivor function . By the delta method, the standard error of is estimated by
where is the first derivative of the function g. The 100(1–)% confidence interval for is given by
where is the inverse function of g. That choices of the transformation g are as follows:
arcsinesquare root transformation: The estimated variance of is The 100(1–)% confidence interval for is given by
linear transformation: This is the same as having no transformation in which g is the identity. The 100(1–)% confidence interval for is given by
log transformation: The estimated variance of is The 100(1–)% confidence interval for is given by
loglog transformation: The estimated variance of is The 100(1–)% confidence interval for is given by
logit transformation: The estimated variance of is
The 100(1–)% confidence limits for are given by
The pointwise confidence interval for the survivor function 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, , for the bands can be specified by the BANDMAX= option; the minimum time, , 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 , 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 ; if the lower bound is increasing from the right, it is made flat from 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 KaplanMeier estimation, let be the D distinct events times, and at time , there are events. Let be the number of individuals who are at risk at time . The variance of , given by the Greenwood formula, is , where
Let be the time range for the confidence band so that is less than or equal to the largest event time. For the HallWellner band, can be zero, but for the equalprecision band, is greater than or equal to the smallest event time. Let
Let be a Brownian bridge.
The 100(1–)% HW band of Hall and Wellner (1980) is
for all , where the critical value is given by
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 and in the former correspond to and in the latter, respectively. You can obtain the other transformations (arcsinesquare root, loglog, log, and logit) for the confidence bands by replacing and in the corresponding pointwise confidence interval formula by and the following , respectively:
The 100(1–)% EP band of Nair (1984) is
for all , where is given by
PROC LIFETEST uses the approximation of Miller and Siegmund (1982, Equation 8) to approximate the tail probability in which is obtained by solving x in
where 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 ( for the pointwise confidence interval and for the band). You can obtain the other transformations (arcsinesquare root, loglog, log, and logit) for the confidence bands by replacing by in the formula of the pointwise confidence intervals.
Kernelsmoothed estimators of the hazard function are based on the NelsonAalen estimator and its variance . Consider the jumps of and at the event times as follows:
where =0.
The kernelsmoothed estimator of is a weighted average of over event times that are within a bandwidth distance b of t. The weights are controlled by the choice of kernel function, , defined on the interval [–1,1]. The choices are as follows:
uniform kernel:
Epanechnikov kernel:
biweight kernel:
The kernelsmoothed hazard rate estimator is defined for all time points on . For time points t for which , the kernelsmoothed estimated of based on the kernel is given by
The variance of is estimated by
For t < b, the symmetric kernels are replaced by the corresponding asymmetric kernels of Gasser and Müller (1979). Let . The modified kernels are as follows:
uniform kernel:
Epanechnikov kernel:
biweight kernel:
For , let . The asymmetric kernels for are used with x replaced by –x.
Using the log transform on the smoothed hazard rate, the 100(1–)% pointwise confidence interval for the smoothed hazard rate is given by
where is the 100(1–)th percentile of the standard normal distribution.
The following mean integrated squared error (MISE) over the range and is used as a measure of the global performance of the kernel function estimator:
The last term is independent of the choice of the kernel and bandwidth and can be ignored when you are looking for the best value of b. The first integral can be approximated by using the trapezoid rule by evaluating at a grid of points . You can specify , and M by using the options GRIDL=, GRIDU=, and NMINGRID=, respectively, of the HAZARD plot. The second integral can be estimated by the RamlauHansen (1983a, 1983b) crossvalidation estimate:
Therefore, for a fixed kernel, the optimal bandwidth is the quantity b that minimizes
The minimization is carried out by the golden section search algorithm.
Let K be the number of groups. Let be the underlying survivor function of the kth group, . The null and alternative hypotheses to be tested are
for all
versus
at least one of the ’s is different for some
respectively, where is the largest observed time.
The likelihood ratio test statistic (Lawless, 1982) for test versus 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
where is the total number of events in the kth group, , is the total time on test in the kth stratum, and . The approximate probability value is computed by treating as having a chisquare distribution with K – 1 degrees of freedom.
Let ( denote an independent sample of rightcensored survival data, where is the possibly rightcensored time, is the censoring indicator (=0 if is censored and =1 if is an event time), and for K different groups. Let be the distinct event times in the sample. At time let be a positive weight function, and let and be the size of the risk set and the number of events in the kth group, respectively. Let , .
The choices of the weight function are given in Table 56.3.
Table 56.3: Weight Functions for Various Tests
Test 


Logrank 
1.0 
Wilcoxon 

TaroneWare 

PetoPeto 

Modified PetoPeto 

HarringtonFleming (p,q) 

In Table 56.3, is the productlimit estimate at t for the pooled sample, and is a survivor function estimate close to given by
The rank statistics (Klein and Moeschberger, 1997, Section 7.3) for testing versus have the form of a Kvector with
and the variance of and the covariance of and are, respectively,
The statistic 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 . The overall test statistic for homogeneity is , where denotes a generalized inverse of . This statistic is treated as having a chisquare distribution with degrees of freedom equal to the rank of for the purposes of computing an approximate probability level.
PROC LIFETEST computes the weighted logrank test (Xie and Liu, 2005, 2011) if you specify the WEIGHT statement. Let ( denote an independent sample of rightcensored survival data, where is the possibly rightcensored time, is the censoring indicator (=0 if is censored and =1 if is an event time), for K different groups, and is the weight from the WEIGHT statement. Let be the distinct event times in the sample. At each , and for each , let
Let and denote the number of events and the number at risk, respectively, in the combined sample at time . Similarly, let and denote the weighted number of events and the weighted number at risk, respectively, in the combined sample at time . The test statistic is
and the variance of and the covariance of and are, respectively,
Let . Under , the weighted Ksample test has a statistic given by
with K – 1 degrees of freedom.
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 (), let be the test statistic (Klein and Moeschberger, 1997, Section 7.5) for the sth stratum, and let be its covariance matrix. Let
A global test statistic is constructed as
Under the null hypothesis, the test statistic has a distribution with the same degrees of freedom as the individual test for each stratum.
Let denote a chisquare random variable with r degrees of freedom. Denote and as the density function and the cumulative distribution function of a standard normal distribution, respectively. Let m be the number of comparisons; that is,
For a twosided test that compares the survival of the jth group with that of lth group, , the test statistic is
and the raw pvalue is
Adjusted pvalues for various multiplecomparison adjustments are computed as follows:
Bonferroni adjustment:
DunnettHsu adjustment: With the first group being the control, let be the matrix of contrasts; that is,
Let and be covariance and correlation matrices of , respectively; that is,
and
The factoranalytic covariance approximation of Hsu (1992) is to find such that
where is a diagonal matrix with the jth diagonal element being and . The adjusted pvalue is
which can be obtained in a DATA step as
Scheffé adjustment:
Šidák adjustment:
SMM adjustment:
which can also be evaluated in a DATA step as
Tukey adjustment:
which can also be evaluated in a DATA step as
Trend tests (Klein and Moeschberger, 1997, Section 7.4) have more power to detect ordered alternatives as
with at least one inequality
or
with at least one inequality
Let be a sequence of scores associated with the k samples. The test statistic and its standard error are given by and , respectively. Under , the zscore
has, asymptotically, a standard normal distribution. PROC LIFETEST provides both onetail and twotail pvalues for the test.
The rank tests for the association of covariates (Kalbfleisch and Prentice, 1980, Chapter 6) are more general cases of the rank tests for homogeneity. In this section, the index is used to label all observations, , and the indices range only over the observations that correspond to events, . The ordered event times are denoted as , the corresponding vectors of covariates are denoted as , and the ordered times, both censored and event times, are denoted as .
The rank test statistics have the form
where n is the total number of observations, are rank scores, which can be either logrank or Wilcoxon rank scores, is 1 if the observation is an event and 0 if the observation is censored, and is the vector of covariates in the TEST statement for the th observation. Notice that the scores, , depend on the censoring pattern and that the terms are summed up over all observations.
The logrank scores are
and the Wilcoxon scores are
where is the number at risk just prior to .
The estimates used for the covariance matrix of the logrank statistics are
where is the corrected sum of squares and crossproducts matrix for the risk set at time ; that is,
where
The estimate used for the covariance matrix of the Wilcoxon statistics is
where
In the case of tied failure times, the statistics are averaged over the possible orderings of the tied failure times. The covariance matrices are also averaged over the tied failure times. Averaging the covariance matrices over the tied orderings produces functions with appropriate symmetries for the tied observations; however, the actual variances of the statistics would be smaller than the preceding estimates. Unless the proportion of ties is large, it is unlikely that this will be a problem.
The univariate tests for each covariate are formed from each component of and the corresponding diagonal element of as . These statistics are treated as coming from a chisquare distribution for calculation of probability values.
The statistic is computed by sweeping each pivot of the matrix in the order of greatest increase to the statistic. The corresponding sequence of partial statistics is tabulated. Sequential increments for including a given covariate and the corresponding probabilities are also included in the same table. These probabilities are calculated as the tail probabilities of a chisquare distribution with one degree of freedom. Because of the selection process, these probabilities should not be interpreted as pvalues.
If desired for data screening purposes, the output data set requested by the OUTTEST= option can be treated as a sum of squares and crossproducts matrix and processed by the REG procedure by using the option METHOD=RSQUARE. Then the sets of variables of a given size can be found that give the largest test statistics. ProductLimit Estimates and Tests of Association illustrates this process.