Pointwise confidence intervals for a survival distribution with small samples or heavy censoring

Abstract

We propose a beta product confidence procedure (BPCP) that is a non-parametric confidence procedure for the survival curve at a fixed time for right-censored data assuming independent censoring. In such situations, the Kaplan–Meier estimator is typically used with an asymptotic confidence interval (CI) that can have coverage problems when the number of observed failures is not large, and/or when testing the latter parts of the curve where there are few remaining subjects at risk. The BPCP guarantees central coverage (i.e. ensures that both one-sided error rates are no more than half of the total nominal rate) when there is no censoring (in which case it reduces to the Clopper–Pearson interval) or when there is progressive type II censoring (i.e. when censoring only occurs immediately after failures on fixed proportions of the remaining individuals). For general independent censoring, simulations show that the BPCP maintains central coverage in many situations where competing methods can have very substantial error rate inflation for the lower limit. The BPCP gives asymptotically correct coverage and is asymptotically equivalent to the CI on the Kaplan–Meier estimator using Greenwood’s variance. The BPCP may be inverted to create confidence procedures for a quantile of the underlying survival distribution. Because the BPCP is easy to implement, offers protection in settings when other methods fail, and essentially matches other methods when they succeed, it should be the method of choice.

1. Introduction

In this paper, we use products of beta random variables to create a new pointwise confidence procedure for the survival distribution from right-censored data. Our beta product confidence procedure (BPCP) is invariant to monotonic transformations and can be applied to data with tied event times. The BPCP is designed to guarantee central coverage, so that a confidence interval (CI) allows one to conclude whether survival at a specific time is higher (or lower) than a reference value, with a type I error rate that is less than or equal to . Theoretically, under progressive type II censoring the BPCP guarantees central coverage. Our simulations show that under independent censoring for small samples the BPCP retains coverage, whereas existing procedures based on the Kaplan–Meier estimator do not. Further asymptotically, both limits of the BPCP are correct and equivalent to the usual confidence limits based on the Kaplan–Meier estimator and Greenwood’s formula.

A common approach to estimating CIs for the survival distribution is to use a normal approximation with the Greenwood formula for the variance of the Kaplan–Meier estimator and the delta method on a transformation (see, e.g. Kalbfleisch and Prentice, 2002; Meeker and Escobar, 1998; SAS, 2011; Therneau, 2012). Strawderman and others (1997) discuss improving the normal CIs using Edgeworth expansions. Barber and Jennison (1999) review several more complex confidence procedures for survival and identify two CIs that perform quite well, the constrained estimator of Thomas and Grunkemeier (1975) and the constrained bootstrap, the latter of which can be computationally time consuming.

Although many of the previously developed CIs are asymptotically consistent when the censoring is independent with a fixed distribution (Andersen and others, 1993), those CIs can have coverage problems at times before very many events have occurred. For example, consider CIs before the first death at 14 days (0.038 years) in the data from Nash and others (2007) (see Figure 1 and Section 8 for details). The Kaplan–Meier estimator before the first death equals 1 and its estimated variance is 0, so the Greenwood 95% CI is [1,1], which is clearly not correct. Andersen and others (1993, p. 214, 268) discuss adjusting this first interval using exact methods, but our BPCP does this naturally, giving a 95% CI of . This interval is the same as the exact Clopper–Pearson interval for 34 out of 34 surviving. The BPCP reduces to this type of exact interval at each time point when there is no censoring.

Fig. 1.

The black middle curve is the Kaplan–Meier estimator of survival for 34 patients with 12 observed deaths from Nash and others (2007). Censored values are denoted by small vertical hashes. Gray lines are previously derived CIs and the black CIs are the BPCP intervals developed in this paper.

A second coverage problem with many traditional CIs occurs when the Kaplan–Meier estimate does not change as subjects are censored, and the Greenwood CIs, for example, also do not change. In Figure 1, we would expect that the variance of the Kaplan–Meier estimate would be larger (and hence the CI should be larger) just before the 12th failure time (6.35 years) when there are only 4 subjects at risk, than just after the 11th failure (4.93 years) when there are 13 subjects at risk. Because of the constancy of the Greenwood CIs between failure times, we have oddities such as the following: the 95% lower confidence limit just before the 12th failure is , but immediately after the 12th failure our estimate of the survival distribution is , even lower than the Greenwood lower limit just prior to the failure. There are several additional methods that allow CIs to change in intervals where only censored observations are observed (Peto and others, 1977; Dorey and Korn, 1987; Borkowf, 2005), but they are all based on asymptotic normal approximations, so their coverage for small samples will not be guaranteed. It appears that the Greenwood and other previously developed lower limits just prior to the 12th failure are too high. In fact, we show through simulations in Section 7 that the lower bound can have very poor one-sided coverage for previously developed methods when there are either few observed failures or when there are few subjects at risk.

A third issue with some previously developed CIs is that the upper limit may increase (see the upper limits in Figure 1), while we know that the survival curve does not increase. Our simulations show that some upper bounds can be needlessly conservative compared with our BPCP.

We can invert the BPCP to obtain confidence procedures for quantiles of the survival distribution (see, e.g. Barber and Jennison, 1999). These BPCP quantile confidence procedures are simpler than the method of Guilbaud (2001), which uses a mixture representation of the order statistics to provide an exact non-parametric confidence procedure for quantiles from progressive type II censoring. The associated median unbiased estimator (MUE) provides estimates of quantiles (e.g. median).

We begin by defining beta product random variables in Section 2, since they are essential in the development of the BPCP. Our development of the BPCP first assumes continuous failure times (Section 3), but then we allow ties through grouping of the failure times (Section 4). The BPCP requires estimation of beta product quantiles described in Section 5. In Section 6, we discuss extensions such as MUEs of survival at t based on the BPCP. We end with simulations, applications, and a discussion.

2. Beta product random variables

Let be a beta random variable with parameters a and b, with mean and variance , and let be the pth quantile of that beta distribution. For notational convenience, we extend the definition of beta distribution to include the limits. Specifically, with and with are random variables with point masses at 1 and 0, respectively, so that and for . We leave undefined.

Let and be vectors with all values ≥0ja_j = b_j = 0BP(a,b), such that

where ∼ denotes “is distributed as” and the terms in the product are all independent. Let be the pth quantile of .

In some cases, a beta product random variable simplifies to a beta random variable:

for constants and c where the beta product variables are defined. To see this for cases with see Casella and Berger (2002, p. 158), and for cases with some parameters equal to 0, we can show the result by inspecting each case. By repeated use of this property, we have

(2.1)

3. Continuous failure with no grouping

3.1. Proposed confidence procedure

We define our procedure first without motivation; in later subsections, we motivate it under various censoring conditions. Throughout Section 3, we assume that the failure time for the ith individual is continuously distributed with random variable X_i and . Suppose that there are n total observations, and we observe the failure time exactly in of those observations. Let the k observed failure times be , and define . Let be the number at risk just before time t, and no more individuals are at risk after , and let

Similarly, define

where . Let be all the data collected up until time t. We define the BPCP for with as

(3.1)

and with as and . Note that and can each be separately interpreted as one-sided % confidence procedures.

3.2. No censoring

We begin motivating and exploring the properties of the BPCP with the simplest case, when there are no censored observations and no tied event times. Suppose that we observe independent observations from the distribution . Then, by the probability integral transformation (see, e.g. Casella and Berger, 2002, p. 54), are uniformly distributed. Let be the order statistics; then is the jth order statistic from a uniform distribution and is distributed (see, e.g. Casella and Berger, 2002, p. 230). The associated survival distribution, is distributed . With no censoring, (2.1) shows that B(n−j+1,j)∼BP{Y(T_j),1(T_j)},1(T_j) = 1₀(T_j) = 1₁(T_j). Thus, the BPCP uses quantiles of that distribution at the observed failures, and in between the observed failures the BPCP just acts conservatively (see Figure 2). A formal proof that the BPCP guarantees central coverage is given in Theorem 1, in which the no censoring setting is just a special case. Note that for t in the intervals between the death times, this confidence procedure is equivalent to performing the Clopper–Pearson interval for a binomial observation based on the fact that the number that survive past t is binomial with parameter , and the Clopper–Pearson interval for out of n is {Q(α/2,n−j,j+1),Q(1−α/2,n−j+1,j)} (see, e.g. Meeker and Escobar, 1998).

Figure 2.

BPCP without censoring. Recall that . Note that the upper limits use the th quantile of in , while the lower limits use the th quantile of in . For any the CI is equivalent to the Clopper–Pearson interval for successes out of n trials with probability .

3.3. Progressive type II censoring

Now assume progressive type II censoring (see Kalbfleisch and Prentice [2002, p. 54]), and so immediately after each failure a fixed proportion of the remaining individuals are randomly selected to be censored. Under this censoring scheme, no censoring occurs before the first event time, and we know from Section 3.2 that the distribution of S(T₁) = S(X₍₁₎)Beta(n,1)T₁Y (T₂) subjects at risk, and the conditional survival distribution, given survival after and , can be shown to be . Using this type of argument, we show formally in supplementary material available at Biostatistics online (Section A) that

This motivates the following theorem.

Theorem 1 —

Under independent progressive type II censoring and assuming a continuous failure time distribution, the BPCP given by (3.1) guarantees central coverage for .

Proof —

For the proof, see supplementary material available at Biostatistics online (Section B).

3.4. Independent censoring

Now suppose that the censoring times are independent of the failure times, and allow censoring to happen at any time. Although we do not formally prove guaranteed central coverage of the BPCP under this scenario, we give some motivation to argue that the BPCP is a reasonable confidence procedure.

Consider first the lower BPCP confidence limit. Suppose that there are subjects censored before and that the observed censoring times before are T₀₁<T₀₂<⋯<T_0k0Y (T_0j)−Y (T_0,j+1)T_0jT₀₁ we have subjects at risk. Consider the lower CI for . For , we know that none of the n at-risk subjects had the event before , and if a subject had an event at , then the lower interval would have been . We let the lower CI be as high as that value, suggesting that we conservatively retain coverage. For the next interval, , imagine that was actually and represented a failure time. Then from the previous section a lower CI for for would be . Each of these modifications ( moving to zero and representing a failure) indicate a lower survival, so we treat as a conservative lower limit. We continue with this reasoning for intervals up until . At the lower limit matches the limit for the associated progressive type II case (i.e. matches the modification that moves all the censored observations before to ). Continuing with this type of reasoning throughout the rest of the sample space suggests the one-sided guaranteed coverage of the lower BPCP.

Now consider the upper limit of the BPCP. The upper BPCP limit under independent censoring matches the upper limit progressive type II case (i.e. matches the case where all censoring times are moved to immediately following the previous failure time). This fact appears to suggest that the upper BPCP is too low, since knowing that the censoring times did in fact happen later than immediately after the last failure would suggest that the survival curve was larger in that interval. But (3.1) implies that the upper limit is constant between observed failure times, and it does not decrease between failures regardless of the censoring pattern between failures. Although we do not formally prove that the BPCP guarantees central coverage, we show in Theorem 2 the asymptotic equivalence of the BPCP to the usual normal theory confidence procedures, and hence the BPCP has correct coverage asymptotically.

The Nelson–Aalen estimator of the cumulative hazard, , and its variance estimate, , are, respectively,

By asymptotic normality, a confidence procedure for the cumulative hazard is where is the pth quantile of the standard normal distribution (see, e.g. Andersen and others, 1993). Since for continuous failure times where is the cumulative hazard, a confidence procedure for is

(3.2)

Another asymptotically equivalent confidence procedure is to use the Kaplan–Meier survival estimator

and the Greenwood formula for its variance (Kaplan and Meier, 1958),

yielding the interval

(3.3)

Other asymptotically equivalent confidence procedures may be constructed using transformations and the delta method (see, e.g. Andersen and others, 1993). All of these confidence procedures are all known to give % central coverage asymptotically, so the following theorem shows that the asymptotic coverage of the BPCP approaches the nominal value.

Theorem 2 —

Under independent censoring with continuous failure time where the censoring distribution is the same for all individuals and for all , where C represents the censoring random variable, the BPCP for for is asymptotically equivalent to the procedures given in (3.2) or (3.3).

Proof —

For the proof, see supplementary material available at Biostatistics online (Section C).

4. Handling ties via grouping

The continuity of the failure times is a vital condition for the motivation of the BPCPs, but ties in failure times are common with real data. Now we formally adjust the BPCP to allow for ties. The basic idea is simple: tied failures can be interpreted as grouped continuous failures, and grouping hides information about the continuous failure time, but we can extend the confidence procedures of (3.1) in a conservative way using the monotonicity of survival distributions.

Under grouping, we only observe the number who are still at risk and the number who were known to have failed at certain assessment times. For full generality, we define those assessment times as . We consider only conventional grouping for right-censored data, where if failure observations and censored observations occur in the same interval, the censored observations are assumed to occur after the failure observations in that interval. Since we still assume continuous survival, the probability that a failure time would occur exactly at some g_j is .

Let represent the information about survival or censoring as in Section 3, only now is unobserved. Let be the observed data up until time g_i after grouping, . Although not measured continuously, it is convenient to use for , with for . Let be the set of all that are consistent with under conventional grouping. Then we extend the BPCP to grouped data. Define the BPCP for applied to grouped data as

(4.1)

where and are given in (3.1). Equivalently, for with ,

(4.2)

and, for , and . Expression (4.2) is more straightforward for implementation and is motivated by the monotonicity of both the upper and lower limits of the continuous BPCP. Expression (4.1) clearly shows that if a BPCP under continuity guarantees coverage, then it will continue to guarantee coverage under conventional grouping. If there are intervals with more than one failure, then since we assume all censoring in an interval happens at the end, the beta product vectors may be collapsed using (2.1); see supplementary material available at Biostatistics online (Section D).

5. Estimation of the beta product quantiles

The calculation of used in the definition of the BPCP is not straightforward (Johnson and others, 1995), so we propose two implementations: the Monte Carlo implementation of the BPCP () and the method of moments implementation of the BPCP (). To estimate in , we take m Monte Carlo replicates, where the ith replicate is with . Then we estimate with the empirical quantile of the m Monte Carlo R_i values.

For the method of moments implementation, we estimate with the pth quantile of a beta distribution with the same mean and variance as . In particular, when no and no we use the pth quantile of a beta distribution with parameters and where and . Fan (1991) has shown that this method works fairly well by comparing the first 10 moments of this beta distribution to the first 10 moments of the true distribution of for several examples. Modification of the method to cases with or is straightforward.

06. Extensions

6.1. Confidence procedures on quantiles

Once we have a CI procedure for for any t, say , we can invert that procedure to provide CIs for quantiles (Barber and Jennison, 1999). A CI for can represent a test of , where we fail to reject the null hypothesis if . For most of this paper we have fixed , but we could alternatively have fixed . Since and are equivalent, a confidence procedure for is the set of all t for which .

6.2. MUE of survival

Because we have a confidence procedure that is defined for all confidence levels, we can create an MUE of the survival from the confidence procedure (Read, 1985). Specifically, the MUE of is motivated by letting , and is defined as

We performed a simple simulation. Let and suppose that the failure times are distributed exponentially with mean 1 and the censoring times are distributed uniformly on . We simulated this times. Because the Kaplan–Meier is not defined after the largest observation if it is censored, we define it in three ways after the last observation: KML defines it as 0, KMH defines it as the Kaplan–Meier at the last value, and KMM = 0.5*KML+0.5*KMH. The simulated mean squared error (MSE) for was between and less than the MSE for all three Kaplan–Meier methods for values not at the extremes (when or ), and was between and less than the Kaplan–Meier methods at ; however, the MSE for at was higher than the three Kaplan–Meier estimators, and at was higher than KML and KMM. Simulated MSE details are in supplementary material available at Biostatistics online (Section E).

To gain intuition for this result, we examine the case of no censoring. Let t∈(X_(j),X_(j+1))t is . The survival estimator that is the best invariant one for monotonic transformations under the squared error loss function is , which pulls the Kaplan–Meier estimator toward (see, e.g. Ferguson, 1967, Section 4.8). Our MUE is

For large n, the beta distribution approaches the normal distribution and the median approaches the mean, so our MUE is approximately

In other words, it is approximately halfway between and the Kaplan–Meier estimator . For small we would expect our estimator to be between those two estimators, and hence extends this appealing shrinkage to censored data (see also Borkowf, 2005, on shrinkage estimators for Kaplan–Meier).

For situations with extreme censoring toward the end of follow-up with few additional failures (see, e.g. Figure 3), the MUE will decrease substantially over time, implying that there is real information about a survival decrease at those time points, when in fact the MUE decrease may be caused only by the lack of confidence in the estimator. However, the MUE can be appealing in the middle part of the survival curve as shown by our simulation.

Figure 3.

National Wilms Tumor Study Group, third and fourth clinical trials (Breslow and Chatterjee, 1999). The black middle curve is the Kaplan–Meier estimator for time to relapse for the children included in the dataset in the R survival package (Therneau, 2012). Thick gray lines are standard CIs (Greenwood variance using log transformation) and the black CIs are the BPCP intervals.

7. Simulations

We conducted a simulation study to evaluate coverage of the BPCP compared with competing methods under a range of scenarios. For this simulation, we studied both BPCP_MMBPCP_MC100 000 Monte Carlo samples). The competing asymptotic methods were as follows: Greenwood log, Greenwood variance assuming normality on (default in Therneau (2012)); Modified GW log, same as previous but for the lower limit multiply the variance at t by a factor of , where T_j is the largest observed survival less than or equal to t (Therneau, 2012, conf. lower option); Borkowf log, a different modification that also gives wider intervals with more censoring and assumes normality on (Borkowf, 2005); Borkowf log,shrink, which uses a shrinkage estimator of the Kaplan–Meier estimator with a hybrid variance estimator (Borkowf, 2005); Strawderman–Wells (see Strawderman and others, 1997, Equation (9)), an Edgeworth expansion method; Thomas–Grunkemeier (Thomas and Grunkemeier, 1975), an empirical likelihood ratio method; Constrained Beta (Barber and Jennison, 1999); Bootstrap, a non-parametric bootstrap method (Efron, 1981; Akritas, 1986); and Constrained Bootstrap (Barber and Jennison, 1999), where the last two methods use 1000 bootstrap samples. Finally, we also considered the inefficient binomial approach which guarantees coverage, but only includes individuals whose censoring times are known to be greater than or equal to the test time (). Unlike the other methods, requires knowledge of the censoring times even for individuals with observed failures. However, there are real-life situations where this information would be available. For example, consider a staggered entry trial where all censoring occurs at the end of the trial and we wish to test at four years. To use the method, we only include patients who began the trial at least 4 years prior to the end, so that censored observations will not be included nor will failures for patients enrolled after 4 years prior to the end. Because all the methods have asymmetric coverage, instead of presenting results in terms of coverage, we present rejection rates for both one-sided tests.

We first simulated the size of the tests. We let subjects, where the failure times are exponential with a mean of 10 and the independent censoring times are uniform on 0–5. We ran 100 000 replications. We tested at times 1–4 and, for each time, list the survival, the simulated average number of observed failures (which we write as ), and the simulated number at risk (). The simulation results are given in Table 1 as the percent error on each side of the 95% CI, so guaranteed central coverage would have or less. Overall, we see that the asymptotic methods (the first nine rows) have some situations where the lower limit is too high, giving error rates larger than 2.5%. At when there are few observed failures, seven of the asymptotic methods have 6.7% error rates for the lower bound. Note the Greenwood log has up to 11.2% simulated lower error rates. Among the asymptotic methods, the Constrained Bootstrap is best (and most computationally time consuming), but it still has a case where the error rate is about twice what it should be (at the simulated lower error rate is ). None of the upper limits appear much higher than 2.5%. Some of the methods which can have much larger upper intervals than the BCPC (see Figure 1), have much lower simulated size on the high end than the BPCP. Importantly, the BPCP and methods are the only ones that retain the proper coverage for both sides.

Table 1.

Simulated size with and

	Low	High	Low	High	Low	High	Low	High
Greenwood log	6.7	0.2	10.0	0.3	9.3	0.2	11.2	0.1
Modified lower	6.7	0.2	7.3	0.3	5.6	0.2	3.7	0.1
Borkowf log	6.7	0.1	6.8	0.2	5.3	0.2	3.9	0.0
Borkowf log shrink	6.7	0.2	4.7	0.2	3.7	0.2	2.8	0.0
Strawderman–Wells	6.7	2.6	4.1	2.0	4.6	1.8	6.5	1.4
Thomas–Grunkemeier	6.7	2.1	3.8	2.3	4.1	2.4	5.6	2.4
Constrained Beta	0.0	1.4	3.6	1.9	4.6	2.1	6.0	2.5
Bootstrap	6.7	1.2	6.4	1.3	6.2	1.5	7.5	1.3
Constrained Bootstrap	0.0	2.7	2.2	2.4	3.2	2.3	4.9	1.9
Binomial	0.0	1.4	0.7	1.4	0.5	1.4	0.1	1.0
BPCP (MM)	0.0	1.3	0.3	1.4	0.1	1.3	0.0	1.1
BPCP (MC)	0.0	1.3	0.3	1.4	0.1	1.5	0.0	1.3

Simulation had 100 000 replications. Percent error on each side of 95% interval (nominal is 2.5%).

We repeated simulations using a mixture of exponentials to mimic the Nash and others (2007) data (see Section 8.1), and the results were similar although slightly less extreme (the maximum lower error rates for the asymptotic methods ranged from 3.8% to 7.0%); see supplementary material available at Biostatistics online (Section F). The problems with a type I error rate are more severe in the exponential simulation than in the mixture of exponentials. This is presumably because the model based on the Nash and others (2007) data has a flatter survival curve during the time when there is much censoring, so that failure to address the censoring between events is less critical than for the simple exponential model.

Since both the BPCP methods and the method have proper simulated coverage for both sides, we compare them by checking the power (i.e. testing null hypotheses that are different from the true values). We used the same simulation set-up as in Table 1, except that we test two different false null hypotheses for each time point t, the null that is exponential with mean equal to either 2.5 (lower than the truth) or 100 (higher than the truth). In Table 2, we give the percent rejection rates for the three methods that have at least proper coverage. The results exhibited in Table 2 show that the BPCP typically has better power than , and, not surprisingly, this advantage improves as censoring increases. We note that when the first year survival is tested against , the binomial has superior power to BPCP methods. This may be a function of luck and due to the fact that the power for exact binomial tests is not monotonically increasing in the sample size due to the discreteness of the tests (see, e.g. Chan and Bohidar, 1998, Table II). Typically, the BPCP methods will have larger power since they will use more of the data.

Table 2.

Simulated power with and

Binomial	73.0	56.5	86.4	65.5	81.1	64.3	53.4	49.3
BPCP (MM)	76.3	50.9	92.5	83.8	90.4	87.0	65.9	90.3
BPCP (MC)	76.1	51.6	92.2	83.7	89.2	88.0	60.6	90.9

Percent rejected for testing H₀ : S(t) =  S₀(t) =  S₀.

Finally, note that in both simulation tables the BPCP (MM) and the BPCP (MC) methods generally match quite well. Thus, the MM implementation can be recommended for routine use since it is much faster computationally.

8. Applications

8.1. Pilot study of treatment in severe systemic sclerosis

Between 1997 and 2005, a cohort of 34 patients with severe systemic sclerosis was enrolled in a single-arm pilot study of high-dose immunosuppressive therapy and autologous hetapoietic cell transplantation (Nash and others, 2007). The entry criteria were such that the expected 5-year mortality rate with conventional treatment would be about 0.50. The median follow-up for this study was 4 years.

Figure 1 presents the Kaplan–Meier estimator for all-cause mortality in this cohort, along with pointwise 95% CIs using , as well as some of the competing methods. Since the patient population was selected to have approximately 0.50 mortality rate at 5 years with conventional therapy, it would be natural to assess how the results at the same time point with this alternate regimen compare. The survival estimate in the new cohort at 5 years is 0.636, suggesting an improved survival. Nonetheless, all methods include 0.50 in their 5-year 95% CIs, and the BP CI has the lowest lower bound of all the methods. The methods are relatively similar at the 5-year point since there was a death just prior at 4.93 years. The methods are especially divergent at 6.3 years, a point where most patients who were at risk at the prior death have now been censored, so there are only four patients remaining at risk; at this time, the BP CI is , whereas the Greenwood is . We know from simulations that only the BPCP is likely to have proper coverage (see, e.g. Table 1). We also note that there is no censoring during the first 2 years, and at any point in that time period the BPCP is a Clopper–Pearson interval. Across time, the upper bound of the BP CI is the lowest of all methods except Strawderman–Wells and Thomas–Grunkemeier, and the lower bound of the BPCP is the lowest throughout.

As shown in simulations, unlike competing methods, our procedure would allow us to test whether the survival at 5 years is greater than with the type I error rate less than or equal to . If we had used the Greenwood CI or other competing CIs, we would not have been assured control of the type I error rate at .

The median survival time estimated by Kaplan–Meier is 6.35 years and the BPCP 95% CI on the median is . The upper limit is infinity because there is no value of t for which U_t<0.5.

8.2. National Wilms Tumor Study

To emphasize that the differences in the and the Greenwood (log) method are slight when the sample size is large, we plot in Figure 3 the Kaplan–Meier estimator with the two types of CIs for 4028 children in the third and fourth clinical trials of the National Wilms Tumor Study Group (Breslow and Chatterjee, 1999; Therneau, 2012). The event is time until relapse and there were 571 who relapsed, but only 1 who relapsed after 10 years. We see that the two types of CIs match well early in the study when there are a large number of individuals at risk, but at the end of the follow-up, when there are fewer individuals still at risk, we see the substantial differences between the methods. This application points to an undesirable aspect of the MUE as discussed at the end of Section 6.2.

9. Discussion

We have proposed a pointwise CI for right-censored data, and have shown that it guarantees central coverage when the data are uncensored or censored with Progressive Type II censoring. For independent censoring, we have shown that our BPCP is asymptotically correct, and simulations have shown that it maintains proper coverage. The BPCP for survival can be inverted to get CIs for quantiles of the survival distribution, and the good coverage properties of the BPCP for survival are expected to carry over to its inversion as well. The calculations needed to do the BPCP (method of moments implementation) are straightforward and can be done quickly using the bpcp R package. The full range of CIs for the National Wilms Tumor Study of Figure 3 took less than a second to calculate on a standard PC (1.8 GHz, 3.25 GB RAM).

The BPCP offers protection in settings where other methods fail, and it essentially matches the other methods when they succeed. Thus, we believe BPCP should be the preferred method for constructing pointwise CIs for the Kaplan–Meier curve.

10. Software

An R package called bpcp is available online at http://cran.r-project.org. The package includes software and documentation to calculate the BPCP methods and most of the competing methods.

Supplementary material

Supplementary material is available at http://biostatistics.oxfordjournals.org.