Kaplan-Meier estimator

Comparison of Kaplan-Meier survival estimates is often called for in clinical trial analysis. With survival analysis, you are trying to determine which treatment group displays a better time-to-event distribution than another. Part of this analysis is the production of Kaplan-Meier estimates plots that show the probability of a given event over time for each treatment group. In the following example you see that New Drug displays better survival estimates over time than either Old Drug or Placebo. ... 

Here is the SAS program that creates this Kaplan-Meier estimates plot. 

Program 6.8 Creating a Kaplan-Meier Estimates Plot Using PROC GPLOT... 

The Kaplan-Meier estimates produce a step function for each group and are plotted over the lifetime of the animals. Planned, accidentally killed, and lost animals are censored. Moribund deaths are considered to be treatment related. A graphical representation of Kaplan-Meier estimates provide excellent interpretation of survival adjusted data except in the cases where the curves cross between two or more groups. When the curves cross and change direction, no meaningful interpretation of the data can be made by any statistical method because proportional odds characteristic is totally lost over time. This would be a rare case where treatment initially produces more tumor or death and then, due to repair or other mechanisms, becomes beneficial. 

Life tables can be constructed to provide estimates of the event time distributions. Estimates commonly used are known as the Kaplan-Meier estimates. 

It is straightforward to obtain the estimated probability of surviving for various key time points from the Kaplan-Meier estimates. In the Packer et al. (2001) example, the estimated survival probability at 12 months in the carvedilol group was 0.886 compared to 0.815 in the placebo group, an absolute difference of 7.1 per cent in the survival rates. A standard error formula provided by Greenwood (1926) enables us to obtain confidence intervals for these individual survival rates and for their differences. 

The cumulative probability of a suicide attempt or hospitalization to prevent a suicide was significantly different by Kaplan Meier estimates (P< 0.02) in favor of clozapine. 

There is every reason to expect that this property would exist with olanzapine. Three fixed-dose ranges of olanzapine (5.0 2.5 mg, 10.0 2.5 mg, 15.0 2.5 mg) and one fixed-dose range of haloperidol (15.0 5 mg) were compared with placebo for up to 52 weeks of therapy (218). Survival analysis of time to rehospitalization for psychotic symptoms indicated that olanzapine was comparable to haloperidol and significantly better than placebo (p = 0.007). Kaplan-Meier estimation showed that 71.5% of olanzapine-treated patients did not relapse, compared with 32.8% for those on placebo. Further, another survival analysis demonstrated that significantly fewer patients in the olanzapine treatment group experienced relapse at any given time than those in the haloperidol group (i.e., p = 0.048 80.9% for olanzapine compared with 72.2% for haloperidol). 

There has been a multicenter, randomized, placebo-controUed, double-blind comparison of aprindine and digoxin in the prevention of atrial fibrillation and its recurrence in 141 patients with symptomatic paroxysmal or persistent atrial fibrillation who had converted to sinus rhythm (7). They were randomized in equal numbers to aprindine 40 mg/day, digoxin 0.25 mg/day, or placebo and followed every 2 weeks for 6 months. After 6 months the Kaplan-Meier estimates of the numbers of patients who had no recurrences with aprindine, digoxin, and placebo were 33, 29, and 22% respectively. The rates of adverse events were similar in the three groups. This suggests that aprindine has a very small beneficial effect in preventing relapse of sjmptomatic atrial fibrillation after conversion to sinus rhythm. Furthermore, recurrence occurred later with aprindine than with placebo or digoxin (about 60% recurrence at 115 days compared with 30 days). 

Figures. Kaplan-Meier estimates of time to first notable serum creatinine increase in patients with multiple myeloma or breast cancer with bone metastases receiving 4 mg zoledronic acid or 90 mg pamidronate and Andersen-Gill multiple event analysis of the risk of elevated serum creatinine between treatment groups. After start of study drug. (Reprinted with permission from [75])...

In the multicentre. Phase 111 comparative trial in patients with >1 bone lesion from breast cancer or multiple myeloma (N=l, 648), there were no significant differences in renal safety profiles between patients given a 15-minute infusion of 4 mg zoledronic acid and those given a 2-hour infusion of 90 mg pamidronate (Table 1) [22]. Kaplan-Meier estimates demonstrated that there were no significant differences in time to first notable serum creatinine increase between treatment groups (HR=1.057 P=0.839 Figure 3) [75],... 

Table 3 Kaplan-Meier Estimates of Proportion with PCP in HIV-Infected Persons... 

Sometimes, these data are presented in a shorter table that displays only those time points at which an individual had an event or was censored, and thus the only values of time for which the probability of survival changes. It is more common, however, to see analyses of this type displayed graphically. The Kaplan-Meier estimate of the survival distribution is displayed for both groups in Figure 8.3. The survival curves displayed in the figure are termed "step functions" because of their appearance. We return to the interpretation of Figure 8.3 after we have fully specified the survival distribution function. 

Figure 8.3 Kaplan-Meier estimate of the survival distribution for adverse event A...

A common measure of central tendency from the Kaplan-Meier estimate is the median survival time (note that this can be estimated only if more than half the participants experience the event). The median survival time is the earliest value of t such that the probability of survival is < 0.5. Note that when observations are censored any estimate of the mean is biased because, technically, the event would eventually occur if we followed participants indefinitely. 

The Kaplan-Meier estimate is a nonparametric method that requires no distributional assumptions. The only assumption required is that the observations are independent. In the case of this example, the observations are event times (or censoring times) for each individual. Observations on unique study participants can be considered independent. The confidence interval approach described here is consistent with the stated preference for estimation and description of risks associated with new treatments. A method for testing the equality of survival distributions is discussed in Chapter 11. 

The distributions of immune function parameters were compared across treatment groups via t-test. Kaplan Meier estimates were used to summarize overall survival (OS) and event free survival (EPS). Survival was calculated from PBSC infusion until death or date of last contact. EPS is defined as the absence of death, relapse, disease progression and de novo secondary cancer. Cox regression models were fitted for the outcomes of OS and EPS. P values associated with regression models were derived from the Wald test. The last possible day of contact was Pebruary 21,2006. 

FIGURE 1.3 A Kaplan -Meier estimates of the rates of cardiac... 

FIGURE 1.4 Kaplan-Meier estimates of the probability of survival in the group assigned to receive an implantable defibrillator and the group assigned to receive conventional therapy in MADIT-II. (From Ref. 22, with permission.)... 

FIGURE 1.6 Kaplan-Meier estimates of death from any cause in patients with both ischemic and nonischemic cardiomyopathy randomized to amiodarone, placebo, or implantable cardioverter-defibrillator therapy in SCD-HeFT. (From Ref. 29, with permission.)... 

Figure 1 TAP study. Kaplan-Meier estimate of the cumulative proportion of eyes with moderate vision loss (>15 letters) in verteporfin-treated and placebo groups at each three-month study visit. Abbreviation TAP, Treatment of Age-Related Macular Degeneration with Photodynamic Therapy. Source From Ref. 9.

For several years, it has been acknowledged that whenever possible bioassay data should be corrected for early death. In both CDC s and EPA s analysis of Kociba et al (6), the individual animal pathology results were not available. Recently, these data were made available and the analysis showed some interesting results (76-77). The nonparametric Kaplan-Meier estimates of the probability of a female rat developing a hepatocellular neoplastic nodule or carcinoma by the end of 25 months, the duration of the Kociba study, are shown in Figure 5. These estimates are computed separately for each dose level and take into account the observation times of each rat. The values of these estimates for the two highest dose levels are 0.81 at... 

Figure 5 The Kaplan-Meier estimates of the probability of a female rat developing a hepatocellular neoplastic nodule or carcinoma in the Kociba et al (1978) study suggest that the dose-response relationship resembles one with a saturation-like phenomenon occurring at the highest experimental dose level (From ref. 76). Reproduced with permission from Paustenbach et al. Copyright 1986,...

Kaplan-Meier estimator. A term from survival analysis. A statistic which was... 

Fig. 1. Kaplan-Meier estimates of time to first CVD event in the intensive therapy and conventional therapy group...

Fig. 2. (A) Distribution of the total number of cardiovascular events in the intensive therapy group in the Steno-2 Study [16] (light gray) and the conventional therapy group (dark gray). (B) Kaplan-Meier estimates of time to first stroke in the intensive therapy and conventional therapy group. HR denotes hazard ratio. 95%C1 denotes confidence interval. (C) Kaplan-Meier estimates of time to first percutaneous coronary intervention (PCI) or coronary bypass graft (CABG) in the intensive therapy and conventional therapy group. (D) Kaplan-Meier estimates of time to first myocardial infarction in the intensive therapy and conventional therapy group (reproduced from ref. 13).

F is the Kaplan-Meier estimator of the marginal distribution 74 based on the random censored data (I(n),Z(nJ). The consistency of F has been studied extensively in the literature. Bitouze, et cd. (1999) derived an exponential-type inequahty for the Kaplan-Meier estimator F which is given as follows. 

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