Comparing Survival of a Sample to That of a Standard Population

Comparing groups on the basis of survival is common in medical research. Survival time data require methods that properly account for the situation when the time of death is not observed because some subjects are still alive at the end of the study (censoring). In addition, methods are required that make no assumptions about the shape of the survival time distribution (nonparametric). There are widely used methods for statistical comparison and graphic display of survival of two samples. The log-rank test (1) provides a comparison of the observed number of deaths in each group versus the number that would be expected if the total mortality were distributed according to the proportion in each group. These statistical comparisons are often accompanied by Kaplan–Meier curves that provide a graphic display of the distribution of survivorship over time (2). This estimator, calculated from samples that are partially censored, is a monotonically non-increasing step function with steps at each observed death time. Although calculated separately for each group, these graphs are displayed simultaneously in a single plot to promote a visual comparison of survival over the entire study period. It is often of interest to compare the survival of a single sample to that of a defined reference population. For example, when a series of patients with a rare, life-threatening disease has been collected, it may be of interest to know if the study sample is experiencing the same survival as the demographically matched standard (general) population, according to actuarial tables. This is especially of interest when the disease is curable or not usually lethal and the age of onset is late in life. It is not appropriate to use methods developed for two-sample comparisons to do this analysis, because the variance would be incorrectly calculated and thus the P value would be invalid. It is possible to provide a single summary measure of the relative survival of a sample compared with a standard population by estimating the standardized mortality ratio (3). However, investigators often want to report a P value from a statistical test that compares the two populations—essentially a one-sample log-rank test. Although such tests are published in the statistical literature (4,5), medical investigators do not generally read this literature, and thus these tests are not widely known and used by this community. In fact, these articles (4,5) have been cited fewer than 10 times in the medical literature over the past 20 years. Some ad hoc methods have been devised for a one-sample survival test. For example, one approach is to use the actuarial tables to determine the expected remaining lifetime at the age of study entry for each of the subjects in the sample and then to treat these times as exact death times of a hypothetical sample of the same number of subjects from the reference population. It is possible then to calculate a two-sample log-rank test and report the resulting P value. However, this test is inappropriate because the variance would be incorrectly calculated and thus the P value would be invalid. Similar pitfalls arise in trying to obtain an accompanying graphical display that would appropriately represent the survival of the standard population in the same manner in which the Kaplan–Meier plot represents the sample. Because there are no methods that are widely cited in the medical literature, there is a tendency to develop ad hoc methods. For example, one approach is to calculate the expected remaining lifetime for each subject in the sample by using the actuarial tables matched by age, sex, and/or race. This set of numbers is then treated as exact observed death times, and the Kaplan–Meier estimator is calculated for this hypothetical population. This calculation results in a step function with the number of steps equal to the size of the sample being studied. This is not correct. As an illustration, suppose everyone in the sample began observation at the same age and, thus, would have the same expected remaining lifetime, say s. The survival distribution for the hypothetical matched sample representing the reference group would then have a value of 1 until s, at which point the curve would drop to 0. The correct method must use the entire remaining survival curve (calculated from the reference actuarial tables) for each subject. The purpose of this commentary is to describe both the simple one-sample log-rank test that is equivalent to the standardized mortality ratio and an estimate for survivorship in the matched standard population that allows a visual comparison of survivorship of the sample and standard populations. We will discuss the issues in designing a study that will rely on one-sample methods. The software to perform analyses discussed in this commentary can be found at our Web site: http://biostatistics.mgh.harvard. edu/biostatistics/resources.html (6). As an illustration, we use these methods to compare the survival of a small cohort of patients diagnosed with extra-mammary Paget’s disease at Massachusetts General Hospital with the survival of the general population (7).