Analysis of a 2 × 2 table implies an examination of the data at a specific point in time. This analysis is satisfactory if we are looking for events that occur within relatively short periods and if all patients have the same duration of follow-up. In longer-term studies, however, we are interested not only in the total number of events but also in their timing. For instance, we may focus on whether therapy for patients with a uniformly fatal condition (unresectable lung cancer, for example) delays death.
When the timing of events is important, investigators could present the results in the form of several 2 × 2 tables constructed at different points of time after the study began. For example, Table 9-2 represents the situation after the study was finished. Similar tables could be constructed describing the fate of all patients available for analysis after their enrollment in the trial for 1 week, 1 month, 3 months, or whatever time we chose to examine. The analysis of accumulated data that takes into account the timing of events is called survival analysis. Do not infer from the name, however, that the analysis is restricted to deaths; in fact, any dichotomous outcome occurring over time will qualify.
The survival curve of a group of patients describes their status at different times after a defined starting point.8 In Figure 9-2, we show the survival curve from the bleeding varices trial. Because the investigators followed up some patients for a longer time, the survival curve extends beyond the mean follow-up of approximately 10 months. At some point, prediction becomes imprecise because there are few patients remaining to estimate the probability of survival. The CIs around the survival curves capture the precision of the estimate.
Survival Curves for Ligation and Sclerotherapy
Even if the true RR, or RRR, is constant throughout the duration of follow-up, the play of chance will ensure that the point estimates differ. Ideally then, we would estimate the overall RR by applying an average, weighted for the number of patients available, for the entire survival experience. Statistical methods allow just such an estimate. The probability of events occurring at any point in each group is referred to as the hazard for that group, and the weighted RR during the entire study duration is known as the hazard ratio.
A major advantage of using survival analysis is the ability to account for differential length of follow-up. In many trials of a fixed duration, some patients are enrolled early and thus have long follow-up and some later with consequently shorter follow-up. Survival analysis takes into account both those with shorter (by a process called censoring) and those with longer follow-up, and all contribute to estimates of hazard and the hazard ratio. Patients are censored at the point at which they are no longer being followed up. Appropriate accounting for those with differential length of follow-up is not possible in 2 × 2 tables that deal only with the number of events.
“Competing risks” is an issue that arises when one event influences the likelihood of another event. The most extreme example is death: if the outcome is stroke, people who die can no longer have a stroke. Competing risks also can arise when there are 2 or more outcome events among living patients (for instance, if a patient has a stroke, the likelihood of a subsequent transient ischemic attack may decrease). Investigators can deal with the problem of competing risks by censoring patients at the time of the “competing” events (death and stroke in the previous examples). The censoring approach, however, has its limitations.9
Specifically, the usual assumption is that the censored events are independent of the main outcome of interest, but in practice this assumption may not be correct. In our example, it is probable that patients who experience myocardial infarction have a higher death rate than those without myocardial infarction, and this would violate the assumption of independence. Investigators also sometimes use censoring for those lost to follow-up. This is much more problematic because the censoring assumes that those with shorter follow-up are similar to those with longer follow-up—the only difference, indeed, being length of follow-up. Because loss to follow-up may be associated with a higher or lower likelihood of events (and thus, those lost differ from those who are followed up), the censoring approach does not deal with the risk of bias associated with loss to follow-up.9