# Introduction to Survival Times

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## Chapter: Biostatistics for the Health Sciences: Analysis of Survival Times

Information from survival analysis is used frequently to assess the efficacy of clinical trials.

Analysis of Survival Times

INTRODUCTION TO SURVIVAL DATA

In survival analysis, we follow patients over time, until the occurrence of a particular event such as death, relapse, recurrence, or some other event that represents a dichotomy. Of special interest to the practitioners of survival analysis is the construction of survival curves, which are based on the time interval between a procedure and an event.

Information from survival analysis is used frequently to assess the efficacy of clinical trials. Researchers follow patients during the trial in order to track events such as a recurrence of an illness, occurrence of an adverse event related to the treatment, or death. The term “survival analysis” came about because often mortali-ty (death) was studied as the outcome; however, survival analysis can be applied more generally to many different types of events.

In a clinical trial, an investigator may want to compare a survival curve for a treatment group with one for a control group to determine whether the treatment is associated with increased longevity; one of the notable examples arises from the area of cancer treatment studies, which focus on five-year survival rates after treatment. A new, specialized area in survival analysis is the estimation of cure rates. The investigator may believe that a certain percentage of patients will be cured by a treatment and, thus, uses survival analysis to estimate the cure rate. Section 15.2.4 will cover cure rate models that use a modification to the survival curve.

Several characteristics of survival data make them different from most data we encounter: (1) patients are in the study for varying amounts of time; (2) because some patients experience the event, these are the ones who provide complete infor mation; and (3) the trial is eventually terminated and the patients who have not ex-perienced the event are “right-censored.” The term right-censored refers to the fact that we do not know how much longer patients who remained in the trial until its end would have gone event-free. The time to the event for them is at least the time from treatment to the end of the study. Right-censoring is the primary characteristic of survival data that makes the analysis unique and different from other methods previously covered in this text.

As noted in point (1) above, a feature of data from survival analyses is that pa-tients typically do not enter the study at the same time. Clinical trials generally have an accrual period that could be six months or longer. Candidates for the study are found and a sufficient number enrolled during the accrual period until statistical power or precision requirements have been met.

Still another factor that produces varying amounts of observation time in the study has to do with the initiation of disease onset. Although the time of occurrence of the event is generally well defined and easily recognized, the onset of the clinical syndrome leading to the event may be ambiguous. Thus, what is called “the starting time” for the time to event is sometimes difficult to define. For example, if we are studying a chronic disease such as cancer, diabetes, or heart disease, the precise time of onset may be impossible to delineate.

A common substitute for date of onset is date of diagnosis. This alternative may be unreliable because of the considerable lag that often exists between the first occurrence of a disease and its diagnosis. This lag may be due to health ser-vice utilization patterns (e.g., lack of health insurance coverage, infrequent doctor visits, and delay in seeking health care) or the natural history of many chronic dis-eases (e.g., inapparent signs and symptoms of the early phases of disease). Some infections, such as HIV or hepatitis C, are associated with an extended latency pe-riod between lodgment of a virus and development of observable symptoms. Consequently, date of diagnosis is used as the best available proxy for date of on-set.

With respect to point (2) above, some patients may be lost to follow-up. For ex-ample, they decide to drop out of the study because they leave the geographic area. Sometimes, statisticians treat this form of censoring differently from right censor-ing. Although start times vary on the actual time scale, in survival analysis we cre-ate a scale that ignores the starting time. We are interested only in the time interval from entry into the study (or treatment time, beginning when the patient is random-ized into a treatment group) until the event or censoring occurs. Thus, we modify the time axis as if all patients start together.

We can use parametric models to describe patients’ survival functions. These models are applicable when each patient is viewed as having a time to event that is similar to a random draw from some survival distribution whose form is known ex-cept for a few parameters (the exponential and Weibull distributions are examples of such parametric models). When the parametric form is difficult to specify, non-parametric techniques can be used to estimate the survival function. Details follow in the next section.

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