ABSTRACT
Statistical analyses of clinical trials involving fatal diseases often focus on overall survival time as the primary outcome. In such settings, typically there are one or more intermediate events that the patient may experience prior to death. Such nonfatal events may be very important in that they characterize morbidity or early treatment effects, and the occurrence of a particular event or the time from the start of treatment to the event may help predict the patient’s subsequent survival time [1-5]. Consequently, such intermediate events often are used by physicians for therapeutic decision-making, and many clinical trial designs base their primary treatment evaluation on intermediate event times rather than overall survival time in order to reduce the trial’s cost and duration. An example of an important intermediate event in oncology settings where patients have active disease at enrollment is disease progression, which is characterized by a specified amount or type of worsening compared to the patient’s baseline disease status. While disease progression increases the patient’s risk of death, a technical complication is that each patient may die either with or without disease progression occurring first. A common practice is to use the minimum of the time to progression and the time to death without progression, so-called
progression-free survival (PFS) time, as the primary endpoint for treatment evaluation. Similarly, if a trial’s entry criteria require that the patient’s disease has been brought into remission previously and that the patient is still disease-free, each patient may die either with or without recurrence of disease. In this setting, the minimum of the time to recurrence and the time to death without recurrence, ‘‘diseasefree survival’’ (DFS) time, commonly is used as the primary endpoint. Both cases may be described generally in terms of three transition times, T1¼ time from the start of treatment to the intermediate event, T2¼ time from the intermediate event to death, and T0¼ time from the start of treatment to death without the intermediate event occurring, sometimes called regimen-related death. Aside from administrative right censoring or drop-out, overall survival time is actually TD¼ T0I(T0< T1)þ (T1þ T2)I (T0> T1), and PFS or DFS is TF¼min{T0,T1}, where I(A) indicates the event A. The ultimate goal of therapy is to cure the disease, that is, to make T2 so large that, for a patient with covariates Z¼ (Z1, . . . , Zq), the distribution of [T2 j Z] is close to that of the survival time of an individual with similar covariates but without the disease. Similarly, an ideal treatment would have no regimen-related death, formally, Pr(T0< T1)¼ 0. A key point is that (T0,T1,T2) are actually potential outcomes, or counterfactuals [6], since either T0 or (T1,T2) but not both are observed for any patient. There also are many examples in oncology of desirable intermediate events that
decrease the risk of death. These include achieving a substantive amount of shrinkage of a solid tumor, achieving complete remission (CR) of acute leukemia, defined in terms of the patient’s circulating blast cells, white blood cells, and platelet count all returning to normal levels and, in stem cell transplantation, achieving engraftment of the transplanted cells in the bone marrow, characterized by an absolute neutrophil count achieving a functional level. The issue noted above, that a second-line treatment may be given when an intermediate event occurs, also arises with desirable intermediate events. For example, once frontline treatment of leukemia achieves CR, a second round of ‘‘consolidation’’ treatment to reduce the risk of disease recurrence may be given [7]. Similarly, once a stem cell transplant from a matched donor has engrafted, the patient may receive prophylactic treatment to reduce the risk of graft-versus-host disease. While the relationships between intermediate events and survival time may seem
intuitively obvious, several technical complications arise when analyzing the multivariate event time data that they motivate. In the simplest case with one possible intermediate event motivating potential outcomes (T0,T1, T2), the times to death with or without the antecedent event, specifically T1þT2 and T0, may have very different distributions. Moreover, the time T1 to the intermediate event and the subsequent survival time T2 typically are not independent. In more complex settings where two or more different intermediate events are possible, the occurrence of one event may preclude the others, that is, the events may be competing risks. This is the case we consider in this chapter. We describe a multivariate parametric model for the times to the nonfatal events, the residual survival times following the events, and the time to death without any preceding intermediate event. We illustrate the model by fitting it to a data set from patients with acute myelogenous leukemia (AML) or myelodyplastic syndrome (MDS) that includes the times to two intermediate events, one desirable and the other undesirable, as well as overall survival time. The general competing risks model structure, and statistical analyses similar to those presented here, are given in Shen and Thall [8] and Estey et al. [9].
