Tests for multivariate recurrent events in the presence of a terminal event

Biostatistics (2004), 5, 1, pp. 129–143 Printed in Great Britain Tests for multivariate recurrent events in the presence of a terminal event S UMMARY In studies involving diseases associated with high rates of mortality, trials are frequently conducted to evaluate the effects of therapeutic interventions on recurrent event processes terminated by death. In this setting, cumulative mean functions form a natural basis for inference for questions of a health economic nature, and Ghosh and Lin (2000) recently proposed a relevant class of test statistics. Trials of patients with cancer metastatic to bone, however, involve multiple types of skeletal complications, each of which may be repeatedly experienced by patients over their lifetime. Traditionally the distinction between the various types of events is ignored and univariate analyses are conducted based on a composite recurrent event. However, when the events have different impacts on patients’ quality of life, or when they incur different costs, it can be important to gain insight into the relative frequency of the specific types of events and treatment effects thereon. This may be achieved by conducting separate marginal analyses with each analysis focusing on one type of recurrent event. Global inferences regarding treatment benefit can then be achieved by carrying out multiplicity adjusted marginal tests, more formal multiple testing procedures, or by constructing global test statistics. We describe methods for testing for differences in mean functions between treatment groups which accommodate the fact that each particular event process is ultimately terminated by death. The methods are illustrated by application to a motivating study designed to examine the effect of bisphosphonate therapy on the incidence of skeletal complications among patients with breast cancer metastatic to bone. We find that there is a consistent trend towards a reduction in the cumulative mean for all four types of skeletal complications with bisphosphonate therapy; there is a significant reduction in the need for radiation therapy for the treatment of bone. The global test suggests that bisphosphonate therapy significantly reduces the overall number of skeletal complications. Keywords: Marginal methods; Multivariate response; Recurrent event; Robust inference; Terminal event. 1. I NTRODUCTION In studies involving diseases associated with high mortality rates, trials are frequently conducted to evaluate the effects of therapeutic interventions on response processes terminated by death. Examples include studies of medical costs (Lin et al., 1997; Bang and Tsiatis, 2000), quality of life (Zhao and Tsiatis, 1997, 1999) and recurrent events (Cook and Lawless, 1997; Li and Lagakos, 1997). Interest typically lies in cumulative aspects of such response processes, such as the cumulative lifetime costs, quality adjusted lifetime, or the cumulative lifetime number of events. Analyses dealing with such ∗ To whom correspondence should be addressed. c Oxford University Press (2004); all rights reserved. Biostatistics 5(1)  BINGSHU ERIC CHEN∗ , RICHARD J. COOK Department of Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1 130 B. E. C HEN AND R. J. C OOK 2. B ISPHOSPHONATES FOR THE TREATMENT OF BONE METASTASES Theriault et al. (1999) report on a multicenter randomized trial designed to investigate the effect of a bisphosphonate, pamidronate, on the development of skeletal complications in breast cancer patients with bone metastases. Patients were accrued from 85 study sites in the United States, Canada, Australia and New Zealand. Patients with stage IV breast cancer receiving cytotoxic chemotherapy with at least two predominantly lytic bone lesions at least one centimeter in diameter were randomized within strata defined by ECOG status. A total of 371 women were enrolled in the study with 182 randomized to receive 90 mg of pamidronate every four weeks and 189 randomized to receive dextrose infusions at the same time points. After completion of the planned one year of follow-up, the follow-up was extended for an additional year to assess long-term effects and survival. At monthly visits patients were assessed and the occurrence of skeletal complications was recorded. Skeletal complications include nonvertebral and vertebral fractures, the need for surgery to treat or prevent fractures, and the need for radiation for the questions must address the dependent censoring of the cumulative response which results from the fact that survival times are typically right-censored for some individuals (Strawderman, 2000). Suitable techniques are frequently formulated in terms of ‘inverse probability of censoring weighted’ analyses, although alternative approaches can also be taken (e.g. Strawderman, 2000). For problems in which a single type of recurrent event is of interest, Cook and Lawless (1997) proposed the use of cumulative mean functions which reflect the marginal cumulative mean number of events experienced per patient over time, accounting for the fact that the recurrent event process is terminated by death. Such mean functions form a natural basis for inference for questions of a health economic nature or for other settings where interest lies in comparing overall disease burden at the population level. If interest lies in testing for differences in cumulative mean functions between groups, a class of test statistics recently developed by Ghosh and Lin (2000) may be used. Trials of patients with cancer metastatic to bone, however, involve multiple types of skeletal complications which may be repeatedly experienced by patients over the course of follow-up (Theriault et al., 1999). Traditionally the distinction between the various types of events is ignored and univariate analyses are conducted based on a composite recurrent event. However, when the events have a different impact on quality of life, or when they incur different costs, it can be important to gain insight into the relative frequency of the various types of events and treatment effects thereon. This may be achieved by conducting separate marginal analyses with each analysis focusing on one type of recurrent event. Global inferences regarding treatment effect can then be conducted by carrying out multiplicity adjusted marginal tests, using more formal multiple testing procedures, or by constructing global test statistics. We describe methods for testing for differences between treatment groups with respect to multiple cumulative mean functions in the presence of a common terminal event (i.e. death). These methods will be shown to be valid in settings where there is a dependence between the recurrent event rate and survival time; naive tests based on rate functions (e.g. Cook et al., 1996) are invalid in such settings. The remainder of the paper is organized as follows. In Section 2 we briefly discuss a motivatin (...truncated)