Statistical Models and Methods for Lifetime Data

histories are grouped to give monthly counts of events, the information needed to compute Nelson’s variance estimate for the sample MCF, described in Chapter 4, has been lost. Instead, Nelson describes and illustrates how to compute a simple “naive” variance estimate and con dence intervals that would be correct under the assumption of an underlying nonhomogeneous Poisson process model. Chapter 6 describes methods for analyzing recurrence data when events can be divided into categories. Examples include different failure modes for a system and gender of a child born. The basic underlying model is that each category has its own population MCF. Under weak assumptions, the MCFs for individual event types can be added to give the MCF for a speci ed set of event types. Knowing the MCF for each category would allow one to answer questions about the MCF for all types of events combined, for a particular type of event, and for any chosen subset of the event types. Each of these questions is illustrated with data on failures of traction motors for subway cars. For example, a reliability analyst can estimate the MCF for a system under the assumption that one or more failure modes can be eliminated. As Nelson points out, the MCF estimators presented in this chapter do not require independence of the underlying stochastic processes that generate the different kinds of events. This is in contrast to the life data competing-risk model where the assumption of independence is critical for making inferences about the effect of removing a failure mode (see, e.g., chapt. 5 of Nelson 1982). For the MCF model, there is, however, the tacit assumption that eliminating a failure mode will not affect the MCF functions of the other failure modes. Although the methods based on the nonparametric recurrence stochastic process model are versatile and require minimal assumptions, in more complicated situations they cannot be applied without careful thought. Consider, for example, a repairable system that has a replaceable unit with two failure modes (A and B), both of which are caused by a common mechanism (e.g., corrosion). Due to the common cause, the times to failure for the two components are highly correlated. When the component fails from either A or B, it is replaced, censoring the other mode. If this censoring of the other mode is naively overlooked, and an engineering change is made to eliminate one of the failure modes (say A), then only the symptom has been  xed and there will be a corresponding increase in the MCF for mode B. Then looking at the past data for the occurrence of mode B to estimate the MCF of mode B alone would be misleading. Chapter 7 presents methods for comparing sample MCFs to see whether they differ statistically. The methods are based on an estimate of the difference between two MCFs of processes to be compared and con dence intervals computed for this difference. The methods are illustrated by comparing treatments for recurrent bladder tumors and replacements of two different batches of locomotive breaking grids. Both pointwise and simultaneous (over time) comparisons are described. There is also brief discussion of multiple comparison methods that would be needed if more than two MCFs are to be compared. Chapter 8 provides a very useful survey of other topics closely related to the methods presented in the main part of the book. Topics described include Poisson process and nonhomogeneous Poisson process models, renewal process models, models with covariates, and other models. As with Wayne Nelson’s other books, this book contains a valuable collection of interesting actual applications and corresponding data that we can expect to be used in future publications by other people doing research in this area. An Excel workbook that contains all of the data in the book is available at http:// www.siam. org/books/sa10. In summary, this is an important and interesting book from which most statisticians and many others who analyze data will derive bene t. As the easyto-use tools described here become more commonly known, I predict that the methods will be much more widely used.