A Cox Model for Biostatistics of the Future

Professor Sir David R. Cox (DRC) is widely acknowledged as among the most important scientists of the second half of the twentieth century. He inherited the mantle of statistical science from Pearson and Fisher, advanced their ideas, and translated statistical theory into practice so as to forever change the application of statistics in many fields, but especially biology and medicine. The logistic and proportional hazards models he substantially developed, are arguably among the most influential biostatistical methods in current practice. This paper looks forward over the period from DRC's 80th to 90th birthdays, to speculate about the future of biostatistics, drawing lessons from DRC's contributions along the way. We consider "Cox's model" of biostatistics, an approach to statistical science that: formulates scientific questions or quantities in terms of parameters gamma in probability models f(y; gamma) that represent in a parsimonious fashion, the underlying scientific mechanisms (Cox, 1997); partition the parameters gamma = theta, eta into a subset of interest theta and other "nuisance parameters" eta necessary to complete the probability distribution (Cox and Hinkley, 1974); develops methods of inference about the scientific quantities that depend as little as possible upon the nuisance parameters (Barndorff-Nielsen and Cox, 1989); and thinks critically about the appropriate conditional distribution on which to base infrences. We briefly review exciting biomedical and public health challenges that are capable of driving statistical developments in the next decade. We discuss the statistical models and model-based inferences central to the CM approach, contrasting them with computationally-intensive strategies for prediction and inference advocated by Breiman and others (e.g. Breiman, 2001) and to more traditional design-based methods of inference (Fisher, 1935). We discuss the hierarchical (multi-level) model as an example of the future challanges and opportunities for model-based inference. We then consider the role of conditional inference, a second key element of the CM. Recent examples from genetics are used to illustrate these ideas. Finally, the paper examines causal inference and statistical computing, two other topics we believe will be central to biostatistics research and practice in the coming decade. Throughout the paper, we attempt to indicate how DRC's work and the "Cox Model" have set a standard of excellence to which all can aspire in the future.

EFFICIENT EVALUATION OF RANKING PROCEDURES WHEN THE NUMBER OF UNITS IS LARGE WITH APPLICATION TO SNP IDENTIFICATION

Simulation-based assessment is a popular and frequently necessary approach to evaluation of statistical procedures. Sometimes overlooked is the ability to take advantage of underlying mathematical relations and we focus on this aspect. We show how to take advantage of large-sample theory when conducting a simulation using the analysis of genomic data as a motivating example. The approach uses convergence results to provide an approximation to smaller-sample results, results that are available only by simulation. We consider evaluating and comparing a variety of ranking-based methods for identifying the most highly associated SNPs in a genome-wide association study, derive integral equation representations of the pre-posterior distribution of percentiles produced by three ranking methods, and provide examples comparing performance. These results are of interest in their own right and set the framework for a more extensive set of comparisons.