Abstract
Parametric statistical problems involving both large amounts of data and models with many parameters raise issues that are explicitly or implicitly differential geometric. When the number of nuisance parameters is comparable to the sample size, alternative approaches to inference on interest parameters treat the nuisance parameters either as random variables or as arbitrary constants. The two approaches are compared in the context of parametric survival analysis, with emphasis on the effects of misspecification of the random effects distribution. Notably, we derive a detailed expression for the precision of the maximum likelihood estimator of an interest parameter when the assumed random effects model is erroneous, recovering simply derived results based on the Fisher information in the correctly specified situation but otherwise illustrating complex dependence on other aspects. Methods of assessing model adequacy are given. The results are both directly applicable and illustrate general principles of inference when there is a high-dimensional nuisance parameter. Open problems with an information geometrical bearing are outlined.
Highlights
Statistical analysis when the number of unknown parameters is comparable with the number of independent observations may demand modification of maximumlikelihood-based methods [7]
We study analyses based on underlying exponential distributions, that is that the observations are in effect the first point events in individual Poisson processes
Equation (4) shows that the gamma random effects formulation is more efficient than the one in which nuisance parameters are treated as arbitrary constants, provided that the random effects specification is reasonable
Summary
Statistical analysis when the number of unknown parameters is comparable with the number of independent observations may demand modification of maximumlikelihood-based methods [7]. Yates [22,23] has discussed these issues in depth both for factorial experiments and for variety trials in connection with balanced and partially balanced incomplete block designs. His development, powerful and almost explanation free, hinges, especially for incomplete block designs, on the geometry of least squares and the distinction between error-estimating and effect-estimating subspaces. Similar forms of argument implicitly underlie the present paper Later discussion of these issues has mostly been either in general terms [6, chapter 2], or has approached them from a more decision-oriented perspective In the present paper we show the considerations involved in the context of parametric analysis of matched pair survival data. The results aim both to be directly applicable and to illustrate general principles
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