Semiparametric regression is a fusion between parametric regression and nonparametric regression that integrates low-rank penalized splines, mixed model and hierarchical Bayesian methodology – thus allowing more streamlined handling of longitudinal and spatial correlation. We review progress in the field over the five-year period between 2003 and 2007. We find semiparametric regression to be a vibrant field with substantial involvement and activity, continual enhancement and widespread application.

https://ro.uow.edu.au/cgi/viewcontent.cgi?article=9857&context=infopapers

Semiparametric regression during 2003–2007

Linear mixed models with normally distributed response are routinely used in longitudinal data. However, the accuracy of the assumed normal distribu- tion is crucial for valid inference of the parameters. We present a new class of asym- metric linear mixed models that provides for an efficient estimation of the parame- ters in the analysis of longitudinal data. We assume that, marginally, the random effects follow a multivariate skew-normal/independent distribution (Branco and Dey (2001)) and that the random errors follow a symmetric normal/independent distribution (Lange and Sinsheimer (1993)), providing an appealing robust alter- native to the usual symmetric normal distribution in linear mixed models. Specific distributions examined include the skew-normal, the skew-t, the skew-slash, and the skew-contaminated normal distribution. We present an efficient EM-type algo- rithm for the computation of maximum likelihood estimation of parameters. The technique for the prediction of future responses under this class of distributions is also investigated. The methodology is illustrated through an application to Fram- ingham cholesterol data and a simulation study.

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Likelihood based inference for skew-normal independent linear mixed models

Statistical models that include random effects are commonly used to analyze longitudinal and correlated data, often with strong and parametric assumptions about the random effects distribution. There is marked disagreement in the literature as to whether such parametric assumptions are important or innocuous. In the context of generalized linear mixed models used to analyze clustered or longitudinal data, we examine the impact of random effects distribution misspecification on a variety of inferences, including prediction, inference about covariate effects, prediction of random effects and estimation of random effects variances. We describe examples, theoretical calculations and simulations to elucidate situations in which the specification is and is not important. A key conclusion is the large degree of robustness of maximum likelihood for a wide variety of commonly encountered situations.

Misspecifying the Shape of a Random Effects Distribution: Why Getting It Wrong May Not Matter

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Most plant and animal populations have substantial interannual variability in survival, growth rate, and fecundity. They also exhibit substantial variability among individuals in traits such as size, age, condition, and disease status that have large impacts on individual fates and consequently on the future of the population. We present here methods for constructing and analyzing a stochastic integral projection model (IPM) incorporating both of these forms of variability, illustrated through a case study of the monocarpic thistle Carlina vulgaris. We show how model construction can exploit the close correspondence between stochastic IPMs and statistical analysis of trait-fate relationships in a ''mixed'' or ''hierarchical'' models framework. This correspondence means that IPMs can be parameter- ized straightforwardly from data using established statistical techniques and software (vs. the largely ad hoc methods for stochastic matrix models), properly accounting for sampling error and between-year sample size variation and with vastly fewer parameters than a conventional stochastic matrix model. We show that the many tools available for analyzing stochastic matrix models (such as stochastic growth rate, kS, small variance approximations, elasticity/sensitivity analysis, and life table response experiment (LTRE) analysis) can be used for IPMs, and we give computational formulas for elasticity/sensitivity analyses. We develop evolutionary analyses based on the connection between growth rate sensitivity and selection gradients and present a new method using techniques from functional data analysis to study the evolution of function-valued traits such as size-dependent flowering probability. For Carlina we found consistent selection against variability in both state-specific transition rates and the fitted functions describing state dependence in demographic rates. For most of the regression parameters defining the IPM there was also selection against temporal variance; however, in some cases the effects of nonlinear averaging were big enough to favor increased temporal variation. The LTRE analysis identified year-to-year variation in survival as the dominant factor in population growth variability. Evolutionary analysis of flowering strategy showed that the entire functional relationship between plant size and flowering probability is at or near an evolutionarily stable strategy (ESS) shaped by the size-specific trade-off between the benefit (fecundity) and cost (mortality) of flowering in a temporally varying environment.

Integral projection models for populations in temporally varying environments

A linear mixed model with a smooth random effects density is proposed. A similar approach to P-spline smoothing of Eilers and Marx (1996, Statistical Science 11, 89-121) is applied to yield a more flexible estimate of the random effects density. Our approach differs from theirs in that the B-spline basis functions are replaced by approximating Gaussian densities. Fitting the model involves maximizing a penalized marginal likelihood. The best penalty parameters minimize Akaike's Information Criterion employing Gray's (1992, Journal of the American Statistical Association 87, 942-951) results. Although our method is applicable to any dimensions of the random effects structure, in this article the two-dimensional case is explored. Our methodology is conceptually simple, and it is relatively easy to fit in practice and is applied to the cholesterol data first analyzed by Zhang and Davidian (2001, Biometrics 57, 795-802). A simulation study shows that our approach yields almost unbiased estimates of the regression and the smoothing parameters in small sample settings. Consistency of the estimates is shown in a particular case.

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Wendimagegn Ghidey

Papers

Smooth random effects distribution in a linear mixed model.