THE DESIGN AND ANALYSIS OF EXPERIMENTS. By Oscar Kempthorne. New York, John Wiley and Sons, Inc., 1952. 631 pp. $8.50. This book by a teacher of statistics (as well as a consultant for \"experimenters\") is a comprehensive study of the philosophical background for the statistical design of experiment. It is necessary to have some facility with algebraic notation and manipulation to be able to use the volume intelligently. The problems are presented from the theoretical point of view, without such practical examples as would be helpful for those not acquainted with mathematics. The mathematical justification for the techniques is given. As a somewhat advanced treatment of the design and analysis of experiments, this volume will be interesting and helpful for many who approach statistics theoretically as well as practically. With emphasis on the \"why,\" and with description given broadly, the author relates the subject matter to the general theory of statistics and to the general problem of experimental inference. MARGARET J. ROBERTSON

The Design and Analysis of Experiments

1. Introduction and Overview. 2. Detecting Influential Observations and Outliers. 3. Detecting and Assessing Collinearity. 4. Applications and Remedies. 5. Research Issues and Directions for Extensions. Bibliography. Author Index. Subject Index.

Regression Diagnostics: Identifying Influential Data and Sources of Collinearity

Summary. Recent work by Reiss and Ogden provides a theoretical basis for sometimes preferring restricted maximum likelihood (REML) to generalized cross-validation (GCV) for smoothing parameter selection in semiparametric regression. However, existing REML or marginal likelihood (ML) based methods for semiparametric generalized linear models (GLMs) use iterative REML or ML estimation of the smoothing parameters of working linear approximations to the GLM. Such indirect schemes need not converge and fail to do so in a non-negligible proportion of practical analyses. By contrast, very reliable prediction error criteria smoothing parameter selection methods are available, based on direct optimization of GCV, or related criteria, for the GLM itself. Since such methods directly optimize properly defined functions of the smoothing parameters, they have much more reliable convergence properties. The paper develops the first such method for REML or ML estimation of smoothing parameters. A Laplace approximation is used to obtain an approximate REML or ML for any GLM, which is suitable for efficient direct optimization. This REML or ML criterion requires that Newton–Raphson iteration, rather than Fisher scoring, be used for GLM fitting, and a computationally stable approach to this is proposed. The REML or ML criterion itself is optimized by a Newton method, with the derivatives required obtained by a mixture of implicit differentiation and direct methods. The method will cope with numerical rank deficiency in the fitted model and in fact provides a slight improvement in numerical robustness on the earlier method of Wood for prediction error criteria based smoothness selection. Simulation results suggest that the new REML and ML methods offer some improvement in mean-square error performance relative to GCV or Akaike's information criterion in most cases, without the small number of severe undersmoothing failures to which Akaike's information criterion and GCV are prone. This is achieved at the same computational cost as GCV or Akaike's information criterion. The new approach also eliminates the convergence failures of previous REML- or ML-based approaches for penalized GLMs and usually has lower computational cost than these alternatives. Example applications are presented in adaptive smoothing, scalar on function regression and generalized additive model selection.

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Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models

On robust estimation of the location parameter

B-splines are attractive for nonparametric modelling, but choosing the optimal number and positions of knots is a complex task. Equidistant knots can be used, but their small and discrete number allows only limited control over smoothness and fit. We propose to use a relatively large number of knots and a difference penalty on coefficients of adjacent B-splines. We show connections to the familiar spline penalty on the integral of the squared second derivative. A short overview of B-splines, of their construction and of penalized likelihood is presented. We discuss properties of penalized B-splines and propose various criteria for the choice of an optimal penalty parameter. Nonparametric logistic regression, density estimation and scatterplot smoothing are used as examples. Some details of the computations are presented.

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Flexible smoothing with B-splines and penalties

Introduction.- Regression Models.- The Classical Linear Model.- Extensions of the Classical Linear Model.- Generalized Linear Models.- Categorical Regression Models.- Mixed Models.- Nonparametric Regression.- Structured Additive Regression.- Quantile Regression.- A Matrix Algebra.- B Probability Calculus and Statistical Inference.- Bibliography.- Index.

Regression: Models, Methods and Applications

Direct generalized additive modeling with penalized likelihood

Penalized splines have gained much popularity as a flexible tool for smoothing and semi-parametric models. Two approaches have been advocated: 1 use a B-spline basis, equally spaced knots, and difference penalties [Eilers PHC, Marx BD. Flexible smoothing using B-splines and penalized likelihood with Comments and Rejoinder. Stat Sci 1996, 11:89-121.] and 2 use truncated power functions, knots based on quantiles of the independent variable and a ridge penalty [Ruppert D, Wand MP, Carroll RJ. Semiparametric Regression. New York: Cambridge University Press; 2003]. We compare the two approaches on many aspects: numerical stability, quality of the fit, interpolation/extrapolation, derivative estimation, visual presentation and extension to multidimensional smoothing. We discuss mixed model and Bayesian parallels to penalized regression. We conclude that B-splines with difference penalties are clearly to be preferred. WIREs Comp Stat 2010 2 637-653 DOI: 10.1002/wics.125

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Splines, knots, and penalties

We consider generalized linear regression with many highly correlated regressors—for instance, digitized points of a curve on a spatial or temporal domain. We refer to this setting as signal regression, which requires severe regularization because the number of regressors is large, often exceeding the number of observations. We solve collinearity by forcing the coefficient vector to be smooth on the same domain. Dimension reduction is achieved by projecting the signal coefficient vector onto a moderate number of B splines. A difference penalty between the B-spline coefficients further increases smoothness-the P-spline framework of Eilers and Marx. The procedure is regulated by a penalty parameter chosen using information criteria or cross-validation.

Brian D. Marx

Papers

Flexible smoothing with B-splines and penalties

Regression: Models, Methods and Applications

Direct generalized additive modeling with penalized likelihood

Splines, knots, and penalties

Generalized linear regression on sampled signals and curves: a P -spline approach