Efficient Bayesian Multivariate Surface Regression

Methods for choosing a fixed set of knot locations in additive spline models are fairly well established in the statistical literature. The curse of dimensionality makes it nontrivial to extend these methods to nonadditive surface models, especially when there are more than a couple of covariates. We propose a multivariate Gaussian surface regression model that combines both additive splines and interactive splines, and a highly efficient Markov chain Monte Carlo algorithm that updates all the knot locations jointly. We use shrinkage prior to avoid overfitting with different estimated shrinkage factors for the additive and surface part of the model, and also different shrinkage parameters for the different response variables. Simulated data and an application to firm leverage data show that the approach is computationally efficient, and that allowing for freely estimated knot locations can offer a substantial improvement in out‐of‐sample predictive performance.     

A note on all-bias designs with applications in spline regression models

If a model is fitted to empirical data, bias can arise from terms which are not incorporated in the model assumptions. As a consequence the commonly used optimality criteria based on the generalized variance of the estimator of the model parameters may not lead to efficient designs for the statistical analysis. In this note some general aspects of all-bias designs are presented, which were introduced in this context by Box and Draper (1959). Using an interesting correspondence between the points of all-bias designs and the knots of quadrature formulas we establish sufficient conditions such that a given design is an all-bias design. The results are illustrated in the special case of spline regression models. In particular our results generalize recent findings of Woods and Lewis (2006).     

The impact of period effects on dose level contrasts in alternating cross‐over designs for first‐time‐in‐human studies

For first‐time‐in‐human studies with small molecules alternating cross‐over designs are often employed and at study end are analyzed using linear models. We discuss the impact of including a period effect in the model on the precision with which dose level contrasts can be estimated and quantify the bias of least squares estimators if a period effect is inherent in the data that is not accounted for in the model. We also propose two alternative designs that allow a more precise estimation of dose level contrasts compared with the standard design when period effects are included in the model. Copyright © 2010 John Wiley & Sons, Ltd.     

Circular balanced repeated measurements designs

The concept of a circular design is defined and when proper balance for various effects is assumed, its universal optimality is proved over the class of all designs with the same set of parameters, Such designs are shown to minimize the variance of the best linear unbiased estimators of contrasts of residual and direct effects over the class of equireplicated designs. All models assume first order residual effects and are of a circular nature. The proofs are presented in a unified manner for several models at a time. They are based on certain matrix domination which occurs when parameters are eliminated from a linear modelj this latter fact is proved for a general linear model.     

Modified spline regression based on randomly right-censored data: A comparative study

ABSTRACT

In this paper, we propose modified spline estimators for nonparametric regression models with right-censored data, especially when the censored response observations are converted to synthetic data. Efficient implementation of these estimators depends on the set of knot points and an appropriate smoothing parameter. We use three algorithms, the default selection method (DSM), myopic algorithm (MA), and full search algorithm (FSA), to select the optimum set of knots in a penalized spline method based on a smoothing parameter, which is chosen based on different criteria, including the improved version of the Akaike information criterion (AICc), generalized cross validation (GCV), restricted maximum likelihood (REML), and Bayesian information criterion (BIC). We also consider the smoothing spline (SS), which uses all the data points as knots. The main goal of this study is to compare the performance of the algorithm and criteria combinations in the suggested penalized spline fits under censored data. A Monte Carlo simulation study is performed and a real data example is presented to illustrate the ideas in the paper. The results confirm that the FSA slightly outperforms the other methods, especially for high censoring levels.     

Asymptotics for penalised splines in generalised additive models

This paper discusses asymptotic theory for penalised spline estimators in generalised additive models. The purpose of this paper is to establish the asymptotic bias and variance as well as the asymptotic normality of the ridge-corrected penalised spline estimator. Furthermore, the asymptotics for the penalised quasi-likelihood fit in mixed models are also discussed.     

Robustness properties of minimally‐supported Bayesian D‐optimal designs for heteroscedastic models

Bayesian D‐optimal designs supported on a fixed number of points were found by Dette & Wong (1998) for estimating parameters in a polynomial model when the error variance depends exponentially on the explanatory variable. The present authors provide optimal designs under a broader class of error variance structures and investigate the robustness properties of these designs to model and prior distribution assumptions. A comparison of the performance of the optimal designs relative to the popular uniform designs is also given. The authors' results suggest that Bayesian D‐optimal designs suported on a fixed number of points are more likely to be globaly optimal among all designs if the prior distribution is symmetric and is concentrated around its mean.     

Optimal regression designs in the presence of random block effects

A- and D-optimal regression designs under random block-effects models are considered. We first identify certain situations where D- and A-optimal designs do not depend on the intra-block correlation and can be obtained easily from the optimal designs under uncorrelated models. For example, for quadratic regression on [−1,1], this covers D-optimal designs when the block size is a multiple of 3 and A-optimal designs when the block size is a multiple of 4. In general, the optimal designs depend on the intra-block correlation. For quadratic regression, we provide expressions for D-optimal designs for any block size. A-optimal designs with blocks of size 2 for quadratic regression are also obtained. In all the cases considered, robust designs which do not depend on the intrablock correlation can be constructed.     

Conditional estimation using prior information in 2‐stage group sequential designs assuming asymptotic normality when the trial terminated early

Two‐stage designs are widely used to determine whether a clinical trial should be terminated early. In such trials, a maximum likelihood estimate is often adopted to describe the difference in efficacy between the experimental and reference treatments; however, this method is known to display conditional bias. To reduce such bias, a conditional mean‐adjusted estimator (CMAE) has been proposed, although the remaining bias may be nonnegligible when a trial is stopped for efficacy at the interim analysis. We propose a new estimator for adjusting the conditional bias of the treatment effect by extending the idea of the CMAE. This estimator is calculated by weighting the maximum likelihood estimate obtained at the interim analysis and the effect size prespecified when calculating the sample size. We evaluate the performance of the proposed estimator through analytical and simulation studies in various settings in which a trial is stopped for efficacy or futility at the interim analysis. We find that the conditional bias of the proposed estimator is smaller than that of the CMAE when the information time at the interim analysis is small. In addition, the mean‐squared error of the proposed estimator is also smaller than that of the CMAE. In conclusion, we recommend the use of the proposed estimator for trials that are terminated early for efficacy or futility.     

Some Restricted randomization rules in sequential designs

This paper presents a new class of designs (Big Stick Designs) for sequentially assigning experimental units to treatments, when only the time covariate is considered. By prescribing the degree of imbalance which the experimenters can tolerate, complete randomization is used as long as the imbalance of the treatment allocation does not exceed the prescribed value. Once it reaches the value, a deterministic assignment is made to lower the imbalance. Such designs can be easily implemented with no programming and little personnel support. They compare favorably with the Biased Coin Designs, the Permuted Black Designs, and the Urn Designs, as far as the accidental bias and selection bias are concerned. Generalizations of these designs are considered to achieve various purposes, e.g., avoidance of deterministic assignments, early balance, etc.     

Robust designs for misspecified logistic models

We develop criteria that generate robust designs and use such criteria for the construction of designs that insure against possible misspecifications in logistic regression models. The design criteria we propose are different from the classical in that we do not focus on sampling error alone. Instead we use design criteria that account as well for error due to bias engendered by the model misspecification. Our robust designs optimize the average of a function of the sampling error and bias error over a specified misspecification neighbourhood. Examples of robust designs for logistic models are presented, including a case study implementing the methodologies using beetle mortality data.     

Multiphase experiments with at least one later laboratory phase. II. Nonorthogonal designs

Principles and laws that apply to nonorthogonal multiphase experiments are developed and illustrated using examples that are nonorthogonal but structure‐balanced, not structure, but first‐order, balanced or unbalanced, thus exposing the differences between the different design types. The design of such experiments using standard designs, a catalogue of designs and computer searches is exemplified. Factor–allocation diagrams are employed to depict the allocations in the examples, and used in producing the anatomies of designs or, when possible, the related skeleton‐analysis‐of‐variance tables, to assess the properties of designs. The formulation of mixed models based on them is also described. Tools used for structure‐balanced experiments are also shown to be applicable to those experiments that are not.     

Block designs robust against the presence of positional effects

The problem considered is to find optimum designs for treatment effects in a block design (BD) setup, when positional effects are also present besides treatment and block effects, but they are ignored while formulating the model. In the class of symmetric balanced incomplete block designs, the Youden square design is shown to be optimal in the sense of minimizing the bias term in the mean squared error (MSE) of the best linear unbiased estimators of the full set of orthonormal treatment contrasts, irrespective of the value of the positional effects.     

Minimum bias designs with constraints

A new class of model-robust optimality criteria, based on the mean squared error, is introduced in this paper. The motivation is to find designs when the researcher is more concerned with controlling the variance than the bias, or vice versa. The set of criteria proposed here is also appealing from a mathematical perspective in the sense that, unlike the Box and Draper (1959, J. Amer. Statist. Assoc. 54, 622–654), criterion, they can be imbedded in the framework of convex design theory and, hence, facilitate the search for globally optimal designs. The basic idea is to minimize a convex function of the bias part of the mean squared error subject to a convex constraint on the variance part, or vice versa. Equivalence theorems are derived and examples for the linear and quadratic regression problems are provided.     

Error variance bias in neighbour balance and evenness of distribution designs

Neighbour balance and evenness of distribution designs help to address user concerns in the two‐dimensional layout of agricultural field trials. This is done by minimising the occurrence of pairwise treatment plot neighbours and ensuring that the replications of treatments are spread out across rows and columns of a trial. Such considerations result in a restriction on the normal randomisation process for a row‐column design which can lead to error variance bias. In this paper, uniformity trial data is used to assess the degree of this bias for both resolvable and non‐resolvable designs. Comparisons are made with a similar investigation using Linear Variance spatial designs.     

Modal non‐linear regression in the presence of Laplace measurement error

In this paper, we propose a robust estimation procedure for a class of non‐linear regression models when the covariates are contaminated with Laplace measurement error, aiming at constructing an estimation procedure for the regression parameters which are less affected by the possible outliers, and heavy‐tailed underlying distribution, as well as reducing the bias introduced by the measurement error. Starting with the modal regression procedure developed for the measurement error‐free case, a non‐trivial modification is made so that the modified version can effectively correct the potential bias caused by measurement error. Large sample properties of the proposed estimate, such as the convergence rate and the asymptotic normality, are thoroughly investigated. A simulation study and real data application are conducted to illustrate the satisfying finite sample performance of the proposed estimation procedure.     

Flexible pair-copula estimation in D-vines using bivariate penalized splines

The paper presents a new method for flexible fitting of D-vines. Pair-copulas are estimated semi-parametrically using penalized Bernstein polynomials or constant and linear B-splines, respectively, as spline bases in each knot of the D-vine throughout each level. A penalty induce smoothness of the fit while the high dimensional spline basis guarantees flexibility. To ensure uniform univariate margins of each pair-copula, linear constraints are placed on the spline coefficients and quadratic programming is used to fit the model. The amount of penalizations for each pair-copula is driven by a penalty parameter which is selected in a numerically efficient way. Simulations and practical examples accompany the presentation.     

Bayesian D-optimal designs for exponential regression models

We consider the Bayesian D-optimal design problem for exponential growth models with one, two or three parameters. For the one-parameter model conditions on the shape of the density of the prior distribution and on the range of its support are given guaranteeing that a one-point design is also Bayesian D-optimal within the class of all designs. In the case of two parameters the best two-point designs are determined and for special prior distributions it is proved that these designs are Bayesian D-optimal. Finally, the exponential growth model with three parameters is investigated. The best three-point designs are characterized by a nonlinear equation. The global optimality of these designs cannot be proved analytically and it is demonstrated that these designs are also Bayesian D-optimal within the class of all designs if gamma-distributions are used as prior distributions.     

Coverage Properties of Confidence Intervals for Generalized Additive Model Components

Abstract. We study the coverage properties of Bayesian confidence intervals for the smooth component functions of generalized additive models (GAMs) represented using any penalized regression spline approach. The intervals are the usual generalization of the intervals first proposed by Wahba and Silverman in 1983 and 1985, respectively, to the GAM component context. We present simulation evidence showing these intervals have close to nominal ‘across‐the‐function’ frequentist coverage probabilities, except when the truth is close to a straight line/plane function. We extend the argument introduced by Nychka in 1988 for univariate smoothing splines to explain these results. The theoretical argument suggests that close to nominal coverage probabilities can be achieved, provided that heavy oversmoothing is avoided, so that the bias is not too large a proportion of the sampling variability. The theoretical results allow us to derive alternative intervals from a purely frequentist point of view, and to explain the impact that the neglect of smoothing parameter variability has on confidence interval performance. They also suggest switching the target of inference for component‐wise intervals away from smooth components in the space of the GAM identifiability constraints.     

Designing field experiments which are subject to representation bias

The term 'representation bias' is used to describe the disparities that exist between treatment effects estimated from field experiments, and those effects that would be seen if treatments were used in the field. In this paper we are specifically concerned with representation bias caused by disease inoculum travelling between plots, or out of the experimental area altogether. The scope for such bias is maximized in the case of airborne spread diseases. This paper extends the work of Deardon et al. (2004), using simulation methods to explore the relationship between design and representation bias. In doing so, we illustrate the importance of plot size and spacing, as well as treatment-to-plot allocation. We examine a novel class of designs, incomplete column designs, to develop an understanding of the mechanisms behind representation bias. We also introduce general methods of designing field trials, which can be used to limit representation bias by carefully controlling treatment to block allocation in both incomplete column and incomplete randomized block designs. Finally, we show how the commonly used practice of sampling from the centres of plots, rather than entire plots, can also help to control representation bias.