Optimal designs for copula models

Copula modelling has in the past decade become a standard tool in many areas of applied statistics. However, a largely neglected aspect concerns the design of related experiments. Particularly the issue of whether the estimation of copula parameters can be enhanced by optimizing experimental conditions and how robust all the parameter estimates for the model are with respect to the type of copula employed. In this paper an equivalence theorem for (bivariate) copula models is provided that allows formulation of efficient design algorithms and quick checks of whether designs are optimal or at least efficient. Some examples illustrate that in practical situations considerable gains in design efficiency can be achieved. A natural comparison between different copula models with respect to design efficiency is provided as well.     

Lattice conditional independence models for seemingly unrelated regressions

Seemingly unrelated regressions (SUR) models appear frequently in econometrics and in the analyses of repeated measures designs and longitudinal data. It is known that iterative algorithms are generally required to obtain the MLEs of the regression parameters. Under a minimal set of lattice conditional independence (LCI) restrictions imposed on the covariance structure, however, closed-form MLEs can be obtained by standard linear regression techniques (Andersson and Perlman, 1993, 1994, 1998). In this paper, simulation is used to study the efficiency of these LCI model-based estimators. We also propose two possible improvements of the usual two-stage estimators for the regression parameters.     

On Block Updating in Markov Random Field Models for Disease Mapping

Gaussian Markov random field (GMRF) models are commonly used to model spatial correlation in disease mapping applications. For Bayesian inference by MCMC, so far mainly single-site updating algorithms have been considered. However, convergence and mixing properties of such algorithms can be extremely poor due to strong dependencies of parameters in the posterior distribution. In this paper, we propose various block sampling algorithms in order to improve the MCMC performance. The methodology is rather general, allows for non-standard full conditionals, and can be applied in a modular fashion in a large number of different scenarios. For illustration we consider three different applications: two formulations for spatial modelling of a single disease (with and without additional unstructured parameters respectively), and one formulation for the joint analysis of two diseases. The results indicate that the largest benefits are obtained if parameters and the corresponding hyperparameter are updated jointly in one large block. Implementation of such block algorithms is relatively easy using methods for fast sampling of Gaussian Markov random fields ( Rue, 2001 ). By comparison, Monte Carlo estimates based on single-site updating can be rather misleading, even for very long runs. Our results may have wider relevance for efficient MCMC simulation in hierarchical models with Markov random field components.     

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Combining non-linear regressions that have unequal error variances and some parameters in common

Methods of estimation and inference are presented for the situation where two non-linear regression models with unequal error variances contain some parameters in common. Such a situation arises in structural chemistry, when bond lengths are available for three nearly collinear atoms in crystals and a model is required to quantify the extent and form of the relationship between the longer and the shorter bond. Some atomic triples are symmetric and require a different model and error variance from those required by the asymmetric triples. The profile likelihood for the regression parameters is a weighted sum of the logarithms of the sums-of-squares functions from each model, and the estimates can be obtained by using a simple modification to a standard non-linear least squares program. A likelihood ratio test for assessing whether the parameters in common are equal is described. When these techniques are applied to two data sets consisting of bond lengths for bromine–tellurium–bromine and sulphur–tellurium–sulphur triples, there is no evidence against the equality hypothesis. An extension to the model to allow for a non-constant variance is required for proper analysis of the sulphur–tellurium–sulphur data.     

Variance of the difference between two estimated responses

Minimization of the maximum and average variance of the difference between estimated responses are taken as design criteria for univariate polynomial regression models. An optimal design under the first criterion is derived for the second-order model and a class of designs nearly optimal under the second criterion is obtained for the general polynomial models.     

Variance-balanced designs under interference for dependent observations

It is shown that variance-balanced designs can be obtained from Type I orthogonal arrays for many general models with two kinds of treatment effects, including ones for interference, with general dependence structures. These designs can be used to obtain optimal and efficient designs. Some examples and design comparisons are given.     

Rational Regression Models with Continuous Chebyshev Design

In rational regression models, the G-optimal designs are very difficult to derive in general. Even when an G-optimal design can be found, it has, from the point of view of modern nonparametric regression, certain drawbacks because the optimal design crucially depends on the model. Hence, it can be used only when the model is given in advance. This leads to the problem of finding designs which would be nearly optimal for a broad class of rational regression models. In this article, we will show that the so-called continuous Chebyshev Design is a practical solution to this problem.     

Finite sample performance of sequential designs for model identification

Classical regression analysis is usually performed in two steps. In the first step, an appropriate model is identified to describe the data generating process and in the second step, statistical inference is performed in the identified model. An intuitively appealing approach to the design of experiment for these different purposes are sequential strategies, which use parts of the sample for model identification and adapt the design according to the outcome of the identification steps. In this article, we investigate the finite sample properties of two sequential design strategies, which were recently proposed in the literature. A detailed comparison of sequential designs for model discrimination in several regression models is given by means of a simulation study. Some non-sequential designs are also included in the study.     

CVX‐based algorithms for constructing various optimal regression designs

CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada     

Standard errors for EM estimation

The EM algorithm is a popular method for computing maximum likelihood estimates. One of its drawbacks is that it does not produce standard errors as a by-product. We consider obtaining standard errors by numerical differentiation. Two approaches are considered. The first differentiates the Fisher score vector to yield the Hessian of the log-likelihood. The second differentiates the EM operator and uses an identity that relates its derivative to the Hessian of the log-likelihood. The well-known SEM algorithm uses the second approach. We consider three additional algorithms: one that uses the first approach and two that use the second. We evaluate the complexity and precision of these three and the SEM in algorithm seven examples. The first is a single-parameter example used to give insight. The others are three examples in each of two areas of EM application: Poisson mixture models and the estimation of covariance from incomplete data. The examples show that there are algorithms that are much simpler and more accurate than the SEM algorithm. Hopefully their simplicity will increase the availability of standard error estimates in EM applications. It is shown that, as previously conjectured, a symmetry diagnostic can accurately estimate errors arising from numerical differentiation. Some issues related to the speed of the EM algorithm and algorithms that differentiate the EM operator are identified.     

Bayesian and maximin optimal designs for heteroscedastic regression models

The authors consider the problem of constructing standardized maximin D‐optimal designs for weighted polynomial regression models. In particular they show that by following the approach to the construction of maximin designs introduced recently by Dette, Haines & Imhof (2003), such designs can be obtained as weak limits of the corresponding Bayesian _q‐optimal designs. They further demonstrate that the results are more broadly applicable to certain families of nonlinear models. The authors examine two specific weighted polynomial models in some detail and illustrate their results by means of a weighted quadratic regression model and the Bleasdale–Nelder model. They also present a capstone example involving a generalized exponential growth model.     

Combinatorial EM algorithms

The complete-data model that underlies an Expectation-Maximization (EM) algorithm must have a parameter space that coincides with the parameter space of the observed-data model. Otherwise, maximization of the observed-data log-likelihood will be carried out over a space that does not coincide with the desired parameter space. In some contexts, however, a natural complete-data model may be defined only for parameter values within a subset of the observed-data parameter space. In this paper we discuss situations where this can still be useful if the complete-data model can be viewed as a member of a finite family of complete-data models that have parameter spaces which collectively cover the observed-data parameter space. Such a family of complete-data models defines a family of EM algorithms which together lead to a finite collection of constrained maxima of the observed-data log-likelihood. Maximization of the log-likelihood function over the full parameter space then involves identifying the constrained maximum that achieves the greatest log-likelihood value. Since optimization over a finite collection of candidates is referred to as combinatorial optimization, we refer to such a family of EM algorithms as a combinatorial EM (CEM) algorithm. As well as discussing the theoretical concepts behind CEM algorithms, we discuss strategies for improving the computational efficiency when the number of complete-data models is large. Various applications of CEM algorithms are also discussed, ranging from simple examples that illustrate the concepts, to more substantive examples that demonstrate the usefulness of CEM algorithms in practice.     

On simulation of weak records

In the present paper, we discuss algorithms of generation of weak records. These generation algorithms are based on two different methods. In the case, when the inverse function for the underlying distribution function can be obtained explicitly, the corresponding generation algorithms are built on the inverse-transform method. In the case, when the inverse function cannot be obtained explicitly, the algorithms are based on the rejection method. Generation algorithms of our paper are supplied with illustrative examples.     

Likelihood analysis for a class of beta mixed models

Beta regression is a suitable choice for modelling continuous response variables taking values on the unit interval. Data structures such as hierarchical, repeated measures and longitudinal typically induce extra variability and/or dependence and can be accounted for by the inclusion of random effects. In this sense, Statistical inference typically requires numerical methods, possibly combined with sampling algorithms. A class of Beta mixed models is adopted for the analysis of two real problems with grouped data structures. We focus on likelihood inference and describe the implemented algorithms. The first is a study on the life quality index of industry workers with data collected according to an hierarchical sampling scheme. The second is a study assessing the impact of hydroelectric power plants upon measures of water quality indexes up, downstream and at the reservoirs of the dammed rivers, with a nested and longitudinal data structure. Results from different algorithms are reported for comparison including from data-cloning, an alternative to numerical approximations which also allows assessing identifiability. Confidence intervals based on profiled likelihoods are compared with those obtained by asymptotic quadratic approximations, showing relevant differences for parameters related to the random effects. In both cases, the scientific hypothesis of interest was investigated by comparing alternative models, leading to relevant interpretations of the results within each context.     

Modelling time series of counts with overdispersion

The time series of counts observed in practice often exhibit overdispersion. The INGARCH(p, q) models are able to describe integer-valued processes with overdispersion. Known properties of these models, however, are nearly exclusively restricted to the special case p = q = 1. In this article, we derive a set of equations from which the variance and the autocorrelation function of the general case can be obtained. We investigate the purely autoregressive INGARCH(p, 0) models and show that they are closely related to the standard AR(p) models. For p = 1, we determine the marginal distribution in terms of its cumulants. A real-data example highlights potential fields of application of the INGARCH(p, 0) models.     

Learning discrete decomposable graphical models via constraint optimization

Statistical model learning problems are traditionally solved using either heuristic greedy optimization or stochastic simulation, such as Markov chain Monte Carlo or simulated annealing. Recently, there has been an increasing interest in the use of combinatorial search methods, including those based on computational logic. Some of these methods are particularly attractive since they can also be successful in proving the global optimality of solutions, in contrast to stochastic algorithms that only guarantee optimality at the limit. Here we improve and generalize a recently introduced constraint-based method for learning undirected graphical models. The new method combines perfect elimination orderings with various strategies for solution pruning and offers a dramatic improvement both in terms of time and memory complexity. We also show that the method is capable of efficiently handling a more general class of models, called stratified/labeled graphical models, which have an astronomically larger model space.     

Constructing Efficient Experimental Designs for Generalized Linear Models

Exchange algorithms are popular for finding optimal or efficient designs for linear models, but there are few discussions of this type of algorithm for generalized linear models (GLMs) in literature. A new algorithm, generalized Coordinate Exchange Algorithm (gCEA), is developed in this article to construct efficient designs for GLMs. We compare the performance of the proposed algorithm with other optimization algorithms, including point exchange algorithm, columnwise-pairwise algorithm, simulated annealing and generic algorithm, and demonstrate the superior performance of this new algorithm.     

Better experimental design by hybridizing binary matching with imbalance optimization

We present a new experimental design procedure that divides a set of experimental units into two groups in order to minimize error in estimating a treatment effect. One concern is the elimination of large covariate imbalance between the two groups before the experiment begins. Another concern is robustness of the design to misspecification in response models. We address both concerns in our proposed design: we first place subjects into pairs using optimal nonbipartite matching, making our estimator robust to complicated nonlinear response models. Our innovation is to keep the matched pairs extant, take differences of the covariate values within each matched pair, and then use the greedy switching heuristic of Krieger et al. (2019) or rerandomization on these differences. This latter step greatly reduces covariate imbalance. Furthermore, our resultant designs are shown to be nearly as random as matching, which is robust to unobserved covariates. When compared to previous designs, our approach exhibits significant improvement in the mean squared error of the treatment effect estimator when the response model is nonlinear and performs at least as well when the response model is linear. Our design procedure can be found as a method in the open source R package available on CRAN called GreedyExperimentalDesign .     

Random effects models for count data

Random effects models are considered for count data obtained in a cross or nested classification. The main feature of the proposed models is the use of the additive effects on the original scale in contrast to the commonly used log scale. The rationale behind this approach is given. The estimation of variance components is based on the usual mean square approach. Directly analogous results to those from the analysis of variance models for continuous data are obtained. The usual Poisson dispersion test procedure can be used not only to test for no overall random effects but also to assess the adequacy of the model. Individual variance component can be tested by using the usual F-test. To get a reliable estimate, a large number of factor levels seem to be required.