Adaptive paired comparison design

An important aspect of paired comparison experiments is the decision of how to form pairs in advance of collecting data. A weakness of typical paired comparison experimental designs is the difficulty in incorporating prior information, which can be particularly relevant for the design of tournament schedules for players of games and sports. Pairing methods that make use of prior information are often ad hoc algorithms with little or no formal basis. The problem of pairing objects can be formalized as a Bayesian optimal design. Assuming a linear paired comparison model for outcomes, we develop a pairing method that maximizes the expected gain in Kullback–Leibler information from the prior to the posterior distribution. The optimal pairing is determined using a combinatorial optimization method commonly used in graph-theoretic contexts. We discuss the properties of our optimal pairing criterion, and demonstrate our method as an adaptive procedure for pairing objects multiple times. We compare the performance of our method on simulated data against random pairings, and against a system that is currently in use in tournament chess.     

A useful approximation for planning block designs with potential missing data

Many empirical studies are planned with the prior knowledge that some of the data may be missed. This knowledge is seldom explicitly incorporated into the experiment design process for lack of a candid methodology. This paper proposes an index related to the expected determinant of the information matrix as a criterion for planning block designs. Due to the intractable nature of the expected determinantal criterion an analytic expression is presented only for a simple 2x2 layout. A first order Taylor series approximation function is suggested for larger layouts. Ranges over which this approximation is adequate are shown via Monte Carlo simulations. The robustness of information in the block design relative to the completely randomized design with missing data is discussed.     

Robust designs for linear mixed effects models

Summary.  In health sciences, medicine and social sciences linear mixed effects models are often used to analyse time-structured data. The search for optimal designs for these models is often hampered by two problems. The first problem is that these designs are only locally optimal. The second problem is that an optimal design for one model may not be optimal for other models. In this paper the maximin principle is adopted to handle both problems, simultaneously. The maximin criterion is formulated by means of a relative efficiency measure, which gives an indication of how much efficiency is lost when the uncertainty about the models over a prior domain of parameters is taken into account. The procedure is illustrated by means of three growth studies. Results are presented for a vocabulary growth study from education, a bone gain study from medical research and an epidemiological decline in height study. It is shown that, for the mixed effects polynomial models that are applied to these studies, the maximin designs remain highly efficient for different sets of models and combinations of parameter values.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

Optimal design in average for inference in generalized linear models

This paper considers the problem of optimal design for inference in Generalized Linear Models, when prior information about the parameters is available. The general theory of optimum design usually requires knowledge of the parameter values. These are usually unknown and optimal design can, therefore, not be used in practice. However, one way to circumvent this problem is through so-called “optimal design in average”, or shortly, “ave optimal”. The ave optimal design is chosen to minimize the expected value of some criterion function over a prior distribution. We focus our interest on the aveD _A-optimality, including aveD- and avec-optimality and show the appropriate equivalence theorems for these optimality criterions, which give necessary conditions for an optimal design. Ave optimal designs are of interest when e.g. a factorial experiment with a binary or a Poisson response in to be conducted. The results are applied to factorial experiments, including a control group experiment and a 2×2 experiment.     

Bayesian analysis of multivariate survival data using Monte Carlo methods

This paper deals with the analysis of multivariate survival data from a Bayesian perspective using Markov-chain Monte Carlo methods. The Metropolis along with the Gibbs algorithm is used to calculate some of the marginal posterior distributions. A multivariate survival model is proposed, since survival times within the same group are correlated as a consequence of a frailty random block effect. The conditional proportional-hazards model of Clayton and Cuzick is used with a martingale structured prior process (Arjas and Gasbarra) for the discretized baseline hazard. Besides the calculation of the marginal posterior distributions of the parameters of interest, this paper presents some Bayesian EDA diagnostic techniques to detect model adequacy. The methodology is exemplified with kidney infection data where the times to infections within the same patients are expected to be correlated.     

On estimation and testing arising from order restricted balanced mixed model

The randomized complete block design is one of the most widely used experimental designs to systematically control the variability arising from known nuisance sources. The balanced mixed effects model is usually appropriate for such an experiment when the blocks used in the experiment are randomly chosen. In applications with k increasing or decreasing treatment levels, there is sometimes prior knowledge about the ordering of the treatment effects. The most commonly seen orderings include simple ordering, simple tree ordering and umbrella orderings with known or unknown peaks. A natural question is how to incorporate the prior ordering information in estimating the parameters in a balanced mixed effects model so that the estimated treatment effects are consistent with the prior information and the estimated variances of the block effects and experiment errors are nonnegative. In this paper we derive the maximum likelihood estimators of the parameters in a balanced mixed model subject to any partial ordering of the treatment effects, which includes the usual maximum likelihood estimators as a special case. An example is provided to illustrate the results.     

A Bayesian semiparametric method for analyzing length-biased data

Survival data obtained from prevalent cohort study designs are often subject to length-biased sampling. Frequentist methods including estimating equation approaches, as well as full likelihood methods, are available for assessing covariate effects on survival from such data. Bayesian methods allow a perspective of probability interpretation for the parameters of interest, and may easily provide the predictive distribution for future observations while incorporating weak prior knowledge on the baseline hazard function. There is lack of Bayesian methods for analyzing length-biased data. In this paper, we propose Bayesian methods for analyzing length-biased data under a proportional hazards model. The prior distribution for the cumulative hazard function is specified semiparametrically using I-Splines. Bayesian conditional and full likelihood approaches are developed for analyzing simulated and real data.     

An approach for finding fully Bayesian optimal designs using normal-based approximations to loss functions

The generation of decision-theoretic Bayesian optimal designs is complicated by the significant computational challenge of minimising an analytically intractable expected loss function over a, potentially, high-dimensional design space. A new general approach for approximately finding Bayesian optimal designs is proposed which uses computationally efficient normal-based approximations to posterior summaries to aid in approximating the expected loss. This new approach is demonstrated on illustrative, yet challenging, examples including hierarchical models for blocked experiments, and experimental aims of parameter estimation and model discrimination. Where possible, the results of the proposed methodology are compared, both in terms of performance and computing time, to results from using computationally more expensive, but potentially more accurate, Monte Carlo approximations. Moreover, the methodology is also applied to problems where the use of Monte Carlo approximations is computationally infeasible.     

A comparative study of the K-means algorithm and the normal mixture model for clustering: Univariate case

This paper gives a comparative study of the K-means algorithm and the mixture model (MM) method for clustering normal data. The EM algorithm is used to compute the maximum likelihood estimators (MLEs) of the parameters of the MM model. These parameters include mixing proportions, which may be thought of as the prior probabilities of different clusters; the maximum posterior (Bayes) rule is used for clustering. Hence, asymptotically the MM method approaches the Bayes rule for known parameters, which is optimal in terms of minimizing the expected misclassification rate (EMCR).     

Priors and component structures in autoregressive time series models

New approaches to prior specification and structuring in autoregressive time series models are introduced and developed. We focus on defining classes of prior distributions for parameters and latent variables related to latent components of an autoregressive model for an observed time series. These new priors naturally permit the incorporation of both qualitative and quantitative prior information about the number and relative importance of physically meaningful components that represent low frequency trends, quasi-periodic subprocesses and high frequency residual noise components of observed series. The class of priors also naturally incorporates uncertainty about model order and hence leads in posterior analysis to model order assessment and resulting posterior and predictive inferences that incorporate full uncertainties about model order as well as model parameters. Analysis also formally incorporates uncertainty and leads to inferences about unknown initial values of the time series, as it does for predictions of future values. Posterior analysis involves easily implemented iterative simulation methods, developed and described here. One motivating field of application is climatology, where the evaluation of latent structure, especially quasi-periodic structure, is of critical importance in connection with issues of global climatic variability. We explore the analysis of data from the southern oscillation index, one of several series that has been central in recent high profile debates in the atmospheric sciences about recent apparent trends in climatic indicators.     

Cost-cautious designs for confirmatory bioassay

Confirmatory bioassay experiments take place in late stages of the drug discovery process when a small number of compounds have to be compared with respect to their properties. As the cost of the observations may differ considerably, the design problem is well specified by the cost of compound used rather than by the number of observations. We show that cost-efficient designs can be constructed using useful properties of the minimum support designs. These designs are particularly suited for studies where the parameters of the model to be estimated are known with high accuracy prior to the experiment, although they prove to be robust against typical inaccuracies of these values. When the parameters of the model can only be specified with ranges of values or by a probability distribution, we use a Bayesian criterion of optimality to construct the required designs. Typically, the number of their support points depends on the prior knowledge for the model parameters. In all cases we recommend identifying a set of designs with good statistical properties but different potential costs to choose from.     

A note on Bayesian nonparametric regression function estimation

In this note the problem of nonparametric regression function estimation in a random design regression model with Gaussian errors is considered from the Bayesian perspective. It is assumed that the regression function belongs to a class of functions with a known degree of smoothness. A prior distribution on the given class can be induced by a prior on the coefficients in a series expansion of the regression function through an orthonormal system. The rate of convergence of the resulting posterior distribution is employed to provide a measure of the accuracy of the Bayesian estimation procedure defined by the posterior expected regression function. We show that the Bayes’ estimator achieves the optimal minimax rate of convergence under mean integrated squared error over the involved class of regression functions, thus being comparable to other popular frequentist regression estimators.     

Bayesian Identification of Multivariate Autoregressive Processes

Identification is one of the most important stages of a time series analysis. This paper develops a direct Bayesian technique to identify the order of multivariate autoregressive processes. By employing the conditional likelihood function and a matrix normal-Wishart prior density, or Jeffrey' vague prior, the proposed identification technique is based on deriving the exact posterior probability mass function of the model order in a convenient form. Then one may easily evaluate the posterior probabilities of the model order and choose the order that maximizes the posterior mass function to be the suitable order of the time series data being analyzed. Assuming the bivariate autoregressive processes, a numerical study, with different prior mass functions, is carried out to assess the efficiency of the proposed technique. The analysis of the numerical results supports the adequacy of the proposed technique in identifying the orders of multivariate autoregressive processes.     

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.     

A method of selecting the levels of regressor variables in lognormal regression model

Kupper and Meydrech and Myers and Lahoda introduced the mean squared error (MSE) approach to study response surface designs, Duncan and DeGroot derived a criterion for optimality of linear experimental designs based on minimum mean squared error. However, minimization of the MSE of an estimator maxr renuire some knowledge about the unknown parameters. Without such knowledge construction of designs optimal in the sense of MSE may not be possible. In this article a simple method of selecting the levels of regressor variables suitable for estimating some functions of the parameters of a lognormal regression model is developed using a criterion for optimality based on the variance of an estimator. For some special parametric functions, the criterion used here is equivalent to the criterion of minimizing the mean squared error. It is found that the maximum likelihood estimators of a class of parametric functions can be improved substantially (in the sense of MSE) by proper choice of the values of regressor variables. Moreover, our approach is applicable to analysis of variance as well as regression designs.     

Generalized case–cohort sampling

A class of cohort sampling designs, including nested case–control, case–cohort and classical case–control designs involving survival data, is studied through a unified approach using Cox's proportional hazards model. By finding an optimal sample reuse method via local averaging, a closed form estimating function is obtained, leading directly to the estimators of the regression parameters that are relatively easy to compute and are more efficient than some commonly used estimators in case–cohort and nested case–control studies. A semiparametric efficient estimator can also be found with some further computation. In addition, the class of sampling designs in this study provides a variety of sampling options and relaxes the restrictions of sampling schemes that are currently available.     

Wavelet thresholding via a Bayesian approach

We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion that is common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relationship between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relationship gives insight into the meaning of the Besov space parameters. Moreover, the relationship established makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might be difficult to elicit; with this in mind, we propose a standard choice of prior hyperparameters that works well in our examples. Several simulated examples are used to illustrate our method, and comparisons are made with other thresholding methods. We also present an application to a data set that was collected in an anaesthesiological study.     

New adaptive designs for delayed response models

Adaptive designs are effective mechanisms for flexibly allocating experimental resources. In clinical trials particularly, such designs allow researchers to balance short- and long-term goals. Unfortunately, fully sequential strategies require outcomes from all previous allocations prior to the next allocation. This can prolong an experiment unduly. As a result, we seek designs for models that specifically incorporate delays.We utilize a delay model in which patients arrive according to a Poisson process and their response times are exponential. We examine three designs with an eye towards minimizing patient losses: a delayed two-armed bandit rule which is optimal for the model and objective of interest; a newly proposed hyperopic rule; and a randomized play-the-winner rule. The results show that, except when the delay rate is several orders of magnitude different than the patient arrival rate, the delayed response bandit is nearly as efficient as the immediate response bandit. The delayed hyperopic design also performs extremely well throughout the range of delays, despite the fact that the rate of delay is not one of its design parameters. The delayed randomized play-the-winner rule is far less efficient than either of the other methods.     

A semi-infinite programming based algorithm for finding minimax optimal designs for nonlinear models

Minimax optimal experimental designs are notoriously difficult to study largely because the optimality criterion is not differentiable and there is no effective algorithm for generating them. We apply semi-infinite programming (SIP) to solve minimax design problems for nonlinear models in a systematic way using a discretization based strategy and solvers from the General Algebraic Modeling System (GAMS). Using popular models from the biological sciences, we show our approach produces minimax optimal designs that coincide with the few theoretical and numerical optimal designs in the literature. We also show our method can be readily modified to find standardized maximin optimal designs and minimax optimal designs for more complicated problems, such as when the ranges of plausible values for the model parameters are dependent and we want to find a design to minimize the maximal inefficiency of estimates for the model parameters.