Small Box-Behnken design

Box-Behnken design has been popularly used for the second-order response surface model. It is formed by combining two-level factorial designs with incomplete block designs in a special manner—the treatments in each block are replaced by an identical design. In this paper, we construct small Box-Behnken design. These designs can fit the second-order response surface model with reasonably high efficiencies but with only a much smaller run size. The newly constructed designs make use of balanced incomplete block design (BIBD) or partial BIBD, and replace treatments partly by 2_III³⁻¹ designs and partly by full factorial designs. It is shown that the orthogonality properties in the original Box and Behnken designs will be kept in the new designs. Furthermore, we classify the parameters into groups and introduce Group Moment Matrix (GMM) to estimate all the parameters in each group. This allows us to significantly reduce the amount of computational costs in the construction of the designs.     

Optimal Pair Matching With Two Control Groups

In an effort to detect hidden biases due to failure to control for an unobserved covariate, some observational or nonrandomized studies include two control groups selected to systematically vary the unobserved covariate. Comparisons of the treated group and two control groups must, of course, control for imbalances in observed covariates. Using the three groups, we form pairs optimally matched for observed covariates—that is, we optimally construct from observational data an incomplete block design. The incomplete block design may use all available data, or it may use data selectively to produce a balanced incomplete block design, or it may be the basis for constructing a matched sample when expensive outcome information is to be collected only for sampled individuals. The problem of optimal pair matching with two control groups is shown by a series of transformations to be equivalent to a particular form of optimal nonbipartite matching, a problem for which polynomial time algorithms exist. In our examples, we implement the procedure using a nonbipartite matching algorithm due to Derigs. We illustrate the method with data from an observational study of the employment effects of the minimum wage.     

A discrete survival model with random effects and covariate-dependent selection

In this paper we present a discrete survival model with covariates and random effects, where the random effects may depend on the observed covariates. The dependence between the covariates and the random effects is modelled through correlation parameters, and these parameters can only be identified for time-varying covariates. For time-varying covariates, however, it is possible to separate regression effects and selection effects in the case of a certain dependene structure between the random effects and the time-varying covariates that are assumed to be conditionally independent given the initial level of the covariate. The proposed model is equivalent to a model with independent random effects and the initial level of the covariates as further covariates. The model is applied to simulated data that illustrates some identifiability problems, and further indicate how the proposed model may be an approximation to retrospectively collected data with incorrect specification of the waiting times. The model is fitted by maximum likelihood estimation that is implemented as iteratively reweighted least squares. © 1998 John Wiley & Sons, Ltd.     

The choice of screening design

A screening design is an experimental plan used for identifying the expectedly few active factors from potentially many. In this paper, we compare the performances of 3 experimental plans, a Plackett‐Burman design, a minimum run resolution IV design, and a definitive screening design, all with 12 and 13 runs, when they are used for screening and 3 out of 6 factors are active. The functional relationship between the response and the factors was allowed to be of 2 types, a second‐order model and a model with all main effects and interactions included. D‐efficiencies for the designs ability to estimate parameters in such models were computed, but it turned out that these are not very informative for comparing the screening performances of the 2‐level designs to the definitive screening design. The overall screening performance of the 2‐level designs was quite good, but there exist situations where the definitive screening design, allowing both screening and estimation of second‐order models in the same operation, has a reasonable high probability of being successful.     

Mixtures of factor analyzers: an extension with covariates

This paper examines the analysis of an extended finite mixture of factor analyzers (MFA) where both the continuous latent variable (common factor) and the categorical latent variable (component label) are assumed to be influenced by the effects of fixed observed covariates. A polytomous logistic regression model is used to link the categorical latent variable to its corresponding covariate, while a traditional linear model with normal noise is used to model the effect of the covariate on the continuous latent variable. The proposed model turns out be in various ways an extension of many existing related models, and as such offers the potential to address some of the issues not fully handled by those previous models. A detailed derivation of an EM algorithm is proposed for parameter estimation, and latent variable estimates are obtained as by-products of the overall estimation procedure.     

Decomposable symmetric designs

The first infinite families of symmetric designs were obtained from finite projective geometries, Hadamard matrices, and difference sets. In this paper we describe two general methods of constructing symmetric designs that give rise to the parameters of all other known infinite families of symmetric designs. The method of global decomposition produces an incidence matrix of a symmetric design as a block matrix with each block being a zero matrix or an incidence matrix of a smaller symmetric design. The method of local decomposition represents incidence matrices of a residual and a derived design of a symmetric design as block matrices with each block being a zero matrix or an incidence matrix of a smaller residual or derived design, respectively.     

Connectedness of PBIB designs having asymmetrical association schemes

Summary A necessary and sufficient condition for the connectedness ofm-associate partially balanced incomplete block (PBIB) designs having an asymmetrical association scheme is given, only in terms of design parameters, without inner structure parameters of designs. Supported in part by Grant 321-6066-58530013 (Japan).     

Partially balanced incomplete block designs from weakly divisible nearrings

In [[6] Riv. Mat. Univ. Parma 11 (2) (1970) 79-96] Ferrero demonstrates a connection between a restricted class of planar nearrings and balanced incomplete block designs. In this paper, bearing in mind the links between planar nearrings and weakly divisible nearrings (wd-nearrings), first we show the construction of a family of partially balanced incomplete block designs from a special class of wd-nearrings; consequently, we are able to give some formulas for calculating the design parameters.     

A comparison of asymptotic covariance matrices of three consistent estimators in the Poisson regression model with measurement errors

We consider a Poisson model, where the mean depends on certain covariates in a log-linear way with unknown regression parameters. Some or all of the covariates are measured with errors. The covariates as well as the measurement errors are both jointly normally distributed, and the error covariance matrix is supposed to be known. Three consistent estimators of the parameters—the corrected score, a structural, and the quasi-score estimators—are compared to each other with regard to their relative (asymptotic) efficiencies. The paper extends an earlier result for a scalar covariate.     

A new family of covariate-adjusted response adaptive designs and their properties

It is often important to incorporate covariate information in the design of clinical trials. In literature there are many designs of using stratification and covariate-adaptive randomization to balance certain known covariate. Recently, some covariate-adjusted response-adaptive （CARA） designs have been proposed and their asymptotic properties have been studied （Ann. Statist. 2007）. However, these CARA designs usually have high variabilities. In this paper, a new family of covariate-adjusted response-adaptive （CARA） designs is presented. It is shown that the new designs have less variables and therefore are more efficient.     

A class of transformation rate models for recurrent event data

Recurrent event data frequently occur in longitudinal studies, and it is often of interest to estimate the effects of covariates on the recurrent event rate. This paper considers a class of semiparametric transformation rate models for recurrent event data, which uses an additive Aalen model as its covariate dependent baseline. The new models are flexible in that they allow for both additive and multiplicative covariate effects, and some covariate effects are allowed to be nonparametric and time-varying. An estimating procedure is proposed for parameter estimation, and the resulting estimators are shown to be consistent and asymptotically normal. Simulation studies and a real data analysis demonstrate that the proposed method performs well and is appropriate for practical use.     

Multivariate logistic regression with incomplete covariate and auxiliary information

In this article, we propose and explore a multivariate logistic regression model for analyzing multiple binary outcomes with incomplete covariate data where auxiliary information is available. The auxiliary data are extraneous to the regression model of interest but predictive of the covariate with missing data. Horton and Laird [N.J. Horton, N.M. Laird, Maximum likelihood analysis of logistic regression models with incomplete covariate data and auxiliary information, Biometrics 57 (2001) 34–42] describe how the auxiliary information can be incorporated into a regression model for a single binary outcome with missing covariates, and hence the efficiency of the regression estimators can be improved. We consider extending the method of [9] to the case of a multivariate logistic regression model for multiple correlated outcomes, and with missing covariates and completely observed auxiliary information. We demonstrate that in the case of moderate to strong associations among the multiple outcomes, one can achieve considerable gains in efficiency from estimators in a multivariate model as compared to the marginal estimators of the same parameters.     

Some results on resolvable incomplete block designs

This paper is concerned with the uniformity of a certain kind of resolvable incomplete block (RIB for simplicity) design which is called the PRIB design here. A sufficient and necessary condition is obtained, under which a PRIB design is the most uniform in the sense of a discrete discrepancy measure, and the uniform PRIB design is shown to be connected. A construction method for such designs via a kind of U-type designs is proposed, and an existence result of these designs is given. This method sets up an important bridge between PRIB designs and U-type designs.     

Optimal balanced 27 fractional factorial designs of resolutionV,withN≦42

Summary In this paper, we present a class of fractional factorial designs of the 2⁷ series, which are of resolutionV. Such designs allow the estimation of the general mean, the main effects and the two factors interactions (29 parameters in all for the 2⁷ factorial) assuming that the higher order effects are negligible. For every value ofN (the number of runs) such that 29≦N≦42, we give a resolutionV design that is optimal (with respect to the trace criterion) within the subclass of balanced designs. Also, for convenience of analysis, we present for each design, the covariance matrix of the estimates of the various parameters. As a by product, we establish many interesting combinatorial theorems concerning balanced arrays of strength four (which are generalizations of orthogonal arrays of strength four, and also of balanced incomplete block designs with block sizes not necessarily equal).     

Bayesian Nonparametric Modeling for Multivariate Ordinal Regression

Univariate or multivariate ordinal responses are often assumed to arise from a latent continuous parametric distribution, with covariate effects that enter linearly. We introduce a Bayesian nonparametric modeling approach for univariate and multivariate ordinal regression, which is based on mixture modeling for the joint distribution of latent responses and covariates. The modeling framework enables highly flexible inference for ordinal regression relationships, avoiding assumptions of linearity or additivity in the covariate effects. In standard parametric ordinal regression models, computational challenges arise from identifiability constraints and estimation of parameters requiring nonstandard inferential techniques. A key feature of the nonparametric model is that it achieves inferential flexibility, while avoiding these difficulties. In particular, we establish full support of the nonparametric mixture model under fixed cut-off points that relate through discretization the latent continuous responses with the ordinal responses. The practical utility of the modeling approach is illustrated through application to two datasets from econometrics, an example involving regression relationships for ozone concentration, and a multirater agreement problem. Supplementary materials with technical details on theoretical results and on computation are available online.     

Nonlinear general path models for degradation data with dynamic covariates

Degradation data have been widely used to estimate product reliability. Because of technology advancement, time‐varying usage and environmental variables, which are called dynamic covariates, can be easily recorded nowadays, in addition to the traditional degradation measurements. The use of dynamic covariates is appealing because they have the potential to explain more variability in degradation paths. We propose a class of general path models to incorporate dynamic covariates for modeling of degradation paths. Physically motivated nonlinear functions are used to describe the degradation paths, and random effects are used to describe unit‐to‐unit variability. The covariate effects are modeled by shape‐restricted splines. The estimation of unknown model parameters is challenging because of the involvement of nonlinear relationships, random effects, and shaped‐restricted splines. We develop an efficient procedure for parameter estimations. The performance of the proposed method is evaluated by simulations. An outdoor coating weathering dataset is used to illustrate the proposed method. Copyright © 2015 John Wiley & Sons, Ltd.     

Analysis of variance and linear contrasts in experimental design with generalized secant hyperbolic distribution

We consider one-way classification model in experimental design when the errors have generalized secant hyperbolic distribution. We obtain efficient and robust estimators for block effects by using the modified maximum likelihood estimation (MML) methodology. A test statistic analogous to the normal-theory F statistic is defined to test block effects. We also define a test statistic for testing linear contrasts. It is shown that test statistics based on MML estimators are efficient and robust. The methodology readily extends to unbalanced designs.     

Self-dual codes from orbit matrices and quotient matrices of combinatorial designs

In this paper we give constructions of self-orthogonal and self-dual codes, with respect to certain scalar products, with the help of orbit matrices of block designs and quotient matrices of symmetric (group) divisible designs (SGDDs) with the dual property. First we describe constructions from block designs and their extended orbit matrices, where the orbit matrices are induced by the action of an automorphism group of the design. Further, we give some further constructions of self-dual codes from symmetric block designs and their orbit matrices. Moreover, in a similar way as for symmetric designs, we give constructions of self-dual codes from SGDDs with the dual property and their quotient matrices.     

A simulation study for a class of central composite designs with nested sub-experiment

This article introduces a class of central composite designs with nested sub-experiment, which allow for the estimation of both response surface effects (fixed effects of crossed factors) and variance components arising from nested random effects. An iterated least squared method using sufficient statistics is given for obtaining maximum likelihood estimates of the parameters in a mixed model. Simulation results show that advantages for unbalanced designs are greatest when error variance is small.