Optimal crossover designs when carryover effects are proportional to direct effects

Crossover designs are used for a variety of different applications. While these designs have a number of attractive features, they also induce a number of special problems and concerns. One of these is the possible presence of carryover effects. Even with the use of washout periods, which are for many applications widely accepted as an indispensable component, the effect of a treatment from a previous period may not be completely eliminated. A model that has recently received renewed attention in the literature is the model in which first-order carryover effects are assumed to be proportional to direct treatment effects. Under this model, assuming that the constant of proportionality is known, we identify optimal and efficient designs for the direct effects for different values of the constant of proportionality. We also consider the implication of these results for the case that the constant of proportionality is not known.     

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.     

A method for screening active effects in supersaturated designs

A supersaturated design (SSD) is a design whose run size is not enough for estimating all the main effects. The goal in conducting such a design is to identify, presumably only a few, relatively dominant active effects with a cost as low as possible. However, data analysis of such designs remains primitive: traditional approaches are not appropriate in such a situation and several methods which were proposed in the literature in recent years are effective when used to analyze two-level SSDs. In this paper, we introduce a variable selection procedure, called the PLSVS method, to screen active effects in mixed-level SSDs based on the variable importance in projection which is an important concept in the partial least-squares regression. Simulation studies show that this procedure is effective.     

Generalized partially linear varying-coefficient models

Generalized varying-coefficient models are useful extensions of generalized linear models. They arise naturally when investigating how regression coefficients change over different groups characterized by certain covariates such as age. In this paper, we extend these models to generalized partially linear varying-coefficient models, in which some coefficients are constants and the others are functions of certain covariates. Procedures for estimating the linear and non-parametric parts are developed and their associated statistical properties are studied. The methods proposed are illustrated using some simulations and real data analysis.     

An alternative form of the Watson efficiency

Watson [1951. Serial correlation in regression analysis. Ph.D. Thesis, Department of Experimental Statistics, North Carolina State College, Raleigh] introduced a relative efficiency, which is often called the Watson efficiency in literatures, to measure the inefficiency of the least squares in linear regression models. The Watson efficiency is defined by determinant, but we shall show by two examples that such a criterion does not always work well in some cases. In this paper, an alternative form based on Euclidean norm of the Watson efficiency is proposed and some examples are given to illustrate superiority of the new relative efficiency.     

Partially linear models and polynomial spline approximations for the analysis of unbalanced panel data

In this paper, we study the estimation of the unbalanced panel data partially linear models with a one-way error components structure. A weighted semiparametric least squares estimator (WSLSE) is developed using polynomial spline approximation and least squares. We show that the WSLSE is asymptotically more efficient than the corresponding unweighted estimator for both parametric and nonparametric components of the model. This is a significant improvement over previous results in the literature which showed that the simply weighting technique can only improve the estimation of the parametric component. The asymptotic normalities of the proposed WSLSE are also established.     

Testing appearance of linear trend

A test for the detection of a gradual change in simple linear regression is studied. Asymptotic distribution of the “maximum type” statistic is derived. Asymptotic critical values are compared with critical values obtained by simulations. The problem was motivated by the effort of meteorologists to discover a change in meteorological measurements.     

D-optimal minimax regression designs on discrete design space

One classical design criterion is to minimize the determinant of the covariance matrix of the regression estimates, and the designs are called D-optimal designs. To reflect the nature that the proposed models are only approximately true, we propose a robust design criterion to study response surface designs. Both the variance and bias are considered in the criterion. In particular, D-optimal minimax designs are investigated and constructed. Examples are given to compare D-optimal minimax designs with classical D-optimal designs.     

Simulation-based sequential Bayesian design

We consider simulation-based methods for exploration and maximization of expected utility in sequential decision problems. We consider problems which require backward induction with analytically intractable expected utility integrals at each stage. We propose to use forward simulation to approximate the integral expressions, and a reduction of the allowable action space to avoid problems related to an increasing number of possible trajectories in the backward induction. The artificially reduced action space allows strategies to depend on the full history of earlier observations and decisions only indirectly through a low dimensional summary statistic. The proposed rule provides a finite-dimensional approximation to the unrestricted infinite-dimensional optimal decision rule. We illustrate the proposed approach with an application to an optimal stopping problem in a clinical trial.     

Inferences from biased samples with a memory effect

Biased sampling occurs often in observational studies. With one biased sample, the problem of nonparametrically estimating both a target density function and a selection bias function is unidentifiable. This paper studies the nonparametric estimation problem when there are two biased samples that have some overlapping observations (i.e. recaptures) from a finite population. Since an intelligent subject sampled previously may experience a memory effect if sampled again, two general 2-stage models that incorporate both a selection bias and a possible memory effect are proposed. Nonparametric estimators of the target density, selection bias, and memory functions, as well as the population size are developed. Asymptotic properties of these estimators are studied and confidence bands for the selection function and memory function are provided. Our procedures are compared with those ignoring the memory effect or the selection bias in finite sample situations. A nonparametric model selection procedure is also given for choosing a model from the two 2-stage models and a mixture of these two models. Our procedures work well with or without a memory effect, and with or without a selection bias. The paper concludes with an application to a real survey data set.     

Collapsibility of regression coefficients and its extensions

Collapsibility with respect to a measure of association implies that the measure of association can be obtained from the marginal model. We first discuss model collapsibility and collapsibility with respect to regression coefficients for linear regression models. For parallel regression models, we give simple and different proofs of some of the known results and obtain also certain new results. For random coefficient regression models, we define (average) A$A$-collapsibility and obtain conditions under which it holds. We consider Poisson regression and logistic regression models also, and derive conditions for collapsibility and A$A$-collapsibility, respectively. These results generalize some of the results available in the literature. Some suitable examples are also discussed.     

Combined multiple testing by censored empirical likelihood

We propose a new procedure for combining multiple tests in samples of right-censored observations. The new method is based on multiple constrained censored empirical likelihood where the constraints are formulated as linear functionals of the cumulative hazard functions. We prove a version of Wilks’ theorem for the multiple constrained censored empirical likelihood ratio, which provides a simple reference distribution for the test statistic of our proposed method. A useful application of the proposed method is, for example, examining the survival experience of different populations by combining different weighted log-rank tests. Real data examples are given using the log-rank and Gehan-Wilcoxon tests. In a simulation study of two sample survival data, we compare the proposed method of combining tests to previously developed procedures. The results demonstrate that, in addition to its computational simplicity, the combined test performs comparably to, and in some situations more reliably than previously developed procedures. Statistical software is available in the R package ‘emplik’.     

The TTT transformation and a new bathtub distribution model

In this paper we use the total time on test transformation to establish a method for construction of parametric models of lifetime distributions having bathtub-shaped failure rate. We study a particular model which is simple compared to the other existing models. We derive expressions for moments and quantiles and treat estimation methods. Particularly, the maximum likelihood method is studied. Consistency proofs are given.     

On statistical models for multimodal optimization

A new approach for the construction of statistical models for multimodal optimization is proposed; the examples of such models are given. The results of the psychological experiment show that the proposed approach is intuitively acceptable.     

Robust estimators and designs for field experiments

We consider the construction of designs for test-control field experiments, with particular attention being paid to the effects of spatial correlation between adjoining plots. In contrast to previous approaches, in which very specific correlation structures were modelled, we explicitly allow a degree of uncertainty on the part of the experimenter. While fitting a particular correlation structure—and variance structure and regression response—the experimenter is thought to be seeking protection against other possible structures in full neighbourhoods of these particular choices. Robustness, in a minimax sense, is obtained through a modification of the kriging estimation procedure, and through the assignment of treatments to field plots.     

Analysis of supersaturated designs via the Dantzig selector

A supersaturated design is a design whose run size is not enough for estimating all the main effects. It is commonly used in screening experiments, where the goals are to identify sparse and dominant active factors with low cost. In this paper, we study a variable selection method via the Dantzig selector, proposed by Candes and Tao [2007. The Dantzig selector: statistical estimation when p$p$ is much larger than n$n$. Annals of Statistics 35, 2313–2351], to screen important effects. A graphical procedure and an automated procedure are suggested to accompany with the method. Simulation shows that this method performs well compared to existing methods in the literature and is more efficient at estimating the model size.     

Robust designs for one-point extrapolation

We consider the construction of designs for the extrapolation of a regression response to one point outside of the design space. The response function is an only approximately known function of a specified linear function. As well, we allow for variance heterogeneity. We find minimax designs and corresponding optimal regression weights in the context of the following problems: (P1) for nonlinear least squares estimation with homoscedasticity, determine a design to minimize the maximum value of the mean squared extrapolation error (MSEE), with the maximum being evaluated over the possible departures from the response function; (P2) for nonlinear least squares estimation with heteroscedasticity, determine a design to minimize the maximum value of MSEE, with the maximum being evaluated over both types of departures; (P3) for nonlinear weighted least squares estimation, determine both weights and a design to minimize the maximum MSEE; (P4) choose weights and design points to minimize the maximum MSEE, subject to a side condition of unbiasedness. Solutions to (P1)–(P4) are given in complete generality. Numerical comparisons indicate that our designs and weights perform well in combining robustness and efficiency. Applications to accelerated life testing are highlighted.     

Estimation in proportional hazard and log-linear models

Proportional hazard models and models where the dependent variables follow a linear model are shown to be equivalent if and only if the hazard rate is the product of a non-negative periodic function and a Weibull factor. Estimates based on rank statistics are proposed for the parameters in the proportional hazard model.     

Joint modeling of degradation and failure time data

This paper surveys some approaches to model the relationship between failure time data and covariate data like internal degradation and external environmental processes. These models which reflect the dependency between system state and system reliability include threshold models and hazard-based models. In particular, we consider the class of degradation–threshold–shock models (DTS models) in which failure is due to the competing causes of degradation and trauma. For this class of reliability models we express the failure time in terms of degradation and covariates. We compute the survival function of the resulting failure time and derive the likelihood function for the joint observation of failure times and degradation data at discrete times. We consider a special class of DTS models where degradation is modeled by a process with stationary independent increments and related to external covariates through a random time scale and extend this model class to repairable items by a marked point process approach. The proposed model class provides a rich conceptual framework for the study of degradation–failure issues.     

Perfect linear models and perfect parametric functions

General linear models with a common design matrix and with various structures of the variance–covariance matrix are considered. We say that a model is perfect for a linearly estimable parametric function, or the function is perfect in the model, if there exists the best linear unbiased estimator. All perfect models for a given function and all perfect functions in a given model are characterized.