Sequential Designs for Binary Data with the Purpose to Maximize the Probability of Response

Two kinds of sequential designs are proposed for finding the point that maximizes the probability of response assuming a binary response variable and a quadratic logistic regression model. One is a parametric optimal design approach, and the other one is a nonparametric stochastic approximation approach. The suggested sequential designs are evaluated and compared in a simulation study. In summary, the parametric approach performed very well whereas its competitor failed in some cases.     

Robust sequential designs for approximately linear models

We consider the problem of the sequential choice of design points in an approximately linear model. It is assumed that the fitted linear model is only approximately correct, in that the true response function contains a nonrandom, unknown term orthogonal to the fitted response. We also assume that the parameters are estimated by M-estimation. The goal is to choose the next design point in such a way as to minimize the resulting integrated squared bias of the estimated response, to order n^-1. Explicit applications to analysis of variance and regression are given. In a simulation study the sequential designs compare favourably with some fixed-sample-size designs which are optimal for the true response to which the sequential designs must adapt.     

Robust sequential designs for nonlinear regression

The authors introduce the formal notion of an approximately specified nonlinear regression model and investigate sequential design methodologies when the fitted model is possibly of an incorrect parametric form. They present small‐sample simulation studies which indicate that their new designs can be very successful, relative to some common competitors, in reducing mean squared error due to model misspecifi‐cation and to heteroscedastic variation. Their simulations also suggest that standard normal‐theory inference procedures remain approximately valid under the sequential sampling schemes. The methods are illustrated both by simulation and in an example using data from an experiment described in the chemical engineering literature.     

Bayesian design for estimating the turning point of a quadratic regression

The problem of designing an experiment to estimate the point at which a quadratic regression is a maximum, or minimum. is studied. The efficiency of a design depends on the value of the unknown parameters and sequential design is, therefore, more efficient than non-sequential design. We use a Bayesian criterion which is a weighted trace of the inverse of the information matrix with the weights depending on a prior distribution. If design occurs sequentially the weights can be updated. Both sequential and non-sequential Bayesian designs are compared to non-Bayesian sequential designs. The comparison is both theoretical and by simulation.     

Design and analysis considerations for comparing dynamic treatment regimens with binary outcomes from sequential multiple assignment randomized trials

In behavioral, educational and medical practice, interventions are often personalized over time using strategies that are based on individual behaviors and characteristics and changes in symptoms, severity, or adherence that are a result of one's treatment. Such strategies that more closely mimic real practice, are known as dynamic treatment regimens (DTRs). A sequential multiple assignment randomized trial (SMART) is a multi-stage trial design that can be used to construct effective DTRs. This article reviews a simple to use ‘weighted and replicated’ estimation technique for comparing DTRs embedded in a SMART design using logistic regression for a binary, end-of-study outcome variable. Based on a Wald test that compares two embedded DTRs of interest from the ‘weighted and replicated’ regression model, a sample size calculation is presented with a corresponding user-friendly applet to aid in the process of designing a SMART. The analytic models and sample size calculations are presented for three of the more commonly used two-stage SMART designs. Simulations for the sample size calculation show the empirical power reaches expected levels. A data analysis example with corresponding code is presented in the appendix using data from a SMART developing an effective DTR in autism.     

Adaptive sequential design for regression on multi-resolution bases

This work investigates the problem of construction of designs for estimation and discrimination between competing linear models. In our framework, the unknown signal is observed with the addition of a noise and only a few evaluations of the noisy signal are available. The model selection is performed in a multi-resolution setting. In this setting, the locations of discrete sequential D and A designs are precisely constraint in a small number of explicit points. Hence, an efficient stochastic algorithm can be constructed that alternately improves the design and the model. Several numerical experiments illustrate the efficiency of our method for regression. One can also use this algorithm as a preliminary step to build response surfaces for sensitivity analysis.     

A comparison of confidence regions and designs in estimation of a ratio

A computer simulation is performed to compare confidence regions arising from Fie1ler's theorem and approximate large sample intervals in estimating the point of extremum in quadratic regression. In addition, two designs, one of which is motivated by optimal design theory, are compared. These comparisons are made by examining the confidence level and accuracy of the regions, as well as the concentration of the standard point estimator, in a variety of settings. The results have implications for the more general problem of estimating a ratio of linear combinations in the general linear model.     

Optimizing trial design in pharmacogenetics research: comparing a fixed parallel group,group sequential,and adaptive selection design on sample size requirements

Two‐stage clinical trial designs may be efficient in pharmacogenetics research when there is some but inconclusive evidence of effect modification by a genomic marker. Two‐stage designs allow to stop early for efficacy or futility and can offer the additional opportunity to enrich the study population to a specific patient subgroup after an interim analysis. This study compared sample size requirements for fixed parallel group, group sequential, and adaptive selection designs with equal overall power and control of the family‐wise type I error rate. The designs were evaluated across scenarios that defined the effect sizes in the marker positive and marker negative subgroups and the prevalence of marker positive patients in the overall study population. Effect sizes were chosen to reflect realistic planning scenarios, where at least some effect is present in the marker negative subgroup. In addition, scenarios were considered in which the assumed ‘true’ subgroup effects (i.e., the postulated effects) differed from those hypothesized at the planning stage. As expected, both two‐stage designs generally required fewer patients than a fixed parallel group design, and the advantage increased as the difference between subgroups increased. The adaptive selection design added little further reduction in sample size, as compared with the group sequential design, when the postulated effect sizes were equal to those hypothesized at the planning stage. However, when the postulated effects deviated strongly in favor of enrichment, the comparative advantage of the adaptive selection design increased, which precisely reflects the adaptive nature of the design. Copyright © 2013 John Wiley & Sons, Ltd.     

The plan of enrichment designs for dealing with high placebo response

To deal with high placebo response in clinical trials for psychiatric and other diseases, different enrichment designs, such as the sequential parallel design, two‐way enriched design, and sequential enriched design, have been proposed and implemented recently. Depending on the historical trial information and the trial sponsors' resources, detailed design elements are needed for determining which design to adopt. To assist in making more suitable decisions, we perform evaluations for selecting required design elements in terms of power optimization and sample size planning. We also discuss the implementation of the interim analysis related to its applicability.     

A Bayesian sequential design with binary outcome

Several researchers have proposed solutions to control type I error rate in sequential designs. The use of Bayesian sequential design becomes more common; however, these designs are subject to inflation of the type I error rate. We propose a Bayesian sequential design for binary outcome using an alpha‐spending function to control the overall type I error rate. Algorithms are presented for calculating critical values and power for the proposed designs. We also propose a new stopping rule for futility. Sensitivity analysis is implemented for assessing the effects of varying the parameters of the prior distribution and maximum total sample size on critical values. Alpha‐spending functions are compared using power and actual sample size through simulations. Further simulations show that, when total sample size is fixed, the proposed design has greater power than the traditional Bayesian sequential design, which sets equal stopping bounds at all interim analyses. We also find that the proposed design with the new stopping for futility rule results in greater power and can stop earlier with a smaller actual sample size, compared with the traditional stopping rule for futility when all other conditions are held constant. Finally, we apply the proposed method to a real data set and compare the results with traditional designs.     

Sequential design approaches for bioequivalence studies with crossover designs

The planning of bioequivalence (BE) studies, as for any clinical trial, requires a priori specification of an effect size for the determination of power and an assumption about the variance. The specified effect size may be overly optimistic, leading to an underpowered study. The assumed variance can be either too small or too large, leading, respectively, to studies that are underpowered or overly large. There has been much work in the clinical trials field on various types of sequential designs that include sample size reestimation after the trial is started, but these have seen only little use in BE studies. The purpose of this work was to validate at least one such method for crossover design BE studies. Specifically, we considered sample size reestimation for a two-stage trial based on the variance estimated from the first stage. We identified two methods based on Pocock's method for group sequential trials that met our requirement for at most negligible increase in type I error rate.     

Nearly optimal truncated group sequential test on binomial proportions

In this article, we systematically study the optimal truncated group sequential test on binomial proportions. Through analysis of the cost structure, average test cost is introduced as a new optimality criterion. According to the new criterion, the optimal tests on different design parameters including the boundaries, success discriminant value, stage sample vector, stage size, and the maximum sample size are defined. Since the computation time in finding optimal designs by exhaustive search is intolerably long, group sequential sample space sorting method and procedures are developed to find the near-optimal ones. In comparison with the international standard ISO2859-1, the truncated group sequential designs proposed in this article can reduce the average test costs around 20%.     

Design and relative efficiency in two-phase studies

Many two-phase sampling designs have been applied in practice to obtain efficient estimates of regression parameters while minimizing the cost of data collection. This research investigates two-phase sampling designs for so-called expensive variable problems, and compares them with one-phase designs. Closed form expressions for the asymptotic relative efficiency of maximum likelihood estimators from the two designs are derived for parametric normal models, providing insight into the available information for regression coefficients under the two designs. We further discuss when we should apply the two-phase design and how to choose the sample sizes for two-phase samples. Our numerical study indicates that the results can be applied to more general settings.     

Simultaneous variable selection and outlier identification in linear regression using the mean-shift outlier model

We provide a method for simultaneous variable selection and outlier identification using the mean-shift outlier model. The procedure consists of two steps: the first step is to identify potential outliers, and the second step is to perform all possible subset regressions for the mean-shift outlier model containing the potential outliers identified in step 1. This procedure is helpful for model selection while simultaneously considering outlier identification, and can be used to identify multiple outliers. In addition, we can evaluate the impact on the regression model of simultaneous omission of variables and interesting observations. In an example, we provide detailed output from the R system, and compare the results with those using posterior model probabilities as proposed by Hoeting et al. [Comput. Stat. Data Anal. 22 (1996), pp. 252-270] for simultaneous variable selection and outlier identification.     

A candidate-set-free algorithm for generating D-optimal split-plot designs

Summary.  We introduce a new method for generating optimal split-plot designs. These designs are optimal in the sense that they are efficient for estimating the fixed effects of the statistical model that is appropriate given the split-plot design structure. One advantage of the method is that it does not require the prior specification of a candidate set. This makes the production of split-plot designs computationally feasible in situations where the candidate set is too large to be tractable. The method allows for flexible choice of the sample size and supports inclusion of both continuous and categorical factors. The model can be any linear regression model and may include arbitrary polynomial terms in the continuous factors and interaction terms of any order. We demonstrate the usefulness of this flexibility with a 100-run polypropylene experiment involving 11 factors where we found a design that is substantially more efficient than designs that are produced by using other approaches.     

A Bayesian sequential design with adaptive randomization for 2‐sided hypothesis test

Bayesian sequential and adaptive randomization designs are gaining popularity in clinical trials thanks to their potentials to reduce the number of required participants and save resources. We propose a Bayesian sequential design with adaptive randomization rates so as to more efficiently attribute newly recruited patients to different treatment arms. In this paper, we consider 2‐arm clinical trials. Patients are allocated to the 2 arms with a randomization rate to achieve minimum variance for the test statistic. Algorithms are presented to calculate the optimal randomization rate, critical values, and power for the proposed design. Sensitivity analysis is implemented to check the influence on design by changing the prior distributions. Simulation studies are applied to compare the proposed method and traditional methods in terms of power and actual sample sizes. Simulations show that, when total sample size is fixed, the proposed design can obtain greater power and/or cost smaller actual sample size than the traditional Bayesian sequential design. Finally, we apply the proposed method to a real data set and compare the results with the Bayesian sequential design without adaptive randomization in terms of sample sizes. The proposed method can further reduce required sample size.     

D-optimal minimax design criterion for two-level fractional factorial designs

A D-optimal minimax design criterion is proposed to construct two-level fractional factorial designs, which can be used to estimate a linear model with main effects and some specified interactions. D-optimal minimax designs are robust against model misspecification and have small biases if the linear model contains more interaction terms. When the D-optimal minimax criterion is compared with the D-optimal design criterion, we find that the D-optimal design criterion is quite robust against model misspecification. Lower and upper bounds derived for the loss functions of optimal designs can be used to estimate the efficiencies of any design and evaluate the effectiveness of a search algorithm. Four algorithms to search for optimal designs for any run size are discussed and compared through several examples. An annealing algorithm and a sequential algorithm are particularly effective to search for optimal designs.     

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.     

Rank-order choice-based conjoint experiments: Efficiency and design

In a rank-order choice-based conjoint experiment, the respondent is asked to rank a number of alternatives of a number of choice sets. In this paper, we study the efficiency of those experiments and propose a D-optimality criterion for rank-order experiments to find designs yielding the most precise parameter estimators. For that purpose, an expression of the Fisher information matrix for the rank-ordered conditional logit model is derived which clearly shows how much additional information is provided by each extra ranking step. A simulation study shows that, besides the Bayesian D-optimal ranking design, the Bayesian D-optimal choice design is also an appropriate design for this type of experiments. Finally, it is shown that considerable improvements in estimation and prediction accuracy are obtained by including extra ranking steps in an experiment.