Algorithmic construction of optimal block designs for two-colour cDNA microarray experiments using the linear mixed effects model

ABSTRACT

In this study, methods for efficient construction of A-, MV-, D- and E-optimal or near-optimal block designs for two-colour cDNA microarray experiments with array as the block effect are considered. Two algorithms, namely the array exchange and treatment exchange algorithms together with the complete enumeration technique are introduced. For large numbers of arrays or treatments or both, the complete enumeration method is highly computer intensive. The treatment exchange algorithm computes the optimal or near-optimal designs faster than the array exchange algorithm. The two methods however produce optimal or near-optimal designs with the same efficiency under the four optimality criteria.     

An Optimal Reliability and Maintainability Design of a Searching System

In this article, we deal with an optimal reliability and maintainability design problem of a searching system with complex structures. The system availability and life cycle cost are used as optimization criteria and estimated by simulation. We want to determine MTBF (Mean Time between Failures) and MTTR (Mean Time to Repair) for all components and ALDT (Administrative and Logistics Delay Times) of the searching system in order to minimize the life cycle cost and to satisfy the target system availability. A hybrid genetic algorithm with a heuristic method is proposed to find near-optimal solutions and compared with a general genetic algorithm.     

An algorithm for constructing optimal resolvable incomplete block designs

This paper describes an efficient algorithm for the construction of optimal or near-optimal resolvable incomplete block designs (IBDs) for any number of treatments v < 100. The performance of this algorithm is evaluated against known lattice designs and the 414 or-designs of Patterson & Williams [36]. For the designs under study, it appears that our algorithm is about equally effective as the simulated annealing algorithm of Venables & Eccleston [42]. An example of the use of our algorithm to construct the row (or column) components of resolvable row-column designs is given.     

Marginally restricted D-optimal designs for correlated observations

Two practical degrees of complexity may arise when designing an experiment for a model of a real life case. First, some explanatory variables may not be under the control of the practitioner. Secondly, the responses may be correlated. In this paper three real life cases in this situation are considered. Different covariance structures are studied and some designs are computed adapting the theory of marginally restricted designs for correlated observations. An exchange algorithm given by Brimkulov's algorithm is also adapted to marginally restricted D–optimality and it is applied to a complex situation.     

AN ALGORITHM FOR CONSTRUCTING OPTIMAL RESOLVABLE ROW-COLUMN DESIGNS

This paper describes an effective algorithm for constructing optimal or near-optimal resolvable row-column designs (RCDs) with up to 100 treatments. The performance of this algorithm is assessed against 20 2-replicate resolvable RCDs of Patterson & Robinson (1989) and 17 resolvable RCDs based on generalized cyclic designs (GCDs) of Ipinyomi & John (1985). The use of the algorithm to construct RCDs with contiguous replicates is discussed.     

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.     

Generating generalized cyclic designs with factorial balance

An algorithm for generating paired comparison factorially balanced generalized cyclic designs is described. The algorithm is based upon the 2 ⁿ ? 1 class association scheme defined by Shah (1960) for n-factor experiments. The algorithm is highly successful in achieving its objective. Firstorder designs with block size greater than two can also be obtained using the algorithm.     

Inference based on progressive Type I interval censored data from log-normal distribution

This article considers inference for the log-normal distribution based on progressive Type I interval censored data by both frequentist and Bayesian methods. First, the maximum likelihood estimates (MLEs) of the unknown model parameters are computed by expectation-maximization (EM) algorithm. The asymptotic standard errors (ASEs) of the MLEs are obtained by applying the missing information principle. Next, the Bayes’ estimates of the model parameters are obtained by Gibbs sampling method under both symmetric and asymmetric loss functions. The Gibbs sampling scheme is facilitated by adopting a similar data augmentation scheme as in EM algorithm. The performance of the MLEs and various Bayesian point estimates is judged via a simulation study. A real dataset is analyzed for the purpose of illustration.     

The Construction of Efficient Two-replicate Row–Column Designs for Use in Field Trials

Two-replicate row–column designs are often used for field trials in multisite tree or plant breeding programmes. With only two replicates for each trial, it is important to use designs with optimal or near optimal efficiency factors. This paper presents an algorithm for generating such designs. The method extends the contraction approach of Bailey and Patterson to any set of parameters and uses the factorial design construction algorithm of Williams and John to generate designs. Our experience with the algorithm is that it produces designs that are at least as good as, and often much better and more quickly generated than, those obtained by other recent computer algorithms.     

Multistage Sampling Designs and Estimating Equations

In some applications it is cost efficient to sample data in two or more stages. In the first stage a simple random sample is drawn and then stratified according to some easily measured attribute. In each subsequent stage a random subset of previously selected units is sampled for more detailed and costly observation, with a unit's sampling probability determined by its attributes as observed in the previous stages. This paper describes multistage sampling designs and estimating equations based on the resulting data. Maximum likelihood estimates (MLEs) and their asymptotic variances are given for designs using parametric models. Horvitz–Thompson estimates are introduced as alternatives to MLEs, their asymptotic distributions are derived and their strengths and weaknesses are evaluated. The designs and the estimates are illustrated with data on corn production.     

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.     

Simulation-Based Inspection Policies for a One-Shot System in Storage Over a Finite Time Span

In this paper, we consider an inspection policy problem for a one-shot system with two types of units over a finite time span and want to determine inspection intervals optimally with given replacement points of Type 2 units. The interval availability and life cycle cost are used as optimization criteria and estimated by simulation. Two optimization models are proposed to find the optimal inspection intervals for the exponential and general distributions. A heuristic method and a genetic algorithm are proposed to find the near-optimal inspection intervals, to satisfy the target interval availability and minimize the life-cycle cost. We study numerical examples to compare the heuristic method with the genetic algorithm and investigate the effect of model parameters to the optimal solutions.     

Statistical Analysis in Constant-stress Accelerated Life Tests for Generalized Exponential Distribution with Progressive Type-I Hybrid Censoring

Based on progressive Type-I hybrid censored data, statistical analysis in constant-stress accelerated life test (CS-ALT) for generalized exponential (GE) distribution is discussed. The maximum likelihood estimates (MLEs) of the parameters and the reliability function are obtained with EM algorithm, as well as the observed Fisher information matrix, the asymptotic variance-covariance matrix of the MLEs, and the asymptotic unbiased estimate (AUE) of the scale parameter. Confidence intervals (CIs) for the parameters are derived using asymptotic normality of MLEs and percentile bootstrap (Boot-p) method. Finally, the point estimates and interval estimates of the parameters are compared separately through the Monte-Carlo method.     

Efficient three-level screening designs using weighing matrices

Screening is the first stage of many industrial experiments and is used to determine efficiently and effectively a small number of potential factors among a large number of factors which may affect a particular response. In a recent paper, Jones and Nachtsheim [A class of three-level designs for definitive screening in the presence of second-order effects. J. Qual. Technol. 2011;43:1–15] have given a class of three-level designs for screening in the presence of second-order effects using a variant of the coordinate exchange algorithm as it was given by Meyer and Nachtsheim [The coordinate-exchange algorithm for constructing exact optimal experimental designs. Technometrics 1995;37:60–69]. Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8] have used conference matrices to construct definitive screening designs with good properties. In this paper, we propose a method for the construction of efficient three-level screening designs based on weighing matrices and their complete foldover. This method can be considered as a generalization of the method proposed by Xiao et al. [Constructing definitive screening designs using conference matrices. J. Qual. Technol. 2012;44:2–8]. Many new orthogonal three-level screening designs are constructed and their properties are explored. These designs are highly D-efficient and provide uncorrelated estimates of main effects that are unbiased by any second-order effect. Our approach is relatively straightforward and no computer search is needed since our designs are constructed using known weighing matrices.     

Near-optimal designs for dual channel microarray studies

Summary.  Much biological and medical research employs microarray studies to monitor gene expression levels across a wide range of organisms and under many experimental conditions. Dual channel microarrays are a common platform and allow two samples to be measured simultaneously. A frequently used design uses a common reference sample to make conditions across different arrays comparable. Our aim is to formulate microarray experiments in the experimental design context and to use simulated annealing to search for near-optimal designs. We identify a subclass of designs, the so-called interwoven loop designs, that seems to have good optimality properties compared with the near-optimal designs that are found by simulated annealing. Commonly used reference designs and dye swap designs are shown to be highly inefficient.     

Evaluation of Beta Generation Algorithms

In this article, we discuss the maximum likelihood estimates (MLEs) for the exponential and Weibull distributions by considering progressive Type-I interval censored data. For exponential distribution, the explicit expression of MLE of failure rate cannot be obtained when the intervals are not equal in length. The direct application of some numerical algorithms, such as the Newton–Raphson algorithm, is non-ideal because of the cumbersome second derivative. We apply some equivalent quantities to obtain the MLE of failure rate of exponential distribution. Based on the equivalent quantities and the Weibull-to-exponential transformation technique, we propose a new algorithm to obtain the MLEs for the parameters of progressive Type-I interval Weibull data. An example reanalysis and some simulation studies are carried out to illustrate the performance of the estimations using the new algorithm.     

Balanced treatment incomplete block designs through integer programming

An algorithm is presented to construct balanced treatment incomplete block (BTIB) designs using a linear integer programming approach. Construction of BTIB designs using the proposed approach is illustrated with an example. A list of efficient BTIB designs for 2 ? v ? 12, v + 1 ? b ? 50, 2 ? k ? min(10, v), r ? 10, r₀ ? 20 is provided. The proposed algorithm is implemented as part of an R package.     

Lee discrepancy on mixed two- and three-level uniform augmented designs

Follow-up experiment is widely applied to various fields such as science and engineering, since it is an indispensable strategy, especially when some additional resources or information become available after the initial design of experiment is carried out. Moreover, some extra factors may be added in the follow-up experiment. One may augment the number of runs and/or factors for the purpose of application. In this paper, the issue of the uniform row augmented designs and column augmented designs with mixed two- and three-level is investigated. The uniformity of augmented designs is discussed under the Lee discrepancy, some lower bounds of Lee discrepancy for the augmented designs are obtained. The construction algorithm of the uniform augmented designs is given. Some numerical examples indicate that uniform augmented designs can be constructed with high efficiency.     

Double-objective economic statistical design of the VP T2 control chart: Wald's identity approach

Research has shown that applying the T² control chart by using a variable parameters (VP) scheme yields rapid detection of out-of-control states. In this paper, the problem of economic statistical design of the VP T²control chart is considered as a double-objective minimization problem with the statistical objective being the adjusted average time to signal and the economic objective being expected cost per hour. We then find the Pareto-optimal designs in which the two objectives are met simultaneously by using a multi-objective genetic algorithm. Through an illustrative example, we show that relatively large benefits can be achieved by applying the VP scheme when compared with usual schemes, and in addition, the multi-objective approach provides the user with designs that are flexible and adaptive.