Efficient run orders for a two-factor response surface experiment on a correlated process

The standard assumption in the design and analysis of response surface experiments is that the errors are uncorrelated with common variance. In practice however there may be serial dependence between the errors, in which case the efficiency of the design depends on the order in which the runs are carried out. We investigate the effect of run order on a two-factor response surface experiment when there is positive serial correlation, and recommend some particular run orders which are efficient under a wide range of conditions.     

Budget constrained run orders in optimum design

When experiments are to be performed in a time sequence, the observed responses may be affected by an unknown time trend. Run orders that are optimally balanced for time trends usually involve huge costs due to the large number of factor level changes. Therefore, trend-free run orders can be of low practical value in view of economical considerations. Tack and Vandebroek (J. Statist. Plann. Inference 98 (2001) 293) recently presented a design algorithm to construct trend-resistant run orders that give maximal information per unit cost. In this paper, the latter approach is extended to the construction of trendresistant run orders for which the total cost has to be lower than a specified budget. A new design algorithm is proposed that offers the experimenter a general method for solving a wide range of practical design problems.     

A-OPTIMAL RUN ORDERS WITH A LINEAR TREND

A-optimal ran orders are considered in the presence of a linear trend with emphasis on non-orthogonal situations where no trend-free run order can be. A-optimal. Some possibilities for further extension are also briefly indicated.     

Experimentation order in factorial designs: new findings

Under some very reasonable hypotheses, it becomes evident that randomizing the run order of a factorial experiment does not always neutralize the effect of undesirable factors. Yet, these factors do have an influence on the response, depending on the order in which the experiments are conducted. On the other hand, changing the factor levels is many times costly; therefore it is not reasonable to leave to chance the number of changes necessary. For this reason, run orders that offer the minimum number of factor level changes and at the same time minimize the possible influence of undesirable factors on the experimentation have been sought. Sequences which are known to produce the desired properties in designs with 8 and 16 experiments can be found in the literature. In this paper, we provide the best possible sequences for designs with 32 experiments, as well as sequences that offer excellent properties for designs with 64 and 128 experiments. The method used to find them is based on a mixture of algorithmic searches and an augmentation of smaller designs.     

Experimentation order in factorial designs with 8 or 16 runs

Randomizing the order of experimentation in a factorial design does not always achieve the desired effect of neutralizing the influence of unknown factors. In fact, with some very reasonable assumptions, an important proportion of random orders afford the same degree of protection as that obtained by experimenting in the design matrix standard order. In addition, randomization can induce a big number of changes in factor levels and thus make experimentation expensive and difficult. This paper discusses this subject and suggests experimentation orders for designs with 8 or 16 runs that combine an excellent level of protection against the influence of unknown factors, with the minimum number of changes in factor levels.     

Specifications of a continual reassessment method design for phase I trials of combined drugs

In studies of combinations of agents in phase I oncology trials, the dose–toxicity relationship may not be monotone for all combinations, in which case the toxicity probabilities follow a partial order. The continual reassessment method for partial orders (PO‐CRM) is a design for phase I trials of combinations that leans upon identifying possible complete orders associated with the partial order. This article addresses some practical design considerations not previously undertaken when describing the PO‐CRM. We describe an approach in choosing a proper subset of possible orderings, formulated according to the known toxicity relationships within a matrix of combination therapies. Other design issues, such as working model selection and stopping rules, are also discussed. We demonstrate the practical ability of PO‐CRM as a phase I design for combinations through its use in a recent trial designed at the University of Virginia Cancer Center. Copyright © 2013 John Wiley & Sons, Ltd.     

Experimentation order with good properties for 2 k factorial designs

Randomizing the order of experimentation in a factorial design does not always achieve the desired effect of neutralizing the influence of unknown factors. In fact, with some very reasonable assumptions, an important proportion of random orders achieve the same degree of protection as that obtained by experimenting in the design matrix standard order. In addition, randomization can induce a large number of changes in factor levels and thus make experimentation expensive and difficult. De Leon et al. [Experimentation order in factorial designs with 8 or 16 runs, J. Appl. Stat. 32 (2005), pp. 297–313] proposed experimentation orders for designs with eight or 16 runs that combine an excellent level of protection against the influence of unknown factors, with the minimum number of changes in factor levels. This article presents a new methodology to obtain experimentation orders with the desired properties for designs with any number of runs.     

A new economic scheme for CCC charts with run rules based on average number of inspected items

New statistical techniques and procedures have been developed to control high-yield processes along with looking for process improvement opportunities and minimizing production cost. Cumulative count of conforming control chart is generally a technique for high-quality processes, when nonconforming items are rarely produced. The objective of this study is to design control chart based on cumulative count of conforming items and run rules that develops an economic model based on the average number of inspected items to design m-of-m CCC chart in order to facilitate minimum average cost per item produced. The optimal design parameters for different values of nonconforming fraction and different cost parameters in each scenario are determined. Finally, to analyze the behavior of optimal economic solutions, sensitivity analysis of the model parameters is performed.     

Trend-Free Designs in Blocked Factorial Experiments

Appropriate run orders can make all estimable effects free of some trends in blocked fractional factorial experiments. We need to design blocked experiments with effects free of trends in blocks. The generalized foldover scheme given by Coster (1993 Coster , D. C. ( 1993 ). Trend-free run orders of mixed-level fractional factorial designs . Ann. Statist. 21 : 2072 – 2086 .[Crossref], [Web of Science ®] , [Google Scholar]) can be used to obtain such designs. In this article, we propose an easy and better approach to deal with this issue when the same trends appear in blocks. We investigate block trend property of columns in the orthogonal plans in s ^k runs for assigning factors to obtain block-trend free designs in a trend-free order. We illustrate our approach with three examples.     

Some results on two-level factorial designs with dependent observations

There are many situations in which observations in factorial experiments may be dependent. When this is so, run orders are needed that result in efficient estimates of contrasts. The Cheng and Steinberg reverse foldover algorithm, which gives a maximal number of level changes, is known to produce very efficient main-effects two-level designs using the D-criterion, but less is known about other designs, models and criteria. We present some further theoretical results, and give another statistic of importance in predicting efficiency under strong dependence. The theory is illustrated using some 16-run designs.     

Sampling design proportional to order statistic of auxiliary variable

The sampling designs dependent on sample moments of auxiliary variables are well known. Lahiri (Bull Int Stat Inst 33:133–140, 1951) considered a sampling design proportionate to a sample mean of an auxiliary variable. Sing and Srivastava (Biometrika 67(1):205–209, 1980) proposed the sampling design proportionate to a sample variance while Wywiał (J Indian Stat Assoc 37:73–87, 1999) a sampling design proportionate to a sample generalized variance of auxiliary variables. Some other sampling designs dependent on moments of an auxiliary variable were considered e.g. in Wywiał (Some contributions to multivariate methods in, survey sampling. Katowice University of Economics, Katowice, 2003a); Stat Transit 4(5):779–798, 2000) where accuracy of some sampling strategies were compared, too.These sampling designs cannot be useful in the case when there are some censored observations of the auxiliary variable. Moreover, they can be much too sensitive to outliers observations. In these cases the sampling design proportionate to the order statistic of an auxiliary variable can be more useful. That is why such an unequal probability sampling design is proposed here. Its particular cases as well as its conditional version are considered, too. The sampling scheme implementing this sampling design is proposed. The inclusion probabilities of the first and second orders were evaluated. The well known Horvitz–Thompson estimator is taken into account. A ratio estimator dependent on an order statistic is constructed. It is similar to the well known ratio estimator based on the population and sample means. Moreover, it is an unbiased estimator of the population mean when the sample is drawn according to the proposed sampling design dependent on the appropriate order statistic.     

An experimental study of thermal energy storage with phase change materials by design of experiments

Accurate theoretical modelling and simulation of thermal energy storage (TES) by means of phase change materials (PCM) is very complex and its results are not close enough to experimental values. This paper presents the empirical study of a thermal storage unit operating with a commercial PCM called RT25. The study is carried out by means of the statistical procedure, Design of Experiments. This methodology has rarely been used in the analysis of heat transfer problems. The present study has allowed us to investigate the phenomena involved and to design an actual system. We show the whole procedure followed in order to design the set-up, to run the experiments with a 2³ factorial design, to compare its results with a numerical simulation and to get the empirical model by regression. Its results have been used to design actual installations aimed at free-cooling or maintaining the temperature constant in rooms where thermal security is necessary.     

An Adaptive Shiryaev–Roberts Procedure for Signalling Varying Location Shifts

This paper proposes a new adaptive chart based on the Shiryaev–Roberts procedure, by updating the reference value in an adaptive way to achieve the aim of overall good performance over a range of future expected but unknown mean shifts. A two-dimensional Markov chain model is developed to analyze the run length performance. The design guidelines are given. The comparisons of run length performance of the proposed scheme and other charts show that the proposed chart provides quite effective detecting ability over a range of mean shift sizes. The implementation of the new chart is illustrated by a real data example.     

Projection estimation capacity of Hadamard designs

In a screening design, often only a few factors among a large number of potential factors are significantly important. Usually, it is not known which factors will be important ones. Thus, it is of practical interest to know if each projection of a design onto a small subset of factors is able to entertain and estimate all two-factor-interactions along with its main effects, assuming higher order interactions are negligible. In this paper, we investigate the estimation capacity of projections of Hadamard designs with run size up to 60. Possible applications of our results to robust parameter designs are also discussed.     

Minimally changed run sequences in factorial experiments

Randomization of run sequences in factorial experiments may result in large number of changes in factor levels which will make the experimentation expensive, time-consuming and difficult. Experiments in which it is difficult to change the levels of factor(s) use of minimally changed run sequences may often be preferable to a random run sequence. In the present paper, we have developed method for obtaining minimally changed run sequences for factorial experiments. The general expression of factor-wise number of level changes for the developed minimally changed run sequences has also been obtained. A relationship has been established between the time count effect of a lower order factorial with minimally changed run sequences and that of a higher order factorial with minimally changed run sequences obtained through the lower order minimally changed run sequences. For providing a readymade solution to the end users, a SAS macro has also been developed for generating these minimally changed run sequences along with its parameters.     

Some new classes of orthogonal Latin hypercube designs

Orthogonal Latin hypercube (OLH) is a good design choice in a polynomial function model for computer experiments, because it ensures uncorrelated estimation of linear effects when a first-order model is fitted. However, when a second-order model is adopted, an OLH also needs to satisfy the additional property that each column is orthogonal to the elementwise square of all columns and orthogonal to the elementwise product of every pair of columns. Such class of OLHs is called OLHs of order two while the former class just possessing two-dimensional orthogonality is called OLHs of order one. In this paper we present a general method for constructing OLHs of orders one and two for n=s^m$n=s^m$ runs, where s and m may be any positive integers greater than one, by rotating a grouped orthogonal array with a column-orthogonal rotation matrix. The Kronecker product and the stacking methods are revisited and combined to construct some new classes of OLHs of orders one and two with other flexible numbers of runs. Some useful OLHs of order one or two with larger factor-to-run ratio and moderate runs are tabulated and discussed.     

Time truncated attribute control chart for the Weibull distribution using multiple dependent state sampling

In this article, an attribute control chart is proposed for time truncated tests using the Weibull distribution. The design of proposed control chart is presented using the multiple dependent state (MDS) sampling. The control chart coefficients are determined for various specified average run length. The efficiency of the proposed control chart is elaborated with the help of a simulation data and a real data. The proposed control chart perform better than the existing control chart in terms of average run length.     

Optimal Statistical Design of a Multivariate EWMA Chart Based on ARL and MRL

Statistical design is applied to a multivariate exponentially weighted moving average (MEWMA) control chart. The chart parameters are control limit H and smoothing constant r. The choices of the parameters depend on the number of variables p and the size of the process mean shift δ. The MEWMA statistic is modeled as a Markov chain and the Markov chain approach is used to determine the properties of the chart. Although average run length has become a traditional measure of the performance of control schemes, some authors have suggested other measures, such as median and other percentiles of the run length distribution to explain run length properties of a control scheme. This will allow a thorough study of the performance of the control scheme. Consequently, conclusions based on these measures would provide a better and comprehensive understanding of a scheme. In this article, we present the performance of the MEWMA control chart as measured by the average run length and median run length. Graphs are given so that the chart parameters of an optimal MEWMA chart can be determined easily.     

Randomized and random run order experiments

Randomization in industrial and scientific experiments on equipment has meant randomizing the order of application of levels of treatments to units. This definition is inadequate because it does not render independent error terms. Randomization also requires independent resettings of treatment levels when the levels for the preceding run are the same. We review how the literature incorrectly explains how randomization is to be carried out. The need to reset levels of a treatment from one run to the next is never emphasized. Using a simple example we show why statistical tests are biased for all treatments even when levels for just one treatment are not independently reset. Even if the expected mean squares recognize the restrictions on randomization, the usual F test will not give predictable results because its numerator and denominator are correlated.Experimental design on equipment includes experiments from the chemical, automobile, pharmaceutical, and aeronautical industries. The statistical interpretation of data from such experiments will be misleading. Books on experimental design must emphasize the independent resetting of levels just as carefully as they emphasize the random assignment of treatment levels.