Comparison of normal probability plots and dot plots in judging the significance of effects in two level factorial designs

In this article, we present a study carried out to compare the effectiveness of the normal probability plot (NPP) and a simple dot plot in assessing the significance of the effects in experimental designs with factors at two levels (2 ^k?p designs). Several groups of students who had just completed a course that covered factorial designs were asked to identify the significant effects in a total of 32 situations, 16 of which were represented using NPPs and the other 16 using dot plots. Although the 32 scenarios were said to be different, there were really only 16 different situations, each of which was represented using the two methods to be compared. A simple graphical analysis shows no evidence that there is a difference between the two procedures. However, in designs with 16 runs there are some cases where NPP seems to give slightly better results.     

Block designs robust against the presence of positional effects

The problem considered is to find optimum designs for treatment effects in a block design (BD) setup, when positional effects are also present besides treatment and block effects, but they are ignored while formulating the model. In the class of symmetric balanced incomplete block designs, the Youden square design is shown to be optimal in the sense of minimizing the bias term in the mean squared error (MSE) of the best linear unbiased estimators of the full set of orthonormal treatment contrasts, irrespective of the value of the positional effects.     

A Semi-Parametric Quantile Function Estimator for Use in Bootstrap Estimation Procedures

In this note we develop a new quantile function estimator called the tail extrapolation quantile function estimator. The estimator behaves asymptotically exactly the same as the standard linear interpolation estimator. For finite samples there is small correction towards estimating the extreme quantiles. We illustrate that by employing this new estimator we can greatly improve the coverage probabilities of the standard bootstrap percentile confidence intervals. The method does not reqiure complicated calculations and hence it should appeal to the statistical practitioner.     

Estimation of the mean of an exponential distribution in the presence of an outlier

Kale and Sinha (1971) have found an estimator of the mean of an exponential distribution in the présence of an outlying observation with higher expected value. Here an alternative estimator of the mean is proposed and it is compared with the estimator of Kale and Sinha (1971) and the maximum likelihood estimator given by Kale (1975). The proposed estimator is found to be more efficient than the latter two estimators in some cases.     

Bounds on the maximum number of clear two‐factor interactions for 2m‐p designs of resolution III and IV

The authors derive upper and lower bounds on the maximum number of clear two‐factor interactions in 2^m?p fractional factorial designs of resolution III and IV. A two‐factor interaction is said to be clear if it is not aliased with any main effect or with any other two‐factor interaction. The lower bounds are obtained by exhibiting specific designs. By comparing the bounds with the values of the maximum number of clear two‐factor interactions in cases where it is known, one concludes that the construction methods perform quite well.     

Effect of missing plots in some response surface designs

The effect of one or more missing observations for response surface designs arranged in blocks are examined in this paper. The results as applied to a central composite design with orthogonal blocking, and an equirdial design with orthogonal blocking, are reported.     

A Bayesian Approach for Multiple Response Surface Optimization in the Presence of Noise Variables

An approach for the multiple response robust parameter design problem based on a methodology by Peterson (2000) is presented. The approach is Bayesian, and consists of maximizing the posterior predictive probability that the process satisfies a set of constraints on the responses. In order to find a solution robust to variation in the noise variables, the predictive density is integrated not only with respect to the response variables but also with respect to the assumed distribution of the noise variables. The maximization problem involves repeated Monte Carlo integrations, and two different methods to solve it are evaluated. A Matlab code was written that rapidly finds an optimal (robust) solution in case it exists. Two examples taken from the literature are used to illustrate the proposed method.     

Bayesian inference for the mean and standard deviation of a normal population when only the sample size,mean and range are observed

Consider a random sample X₁, X₂,…, X_n, from a normal population with unknown mean and standard deviation. Only the sample size, mean and range are recorded and it is necessary to estimate the unknown population mean and standard deviation. In this paper the estimation of the mean and standard deviation is made from a Bayesian perspective by using a Markov Chain Monte Carlo (MCMC) algorithm to simulate samples from the intractable joint posterior distribution of the mean and standard deviation. The proposed methodology is applied to simulated and real data. The real data refers to the sugar content (^oBRIX level) of orange juice produced in different countries.     

Quantile dispersion graphs to compare the efficiencies of cluster randomized designs

The purpose of this article is to compare efficiencies of several cluster randomized designs using the method of quantile dispersion graphs (QDGs). A cluster randomized design is considered whenever subjects are randomized at a group level but analyzed at the individual level. A prior knowledge of the correlation existing between subjects within the same cluster is necessary to design these cluster randomized trials. Using the QDG approach, we are able to compare several cluster randomized designs without requiring any information on the intracluster correlation. For a given design, several quantiles of the power function, which are directly related to the effect size, are obtained for several effect sizes. The quantiles depend on the intracluster correlation present in the model. The dispersion of these quantiles over the space of the unknown intracluster correlation is determined, and then depicted by the QDGs. Two applications of the proposed methodology are presented.     

Modified tests for change points in variance in the possible presence of mean breaks

This paper considers the detection problem of variance changes for the time series involving abrupt and/or smooth breaks in mean. Often, in these situations, the tests of choice are based on cumulative sum of squares statistics. We show that the test statistics are not robust in the presence of broken mean and their sizes suffer severe distortions. The adjusted residual-based method is then proposed to eliminate these deficiencies and makes a significant improvement. Finally, simulation results confirm the validity of these modified test statistics, and an empirical data analysis using some stock price series from the Shanghai Stock Exchange is reported.     

A proposal for plotting positions in probability plots

Probability plots allow us to determine whether a set of sample observations is distributed according to a theoretical distribution. Plotting positions are fundamental elements in statistics and, in particular, for the construction of probability plots. In this paper, a new plotting position to construct different probability plots, such as Q–Q Plot, P–P Plot and S–P Plot, is proposed. The proposed definition is based on the median of the ith order statistic of the theoretical distribution considered. The main feature of this plotting position formula is that it is independent of the theoretical distribution selected. Moreover, the procedure developed is ‘almost’ exact, reaching, without a high cost of time, an accuracy as great as we want, which avoids using approximations (proposed by other authors).     

The mean and variance of rank and exponential score product moment statistics in the non-circular case

This paper gives a general formula for the first two moments of the non-circular product moment statistic, which is used as a test for the statistical independence of the interarrival times of certain time series data, e.g., neuronal spike discharges. For the special cases of rank and exponential score product moment statistics, the mean and variance are formulated.     

Estimates for mean and dispersion effects in unreplicated factorial designs

One important goal of experimentation in quality improvement is to minimize the variability of a product or process around a target mean value. Factors which affect variances as well as factors that affect the mean can be identified using the analysis of mean and dispersion. Box and Meyer (1986b) proposed a method of model identification and maximum likelihood estimation for mean and dispersion effects from unreplicated designs. In this article, we address two problems associated with MLE’s. First, asymptotic variance of MLE's for dispersion effects which can be used to judge the significance of factors can be misleading. A possible explanation is provided; simulation results also indicate that the asymptotic, variance underestimates.     

Error variance bias in neighbour balance and evenness of distribution designs

Neighbour balance and evenness of distribution designs help to address user concerns in the two‐dimensional layout of agricultural field trials. This is done by minimising the occurrence of pairwise treatment plot neighbours and ensuring that the replications of treatments are spread out across rows and columns of a trial. Such considerations result in a restriction on the normal randomisation process for a row‐column design which can lead to error variance bias. In this paper, uniformity trial data is used to assess the degree of this bias for both resolvable and non‐resolvable designs. Comparisons are made with a similar investigation using Linear Variance spatial designs.     

Algebraic statistics of level codings for fractional factorial designs

We discuss the applications of algebraic statistics to fractional factorial design with special emphasis on the choice of level coding. In particular, we deal with the theory of Bayley's level codings in that framework.     

Graphical evaluation of robust parameter designs based on extended scaled prediction variance and extended spherical average prediction variance

For any response surface design, there are locations in the design region where responses are estimated well and locations where estimation is relatively poor. Consequently, graphical evaluation—such as variance dispersion graphs and the fraction of design space—is used as an alternative to a single-valued criterion. Such plots are used to investigate and compare the prediction capabilities of certain response surface designs currently available to the researcher. In this article, we propose the extended scaled prediction variance and extended spherical average prediction variance as prediction methods. We also illustrate how graphical methods can be employed to evaluate robust parameter designs.     

Difference variance dispersion graphs for comparing response surface designs with applications in food technology

Variance dispersion graphs have become a popular tool in aiding the choice of a response surface design. Often differences in response from some particular point, such as the expected position of the optimum or standard operating conditions, are more important than the response itself. We describe two examples from food technology. In the first, an experiment was conducted to find the levels of three factors which optimized the yield of valuable products enzymatically synthesized from sugars and to discover how the yield changed as the levels of the factors were changed from the optimum. In the second example, an experiment was conducted on a mixing process for pastry dough to discover how three factors affected a number of properties of the pastry, with a view to using these factors to control the process. We introduce the difference variance dispersion graph (DVDG) to help in the choice of a design in these circumstances. The DVDG for blocked designs is developed and the examples are used to show how the DVDG can be used in practice. In both examples a design was chosen by using the DVDG, as well as other properties, and the experiments were conducted and produced results that were useful to the experimenters. In both cases the conclusions were drawn partly by comparing responses at different points on the response surface.     

Two identities involving order statistics in the presence of an outlier

In this note, we derive two simple identities involving order statistics from a sample of size n in the presence of an outlier. These generalize the results of Joshi (1973). These identities will be quite useful in checking the computation of the single moments of order statistics from an outlier model.     

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.