The present work investigates the effects of diesel contaminants and their interaction on tribological properties for bearing steel (En31) and ceramic (Si3N4) sliding contacts using a factorial study. The contaminants are soot, sulphuric acid, moisture and oxidation, and each contaminant has three different level of concentration (low, medium and high) in the test matrix. The factorial test matrix consisted of 20 tests, constructed from a quarter fractional factorial test matrix with four points at the medium values for the contaminants. Results from this matrix required six further tests to elucidate aliased pairs of interactions using Bayesian model selection. A pin-on-disc tribometer was used to carry out all the experiments. All tests were carried out under ambient conditions at 5 m/s sliding speed and contact stress of 1.5–2.05 GPa to simulate a valve-train in a diesel engine with fully formulated heavy-duty diesel engine oil used as lubricant. Four different tribological properties were studied. The factorial study showed that charge was influenced by tribocouple material; the silicon nitride discs produced higher charge than steel discs. However, it was opposite for friction; the silicon nitride disc gave lower friction and the pins showed higher friction than their steel counterparts. For wear scar and temperature, soot contaminant was found to be important. The two important interactions were found for the charge response, with the interaction between sulphuric acid and pin material being more important than sulphuric acid–oxidation interaction. Similarly to charge, an interaction between sulphuric acid and pin material interaction was found for friction.

The aim of this paper is to review methods of designing screening experiments, ranging from designs originally developed for physical experiments to those especially tailored to experiments on numerical models. The strengths and weaknesses of the various designs for screening variables in numerical models are discussed. First, classes of factorial designs for experiments to estimate main effects and interactions through a linear statistical model are described, specifically regular and nonregular fractional factorial designs, supersaturated designs and systematic fractional replicate designs. Generic issues of aliasing, bias and cancellation of factorial effects are discussed. Second, group screening experiments are considered including factorial group screening and sequential bifurcation. Third, random sampling plans are discussed including Latin hypercube sampling and sampling plans to estimate elementary effects. Fourth, a variety of modelling methods commonly employed with screening designs are briefly described. Finally, a novel study demonstrates six screening methods on two frequently-used exemplars, and their performances are compared.

The first investigation is made of designs for screening experiments where the response variable is approximated by a generalised linear model. A Bayesian information capacity criterion is defined for the selection of designs that are robust to the form of the linear predictor. For binomial data and logistic regression, the effectiveness of these designs for screening is assessed through simulation studies using all-subsets regression and model selection via maximum penalised likelihood and a generalised information criterion. For Poisson data and log-linear regression, similar assessments are made using maximum likelihood and the Akaike information criterion for minimally-supported designs that are constructed analytically. The results show that effective screening, that is, high power with moderate type I error rate and false discovery rate, can be achieved through suitable choices for the number of design support points and experiment size. Logistic regression is shown to present a more challenging problem than log-linear regression. Some areas for future work are also indicated.

The mean of the output of interest obtained from a run of a computer simulation model of a system or process often depends on many factors; many times, however, only a few of these factors are important. Sequential bifurcation is a method that has been considered by several authors for identifying these important factors using as few runs of the simulation model as possible. In this article, we propose a new sequential bifurcation procedure whose steps use a key stopping rule that can be calculated explicitly, something not available in the best methods previously considered. Moreover, we show how this stopping rule can also be easily modified to efficiently identify those factors that are important in influencing the variability rather than the mean of the output. In empirical studies, the new method performs better than previously published fully sequential bifurcation methods in terms of achieving the prescribed Type I error. It also achieves higher power for detecting moderately large effects using fewer replications than earlier methods. To achieve this control for midrange effects, the new method sometimes requires more replications than other methods in the case where there are many very large effects.

Product and process improvement can involve a large number of factors that must be varied simultaneously. Understanding how factors interact is a key step in identifying those factors that have a substantial impact on the response. This article gives the first comprehensive assessment and comparison of screening strategies for interactions using two-level supersaturated designs, group screening, and a variety of data analysis methods including shrinkage regression and Bayesian methods. We develop novel methodology to allow application of Bayesian methods in two-stage group screening. Insights on using the strategies are provided through a variety of simulation scenarios and open issues are discussed. Supplementary materials are available online.

When it is anticipated that data to be collected from an experiment cannot be adequately described by a low-order polynomial, alternative modelling and new design methods are required. Local linear regression, where the response is approximated locally by a series of weighted linear regressions, is an effective nonparametric smoothing method that makes few assumptions about the functional form of the response. We present new methods for the optimal design of experiments for local linear regression, including a new criterion, called D-SI-optimality, to find designs that enable precise prediction across a continuous interval. Designs are found numerically for weights defined through the Gaussian and uniform kernels. Theoretical results are presented for the uniform kernel and the special case of prediction at a single point. The sensitivity of the designs to the choice of bandwidth in the local linear regression is studied, and it is found that designs for the Gaussian kernel with large bandwidth have a small number of distinct design points. The methodology is motivated by, and demonstrated on, an experiment from Tribology.

We propose a general design selection criterion for experiments where a generalized linear model describes the response. The criterion allows for several competing aims, such as parameter estimation and model discrimination, and also for uncertainty in the functional form of the linear predictor, the link function and the unknown model parameters. A general equivalence theorem is developed for this criterion. In practice, an exact design is required by experimenters and can be obtained by numerical rounding of a continuous design. We derive bounds on the performance of an exact design under this criterion which allow the efficiency of a rounded continuous design to be assessed.

We consider the problem of finding an optimal design under a Poisson regression model with a log link, any number of independent variables, and an additive linear predictor. Local D-optimality of a class of designs is established through use of a canonical form of the problem and a general equivalence theorem. The results are applied in conjunction with clustering techniques to obtain a fast method of finding designs that are robust to wide ranges of model parameter values. The methods are illustrated through examples.

Inference for a generalized linear model is generally performed using asymptotic approximations for the bias and the covariance matrix of the parameter estimators. For small experiments, these approximations can be poor and result in estimators with considerable bias. We investigate the properties of designs for small experiments when the response is described by a simple logistic regression model and parameter estimators are to be obtained by the maximum penalized likelihood method of Firth [Firth, D., 1993, Bias reduction of maximum likelihood estimates. Biometrika, 80, 27-38]. Although this method achieves a reduction in bias, we illustrate that the remaining bias may be substantial for small experiments, and propose minimization of the integrated mean square error, based on Firth's estimates, as a suitable criterion for design selection. This approach is used to find locally optimal designs for two support points.

Screening is the process of sifting through a set of factors through experimentation to determine the few important factors that have a substantial effect on a response. When the set of factors is large and interactions are anticipated, screening methods using single fractional factorial designs may require too many observations to be feasible. The methodology of two-stage group screening has been suggested as an alternative. This article gives the first description of practical aspects involved in running a two-stage group screening experiment for investigating interactions. Issues involved in the design and analysis of such an experiment are discussed in the context of a study run at Jaguar Cars on cold start optimization. The analysis of this experiment provides insight into how group screening works in practice and how the factorial effects of the individual factors relate to those of the grouped factors. Elicitation of information from subject specialists, choice of factor groups, and selection of designs for two-stage group screening are discussed. Through the analysis of the experimental data, it is shown that the process of group screening can provide an efficient method of detecting interactions among large numbers of factors.

Low, Lewis and Prescott (1999) showed that a crossover design based on a Williams Latin square of order 4 can suffer substantial loss of efficiency if some observations in the final period are unavailable. Indeed, if all observations are missing, the design becomes disconnected. We derive the information matrix for the direct effects of a Uniformly Balanced Repeated Measurements Design (UBRMD) in t periods when subjects may drop out before the end of the study, and examine the maximum loss of information. The special case of loss of observations in the final period only is examined in detail. In particular we show that a UBRMD in t >= 5 periods remains connected when some or all observations in the final period are unavailable.

An eLearning website is described for the design and analysis of experiments, with particular application to chemistry research. Interactive learning content is provided via the R statistical software. Authentic chemistry examples, which make use of simulation and data analysis routines in the software, demonstrate the application of the statistical methods. We outline the content of the website, the scope of the interactive examples and the interface linking the web-browser and the R system.

Designing experiments to identify improvement in products that are assembled from manufactured components does not readily fit into conventional design of experiments methods and can be costly. Efficient methods are explored for determining designs for engineering problems where some, or all, of the factors of interest are (a) not easily set to prescribed values and (b) are dependent on a combination of properties of several components. The methods involve taking a sample of each type of component, measuring the relevant features and then finding a design that specifies an optimal set of assembled products for experiment. Three examples from manufacturing industry are presented to illustrate the approach. Two different algorithms for finding designs are described, an exchange algorithm and a genetic algorithm, and a comparison of their performances is made on the three examples.

Consider the problem of identifying important factors influencing a response in a simulation experiment where the number of factors is large. When the direction of the effect of factors is known, the method of sequential bifurcation is effective for quickly removing non-influential factors. Though good, the method is not fully efficient in that not all the information available is fully utilized. We present a method based on a polytope construction that makes use of all available information and which is therefore more efficient. In this paper we focus on the deterministic case to highlight its theoretical foundation. The method can however be extended to the stochastic case. Numerical examples are given comparing the new method with sequential bifurcation showing its improved performance

The aim of an experiment is often to enable discrimination between competing forms for a response model. We consider this problem when there are two competing generalized linear models (GLMs) for a binomial response. These models are assumed to have a common link function with the linear predictor of one model nested within that of the other. We consider selection of a continuous design for use in a non-sequential strategy and investigate a new criterion, TE-optimality, based on the difference in the deviances from the two models. A comparison is made with three existing design selection criteria, namely T-, Ds- and D-optimality. Issues are raised through the study of two examples in which designs are assessed using simulation studies of the power to reject the null hypothesis of the simpler model being correct, when the data are generated from the larger model. Parameter estimation for these designs is also discussed and a simple method is investigated of combining designs to form a hybrid design to achieve both model discrimination and estimation. Such a method may offer a computational advantage over the use of a compound criterion and the similar performance of the resulting designs is illustrated in an example. MSC: primary 62K05, 62J12; secondary 62K20

Standard factorial designs sometimes may be inadequate for experiments that aim to estimate a generalized linear model, for example, for describing a binary response in terms of several variables. A method is proposed for finding exact designs for such experiments that uses a criterion allowing for uncertainty in the link function, the linear predictor, or the model parameters, together with a design search. Designs are assessed and compared by simulation of the distribution of efficiencies relative to locally optimal designs over a space of possible models. Exact designs are investigated for two applications, and their advantages over factorial and central composite designs are demonstrated.