The analysis and interpretation of interactions

In many individual experiments, and especially when combining results from several experiments, interactions are observed. Various models have been proposed for the analysis of interactions, and product models are often useful. Work in this area is reviewed, starting with the 1920's and going on to recent studies, including some in Australia and Poland.     

Orthogonal Latin hypercube designs for Fourier-polynomial models

Latin hypercube designs (LHDs) are widely used in computer experiments because of their one-dimensional uniformity and other properties. Recently, a number of methods have been proposed to construct LHDs with properties that all linear effects are mutually orthogonal and orthogonal to all second-order effects, i.e., quadratic effects and bilinear interactions. This paper focuses on the construction of LHDs with the above desirable properties under the Fourier-polynomial model. A convenient and flexible algorithm for constructing such orthogonal LHDs is provided. Most of the resulting designs have different run sizes from that of Butler (2001), and thus are new and very suitable for factor screening and building Fourier-polynomial models in computer experiments as discussed in Butler (2001).     

Resolving the Mixed Models Controversy

Two standard mixed models with interactions are discussed. When each is viewed in the context of superpopulation models, the mixed models controversy is resolved. The tests suggested by the expected mean squares under the constrained-parameters model are correct for testing the main effects and interactions under both the unconstrained-and constrained-parameters models.     

Long-term demographic interactions in precensus England

"Modern time series methods are applied to the analysis of annual demographic data for England, 1541-1800. Evidence is found of non-stationarity in the series and of co-integration among the series. Building on economic models of historical demography, optimal inferential procedures are implemented to estimate the structural parameters of long-term equilibria among the variables. Evidence is found for a small, but significant, Malthusian 'preventive check' as well as interactions between fertility, mortality and nuptiality that are consistent with the predictions often made in demographic studies. Tentative experiments to detect the influence of environmental factors fail to reveal any significant impact on the estimates obtained."     

Circuit optimization via sequential computer experiments: design of an output buffer

In electrical engineering, circuit designs are now often optimized via circuit simulation computer models. Typically, many response variables characterize the circuit's performance. Each response is a function of many input variables, including factors that can be set in the engineering design and noise factors representing manufacturing conditions. We describe a modelling approach which is appropriate for the simulator's deterministic input–output relationships. Non-linearities and interactions are identified without explicit assumptions about the functional form. These models lead to predictors to guide the reduction of the ranges of the designable factors in a sequence of experiments. Ultimately, the predictors are used to optimize the engineering design. We also show how a visualization of the fitted relationships facilitates an understanding of the engineering trade-offs between responses. The example used to demonstrate these methods, the design of a buffer circuit, has multiple targets for the responses, representing different trade-offs between the key performance measures.     

Orthogonal main-effect plans in row–column designs for two-level factorial experiments

Row–column designs for two-level factorial experiments are constructed to estimate all the main effects. We give the interactions for row and column blockings. Based on these blockings, independent treatment combinations are proposed to establish the whole design so that practitioners can easily apply it to their experiments. Some examples are given for illustrations. The estimation of two-factor interactions in these designs is discussed.     

Data-driven Kriging models based on FANOVA-decomposition

Kriging models have been widely used in computer experiments for the analysis of time-consuming computer codes. Based on kernels, they are flexible and can be tuned to many situations. In this paper, we construct kernels that reproduce the computer code complexity by mimicking its interaction structure. While the standard tensor-product kernel implicitly assumes that all interactions are active, the new kernels are suited for a general interaction structure, and will take advantage of the absence of interaction between some inputs. The methodology is twofold. First, the interaction structure is estimated from the data, using a first initial standard Kriging model, and represented by a so-called FANOVA graph. New FANOVA-based sensitivity indices are introduced to detect active interactions. Then this graph is used to derive the form of the kernel, and the corresponding Kriging model is estimated by maximum likelihood. The performance of the overall procedure is illustrated by several 3-dimensional and 6-dimensional simulated and real examples. A substantial improvement is observed when the computer code has a relatively high level of complexity.     

Optimum Mixture Design for the Detection of Synergistic Effects

In mixture experiments, optimal designs for the estimation of parameters, both linear and non-linear, have been discussed by several authors. Optimal designs for the estimation of a subset of parameters have also been investigated. However, designs for testing the effects of certain factors and interactions have been studied only in the context of response surface models. In this article, we attempt to find the optimum design for testing the presence of synergistic effects in a mixture model. The classical F-test has been considered and the optimum design has been obtained so as to maximize the power of the test. It is observed that the barycenters are necessarily the support points of the trace-optimal design.     

A modification of the Box–Meyer method for finding the active factors in screening experiments

Screening experiments are conducted to identify a few active factors among a large number of factors. For the objective of identifying active factors, Box and Meyer provided an innovative approach, the Box–Meyer method (BMM). With the use of means models, we propose a modification of the BMM in this paper. Compared with the original BMM, the modified BMM (MBMM) can circumvent the problem that the original BMM runs into, namely that it may fail to identify some active factors due to the ignorance of higher order interactions. Furthermore, the number of explanatory variables in the MBMM is smaller. Therefore, the computational complexity is reduced. Finally, three examples with different types of designs are used to demonstrate the wide applicability of the MBMM.     

Semiparametric partially linear varying coefficient models with panel count data

This paper studies semiparametric regression analysis of panel count data, which arise naturally when recurrent events are considered. Such data frequently occur in medical follow-up studies and reliability experiments, for example. To explore the nonlinear interactions between covariates, we propose a class of partially linear models with possibly varying coefficients for the mean function of the counting processes with panel count data. The functional coefficients are estimated by B-spline function approximations. The estimation procedures are based on maximum pseudo-likelihood and likelihood approaches and they are easy to implement. The asymptotic properties of the resulting estimators are established, and their finite-sample performance is assessed by Monte Carlo simulation studies. We also demonstrate the value of the proposed method by the analysis of a cancer data set, where the new modeling approach provides more comprehensive information than the usual proportional mean model.     

Spatial effects in dynamic conditional correlations

The recent literature on time series has developed a lot of models for the analysis of the dynamic conditional correlation, involving the same variable observed in different locations; very often, in this framework, the consideration of the spatial interactions is omitted. We propose to extend a time-varying conditional correlation model (following an autoregressive moving average dynamics) to include the spatial effects, with a specification depending on the local spatial interactions. The spatial part is based on a fixed symmetric weight matrix, called Gaussian kernel matrix, but its effect will vary along the time depending on the degree of time correlation in a certain period. We show the theoretical aspects, with the support of simulation experiments, and apply this methodology to two space–time data sets, in a demographic and a financial framework, respectively.     

Determination of optimum experimental conditions using dispersion main effects and interactions of factors in replicated factorial experiments

Dispersion main effects and two-factor interactions are first defined and then estimated in replicated factorial experiments. A method based on Union-Intersection rules, using dispersion main effects and two factor interactions is proposed for finding the level combinations of control factors so that the response variability due to noise is minimum. Illustrative examples are also given.     

Parallelizing particle filters with butterfly interactions

The bootstrap particle filter (BPF) is the cornerstone of many algorithms used for solving generally intractable inference problems with hidden Markov models. The long-term stability of the BPF arises from particle interactions that typically make parallel implementations of the BPF nontrivial. We propose a method whereby particle interaction is done in several stages. With the proposed method, full interaction can be accomplished even if we allow only pairwise communications between processing elements at each stage. We show that our method preserves the consistency and the long-term stability of the BPF, although our analysis suggests that the constraints on the stagewise interactions introduce errors leading to a lower convergence rate than standard Monte Carlo. The proposed method also suggests a new, more flexible, adaptive resampling scheme, which, according to our numerical experiments, is the method of choice, displaying a notable gain in efficiency in certain parallel computing scenarios.     

Switching-one-column follow-up experiments for Plackett-Burman designs

Industrial experiments are frequently performed sequentially using two-level fractional factorial designs. In this context, a common strategy for the design of follow-up experiments is to switch the signs in one column. It is well known that this strategy, when applied to two-level fractional factorial resolution III designs, will clear the main effect, for which the switch was performed, from any confounding with any other two-factor interactions and will also clear all the two-factor interactions between that factor and the other main effects from any confounding with other two-factor interactions. In this article, we extend this result and show that this strategy applies to any orthogonal two-level resolution III design and therefore specifically to any two-level Plackett- Burman design .     

Horwitz's rule, transforming both sides and the design of experiments for mechanistic models

Summary. The paper develops methods for the design of experiments for mechanistic models when the response must be transformed to achieve symmetry and constant variance. The power transformation that is used is partially justified by a rule in analytical chemistry. Because of the nature of the relationship between the response and the mechanistic model, it is necessary to transform both sides of the model. Expressions are given for the parameter sensitivities in the transformed model and examples are given of optimum designs, not only for single-response models, but also for experiments in which multivariate responses are measured and for experiments in which the model is defined by a set of differential equations which cannot be solved analytically. The extension to designs for checking models is discussed.     

Cross-over Designs in the Presence of Higher Order Carry-overs

In cross-over experiments, where different treatments are applied successively to the same experimental unit over a number of time periods, it is often expected that a treatment has a carry-over effect in one or more periods following its period of application. The effect of interaction between the treatments in the successive periods may also affect the response. However, it seems that all systematic studies of the optimality properties of cross-over designs have been done under models where carry-over effects are assumed to persist for only one subsequent period. This paper proposes a model which allows for the possible presence of carry-over effects up to k subsequent periods, together with all the interactions between treatments applied at k + 1 successive periods. This model allows the practitioner to choose k for any experiment according to the requirements of that particular experiment. Under this model, the cross-over designs are studied and the class of optimal designs is obtained. A method of constructing these optimal designs is also given.     

Alternative modeling techniques for the quantal response data in mixture experiments

Mixture experiments are commonly encountered in many fields including chemical, pharmaceutical and consumer product industries. Due to their wide applications, mixture experiments, a special study of response surface methodology, have been given greater attention in both model building and determination of designs compared with other experimental studies. In this paper, some new approaches are suggested on model building and selection for the analysis of the data in mixture experiments by using a special generalized linear models, logistic regression model, proposed by Chen et al. [7]. Generally, the special mixture models, which do not have a constant term, are highly affected by collinearity in modeling the mixture experiments. For this reason, in order to alleviate the undesired effects of collinearity in the analysis of mixture experiments with logistic regression, a new mixture model is defined with an alternative ratio variable. The deviance analysis table is given for standard mixture polynomial models defined by transformations and special mixture models used as linear predictors. The effects of components on the response in the restricted experimental region are given by using an alternative representation of Cox's direction approach. In addition, odds ratio and the confidence intervals of odds ratio are identified according to the chosen reference and control groups. To compare the suggested models, some model selection criteria, graphical odds ratio and the confidence intervals of the odds ratio are used. The advantage of the suggested approaches is illustrated on tumor incidence data set.     

Multiplicative-interaction logit models for i×j×2 three-way tables

Multiplicative-interaction (M-I) logit models are proposed for three-way IxJx2 contingency tables where the third variable constitutes a binary response. Models are derived by assigning unknown scores to the categories and forming product interactions from them. Asymptotic results under special sampling constraints are derived for maximum likelihood estimates and the goodness-of-fit statistics. The class of models proposed in this paper are found to be useful when no obvious scores are available. An example is included.     

A unified approach to rank tests for mixed models

The nonparametric version of the classical mixed model is considered and the common hypotheses of (parametric) main effects and interactions are reformulated in a nonparametric setup. To test these nonparametric hypotheses, the asymptotic distributions of quadratic forms of rank statistics are derived in a general framework which enables the derivation of the statistics for the nonparametric hypotheses of the fixed treatment effects and interactions in an arbitrary mixed model. The procedures given here are not restricted to semiparametric models or models with additive effects. Moreover, they are robust to outliers since only the ranks of the observations are needed. They are also applicable to pure ordinal data and since no continuity of the distribution functions is assumed, they can also be applied to data with ties. Some approximations for small sample sizes are suggested and analyzed in a simulation study. The application of the statistics and the interpretation of the results is demonstrated in several worked-out examples where some data sets given in the literature are re-analyzed.     

Design two-level factorial experiments in blocks of size four

Recently, many researchers have devoted themselves to the investigation on the number of replicates needed for experiments in blocks of size two. In practice, experiments in blocks of size four might be more useful than those in blocks of size two. To estimate the main effects and two-factor interactions from a two-level factorial experiment in blocks, we might need many replicates. This article investigates designs with the least number of replicates for factorial experiments in blocks of size four. The methods to obtain such designs are presented.