Unreplicated Factorial Experiments Research Articles

A more powerful test method for analyzing unreplicated factorial two-level experiments

Most of existing methods for analyzing unreplicated two-level factorial designs need the assumption of effect sparsity and only perform well when the effectssparsity assumption holds. The effects-sparsity assumption is frequently true, but it is not always true. This paper proposes a new method that needs not the effects-sparsity assumption and has the potential for detecting up to m-1 active effects, in which m is the number of effects under study. The proposed method is illustrated by two unreplicated factorial designs appeared in the literature. An extensive Monte Carlo comparison study, based on 100,000 replications, is carried out to compare the proposed method with another four popular and powerful existing objective methods. The simulation study shows that the proposed method has more power than the remainder four compared methods in most of the conditions under study.

Improved Approximation Scales for Unreplicated Factorial Experiments

Assessing the sizes of active contrasts in un-replicated factorial and fractional factorial experiments by quick and powerful methods are required in analyzing the big data in various research areas of Human endeavors. One of the old methods based on Lenth (1989) is being used in some statistical and data analytical applications which is fast and less efficient. We propose a new class of tests which are simpler, faster, and more powerful using the location median-function ({psi }_{med}left({varvec{x}}right)) after being skipped one and/or two times. An empirical study of simulation experiments to compute the critical values, sizes and powers using various sample sizes demonstrate the superiority of our methods. The proposed methods are illustrated in examples which can be employed in various fields of research in conducting data analytics using high computing power and machine learning.

A proposed method of identifying significant effects in unreplicated factorial experiments

In many areas of research/ production, a lot of factors are combined to obtain a desired product. To be able to analyze which factors (or combinations of factors and at what level) are significant, the experiment has to be replicated. For economic or practical reasons, it may not be feasible to perform the experiment more than once therefore unreplicated factorial designs are often employed. This is especially true in the field of Medicine, Pharmacy and Industrial production units. The traditional method of analysis of variance (ANOVA) cannot be employed in unreplicated factorial designs, therefore many methods have been proposed in literature. In this paper, a new method of analyzing unreplicated factorial designs is proposed and was compared with some of the existing methods. The four existing methods considered were: Lenth, Berk and Picard, Juan and Pena, and Dong. The comparison was performed using Monte Carlo simulation method. The criteria used in evaluating the performances of the methods are Power and Individual Error Rate (IER). Using these criteria of evaluation, the results showed that on overall performance, Dong method is the best among the four existing methods considered and was closely followed by Berk and Picard, Lenth, then Juan and Pena methods in that order. It was also found that not only is the proposed method simpler to compute, it competed favourably with Dong and even performed better than all the others when IER is used for assessment.

Unbiased condition of the dispersion effects estimator in unreplicated two-level factorial experiments

This article presents a general form of the estimator for identifying dispersion effects from unreplicated two-level factorial experiments, and shows that the widely used estimators such as the BH, MH, and AMH estimators are all special cases of the proposed one, designated as the G estimator. The unbiased condition of the G estimator is proved, and a lower bound of variance of the G estimator is provided. A simulation based on a realistic design illustrates the variation of the variance and MSE (mean square error) of the G estimator on different coefficients. This estimator may be more flexible and has better performance than other methods such as the BH and MH estimators by appropriately selecting the coefficients.

A Bayesian Perspective on the Analysis of Unreplicated Factorial Experiments Using Potential Outcomes

Unreplicated factorial designs have been widely used in scientific and industrial settings, when it is important to distinguish “active” or real factorial effects from “inactive” or noise factorial effects used to estimate residual or “error” terms. We propose a new approach to screen for active factorial effects from such experiments that uses the potential outcomes framework and is based on sequential posterior predictive model checks. One advantage of the proposed method is its ability to broaden the standard definition of active effects and to link their definition to the population of interest. Another important aspect of this approach is its conceptual connection to Fisherian randomization tests. Extensive simulation studies are conducted, which demonstrate the superiority of the proposed approach over existing ones in the situations considered.

Generalized maximum entropy approach to unreplicated factorial experiments

Generalized maximum entropy approach to unreplicated factorial experiments

Clustering Effects in Unreplicated Factorial Experiments

The analysis of unreplicated factorial designs concentrates much attention since there are no degrees of freedom left to estimate the error variance. In this article, we propose clustering the factorial estimates in two groups, one containing the active effects and one containing the inactive effects. The powerfulness of the proposed method is revealed via a comparative simulation study.

An Adaptation of the Bergman–Hynén Test for Dispersion Effects in Unreplicated Two-Level Factorial Designs When the Location Model May Be Incorrect

In the analysis of unreplicated two-level factorial experiments, it is well established that pairs of active location effects that are not identified as active can cause a disturbance in the data that interferes with standard tests and estimates for dispersion effects. An amendment to the dispersion-effect test of Bergman and Hynén (1997) is proposed to make it robust against confounding by location effects. The cost of this robustness is only one degree of freedom from the numerator and denominator of the reference F distribution. Simulations demonstrate a newfound problem with the Bergman–Hynén test and show that the proposed amendment fixes this problem.

THE ML ESTIMATOR OF DISPERSION EFFECTS IN UNREPLICATED FACTORIAL EXPERIMENTS

In unreplicated factorial experiments,if the dispersion effects of factors A,B are active,the existing estimators of dispersion effects are often biased whether or not the dispersion effect of interaction factor AB is active.This results in AB be spuriously identified active factor.In this paper,we propose a new estimator of dispersion effects(called the ML estimator),and give the exact expression of the variance of ML estimator.We prove that the ML estimator of dispersion effect of interaction factor AB is unbiased in a class of models. Finally,a comparison is given between the ML estimator and the existing and commonly used estimators.

Replicated versus un-replicated factorial experiments for preliminary investigation of seed germination and dormancy: alternative approaches using fewer seeds

A 2(3) full factorial, using a total of 3,200 seeds per species, investigated the effects of GA(3), smoke-water, heat-shock and/or light on the germination of Actinobole uliginostan (Asteraceae) and Goodenia fascicularis (Goodeniaceae), following dry after-ripening (DAR) at 15 or 40 degrees C for two or four weeks. The 2(3) factorial experiment was nested within DAR environments. The power analysis (the probability of detecting a true effect at a specified significance), was performed for each environment by using the error estimated from the fully replicated factorial experiments with five replicates. The power was estimated for each main effect and any interactions for two, three, four and five replicates as well as for single replicates. For A. uliginosum data, the power was sufficient with three replicates for main effects and interactions when compared to five replicates, and would have saved 1,280 seeds. However, both main effects and interactions for G. fascicularis, as determined from five-replicate data, were lost when three replicates were analysed. Analysis of single replicates (1/5 of the total number of seeds per species) gave consistent conclusions regarding GA(3) and light main effects upon A. uliginostan. The effect of light upon G. fascicularis was consistent; however the effect of GA(3) and smoke-water were lost. In conclusion, an un-replicated full factorial can give robust conclusions regarding strong main effects upon seed germination, particularly for experiments containing little variation in the percentage of germination within treatments, in order to reduce the number of seeds sacrificed in preliminary screening experiments.

Generalized linear models and transformations for unreplicated factorial experiments in the presence of dispersion effects

For the analysis of replicated designs, many different methods have been suggested. These allow for the estimation of functional dependencies between mean and variance as well as possible dispersion effects within the same model framework. However, in the situation of unreplicated designs, most methods known so far rely on the assumption of constant variances, or a functional relationship between mean and variance as the only source of heteroscedasticity. In this paper, we propose two methods for dealing with unreplicated data, when dispersion effects might also be of importance. One of these is an extension of the Box–Cox-method [Box, G.E.P., Cox, D.R., 1964. An analysis of transformations. Journal of the Royal Statistical Society B 26, 211–252], the other is based on double generalized linear models. Both these methods turn out to yield approximately equivalent results in the case of comparable assumptions, whereas the double generalized linear model is the more general one and allows further extensions. If this class of models is assumed, consistency, asymptotic efficiency and normality of the resulting estimates are shown.

On the equivalence of three estimators for dispersion effects in unreplicated two-level factorial designs

Box and Meyer [1986. Dispersion effects from fractional designs. Technometrics 28(1), 19–27] were the first to consider identifying both location and dispersion effects from unreplicated two-level fractional factorial designs. Since the publication of their paper a number of different procedures (both iterative and non-iterative) have been proposed for estimating the location and dispersion effects. An overview and a critical analysis of most of these procedures is given by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405]. Under a linear structure for the dispersion effects, non-iterative estimation methods for the dispersion effects were proposed by Brenneman and Nair [2001. Methods for identifying dispersion effects in unreplicated factorial experiments: a critical analysis and proposed strategies. Technometrics 43(4), 388–405], Liao and Iyer [2000. Optimal 2 n - p fractional factorial designs for dispersion effects under a location-dispersion model. Comm. Statist. Theory Methods 29(4), 823–835] and Wiklander [1998. A comparison of two estimators of dispersion effects. Comm. Statist. Theory Methods 27(4), 905–923] (see also Wiklander and Holm [2003. Dispersion effects in unreplicated factorial designs. Appl. Stochastic. Models Bus. Ind. 19(1), 13–30]). We prove that for two-level factorial designs the proposed estimators are different representations of a single estimator. The proof uses the framework of Seely [1970a. Linear spaces and unbiased estimation. Ann. Math. Statist. 41, 1725–1734], in which quadratic estimators are expressed as inner products of symmetric matrices.

A two-level unreplicated factorial experiment to determine the effect of organic and inorganic fertilizers on dry matter yield of permanent pasture

In organic pasture systems, the balance of soil fertility is maintained by maximizing the cycling of on-farm nutrients. In this study, the effect of fertility management on dry matter yield in organic pasture systems was examined using an unreplicated 25 factorial design. The factors were nitrogen (N), phosphorus (P), potassium (K), compost and seaweed extract, each at two levels (with [+] and without [−]). The trials were conducted in 2000 and 2001 at two fields on an organic dairy farm in Nova Scotia, Canada. The pasture mix is predominantly perennial ryegrass, orchardgrass and white clover. The fertility treatments were applied in spring, and forage yield was collected from early-, mid- and late-summer harvests. The analysis revealed the significance of several interaction effects, which varied considerably from harvest to harvest. Both +N and +Compost treatments were associated with increased yield. High levels of P and K already present in the soil either diminished or varied the effects of P and K treatments. Seaweed extract enhanced the effectiveness of N and compost, but reduced yield when used alone. The results suggest that the commonly recommended use of seaweed extract for organic pasture management can be detrimental to biomass production unless sufficient nitrogen is provided. Despite the unusually dry weather in the 2 years, the study demonstrated the applicability of unreplicated two-level factorial experiments in agricultural field studies.

On the Exact Null Distribution of the Generalised Likelihood Ratio Test for Analysing Unreplicated Factorial Designs

Al-Shiha and Yang (1999) proposed a multistage procedure for analysing unreplicated factorial experiments, which is based on the statistic that is derived from the generalised likelihood ratio test statistic under the assumption of normality. It was shown by their simulation study that the method is quite competitive with Lenth's (1989) method. In their paper, because of the difficulty of determining the null distribution analytically, the quantiles of the null distribution were empirically simulated. In this paper, we give the exact null distribution of their test statistic, which makes it possible to calculate the critical values of the test.

Two-level factorial designs for searching dispersion factors and estimating location main effects

In factorial experiments, changing the level of a factor might, in certain applications, change the variance of the response variable. Such factors are called dispersion factors in this article. There have been several publications on the analysis methodology for identifying dispersion factors in unreplicated factorial experiments. However, the design optimality issues for the search and identification of dispersion factors seem to be relatively unexplored. In this study, we focus on the two-level factorial experiment in which there is at most one dispersion factor whose identity is not known a priori. The task is to estimate all location main effects and simultaneously identify the dispersion factor, if any. We propose a likelihood-based criterion for selecting the design from a class of competing designs using which one can estimate all location main effects and also correctly identify, with as high a probability as possible, the identity of the dispersion factor (if one exists). It is shown that, if attention is restricted to the class of regular 2 n - p fractional factorial designs, then a large run size design of high resolution may be required to achieve our goal. As an alternative, we consider the class of simple arrays due to its appealing symmetry property and flexibility in run size. Additionally, the trade-off of efficiency between identification of the dispersion factor and estimation of the location main effects is under investigation. The situation involving multiple dispersion factors is also discussed.

The Analysis of Unreplicated Factorial Experiments Using All Possible Comparisons

This article proposes a new procedure for analyzing small unreplicated factorial experiments. This procedure is based on using likelihood ratio tests to compare competing models. An easy method of implementing the procedure is presented and then demonstrated on a real set of data. Results of a simulation study are presented that indicate that the new procedure compares favorably with Lenth's method. Tables of constants are supplied that allow the new procedure to be easily applied to the analysis of 8-run, 12-run, and 16-run experiments.

Identificação de efeitos de dispersão em experimentos fatoriais 2K e 2K-p

Este artigo apresenta uma estratégia para identificar efeitos de locação e efeitos de dispersão em experimentos fatoriais não-replicados, fracionados ou não. A estratégia é adaptada para projetos experimentais com poucas replicações, sendo ilustrada em um exemplo prático. Além da estratégia proposta, outros métodos típicos de experimentos com replicações foram aplicados nesse exemplo e os resultados encontrados foram bastante parecidos.

The analysis of unreplicated factorial experiments from a geometric perspective

AbstractThe author investigates the analysis of unreplicated factorial experiments from a geometric perspective. He considers more specifically a (k + 1)‐run experiment used to estimate k orthogonal contrasts. He observes that once centered and scaled to unit length, the response vector can be viewed as a point on the unit sphere in the vector space spanned by the contrasts. In this context, a model selection procedure is equivalent to a partition of the unit sphere into regions corresponding to the different models considered. The author exploits this approach to gain useful insights into the analysis of such experiments.

Methods for Identifying Dispersion Effects in Unreplicated Factorial Experiments

There has been considerable interest recently in the use of statistically designed experiments to identify both location and dispersion effects for quality improvement. Analysis of dispersion effects usually requires replications that can be expensive or time consuming. Several recent articles have considered identification of both location and dispersion effects from unreplicated fractional factorial experiments. In this article, we provide a systematic study of various methods that are commonly used or have been proposed recently. Both theoretical and simulation results are used to characterize the properties. Although all methods suffer from some degree of bias, some have serious problems when the bias remains large even as the design run size increases to infinity. Based on these analyses, we propose some iterative strategies for model selection and estimation of the dispersion effects. A real example and simulations as well are used to illustrate the results.

Identifying Significant Effects in Fractional Factorial Multiresponse Experiments

A forward selection procedure for identifying active contrasts in unreplicated factorial multiresponse experiments is presented. To test the multivariate effects, p values are calculated by Monte Carlo sampling; the ordinary Hotelling's T2 test is a special case. Collinear responses are effectively handled by successive principal-component decompositions, as when, for instance, the number of response variables exceeds the number of observations. In the univariate case, the test statistic pools the smallest effects as error. Independent sources for error degrees of freedom can be incorporated. The method is illustrated with examples from baguette and mayonnaise production.