On the construction of trend free row-column 2-level factorial experiments

In this paper we consider experimental situations where a complete or fractional factorial experiment having all factors at 2 levels is to be conducted using a 2 ^m × 2 ⁿ row-column design and where there may be an unknown trend effect that can be expressed as a polynomial function of the position in which observations are obtained in the row-column design. Methods are given for allocating the treatments from a complete or fractional 2-level factorial experiment to rows and columns so that the resulting design yields estimates for main effects that have a high level of resistance against trend effects. Research supported by NSF Grant No. DMS-8700945.     

Small asymmetric fractional factorial plans for main effects and specified two-factor interactions

Fractional factorial plans represented by orthogonal arrays of strength two are known to be optimal in a very strong sense under a model that includes the mean and all the main effects, when all interactions are assumed to be absent. When a fractional factorial plan given by an orthogonal array of strength two is not saturated, one might think of entertaining some two-factor interactions also in the model. In such a situation, it is of interest to examine which of the two-factor interactions can be estimated via a plan represented by an orthogonal array, as also to study the overall efficiency of the plan when some interactions are in the model alongwith the mean and all main effects. In this paper, an attempt has been made to examine these issues by considering some practically useful plans for asymmetric (mixed level) factorials with small number of runs.     

An information theoretical algorithm for analyzing supersaturated designs for a binary response

A supersaturated design is a factorial design in which the number of effects to be estimated is greater than the number of runs. It is used in many experiments, for screening purpose, i.e., for studying a large number of factors and identifying the active ones. In this paper, we propose a method for screening out the important factors from a large set of potentially active variables through the symmetrical uncertainty measure combined with the information gain measure. We develop an information theoretical analysis method by using Shannon and some other entropy measures such as Rényi entropy, Havrda–Charvát entropy, and Tsallis entropy, on data and assuming generalized linear models for a Bernoulli response. This method is quite advantageous as it enables us to use supersaturated designs for analyzing data on generalized linear models. Empirical study demonstrates that this method performs well giving low Type I and Type II error rates for any entropy measure we use. Moreover, the proposed method is more efficient when compared to the existing ROC methodology of identifying the significant factors for a dichotomous response in terms of error rates.     

Asymmetrical Factorial Designs Containing Clear Effects

The asymmetrical or mixed-level factorial design is a kind of important design in practice. There is a natural problem on how to choose an optimal (s ²)s ⁿ design for the practical need, where s is any prime or prime power. This paper considers the clear effects criterion for selecting good designs. It answers the questions of when an (s ²)s ⁿ design with fixed number of runs contains clear two-factor interaction (in brief 2fi) components and when it contains clear main effects or clear 2fis. It further gives the complete classification of (s ²)s ⁿ designs according to the clear 2fi components, main effects and 2fis they have.     

On a property of orthogonal arrays and optimal blocking of fractional factorial plans

It is shown that fractional factorial plans represented by orthogonal arrays of strength three are universally optimal under a model that includes the mean, all main effects and all two-factor interactions between a specified factor and each of the other factors. Thus, such plans exhibit a kind of model robustness in being universally optimal under two different models. Procedures for obtaining universally optimal block designs for fractional factorial plans represented by orthogonal arrays are also discussed. Acknowledgements. The authors wish to thank the referees for making several useful comments on a previous version.     

On a heuristic analysis of highly fractionated 2 n factorial experiments

The paper deals with a method for the analysis of highly fractionated factorial designs proposed by Raghavarao and Altan in Metrika 58:185–191 (2003). We show that the method will find “active” factors with almost any set of random numbers. Once that an alias set is found active, Raghavarao and Altan claim that their method can resolve the alias structure of the design and identify which of several confounded effects is active. We show that their method cannot do that. The error in Raghavarao and Altan’s arguments lies in the fact that they treat a set of highly dependent (sometimes even identical) F-statistics as if they were independent.     

Mixed-level fractional factorial split-plot designs containing clear effects

Mixed-level designs are widely used in the practical experiments. When the levels of some factors are difficult to be changed or controlled, fractional factorial split-plot (FFSP) designs are often used. This paper investigates the sufficient and necessary conditions for a ${2^{(n_{1}+n_{2})-(k_1+k_2)}4_s^{1}}$ FFSP design with resolution III or IV to have various clear factorial effects, including two types of main effects and three types of two-factor interaction components. The structures of such designs are shown and illustrated with examples.     

On the construction of <Emphasis Type="Italic">E</Emphasis>(<Emphasis Type="Italic">s</Emphasis><Superscript>2</Superscript>)-optimal supersaturated designs

Supersaturated designs are an important class of factorial designs in which the number of factors is larger than the number of runs. These designs supply an economical method to perform and analyze industrial experiments. In this paper, we consider generalized Legendre pairs and their corresponding matrices to construct E(s ²)-optimal two-level supersaturated designs suitable for screening experiments. Also, we provide some general theorems which supply several infinite families of E(s ²)-optimal two-level supersaturated designs of various sizes.      

The analysis of ranked data in blocked factorial experiments

A non-parametric method for the analysis of blocked factorial experiments, based on ranking within blocks, is proposed and shown to be equivalent to partitioning Friedman's test statistic into a set of contrasts reflecting polynomial components of the main effects and interaction. A slightly modified version of the procedure is suggested to partially overcome the problem of loss of power to detect one component when the model includes other components. This alternative procedure is shown to be equivalent to applying a standard normal theory analysis of variance to the ranks. The null distributions and power comparisons are investigated using simulation methods, and it is shown that the non-parametric methods are almost as powerful as the analysis of variance. Received: February 1999     

A note on the projection properties of Hadamard matrices obtained by the first Payley construction

We show that the projections on four factors of an arbitrary orthogonal array of strength 2 allow the estimation of main effects and two-factor interactions when all other effects are assumed to be zero, if those projections satisfy the bounds given by Weils theorem. The only exceptions are the Hadamard matrices of orders 16 and 24. A consequence is again the estimability of main effects and two-factor interactions for the projections on four factors of the first Payley construction for arbitrary run size.     

Influence of equal or unequal comparison group sample sizes on the detection of differential item functioning using the Mantel–Haenszel and logistic regression techniques

In recent decades several methods have been developed for detecting differential item functioning (DIF), and many studies have aimed to identify both the conditions under which items may or may not be adequate and the factors which affect their power and Type I error. This paper describes a Monte Carlo experiment that was carried out in order to analyse the effect of reference group sample size, focal group sample size and the interaction of the two on the power and Type I error of the Mantel–Haenszel (MH) and Logistic regression (LR) procedures. The data were generated using a three-parameter logistic model, the design was fully-crossed factorial with 12 experimental conditions arising from the crossing of the two main factors, and the dependent variables were power and the rate of false positives calculated across 100 replications. The results enabled the significant factors to be identified and the two statistics to be compared. Practical recommendations are made regarding use of the procedures by psychologists interested in the development and analysis of psychological tests.     

Orthogonal main effect plans for two and three factors in small blocks

Orthogonal main-effect plans for two and three factors in small blocks are obtained from the dual of adjusted orthogonal row-column designs. The method for constructing efficient plans is presented, and a relationship between the average efficiency factors of the row-column design and the corresponding main effects is given for the two-factor case. Orthogonal Main Effect.     

层次分析在业主预测风险中的应用

在工程建设中,业主面临的索赔主要来自承包商,造成索赔的原因有很多种。本文分析了各种可能导致索赔发生的风险因素并应用层次分析法加以量化,然后根据量化的结果对各因素进行风险大小排序;根据排序结果找出主要的风险因素,进而达到对索赔风险进行预测和力求加大主动控制的目的。     

Approximations for densities of sufficient estimators

A simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described. It is an extension of the one developed by O. Barndorff-Nielsen and D.R. Cox (1979) for exponential families. A series expansion in powers of n^?1 is derived of which the first term has an error of order n^?1 which can effectively be reduced to $\text{n}{}^{\mathrm{<mtext>-</mtext><mtext>?3</mtext><mtext>2</mtext>}}$ by renormalization. The results obtained are similar to those given by H.E. Daniels's (1954) saddlepoint method but the derivations are simpler. A brief treatment of approximations to conditional densities is given. Theorems are proved which extend the validity of the multivariate Edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors. These extensions permit the treatment of problems arising in time series analysis. The technique is used by J. Durbin (1980) to obtain approximations to the densities of partial serial correlation coefficients.     

Weighted rank statistics in factorial designs with fixed effects

Nonparametric methods for the analysis of factorial designs with two fixed factors are considered. The sample sizes may be unequal and the distribution functions are not assumed to be continuous. Nonparametric hypotheses for the main effects and for the interaction are tested by ranking procedures where the statistics are weighted according to the different sample sizes within the levels of one factor. Simulations show that the approximations by the limiting normal distribution and by the t - and F -distributions are quite accurate if the samples sizes within the cells are at least 7 for all treatment combinations. Moreover, it turns out that the power for the weighted statistics is much higher than for the unweighted statistics. The application of the suggested procedures is demonstrated by the analysis of a data set from a clinical trial with ordered categorical data.     

Designs for smoothing spline ANOVA models

Smoothing spline estimation of a function of several variables based on an analysis of variance decomposition (SS-ANOVA) is one modern nonparametric technique. This paper considers the design problem for specific types of SS-ANOVA models. As criteria for choosing the design points, the integrated mean squared error (IMSE) for the SS-ANOVA estimate and its asymptotic approximation are derived based on the correspondence between the SS-ANOVA model and the random effects model with a partially improper prior. Three examples for additive and interaction spline models are provided for illustration. A comparison of the asymptotic designs, the 2^d factorial designs, and the glp designs is given by numerical computation. Received May 2000     

U-type and column-orthogonal designs for computer experiments

U-type designs and orthogonal Latin hypercube designs (OLHDs) have been used extensively for performing computer experiments. Both have good spaced filling properties in one-dimension. U-type designs may not have low correlations among the main effects, quadratic effects and two-factor interactions. On the other hand, OLHDs are hard to be found due to their large number of levels for each factor. Recently, alternative classes of U-type designs with zero or low correlations among the effects of interest appear in the literature. In this paper, we present new classes of U-type or quantitative \(3\) -orthogonal designs for computer experiments. The proposed designs are constructed by combining known combinatorial structures and they have their main effects pairwise orthogonal, orthogonal to the mean effect, and orthogonal to both quadratic effects and two-factor interactions.     

One-stage estimation of the effects of operational conditions and practices on productive performance: asymptotically normal and efficient,root-<Emphasis Type="Italic">n</Emphasis> consistent StoNEZD method

Understanding the effects of operational conditions and practices on productive efficiency can provide valuable economic and managerial insights. The conventional approach is to use a two-stage method where the efficiency estimates are regressed on contextual variables representing the operational conditions. The main problem of the two-stage approach is that it ignores the correlations between inputs and contextual variables. To address this shortcoming, we build on the recently developed regression interpretation of data envelopment analysis (DEA) to develop a new one-stage semi-nonparametric estimator that combines the nonparametric DEA-style frontier with a regression model of the contextual variables. The new method is referred to as stochastic semi-nonparametric envelopment of z variables data (StoNEZD). The StoNEZD estimator for the contextual variables is shown to be statistically consistent under less restrictive assumptions than those required by the two-stage DEA estimator. Further, the StoNEZD estimator is shown to be unbiased, asymptotically efficient, asymptotically normally distributed, and converge at the standard parametric rate of order n ^−1/2. Therefore, the conventional methods of statistical testing and confidence intervals apply for asymptotic inference. Finite sample performance of the proposed estimators is examined through Monte Carlo simulations.     

On the Comparison of Run Orders of Unreplicated 2<Superscript><Emphasis Type="Italic">k</Emphasis>–<Emphasis Type="Italic">p</Emphasis></Superscript> Designs in the Presence of a Time Trend

The response from a factorial experiment carried out in a time sequence may be affected by uncontrollable variables that are highly correlated with the time in which they occur. In such a situation, one possibility is to randomize the run order of the experiment. Another possibility is to use a systematic run order that is robust against time trends. Since randomized run orders make the time trend part of the error, it can be hoped that systematic run orders will be more effective to identify truly active factors. In this paper, a simulation study is used to compare the performances of the randomized and the systematic run orders. The response from an experiment where we have observed a strong time trend is used to demonstrate the influence of a realistic time trend on the run orders under consideration. The performance of the run orders is then measured by taking the probabilities of false rejection and the probabilities of detection of active contrasts. Our results show that the randomized run order managed to keep the nominal level, while the systematic did not. Additionally, when there were active factors, then the systematic run orders did not achieve more power than did the randomized run order.