post fisherian experimentation from physical to virtual fisher s pioneering work in design of experiments has inspired further work with broader applications especially in industrial experimentation this article discusses three topics in physical experiments principles of effect hierarchy sparsity and heredity for factorial designs a new method called conditional main effect cme for de aliasing aliased effects and robust parameter design i also review the recent emergence of virtual experiments on a computer some major challenges in computer experiments which must go beyond fisherian principles are outlined

two level multifactor designs for detecting the presence of interactions a design optimality criterion tr l optimality is applied to the problem of designing two level multifactor experiments to detect the presence of interactions among the controlled variables we give rules for constructing tr l optimal foldover designs and tr l optimal fractional factorial designs some results are given on the power of these designs for testing the hypothesis that there are no two factor interactions augmentation of the tr l optimal designs produces designs that achieve a compromise between the criteria of d optimality for parameter estimation in a first order model and tr l optimality for detecting lack of fit we give an example to demonstrate an application to the sensitivity analysis of a computer model

constructing tables of minimum aberration pn m designs fries and hunter 1980 presented a practical algorithm for selecting standard 2 n m fractional factorial designs based on a criterion they called minimum aberration in this article some simple results are presented that enable the fries hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p 2 is prime examples of minimum aberration 2 n m designs with resolution r 4 are given for m n m 9 a matrix is given for generating 3 n m designs with m n m 6 which have or nearly have minimum aberration

algorithmic construction of efficient fractional factorial designs with large run sizes fractional factorial ff designs are widely used in practice and typically are chosen according to the minimum aberration criterion a sequential algorithm is developed for constructing efficient ff designs a construction procedure is proposed that allows a design to be constructed only from its minimum aberration projection in the sequential buildup process to efficiently identify nonisomorphic designs designs are categorized according to moment projection pattern a fast isomorphism checking procedure is developed by matching the factors using their delete one factor projections this algorithm is used to completely enumerate all 128 run designs of resolution 4 all 256 run designs of resolution 4 up to 17 factors all 512 run designs of resolution 5 all 1024 run designs of resolution 6 and all 2048 and 4 096 run designs of resolution 7 a method is proposed for constructing minimum aberration ma designs using only a partial catalog of some good designs three approaches to constructing good de

regular fractional factorial designs with minimum aberration and maximum estimation capacity using the approach of finite projective geometry we make a systematic study of estimation capacity a criterion of model robustness under the absence of interactions involving three or more factors some general results providing designs with maximum estimation capacity are obtained in particular for two level factorials it is seen that constructing a design with maximum estimation capacity calls for choosing points from a finite projective geometry such that the number of lines is maximized and the distribution of these lines among the chosen points is as uniform as possible we also explore the connection with minimum aberration designs under which the sizes of the alias sets of two factor interactions which are not aliased with main effects are the most uniform possible

quarter fraction factorial designs constructed via quaternary codes submitted quarter fraction factorial designs constructed via quaternary codes by frederick k h phoa and hongquan xu university of california los angeles the research of developing a general methodology for the con struction of good nonregular designs has been very active in the last decade recent research by xu and wong 2007 suggested a new class of nonregular designs constructed from quaternary codes this paper explores the properties and uses of quaternary codes towards the construction of quarter fraction nonregular designs some theo retical results are obtained regarding the aliasing structure of such designs optimal designs are constructed under the maximum resolu tion minimum aberration and maximum projectivity criteria these designs often have larger generalized resolution and larger projectiv ity than regular designs of the same size it is further shown that some of these designs have generalized minimum aberration and maximum projectivity among all possible designs 1 introduction in many scienti c researches and investigations the interests lie in the study of e ects of many factors simultaneously frac tional factorial designs especially two level fractional factorial designs are the most commonly used experimental plans for this type of investigations designs that can be constructed through de ning relations among factors are called regular designs any two factorial e ects in a regular design are supported in part by nsf grant dms 05 05728 ams 2000 subject classi cations primary 62k15 keywords and phrases aliasing index fractional factorial design generalized minimum aberration generalized resolution nonregular design projectivity

blocking schemes for definitive screening designs in earlier work jones and nachtsheim proposed a new class of screening designs called definitive screening designs as originally presented these designs are three level designs for quantitative factors that provide estimates of main effects that are unbiased by any second order effect and require only one more than twice as many runs as there are factors definitive screening designs avoid direct confounding of any pair of second order effects and for designs that have more than five factors project to efficient response surface designs for any two or three factors recently jones and nachtsheim expanded the applicability of these designs by showing how to include any number of two level categorical factors however methods for blocking definitive screening designs have not been addressed in this article we develop orthogonal blocking schemes for definitive screening designs we separately consider the cases where all of the factors are quantitative and where there is a mix of quantitative and tw

volterra system identification and kramer s sampling theorem nonlinear systems increase signal bandwidth but volterra systems may be identified uniquely even when the output is sampled at the nyquist rate of the input we present a more transparent and general proof using the theory of sampling expansions for general integral transforms we permit the sampling to be nonuniform

some projection properties of orthogonal arrays the definition of an orthogonal array imposes an important geometric property the projection of an oa 2 t 2 k t a 2 t run orthogonal array with k two level factors and strength t onto any t factors consists of copies of the complete 2 t factorial in this article projections of an oa n 2 k t onto t 1 and t 2 factors are considered the projection onto any t 1 factors must be one of three types one or more copies of the complete 2 t 1 factorial one or more copies of a half replicate of 2 t 1 or a combination of both it is also shown that for k t 2 only when n is a multiple of 2 t 1 can the projection onto some t 1 factors be copies of a half replicate of 2 t 1 therefore if n is not a multiple of 2 t 1 then the projection of an oa n 2 k t with k t 2 onto any t 1 factors must contain at least one complete 2 t 1 factorial some properties of projections onto t 2 factors are established and are applied to show that if n is not a multiple of 8 then for any oa n 2 k 2 with k 4 the projection onto any four factors has the property that all the main effects and two factor interactions of these four factors are estimable when the higher order interactions are negligible

sequentially refined latin hypercube designs reusing every point the use of iteratively enlarged latin hypercube designs for running computer experiments has recently gained popularity in practice this approach conducts an initial experiment with a computer code using a latin hypercube design and then runs a follow up experiment with additional runs elaborately chosen so that the combined design set for the two experiments forms a larger latin hypercube design this augmenting process can be repeated multiple stages where in each stage the augmented design set is guaranteed to be a latin hypercube design we provide a theoretical framework to put this approach on a firm footing numerical examples are given to corroborate the derived theoretical results supplementary materials for this article are available online

multi layer designs for computer experiments space filling designs such as latin hypercube designs lhds are widely used in computer experiments however finding an optimal lhd with good space filling properties is computationally cumbersome on the other hand the well established factorial designs in physical experiments are unsuitable for computer experiments owing to the redundancy of design points when projected onto a subset of factor space in this work we present a new class of space filling designs developed by splitting two level factorial designs into multiple layers the method takes advantage of many available results in factorial design theory and therefore the proposed multi layer designs mlds are easy to generate moreover our numerical study shows that mlds can have better space filling properties than optimal lhds

the efficiencies of fractional factorial designs a frequently stated advantage of fractional factorial ff designs over one factor at a time 1fat designs is their high relative efficiency we study k factor 2k run designs where k is a power of 2 or is divisible by 4 for which the usually stated relative efficiency is k in favor of the resolution iv ff design over an orthogonal 1fat design we examine other ways to measure efficiency under main effects only modeling in the case when all factors are quantitative which enables rescaling of their settings if both designs are restricted to lie in smallest sized hypercubes or hyperspheres of equal volume then the relative efficiency of the ff design is 1 under the full model k effects active we also show that the designs are identical in an important sense the 1fat design can be rescaled and rotated to become a resolution iv ff design the primary advantage of the ff design lies in its projection properties if at most k factors are expected to be active then under equal volume designs the ef

the construction of trend free run orders of two level factorial designs abstract in experimental situations where a factorial design with all factors occurring at two levels is to be run in a time sequence the usual advice given to the experimenter is that the order of runs should be randomized before the experiment is performed however randomization may lead to an undesirable run order for example in a factory experiment there may be a certain learning process that occurs over time as a result of making level changes in the factors being studied or there may be equipment wear out in either case the observations obtained will be affected by uncontrollable variables that are highly correlated with the time or position in which they occur and randomization may lead to a run order in which the estimates of factor effects are adversely effected by the presence of the trend joiner and campbell 1976 gave some more specific examples as to when trend effects can occur in sequential experiments therefore it is important to consider systematic designs in which the estimat

some theory for constructing minimum aberration fractional factorial designs summary minimum aberration is the most established criterion for selecting a regular fractional factorial design of maximum resolution minimum aberration designs for n runs and n 2 m n factors have previously been constructed using the novel idea of complementary designs in this paper an alternative method of construction is developed by relating the wordlength pattern of designs to the so called confounding between experimental runs this allows minimum aberration designs to be constructed for n runs and 5n 16 m n 2 factors as well as for n 2 m n

time trends and factorial experiments when factorial experiments are carried out over a period of time the response may be subject to time trends this might happen for example with the steady buildup of deposits in a test engine this article shows how to arrange the runs in a factorial experiment in sequences so that the main effects and sometimes the two factor interactions are uncorrelated with linear or quadratic time trends this is achieved by using the principle of foldover designs trendfree sequences are obtained for both 2 n and 3 n factorials

using the folded over 12 run plackett burman design to consider interactions this article demonstrates that the folded over 12 run plackett burman design is useful for considering up to 12 factors in 24 runs even if one anticipates that some two factor interactions may be significant the properties of this design are investigated and a sequential procedure for analyzing the data from such a design is proposed the performance of the procedure is investigated through the analysis of real and constructed examples and through a small simulation study applications to other folded over plackett burman designs are also briefly discussed

small composite designs small second order composite designs were suggested by hartley 1959 westlake 1965 provided even smaller designs for k 5 7 and 9 factors for which intricate construction methods were needed here simple designs formed using plackett and burman 1946 designs are offered for k 5 7 and 9 designs with one run fewer than westlake s for k 5 and 7 and three fewer for k 9 are feasible by deleting repeat points that occur in some of the designs

trend robust two level factorial designs summary limiting the possible influence of time order effects may be an important consideration in designing a sequential experiment we consider the problem of finding trend robust run orders of two level factorials when the time effects are modelled via a first order autoregressive error model or via a time series model in both cases run orders are trend robust if the factors change level many times other run orders can be quite inefficient even if they are free of low degree polynomial trends we present a simple algorithm for constructing run orders of two level factorial designs with many level changes

on the baseband representation of a bandpass nonlinearity modeling analysis and compensation of nonlinearities in the transmitter the power amplifier in particular has attracted a lot of attention recently given the same bandpass polynomial or volterra representation of the nonlinear system two different baseband formulations have appeared in the literature the purpose of this correspondence is to examine the discrepancy between the two and to affirm that proper conjugation must be applied in the baseband representation of bandpass nonlinearities a predistortion linearization test bed experiment is conducted to illustrate the concepts

predicting spectral regrowth of nonlinear power amplifiers power amplifiers pas are important elements in communications systems and they are inherently nonlinear nonlinearity generates spectral regrowth i e spectral broadening in digitally modulated signals which causes adjacent channel interference in this paper we present a closed form expression for the auto covariance function of the pa output whose fourier transform yields the output power spectral density psd the pa input does not need to be gaussian and the psd calculation can be carried out using common pa descriptions such as the am am and am pm characteristics we assume that the input is narrow band thus pa memory effects on the order of a symbol period are not present the analytical results allow us to predict spectral regrowth without running expensive or time consuming time domain simulations they also lead to a variety of optimization possibilities in transmitter design that includes the pas

construction of two level split lot fractional factorial designs for multistage processes split lot designs also known as multiway split unit designs are useful in factorial experiments where there are multiple processing stages such experiments occur for example in the fabrication of integrated circuits in the semiconductor industry and in product assembly the designs have a split plot structure at each stage so that settings of the factors are applied to sublots of experimental runs rather than to individual runs this allows experimental runs to be processed at much less expense than if the runs were completely randomized in this article split lot designs are constructed for two three and four processing stages the designs have minimum aberration under the split lot structure and minimize in various senses the confounding of main effects and two factor interactions with the sublots at each stage

one hundred years of the design of experiments on and off the pages of biometrika the earliest important papers on planned experiments in biometrika appeared within a year of each other in 1917 and 1918 roughly speaking one was concerned with design optimality and industrial experiments the other with agricultural trials and blocking we find this approximate division of the subject into two parts helpful in describing its development and growth as a result of the hostility between fisher and karl pearson much of the development of designs for agriculture in the 1920s and 1930s occurred off the pages of biometrika despite this we are able to trace a coherent history of the development of the subject from field trials and response surface methods to clinical trials and bayesian versions of design optimality

nonregular designs with desirable projection properties in many industrial applications the experimenter is interested in estimating some of the main effects and two factor interactions in this article we rank two level orthogonal designs based on the number of estimable models containing a subset of main effects and their associated two factor interactions by ranking designs in this way the experimenter can directly assess the usefulness of the experimental plan for the purpose in mind we apply the new ranking criterion to the class of all non isomorphic two level orthogonal designs with 16 and 20 runs and introduce a computationally efficient algorithm based on two theoretical results which will aid in finding designs with larger run sizes catalogs of useful designs with 16 20 24 and 28 runs are presented

minimum aberration and model robustness for two level fractional factorial designs summary the performance of minimum aberration two level fractional factorial designs is studied under two criteria of model robustness simple sufficient conditions for a design to dominate another design with respect to each of these two criteria are derived it is also shown that a minimum aberration design of resolution ilil or higher maximizes the number of two factor interactions which are not aliases of main effects and subject to that condition minimizes the sum of squares of the sizes of alias sets of two factor interactions this roughly says that minimum aberration designs tend to make the sizes of the alias sets very uniform it follows that minimum aberration is a good surrogate for the two criteria of model robustness that are studied here examples are given to show that minimum aberration designs are indeed highly efficient

a universal formula for the complete bandpass sampling requirements of nonlinear systems a nonlinear system driven by a bandpass input signal may produce an output signal that occupies multiple frequency bands this makes bandpass sampling the output signal of the nonlinear system without causing aliasing a challenging task although the bandpass sampling theory for linear systems is well developed its counterpart for nonlinear systems is relatively immature in the sense that complete bandpass sampling requirements have only been developed up to third order nonlinear systems in this paper a novel method is used to derived the complete bandpass sampling requirements for nonlinear systems of an arbitrary order the strategy used by this method is to build the constraints on the sampling frequency for n th order nonlinear systems upon those for n 1 th order nonlinear systems this process makes the derivation easily extendable to nonlinear systems of any order the derivation method also gives an insight into the mechanism for formularizing the constraints on the bandpass sampling frequency as a result a universal formula for the complete bandpass sampling requirements for nonlinear systems of any order is derived this formula consolidates the bandpass sampling theory for nonlinear systems and facilitates the selection of the bandpass sampling frequency for nonlinear systems driven by bandpass input signals

bandpass sampling criteria for nonlinear systems sampling criteria for nonlinear systems with a band pass input are developed in this paper it is well known that nonlinear systems may produce an output signal with a larger bandwidth than that of their input signal according to the nyquist sampling theorem the sampling rate needs to be at least twice the maximum frequency of the output signal otherwise the sampled output would be aliased however if the input is a bandpass signal the spectrum of the output signal often occupies multiple frequency bands in this case it is possible by using the bandpass sampling concept to sample the output signal at a rate much lower than the nyquist sampling frequency in this paper all conditions in which bandpass sampling can be achieved are derived for nonlinear systems up to the third order furthermore for nonlinear systems higher than the third order some conditions in which bandpass sampling can be guaranteed are derived the result can be used to choose an appropriate sampling frequency for nonlinear systems of an arbitrary order

the potential for parallelism in runge kutta methods part 1 rk formulas in standard form the authors examine the potential for parallelism in runge kutta rk methods based on formulas in standard one step form both negative and positive results are presented many of the negative results are based on a theorem that bounds the order of an rk formula in terms of the minimal polynomial associated with its coefficient matrix the positive results are largely examples of prototypical formulas that offer a potential for effective coarse grain parallelism on machines with a few processors

center points in second order response surface designs previous approaches for selecting the number of center points in a second order response surface design are explained and an integrated variance criterion is offered calculations extending results in the literature are made for various composite and box behnken designs for 2 k 8 the results are compared and discussed and indicate that fewer center points than initially recommended are generally appropriate

a general criterion for factorial designs under model uncertainty motivated by two industrial experiments in which rather extreme prior knowledge was used to choose the design we show that the qb criterion which aims to improve the estimation in as many models as possible by incorporating experimenters prior knowledge along with an approximation to the as criterion is more general and has a better statistical interpretation than many standard criteria the generalization and application of the criterion to different types of designs are presented the relationships between qb and other criteria for different situations are explored it is shown that the e s2 criterion is a special case of qb and several aberration type criteria are limiting cases of our criterion so that qb provides a bridge between alphabetic optimality and aberration the two case studies illustrate the potential benefits of the qb criterion r programs for calculating qb are available online as supplemental materials

blocked regular fractional factorial designs with maximum estimation capacity in this paper the problem of constructing optimal blocked regular fractional factorial designs is considered the concept of minimum aberration due to fries and hunter is a well accepted criterion for selecting good unblocked fractional factorial designs cheng steinberg and sun showed that a minimum aberration design of resolution three or higher maximizes the number of two factor interactions which are not aliases of main effects and also tends to distribute these interactions over the alias sets very uniformly we extend this to construct block designs in which i no main effect is aliased with any other main effect not confounded with blocks ii the number of two factor interactions that are neither aliased with main effects nor confounded with blocks is as large as possible and iii these interactions are distributed over the alias sets as uniformly as possible such designs perform well under the criterion of maximum estimation capacity a criterion of model robustness which has a direct statistical meaning some general results on the construction of blocked regular fractional factorial designs with maximum estimation capacity are obtained by using a finite projective geometric approach

measures of lack of fit for response surface designs and predictor variable transformations some first order 2 k p two level factorials and fractional factorials plus center points and second order cube plus star plus center point composite response surface designs are discussed from the point of view of their ability to detect certain likely kinds of lack of fit of degree one higher than has been fitted this leads to consideration of conditions for representationa adequacy of first and second order models in transformed predictor variables it is shown how to use the estimated regression coefficients from the higher degree model to check if power transformations of the predictor variables could eliminate the lack of fit and also actually to estimate the transformations

trend free block designs for varietal and factorial experiments some general results on the existence of trend free and partially trend free designs are given for both varietal and factorial experiments in particular trend free properties of cyclic and generalized cyclic designs are investigated it is shown that for factorial experiments certain designs which are not completely trend free are nevertheless trend free for estimating a subset of the main effect and interaction contrasts

a trigonometric approach to quaternary code designs with application to one eighth and one sixteenth fractions the study of good nonregular fractional factorial designs has received significant attention over the last two decades recent research indicates that designs constructed from quaternary codes qc are very promising in this regard the present paper shows how a trigonometric approach can facilitate a systematic understanding of such qc designs and lead to new theoretical results covering hitherto unexplored situations we focus attention on one eighth and one sixteenth fractions of two level factorials and show that optimal qc designs often have larger generalized resolution and projectivity than comparable regular designs moreover some of these designs are found to have maximum projectivity among all designs

fractional factorial designs with admissible sets of clear two factor interactions we consider the problem of selecting two level fractional factorial designs that allow joint estimation of all main effects and some specified two factor interactions 2fi s without aliasing from other 2fi s this problem is to find among all 2 m p designs with given m and p those resolution iv designs whose sets of clear 2fi s contain the specified 2fi s as subsets we use a linear graph to represent the set of clear 2fi s for a resolution iv design where each line connecting two vertexes represents a clear 2fi between the factors represented by the two vertexes we call a 2 m p resolution iv design admissible if its graph is not isomorphic to any proper subgraph of the graph of any other 2 m p resolution iv design we show that all even resolution iv designs are inadmissible we then use a classical subgraph isomorphism algorithm to determine all admissible designs of 32 64 and 128 runs this leads to a concise catalog of all admissible designs of 32 and 64 runs and a lengthy but substantially re

input nyquist sampling suffices to identify and compensate nonlinear systems nonlinear systems usually cause spectral spreading resulting in an output signal bandwidth greater than that of the input signal when identifying and compensating nonlinear systems by digital methods it has been common practice to set the sampling frequency at the nyquist rate of the output signal this correspondence presents experimental evidence for the author s proposal that nonlinear systems can be identified and compensated at the lowest possible sampling rate the nyquist rate of the input signal

blocked regular fractional factorial designs with minimum aberration this paper considers the construction of minimum aberration ma blocked factorial designs based on coding theory the concept of minimum moment aberration due to xu 2003 for unblocked designs is extended to blocked designs the coding theory approach studies designs in a row wise fashion and therefore links blocked designs with nonregular and supersaturated designs a lower bound on blocked wordlength pattern is established it is shown that a blocked design has ma if it originates from an unblocked ma design and achieves the lower bound it is also shown that a regular design can be partitioned into maximal blocks if and only if it contains a row without zeros sufficient conditions are given for constructing ma blocked designs from unblocked ma designs the theory is then applied to construct ma blocked designs for all 32 runs 64 runs up to 32 factors and all 81 runs with respect to four combined wordlength patterns

projective properties of certain orthogonal arrays summary a question of importance in factor screening is when a two level orthogonal design for a multifactor experiment can be projected into lower dimension typically p 2 or 3 new results relate to the projectivity p of saturated designs in which n 1 factors are tested in n runs it is shown that a design obtained by doubling an n x n orthogonal array is always of projectivity p 2 a twolevel cyclic design is either a factorial array and hence has p 2 or it has p 3 a two level orthogonal design with 4m runs m odd has p 3 in particular these results allow the designs derived by plackett burman 1946 to be categorised in terms of these projective properties

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

theory of optimal blocking of 2 n m designs in this paper we define the blocking wordlength pattern of a blocked fractional factorial design by combining the wordlength patterns of treatment defining words and block defining words the concept of minimum aberration can be defined in terms of the blocking wordlength pattern and provides a good measure of the estimation capacity of a blocked fractional factorial design by blending techniques of coding theory and finite projective geometry we obtain combinatorial identities that govern the relationship between the blocking wordlength pattern of a blocked 2 n m design and the split wordlength pattern of its blocked residual design based on these identities we establish general rules for identifying minimum aberration blocked 2 n m designs in terms of their blocked residual designs using these rules we study the structures of some blocked 2 n m designs with minimum aberration

stability of parallel explicit ode methods general purpose numerical methods for ordinary initial value problems are usually found in the class of linear multistage multivalue methods first formulated by j c butcher among these the explicit methods are easiest to implement for this reason there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in par allel each such group of concurrent function evaluations can be regarded as a stage of the method it is desirable that methods have a large region of stability so that stability dictated restrictions on the stepsize will occur less frequently in order to do a fair comparison of stability regions one should consider the scaled stability region this is defined using not the nominal stepsize but rather the stepsize per unit of work each stage counts as one work unit in the context of parallel computing each stage may involve any number of independent function evaluations very important results on scaled stability regions were obtained by jeltsch and nevanlinna in the context of serial computation they identify a type of method actually a type of alge braic function whose scaled stability region is optimal this means that the region is not a proper subset of some other scaled stability region the distinguished type of algebraic functions is one that satisfies property c which requires that the principal branch of the algebraic function be analytic outside the stability region except for a pole at infinity whose multiplicity equals the number of work units of the method it is shown that by modifying property c the result on optimal stability regions holds also in the context of parallel computa tion an important consequence is that although the use of parallelism enlarges the class of scaled stability regions it does not result in scaled stability regions that properly contain those obtained without parallelism roughly stated parallelism cannot improve the stability of an explicit linear method but it can even for the standard test problem improve the accuracy

on the construction of trend resistant mixed level factorial run orders in this paper we consider experimental situations where treatments from a mixed level factorial design are to be applied to experimental units over space or time and where there may be an unknown trend effect which can be approximated by some polynomial function of the order in which the observations are obtained a method of constructing run orders of treatments for such situations is given which generally yields least squares estimators for main effects that have a higher degree of trend resistance than the least squares estimators of main effects that come from run orders obtainable by other methods of construction previously given in the literature

connections between two level designs of resolutions iii and v resolution iii designs are resolution iii designs in which no two factor interactions are confounded with one another they arise naturally in the search for small composite designs and were first suggested by hartley 1959 new results given here link resolution iii designs to resolution v designs enabling passage between the two types of designs to be made this means that previously untabulated resolution iii designs can be derived directly from known resolution v designs for example a hartley type composite design can be based on a 27 1 iii design derived from a 26 1 v design the usual composite design is based on a 27 1 v design double the number of factorial runs the resolution iii designs can also be usefully modified for specific industrial applications box and jones 1989 moreover the maximum number of factors it is possible to accommodate in a resolution iii design of a given size can be obtained directly from the known maximum for resolution v designs of the same size

fractional resolution and minimum aberration in blocked 2 n k designs systematic sources of variation can be reduced in fractional factorial experiments by grouping the runs into blocks this is accomplished through the use of blocking factors the concepts of resolution and minimum aberration design optimization criteria ordinarily used to rank unblocked fractional factorial designs are extended to such blocked fractional factorial designs by treating the treatment and blocking factors differently in terms of their contribution to word length in the defining contrast subgroup some limited theoretical results are derived and tables of minimumaberration blocked two level fractional factorial designs are presented and considered the relationship between clear effects effects that are estimable when higher order effects are assumed negligible and minimum aberration in the presence of blocking is discussed

optimal foldover plans for two level nonregular orthogonal designs this article considers optimal foldover plans for nonregular designs by using the indicator function we define words with fractional lengths the extended word length pattern is then used to select among nonregular foldover designs some general properties of foldover designs are obtained using the indicator function we prove that the full foldover plan that reverses the signs of all factors is optimal for all 12 run and 20 run orthogonal designs the optimal foldover plans for all 16 run regular and nonregular orthogonal designs are constructed and tabulated for practical use optimal foldover plans for higher order orthogonal designs can be constructed in a similar manner

minimum aberration construction results for nonregular two level fractional factorial designs summary nonregular two level fractional factorial designs are designs which cannot be specified in terms of a set of defining contrasts the aliasing properties of nonregular designs can be compared by using a generalisation of the minimum aberration criterion called minimum g2 aberration until now the only nontrivial designs that are known to have minimum g2 aberration are designs for n runs and m n 5 factors in this paper a number of construction results are presented which allow minimum g2 aberration designs to be found for many of the cases with n 16 24 32 48 64 and 96 runs and m n 2 2 factors