Existence Conditions for Balanced Fractional 2m Factorial Designs of Resolution 2l + 1 Derived from Simple Arrays

We consider a fractional 2^m factorial design derived from a simple array (SA) such that the (? + 1)-factor and higher-order interactions are negligible, where 2? ? m. The purpose of this article is to give a necessary and sufficient condition for an SA to be a balanced fractional 2^m factorial design of resolution 2? + 1. Such a design is concretely characterized by the suffixes of the indices of an SA.     

Search designs for 2m factorials derived from balanced arrays of strength 2(l+1) and ad-optimal search designs

We consider a search design for the 2^m type such that at most knonnegative effects can be searched among (l+1)-factor interactions and estimated along with the effects up to l- factor interactions, provided (l+1)-factor and higher order interactions are negligible except for the k effects. We investigate some properties of a search design which is yielded by a balanced 2^m design of resolution 2l+1 derived from a balanced array of strength 2(l+1). A necessary and sufficient condition for the balanced design of resolution 2l+1 to be a search design for k=1 is given. Optimal search designs for k=1 in the class of the balanced 2^m designs of resolution V (l=2), with respect to the AD-optimality criterion given by Srivastava (1977), with N assemblies are also presented, where the range of (m,N) is (m=6; 28≤N≤41), (m=7; 35≤N≤63) and (m=8; 44≤N≤74).     

Characterization of information matrices for balanced two-level fractional factorial designs of odd resolution derived from two-symbol simple arrays

We consider a balanced fractional 2^m factorial design of resolution 2?+1 which permits estimation of all factorial effects up through ?-factor interactions under the situation in which all (?+1)-factor and higher order interactions are to be negligible for an integer satisfying [m/2]<lE;?m, where [x] denotes the greatest integer not exceeding x. This paper investigates algebraic structure of the information matrix of such a design derived from a simple array through that of an atomic array. We obtain an explicit expression for the irreducible matrix representation based on the above design and its algebraic properties. The results in this paper will be useful to characterize the designs under consideration.     

CONSTRUCTION OF BALANCED TERNARY DESIGNS

In this paper variance balanced ternary designs are constructed in unequal block sizes for situations when suitable BIB designs do not exist for a given number of treatments because of the constraints bk=vr,and λ(v - 1) =r(k- 1).     

MULTI-FACTOR BALANCED BLOCK DESIGNS WITH COMPLETE ADJUSTED ORTHOGONALITY FOR ALL PAIRS OF TREATMENT FACTORS

A new series of multi-factor balanced block designs is introduced. Each of these designs has the following properties: (i) each of its k– 1 treatment factors is disposed in a cyclic or multi-cyclic balanced incomplete block design with parameters (v,b,r,k,Λ) = (a(k-l) + 1,a²(k-1) +a, ak, k, k) (a > 1); (ii) the incidence of any one of the treatment factors on any other is balanced; and (iii) after adjustment for blocks only, the relationship between any two of the treatment factors is that of adjusted orthogonality. The treatment factors are thus orthogonal to one another in the within-blocks stratum of the analysis of variance. The designs provide a benchmark with which other designs may be compared.     

A NOTE ON THE ANALYSIS OF COVARIANCE IN BALANCED INCOMPLETE BLOCK DESIGNS

The Williams & Ratcliff (1980) and Zelen (1957) methods for the analysis of covariance in incomplete block designs with recovery of inter-block information are compared using data from a balanced incomplete block design. The former is shown to be more efficient. Other advantages of the Williams & Ratcliff formulation are also presented.     

Simulation-based Bayesian inferences for two-variance components linear models

We present a Bayesian analysis of variance component models via simulation. In particular, we study the 2-component hierarchical design model under balanced and unbalanced experiments. Also, we consider 2-factor additive random effect models and mixed models in a cross-classified design. We assess the sensitivity of inference to the choice of prior by a sampling/resampling technique. Finally, attention is given to non-normal error distributions such as the heavy-tailed t distribution.     

Balanced fractional factorial designs of resolution 2l+1 for interesting effects orthogonal to some nuisance parameters: 2m1+m2 series

For 2^m₁+m₂ factorial designs, this paper investigates balanced fractional 2^m₁ factorial designs of resolution 2l+1 with some nuisance parameters concerning the second factors. They are derivable from partially balanced arrays and further permit estimation of the effects up to the l-factor interactions concerning the first factors orthogonally to the nuisance parameters.     

THE ROBUSTNESS OF BINARY AND NON-BINARY BALANCED NESTED ROW-COLUMN DESIGNS UNDER THE UNAVAILABILITY OF BLOCKS: A COMPARISON

In cases where both exist, the balanced, binary nested row-column designs are known to be inferior to a class of balanced non-binary designs. However, if it is possible for blocks of observations to become unavailable after an experiment has commenced, a binary nested row-column design may possibly be better than a non-binary one. This paper investigates the robustness of binary and non-binary variance-balanced nested row-column designs to the unavailability of one or more blocks of observations. Robustness is measured through the C-matrices of the designs resulting from removing blocks, using optimality criteria such as A-, D-, E- and MV-optimality.     

Resolution v.2 search designs

A fractional factorial design is called a resolution V.2 plan if it is capable of estimating all main effects and two-factor interaction effects, plus two three-factor interaction effects, In this paper, a necessary and sufficient condition for such a resolution V.2 plan is given, Furthermore, a new class of two-level resolution V.2 designs is proposed, We prove that the proposed design always satisfies such a necessary and sufficient condition, A comparison of run size between designs of resolutions VII and V.2 is made, It is shown that run size for design of resolution V.2 is significantly smaller.     

Optimal partially balanced nested row-column designs

A partially balanced nested row-column design, referred to as PBNRC, is defined as an arrangement of v treatments in b p × q blocks for which, with the convention that p q, the information matrix for the estimation of treatment parameters is equal to that of the column component design which is itself a partially balanced incomplete block design. In this paper, previously known optimal incomplete block designs, and row-column and nested row-column designs are utilized to develop some methods of constructing optimal PBNRC designs. In particular, it is shown that an optimal group divisible PBNRC design for v = mn kn treatments in p × q blocks can be constructed whenever a balanced incomplete block design for m treatments in blocks of size k each and a group divisible PBNRC design for kn treatments in p × q blocks exist. A simple sufficient condition is given under which a group divisible PBNRC is Ψ_f-better for all f> 0 than the corresponding balanced nested row-column designs having binary blocks. It is also shown that the construction techniques developed particularly for group divisible designs can be generalized to obtain PBNRC designs based on rectangular association schemes.     

BALANCED CHANGE-OVER DESIGNS FOR AUTOCORRELATED OBSERVATIONS

Some aspects of design of change-over experiments with autocorrelated observations are investigated. Williams' designs (1949, 1950) are seen to be balanced for autocorrelated observations.     

The mad estimator: when is it non-linear? applications to two-way design models

The mean absolute deviation (MAD) estimator has recently received a great deal of attention as applied to full-rank linear regression models. This paper provides a necessary and sufficient condition for the MAD estimator to be a non-linear estimator, in which case conditions for the variance of the MAD estimator to be larger or smaller than those for OLS are, in general, unknown. The non-linearity of the MAD estimator is examined for several two-way designs; in particular (1) randomized block design (2) two-way nested design (3) two-way classification with interaction and (4) partially balanced incomplete block design     

BALANCED ARRAYS AND WEIGHING DESIGNS

Dey (19711, Saha (1975), Kageyama & Saha (1983) and others have shown how optimum chemical balance weighing designs can be constructed from the incidence matrices of balanced incomplete block (BIB) designs. In this paper, it is shown that weighing designs can be constructed from some suitably chosen two-symbol balanced arrays of strength two, which need not always be incidence matrices of BIB designs. The findings lead us to construct new optimum chemical balance weighing designs from incidence matrices of BIB designs.     

A Test to Detect Inadequacy of Satterthwaite's Approximation in Balanced Mixed Models

Satterthwaite's approximation of the distribution of a nonnegative linear combination of independent mean squares is addressed in this article. A necessary and sufficient condition for the approximation to be exact is presented for the case of a general balanced mixed model. A test is subsequently developed for detecting any significant departure from this condition using the data under consideration. An example is given to illustrate the proposed methodology.     

Three-factor symmetrical balanced designs with full efficiency for main effects

Some methods for constructing balanced design for 3-factor symmetrical factorial experiments in which all the main effects are completely unconfounded by using balanced arrays and BIB designs are proposed. The method is flexible in terms of selecting block size.     

A note on the connectivity of generalized cyclic designs

A necessary and sufficient condition is given to ensure that a generating blook of a generalized cyclic design will give rise to a connected design. The use of disconnected designs is briefly discussed.     

Characterization of Balanced Fractional 3 m Factorial Designs of Resolution III

We consider a fractional 3 ^m factorial design derived from a simple array (SA), which is a balanced array of full strength, where the non negligible factorial effects are the general mean and the linear and quadratic components of the main effect, and m ≥ 2. In this article, we give a necessary and sufficient condition for an SA to be a balanced fractional 3 ^m factorial design of resolution III. Such a design is characterized by the suffixes of indices of an SA.     

ON THE ESTIMATION OF TOTAL WEIGHT IN SINGULAR SPRING BALANCE WEIGHING DESIGNS UNDER THE CO VARIANCE MATRIX OF ERRORS s2G

The problem of the estimation of the linear combination of weights, c′w, in a singular spring balance weighing design when the error structure takes the form E(ee′) =s?²G has been studied. A lower bound for the variance of the estimated linear combination of weights is obtained and a necessary and sufficient condition for this lower bound to be attained is given. The general results are applied to the case of the total of the weights. For a specified form for G, some optimum spring balance weighing designs for the estimated total weight are found.     

Some results on the regularity in a single replicate design

A necessary and sufficient condition is given for a single replicate design to be regular with respect to a given factorial effect. This condition is applied in establishing necessary and sufficient condition for the regularity of an i order factorial effectl > 1, when a subset of factorial effects are known to be regular.