Jaya Srivastava

Balanced 2m factorial designs of resolution v which allow search and estimation of one extra unknown effect, 4 ≤ m ≤ 8

In this paper, we obtain balanced resolution V plans for 2m factorial experiments (4 ≤ m ≤ 8), which have an additional feature. Instead of assuming that the three factor and higher order effects are all zero, we assume that there is at most one nonnegligible effect among them; however, we do not know which particular effect is nonnegligible. The problem is to search which effect is non-negligible and to estimate it, along with estimating the main effects and two factor interactions etc., as in an ordinary resolution V design. For every value of N (the number of treatments) within a certain practical range, we present a design using which the search and estimation can be carried out. (Of course, as in all statistical problems, the probability of correct search will depend upon the size of “error” or “noise” present in the observations. However, the designs obtained are such that, at least in the noiseless case, this probability equals 1.) It is found that many of these designs are identical with optimal b...

Minimal 2-coverings of a finite affine space based on GF(2)

Abstract Let EG( m , 2) denote the m -dimensional finite Euclidean space (or geometry) based on GF(2), the finite field with elements 0 and 1. Let T be a set of points in this space, then T is said to form a q -covering (where q is an integer satisfying 1⩽ q ⩽ m ) of EG( m , 2) if and only if T has a nonempty intersection with every ( m - q )-flat of EG( m , 2). This problem first arose in the statistical context of factorial search designs where it is known to have very important and wide ranging applications. Evidently, it is also useful to study this from the purely combinatorial point of view. In this paper, certain fundamental studies have been made for the case when q =2. Let N denote the size of the set T . Given N , we study the maximal value of m .

Main effect plan for 2m factorials which allow search and estimation of one unknown effect

Abstract In this paper, we discuss resolution III plans for 2 m factorial experiments which have an additional property. We relax the classical assumption that all the interactions are negligible by assuming that (at most) one of them may be nonnegligible. Which interaction is nonnegligible is unknown. We discuss designs which allow the search and estimation of this interaction, along with the estimation of the general mean and the main effects as in the classical resolution III designs.

Theory of factorial designs of the parallel flats type. I: the coefficient matrix

Abstract Let GF(s) be the finite field with s elements.(Thus, when s=3, the elements of GF(s) are 0, 1 and 2.)Let A(r×n), of rank r, and ci(i=1,…,f), (r×1), be matrices over GF(s). (Thus, for n=4, r=2, f=2, we could have A=[11100121], c1=[10], c2=[02].) Let Ti (i=1,…,f) be the flat in EG(n, s) consisting of the set of all the sn−r solutions of the equations At=ci, wheret′=(t1,…,tn) is a vector of variables.(Thus, EG(4, 3) consists of the 34=81 points of the form (t1,t2,t3,t4), where ts take the values 0,1,2 (in GF(3)). The number of solutions of the equations At=ci is sn−r, where r=Rank(A), and the set of such solutions is said to form an (n−r)-flat, i.e. a flat of (n−r) dimensions. In our example, both T1 and T2 are 2-flats consisting of 34−2=9 points each. The flats T1,T2,…,Tf are said to be parallel since, clearly, no two of them can have a common point. In the example, the points of T1 are (1000), (0011), (2022), (0102), (2110), (1121), (2201), (1212) and (0220). Also, T2 consists of (0002), (2010), (1021), (2101), (1112), (0120), (1200), (0211) and (2222).) Let T be the fractional design for a sn symmetric factorial experiment obtained by taking T1,T2,…,Tf together. (Thus, in the example, 34=81 treatments of the 34 factorial experiment correspond one-one with the points of EG(4,3), and T will be the design (i.e. a subset of the 81 treatments) consisting of the 18 points of T1 and T2 enumerated above.) In this paper, we lay the foundation of the general theory of such ‘parallel’ types of designs. We define certain functions of A called the alias component matrices, and use these to partition the coefficient matrix X (n×v), occuring in the corresponding linear model, into components X.j(j=0,1,…,g), such that the information matrix X is the direct sum of the X′.jX.j. Here, v is the total number of parameters, which consist of (possibly μ), and a (general) set of (geometric) factorial effects (each carrying (s−1) degrees of freedom as usual). For j≠0, we show that the spectrum of X′.jX.j does not change if we change (in a certain important way) the usual definition of the effects. Assuming that such change has been adopted, we consider the partition of the X.j into the Xij (i=1,…,f). Furthermore, the Xij are in turn partitioned into smaller matrices (which we shall here call the) Xijh. We show that each Xijh can be factored into a product of 3 matrices J, ζ (not depending on i,j, and h) and Q(j,h,i)where both the Kronecker and ordinary product are used. We introduce a ring R using the additive groups of the rational field and GF(s), and show that the Q(j,h,i) belong to a ring isomorphic to R. When s is a prime number, we show that R is the cyclotomic field. Finally, we show that the study of the X.j and X′.jX.j can be done in a much simpler manner, in terms of certain relatively small sized matrices over R.

On the inadequacy of the customary orthogonal arrays in quality control and general scientific experimentation, and the need of probing designs of higher revealing power ∗

This paper breaks new ground concerning the general problem of factorial experimentation, namely, the identification and estimation of nonnegligible factorial effects (with minimal number of runs), without making the usual unrealistic and artificial assumptions concerning the negligibility of higher order interactions. (In other words, we consider the general factor screening problem when interactions may be present.) Through an example, it is shown that the customary orthogonal arrays fall short of the need. New principles for sieving the set of factorial effects to determine the large ones, are introduced. The concept of ‘revealing power’of designs, i.e. of their ability to help identify nonnegligible parameters is developed, and the usefulness in this direction of balanced arrays of full strength is studied.

A Comparison of the Determinant, Maximum Root, and Trace Optimality Criteria

Three basic criteria, determinant, trace and maxim-urn root, are in common use for determining optimality of experimental designs. Here examples are presented where the three criteria give rise to different designs. The examples are balanced resolution IV* of the 2m series and are particularly insightful with respect to the dependence of the criteria on the correlation between estimators of the parameter.

On the superiority of the nested multi-dimensional block designs, relative to the classical incomplete block designs☆

Abstract Nested multidimensional equal block designs (NMEBD), introduced in Srivastava (1981), involved firstly, a nuisance factor, giving rise to a set of b blocks. Secondly, nested within each block, there are g nuisance factors, the jth factor having sj levels. Thus, each block has (s1 × s2 × · × sg) possible classes of units. In this paper for the above designs, the problem of recovery of the information existing between the levels of each factor nested inside a block is considered. The discussion is restricted to a special case where the information arising out of the set of all intrablock contrasts gives rise to exactly two sets of normal equations, each having a different error variance associated with it. The problem of combining the information arising from the two sets of normal equations is considered. Three cases arise. The first one is to ignore the nuisance factors, thus effectively reducing the situation to that of a classical incomplete or complete block design. The second one is to use a guess estimate of δ (the ratio of the two variances) and the third is to estimate δ from the data. The efficiency of the last two cases (with respect to the first one) is studied by obtaining the (unconditional) variance of treatment contrasts. Tables are obtained. The results suggest that in practice the last two approaches would usually be moderately to greatly superior to the first one.

Advances in the statistical theory of comparison of lifetimes of machines under the generalized weibull distribution

Some new censoring schemes for comparing lifetimes of machines were introduced in Srivastava (1987), wherein it was shown that in general, these improve on the accuracy and also reduce the total expected time under experiment as well as the expected length of the experiment. Although the above studies were initiated under the generalized WeibuU distribution, the detailed development of the theory for the expected time or length were made under the Weibull only. In this paper, for a certain important special case, this development is extended to the generalized Weibull. Also, an error in the above paper is pointed out, and corrected, showing that the new schemes are actually superior to what they appeared to be there. Furthermore, under a realistic cost function, the problem of planning such experiments is discussed. An example for machines with a series connection is provided.

More efficient and less time-consuming censoring designs for life testing

Abstract The study of self-relocating designs, a class of which was first introduced in Srivastava (1985), is continued. New classes of such (censoring) designs are introduced, and are studied under the Weibull distribution, and partly under certain generalizations of the same. It is shown that these designs are superior with respect to the ‘total expected time under experiment’, as well as the ‘asymptotic variance’ (in a certain sense), compared to the classical type II censoring designs, and thus offer a breakthrough in the field.

Multistage design procedures for identifying two-factor interactions, when higher effects are negligible

Abstract Consider a 2m factorial experiment. Let L(ν×1) be the vector of factorial effects which are nonnegligible. Let μ denote the general mean. It will always be assumed that μ∈L, even though in some situations μ may be zero. All factorial effects which are not in L are thus assumed zero. We do not know what parameters are elements of L, and furthermore, we do not know the value of the various elements of L. The problem of model identification is to determine the elements of L, and also estimate the same. In this paper we shall investigate this problem in a certain limited context. We shall assume that all elements of L are included in the vector of parameters L ∗ (ν ∗ ×1) , except possibly for a set of p parameters. In this paper we shall take L ∗ to consist of the set of main effects, and the general mean μ. We assume p to be small and L to contain possibly a set of p 2-factor interactions. Three-factor and higher interactions are assumed negligible. We start with a saturated resolution III design of the 2m−q type, where m=2k−1. For a given value of (m−q) we shall also consider values of m smaller than the maximum. The purpose of the experiment is to estimate μ and the main effects, and furthermore identify all the p two-factor interactions which are nonnegligible, and also estimate the same. For this purpose, we consider using a multistage design, i.e., a design which allows the experiment to be done in several parts or stages. In the first stage, the resolution three design plus possibly a few extra treatments may be used. After looking at the resulting data, a further set of treatments is selected and observed. This may resolve the problem. If the problem is still not resolved we continue the same way, that is, we take a few more treatments and see if we can resolve the problem. The attempt is to have a design in which the average sample number is as small as possible. Examples are given for k=3,4.