Resistant and Susceptible BIB Designs

Abstract

Let $D$ be a BIB $(\nu, b, r, k, \lambda)$ design on $\Omega$. Let also $L \subset \Omega$ with cardinality of $L$ being $n \leqq \nu - 2$. We define $D$ to be locally resistant of degree $n$ if upon deletion of all the experimental units in $D$ assigned to the treatments in $L$ the remaining structure is variance balanced in the sense that under the usual homoscedastic additive linear model every normalized estimable linear function of the treatment effects are estimable with the same variance. $D$ is defined to be globally resistant of degree $n$ if it has the above property with respect to any subset $L \subset \Omega$ as long as its cardinality is $n. D$ is said to be susceptible if it is not resistant to any nonempty set $L$. Application of these concepts in various branches of sciences and engineering has been indicated. In this paper we have characterized all locally and globally resistant designs of degree one in two different ways. Through one of these characterizations we have been able to relate our theory to the theory of $t$-designs or tactical configurations. Methods for constructing some families of locally and globally resistant designs of degree one are provided. We have also shown that the property of being resistant depends not only on the parameters of $D$ but also depends on the way $D$ has been constructed. To illustrate this we have given three BIB (10, 30, 12, 4, 4) designs; the first design is susceptible, the second is locally resistant to the deletion of a single treatment and the third design is globally resistant. We have also indicated that every BIB $(\nu, b, r, k, \lambda)$ design is locally resistant of degree $k$ if $b = \nu$. Several miscellaneous results are also given, among which a locally resistant design of degree two is included. Several unsolved problems are indicated in the final section.