Robustness Aspects of Experimental Designs

1. Introduction Design of Experiments forms a fascinating branch of Statistics, which owes its origin to agricultural experiments. During 1930's to 1950's there have been major contributions from Fisher and his co-workers in the development of classical theory of design of experiments. Last five decades have witnessed a quantum jump in the development of this branch of Statistics. A careful glance in classical theory reveals that the designs developed have inherent simplicity, which helps in analyzing the data easily. The principal reason for this was lack of computational facilities available in the earlier days. The emphasis on simplicity, with the advent of high-speed computers, shifted to designs, which have some statistical properties or are optimal in some statistical sense. The optimal design theory has been developed under various assumptions like homoscedasticity, independence and normality of errors, etc. The optimal design theory assumes absence from disturbances like missing observations, outlying observations or inadequacy of assumed model, etc. These assumptions may, however, be violated in real life; thus rendering even an optimal design poor. In order to overcome such a situation we have to think of designs, which are insensitive or robust against such disturbances. It may be emphasized that in the literature robustness has been defined with reference to a particular disturbance and broadly speaking the following disturbances have been identified which tend to spoil the ideal structure of the design. (i) Presence of one or more outliers, (ii) Missing data, (iii) Systematic trend in blocks in a block design, (iv) Model inadequacy, (v) Disturbance due to mechanical handling like interchange and exchange of treatments. A design d is said to be robust against one or more of the above disturbances if it remains insensitive to presence of one or more of the above disturbances in terms of design properties. For a review on robust designs one may refer to Dey, Srivastava and Gupta (1991). Consider a design d having some property 'A'. Let d * be the resulting design after some disturbance(s) η has taken place. Then the design d is said to be robust against disturbance η if d * possesses property A .For example suppose a design d is connected i.e., all the elementary treatment contrasts are estimable, then the design d 1 obtained after losing some observation(s) remains connected then we say that the design d is robust against loss of observation(s) as per the connectedness property …