Comparison of New Techniques to Identify Significant Effects in Unreplicated Factorial Designs

Abstract

Industrial experimenters use unreplicated fractional factorial designs whenever, for economical or technical reasons, it is impossible to obtain more than one response for each configuration of the design factors. These designs have usually shared the problem of proper identification of significant effects. This is due to the lack of an independent noise estimate since the variance of the response measure cannot be assessed with just one data point. Also, the use of the multiple interaction estimates to assess the noise is often hindered by the fact that these interactions are, in fractional factorials; confounded with some of the single factors or double interactions one wishes to evaluate. Since an effect is called significant whenever its value is rejected as coming from the same distribution as the noise, the problem is not a trivial one. The classical approach to this problem, first suggested by Daniel in 1959 [1], is based on the usual assumption that all non significant effects are samples of the same normal noise distribution. A further assumpion, the sparsity hypothesis, is that only a small, but unknown, fraction of the computed effects is actually significant. Thus a normal plot of the computed effects should exhibit a straight line behavior except for the significant effects which will appear deviant.