A Significance Test in Multi-Level Orthogonal Designs with Only One Replicate

The ordinary ANOVA has been playing an important role in testing significant factors in experimental designs since it was developed. Especially for orthogonal designs, the ANOVA is almost the unique method of analyzing them. When the columns of an orthogonal array all are allocated by factors or interactions and it is impossible to do replicates, however, the ordinary ANOVA is no more applicable because there are no degrees of freedom left to estimate the error variance. This paper proposes a new test method for analyzing the orthogonal designs with only one replicate using complete orthogonal arrays. It is illustrated with some examples that the proposed method is practicable when the ANOVA is invalid. When there are a few empty columns of the orthogonal array used so that the ANOVA is applicable, the proposed method is more powerful than the ordinary ANOVA for some parameter configurations.