ABSTRACT
In this book we restrict ourselves to eliminating one noise factor by blocking or stratification. One method of eliminating two noise factors is the use of socalled row-and-column designs, where the rows correspond to different levels of one blocking factor and the columns to levels of the other blocking factor (for details see 1/21/4200-1/21/4250 in Rasch et al. 2008). Special cases are Latin Squares and Latin Rectangles or change over designs. Blocking is especially useful when the noise factor is qualitative, since analysis of covariance is not then available, but blocking is also possible when the noise (blocking) factor is quantitative. In this case we can group its values into classes and use these classes to define the levels of the noise factor. Of course we need a priori information in order to choose the size and number of the blocks before the experiment begins (which should be the rule). General principles of blocking have been already described in Section 1.3. We only repeat here some notations. The number b of the blocks is determined as part of the experimental design process as described in Chapter two. In this book we assume that all b blocks have the same block size k and further that each of the v levels of the treatment factor, which will be called treatments, occur exactly r times. In place of b we can also determine r by the methods of Chapter two, due to the relation (6.4) below. We assume in this chapter that we are faced with the following situation: the block size k as well as the number v of treatments is given in advance. We write R-programs which help us to find the smallest design which exists for this (v, k) pair especially in the case when k < v which means that not all treatments can occur in any block together. Block designs in this case have been called incomplete block designs. An early reference on incomplete block designs is Yates (1936). Methods of construction are discussed in many articles and books, for instance, Bailey (2008), Hall (1986), Hanani (1961, 1975), Hedayat and John (1974), John (1987), Kageyama (1983), Kirkman
et al. (1985), Street and Street (1987), Vanstone (1975), Williams (2002), and many others. Further, we assume that some kind of balance is fulfilled. To explain the need for those balances, we first look at the model of a block design.
