on X where, if elements x and y are ith associates, 1 ≤ i ≤ n, then they are together in precisely λi blocks. A PBIBD(n) determines an association scheme but the converse is false. Example Let A(3) be the following association scheme with three associate classes on the set X = {1,2,3,4,5,6}. The (i,j) entry is s if elements i and j are in relation Rs. The blocks of a PBIBD(3) based on A(3) are: The parameters of this PBIBD(3) are: v = 6, b = 8, k = 3, r = 4 and λ1 = λ2 = 2 and λ3 = 1. Also, for the association scheme we have n0 = n2 = 1 and n1 = n3 = 2. The incidence matrix M is and the concurrence matrix MMT is from which we can recover the λ and r values. Properties The parameters of a PBIBD(m) satisfy: A PBIBD(1) is a BIBD and a PBIBD(2) in which λ1 = λ2 is a BIBD. Two associate class PBIBDs PBIBD(2)s have been studied the most since they are the simplest and most useful of the PBIBDs. They fall into six types based on a classification of the then known PBIBD(2)s by : group divisible; triangular; Latin square type; cyclic; partial geometry type; miscellaneous. Applications The mathematical subject of block designs originated in the statistical framework of design of experiments. These designs were especially useful in applications of the technique of analysis of variance (ANOVA). This remains a significant area for the use of block designs. While the origins of the subject are grounded in biological applications (as is some of the existing terminology), the designs are used in many applications where systematic comparisons are being made, such as in software testing. The incidence matrix of block designs provide a natural source of interesting block codes that are used as error correcting codes. The rows of their incidence matrices are also used as the symbols in a form of pulse-position modulation. Statistical application Suppose that skin cancer researchers want to test three different sunscreens. They coat two different sunscreens on the upper sides of the hands of a test person. After a UV radiation they record the skin irritation in terms of sunburn. The number of treatments is 3 (sunscreens) and the block size is 2 (hands per person). A corresponding BIBD can be generated by the R-function design.bib of