Intra- and Inter-Block Analysis for Factorials in Incomplete Block Designs

Two-Factor Experiments in Non-Orthogonal Designs

“…The most commonly used designs are balanced incomplete block designs for which an orthogonal analysis of the factorial effects is available once the block effects have been eliminated. Harshbarger (1954), Kramer and Bradley (1957a, b), Zelen (1958), Bradley et al (1960), Brenna and Kramer (1961) considered certain classes of incomplete block designs which admit an orthogonal analysis of the factorial effects. Kurkjian andZelen (1962, 1963) introduced a special calculus for the analysis of factorial experiments and gave a class of designs, which we shall call Type A, having an orthogonal structure.…”

“…Kramer and Bradley (1957a, b) and Zelen (1958) considered the group-divisible partially balanced incomplete block (PBIB) designs with two associate classes. Type F designs also include the Latin square PBIB designs with association scheme LS 2 , considered by Bradley et al (1960), and the rectangular lattice designs generated from sets of orthogonal Latin squares, considered by Brenna and Kramer (1961). They do not include, however, the Latin square PBIB designs with association scheme LS 3 which were shown by Bradley et al (1960) to have a two-factor orthogonal analysis.…”

Multi-Factor Experiments in Non-Orthogonal Designs

“…Type A designs satisfy the conditions of Theorem 1. Bradley et al (1960) have shown that LS 2 and LS a designs constructed from n x n Latin squares can be used for 11x n factorial experiments. LS 2 designs are of Type A and, therefore, can be used for multi-factor experiments with n~~1 mi = n 2 • LS a designs are not of Type A but do have the structure of Theorem 1 if at least one factor is a multiple of the side of the Latin square used to generate the design.…”

“…WHEN a factorial experiment is conducted in a non-orthogonal block design it is often possible to obtain an orthogonal partitioning of the adjusted treatment sum of squares into components measuring the adjusted factorial effects. Harshbarger (1954), Kramer and Bradley (1957a, b), Zelen (1958), Bradley et al (1960), Brenna and Kramer (1961) and Kurkjian andZelen (1962, 1963) have considered certain classes of incomplete block designs which admit such an orthogonal analysis of the factorial effects. Such designs are said to have factorial structure.…”

Generalized Cyclic Designs in Factorial Experiments

“…In the class of partially balanced incomplete block designs with two associate classes Kramer & Bradley (1957a, b) and Zelen (1958) considered group divisible designs and Bradley, Walpole & Kramer (1960) considered the Latin square designs with association schemes L82 and LS3. Brenna & Kramer (1961) considered the use of factorials in rectangular lattice designs.…”