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1. Lattice Design

What is Incomplete Block? Design & Analysis Web Guide (DAWG), University of TennesseeIncomplete Block Design'Incomplete' in this design simply means all treatments do not occur within the same block. Compare this to the Randomized Complete Block design, where all treatments must be tested in every block.

Otherwise the IBD and RBD are identical, with experimental units divided into groups (blocks) that are similar, but differing from group to group. This design should only be used if the experimental situation forces blocks to be too small, but also is recommended if more than 10 treatments are being tested. In the latter case, experience suggests that complete blocks of 10 or more exp.

Lattice Design

Units generally are so large that they will contain exp. Units that are not similar. Such blocks should be divided into more homogeneous groups, producing an incomplete block design, because the divided complete block will now be too small to hold all treatments.Plant Example: You are working in a greenhouse, and want to compare 9 varieties of flowers. The benches in this greenhouse are only large enough to hold 6 pots, and you know benches differ and so must be blocked on. Here the block size of 6 is too small to hold all 9 treatments, making the blocks 'incomplete'.Animal Example: You are testing 4 meat storage methods, which must be applied to entire cuts of meat. One animal can only produce 2 such cuts, one from each side of the animal. Individual animals differ in meat quality, so must be blocked on.

An incomplete design is produced by having blocks of size 2 that can not hold all 4 treatments.

What is a Counterbalanced Measures Design?The simplest type of counterbalanced measures design is used when there are two possible conditions, A and B. As with the standard repeated measures design, the researchers want to test every subject for both conditions. They divide the subjects into two groups and one group is treated with condition A, followed by condition B, and the other is tested with condition B followed by condition A.Three ConditionsIf you have three conditions, the process is exactly the same and you would divide the subjects into 6 groups, treated as orders ABC, ACB, BAC, BCA, CAB and CBA.Four ConditionsThe problem with complete counterbalancing is that for complex experiments, with multiple conditions, the permutations quickly multiply and the research project becomes extremely unwieldy.

For example, four possible conditions requires 24 orders of treatment (4x3x2x1), and the number of participants must be a multiple of 24, due to the fact that you need an equal number in each group.More Than Four ConditionsWith 5 conditions you need multiples of 120 (5x4x3x2x1), with 7 you need 5040! Therefore, for all but the largest research projects with huge budgets, this is impractical and a compromise is needed. Incomplete Counterbalanced Measures DesignsIncomplete counterbalanced measures designs are a compromise, designed to balance the strengths of counterbalancing with financial and practical reality. One such incomplete counterbalanced measures design is the Latin Square, which attempts to circumvent some of the complexities and keep the experiment to a reasonable size.With Latin Squares, a five-condition research program would look like this:Position 1Position 2Position 3Position 4Position 5Order 1ABCDEOrder 2BCDEAOrder 3CDEABOrder 4DEABCOrder 5EABCDThe Latin Square design has its uses and is a good compromise for many research projects. However, it still suffers from the same weakness as the standard repeated measures design in that carryover effects are a problem.

In the Latin Square, A always precedes B, and this means that anything in condition A that potentially affects B will affect all but one of the orders. In addition, A always follows E, and these interrelations can jeopardize the of the experiment.The way around this is to use a balanced Latin Square, which is slightly more complicated but ensures that the risk of carryover effects is much lower. For experiments with an even number of conditions, the first row of the Latin Square will follow the formula 1, 2, n, 3, n-1, 4, n-2, where n is the number of conditions. For subsequent rows, you add one to the previous, returning to 1 after n.Sounds complicated, so it is much easier to look at an example for a six condition experiment. The subject groups are labelled A to F, the columns represent the conditions tested, and the rows represent the subject groups: Subjects1st2nd3rd4th5th6thA165C321E543As you can see, this ensures that every single condition follows every other condition once, allowing the researchers to pick out any carryover effects during the statistical analysis.When an experiment with an odd number of conditions is designed, the process is slightly more complex and two Latin Squares are needed to avoid carryover effects.

The first is created in exactly the same way and the second is a mirror image: With this design, every single condition follows another two times, and statistical tests allow researchers to analyse the data. This balanced Latin Square is a commonly used instrument to perform large repeated measured designs and is an excellent compromise between maintaining and practicality. There are other variations of counterbalanced measures designs, but these variations are by far the most common.

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