What are randomized block designs and Latin square designs?

Some designed experiments can effectively provide information when measurements are difficult or expensive to make or can minimize the effect of unwanted variability on treatment inference. The following is a brief discussion of two commonly used designs. To show these designs, two treatment factors (A and B) and their interaction (A*B) are considered. These designs are not restricted to two factors, however. If your design is balanced, you can use Balanced ANOVA to analyze your data. If it is not, use GLM.

Randomized block design

A randomized block design is a commonly used design for minimizing the effect of variability when it is associated with discrete units (e.g. location, operator, plant, batch, time). The usual case is to randomize one replication of each treatment combination within each block. There is usually no intrinsic interest in the blocks and these are considered to be random factors. The usual assumption is that the block by treatment interaction is zero and this interaction becomes the error term for testing treatment effects. If you name the blocking variable as Block, the terms in the model would be Block, A, B, and A*B. You would also specify Block as a random factor.

Latin square with repeated measures design

A repeated measures design is a design where repeated measurements are made on the same subject. There are a number of ways in which treatments can be assigned to subjects. With living subjects especially, systematic differences (because of learning, acclimation, resistance, and so on) between successive observations can be suspected. One common way to assign treatments to subjects is to use a Latin square design. An advantage of this design for a repeated measures experiment is that it ensures a balanced fraction of a complete factorial (that is, all treatment combinations represented) when subjects are limited and the sequence effect of treatment can be considered to be negligible.

A Latin square design is a blocking design with two orthogonal blocking variables. In an agricultural experiment there might be perpendicular gradients that might lead you to choose this design. For a repeated measures experiment, one blocking variable is the group of subjects and the other is time. If the treatment factor B has three levels, b1, b2, and b3, then one of twelve possible Latin square randomizations of the levels of B to subjects groups over time is:
  Time 1 Time 2 Time 3
Group 1 b2 b3 b1
Group 2 b3 b1 b2
Group 3 b1 b2 b3

The subjects receive the treatment levels in the order specified across the row. In this example, group 1 subjects would receive the treatments levels in order b2, b3, b1. The interval between administering treatments should be chosen to minimize carryover effect of the previous treatment.

This design is commonly modified to provide information about one or more additional factors. If each group was assigned a different level of factor A, then information about the A and A*B effects could be made available with minimal effort if an assumption about the sequence effect given to the groups can be made. If the sequence effects are negligible compared to the effects of factor A, then the group effect could be attributed to factor A. If interactions with time are negligible, then partial information about the A * B interaction can be obtained. In the language of repeated measures designs, factor A is called a between-subjects factor and factor B a within-subjects factor.

It is not necessary to randomize a repeated measures experiments with a Latin square design.