Usefulness of Latin Square Design

Question

I want to understand the usefulness of the Latin Square Design. Suppose there are two block factors, each with three blocks. There are three experimental groups: A, B, C. One realization of the Latin Square design would be:

> design1 <- c("A", "B", "C", "B", "C", "A", "C", "A", "B")
> matrix(design1,3,3)
     [,1] [,2] [,3]
[1,] "A"  "B"  "C" 
[2,] "B"  "C"  "A" 
[3,] "C"  "A"  "B" 

In this design, rows and columns represent the block factors. This design assumes that there are no interactions among the three factors. However, if we assume that there are no interactions, we can estimate the effects of the three factors without using a Latin Square design, as shown in design2

> design2 <- c("A", "B", "C", "B", "C", "A", "C", "A", "A")
> matrix(design2,3,3)
     [,1] [,2] [,3]
[1,] "A"  "B"  "C" 
[2,] "B"  "C"  "A" 
[3,] "C"  "A"  "A" 

In design2, experimental group B is not represented in the third row and third column, but we can still estimate the block effects and experimental effect. I would like to know the main advantage of the Latin Square design.

> y <- rnorm(9,1,1)
> design1 <- factor(c("A", "B", "C", "B", "C", "A", "C", "A", "B"))
> design2 <- factor(c("A", "B", "C", "B", "C", "A", "C", "A", "A"))
> design3 <- factor(c("A", "B", "C", "B", "C", "A", "A", "A", "A"))
> blk1 <- factor(c("T1", "T2", "T3", "T1", "T2", "T3", "T1", "T2", 
                   "T3"))
> blk2 <- factor(c("A1", "A1", "A1", "A2", "A2", "A2", "A3", "A3", 
                   "A3"))
> 
> anova(lm(y ~ design1 + blk1 + blk2))
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value Pr(>F)
design1    2 4.2330 2.11652  2.4836 0.2871
blk1       2 0.2355 0.11775  0.1382 0.8786
blk2       2 0.6779 0.33897  0.3978 0.7154
Residuals  2 1.7044 0.85221               
> anova(lm(y ~ design2 + blk1 + blk2))
Analysis of Variance Table

Response: y
          Df  Sum Sq Mean Sq F value Pr(>F)
design2    2 2.97039 1.48520  1.0063 0.4984
blk1       2 0.49651 0.24825  0.1682 0.8560
blk2       2 0.43222 0.21611  0.1464 0.8723
Residuals  2 2.95177 1.47589               
> anova(lm(y~design3+blk1+blk2))
Analysis of Variance Table

Response: y
          Df Sum Sq Mean Sq F value Pr(>F)
design3    2 2.8600 1.43001  0.8500 0.5405
blk1       2 0.1706 0.08530  0.0507 0.9517
blk2       2 0.4554 0.22772  0.1354 0.8808
Residuals  2 3.3648 1.68241 

I guess deign1 (LSD) may be more powerful than deign2 or deign3. But statistical power is not discussed as an advantage of LSD in textbooks. Textbooks usually talk about reducing samples, but reducing samples can be accomplished without using LSD.

Answer 1

The main idea with latin square design is that it can accommodate two separate blocking designs. The main example being agricultural field experiments where there might be soil gradients in two different directions. With only one blocking system the need is not so great.

Some similar posts with helpful discussions:

• 3 Treatment Agronomic Experiment: Latin Square or Randomized Complete Block Design with 4 replicates?
• Experimental design – Reducing the number of conditions per participant

As for your design 2, it has unequal replication. You would need a good reason to prefer that ... but it is still a connected design, which means that all pairwise contrasts (A-B, A-C, B-C) can be estimated within blocks. For importance (and illustrations) of connectedness see Examples of connected designs in DOE.

But, if you have only one blocking system, you can still use design 1 a RCBD (randomized complete block design), which will give more degrees of freedom for error than a latin square design (as discussed in my first linked post above).