3 Treatment Agronomic Experiment: Latin Square or Randomized Complete Block Design with 4 replicates? I need help with designing an experiment. Suppose I am testing 2 fertility programs against a control in Lettuce, for a total of 3 groups (Program A, Program B, and Untreated Control). Let's assume that the field has no noticeable differences in factors that could influence yield.
If I were to do a Latin square, that would be 9 experimental units with 3 replications for each group. Now, compare that to a randomized complete block design (RCBD) with 4 replications of each group.
Would the Latin Square actually be more robust because I would be evaluating the treatments against each other both by row and column, even though there are less replicates? Seems to me that the row and column evaluations for 3 treatments in a latin square would be allow for more comparisons between treatments (row x 3 and column x 3) than a 4-repetition RCBD.
Am I missing something? If you were a researcher or evaluating this experiment, which would you consider more robust and why?
 A: It is more natural to compare designs with equal number of observations, so I will compare a $3\times 3$ latin square (LSQ) with a thrice replicated RCBD. The LSQ leaves 2 df (defgrees of freedom) for error, while the RCBD leaves 4 df for error. So the advantage of the RCBD is more df for error, while the LSQ possibly can remove more variation, so give a lower variance. What is more important?
If you make inference with (say) a 95% confidence interval (CI) for effects of interest, those will have the form
$$ \text{estimate}\pm \hat{\sigma} t_{\nu,0.975}/\sqrt{n} $$
Compare those t quantiles: $t_{2,0.975}=4.30, t_{4,0.975}=2.78$ so the variance reduction must be large, at least a factor of $\left( 2.78/4.30 \right)^2 = 0.42$ to get more effective inference.
How does this change with more replicas? Say we double the number of observations above, then the LSQ design gives 4 df for error, while the RCBD gives 10. You can redo the calculation above and make your conclusions.
But the general conclusion will be that with a low $n$, very few plots, it might not be an advantage with a latin square design over a RCBD.
