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?

• This lacks answers so far because some information is missing: How many (land) plots can you use, in total? Your RCBD with 4 replicates would need 12 plots, while the latin square would need 9. Since 12 x 3 = 9 x 4, with 36 plots you could use 4 latin squares or 3 RCBD's. I would then probably go for the latin squares. To assume the field has no noticeable differences in factors that could influence yield seems risky, and what about unnoticable? – kjetil b halvorsen Nov 2 '20 at 20:57
• So can you edit your post with answers to some of my implicit Qs? – kjetil b halvorsen Nov 3 '20 at 16:40
• Sorry I hit return on accident before finishing my post. My main constraint with doing more plots is cost. The lab analysis I would like to do is quite expensive per sample. As for the assumption, yes, it's impossible to control for everything and there will always be influences of other factors, even those that are difficult to notice. – Juan Nov 3 '20 at 16:40
• If you can do so, I would go for at least two replicated latin squares. – kjetil b halvorsen Nov 3 '20 at 16:42
• Ok thank you, so would you suggest 2 squares in the same field or different fields? And in your opinion would that be more robust than 2 RCBDs with 4 replications because you are able to compare means by row and by column, or is the RCBD superior because it would have more data points? – Juan Nov 3 '20 at 16:46

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.
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.