How do I analyse a two-way ANOVA with two blocking factors

[#1]

I have an experiment set up as a randomized complete blocks, where replicate groups of plants are planted in trays. The tray is the blocking factor, and within each tray there are 24 cells, where I planted groups of plants. In each cell of a tray, there is a focal plant and neighbour plants.  Focal plants may be either Tall or Short, and neighbrous (collectively) can be Tall or Short.  So treatment 1 is focal plant size and treatment 2 is neighbour plant size.  All treatment level combinations appeared in each tray 6 times.  At the tray level, some trays are all of a particular genotype, and other trays are of a different genotype (replicated 2-4 times for each genotype).  So I'm wondering if I have to treat all genotypes separately (a series of RCB designs), or if I can include them together in a grand anlaysis, where 'genotype' is considered either a blocking factor, or even a treatment at the tray level (whereas the size treatments were randomized within trays). The response is seed number. I expect genotypes to differ overall in seed number, but I'm not really interested in this -- I'm only interested in the focal/neighbour size treamtments.

If anyone can suggest an analysis, with reference to the Fit Model platform (interactive explanation), that would be much apprecited!

[#2 · Accepted solution]

Your description is clear but correct me if I got something wrong. Let's clarify some important points about the design after saying that I think you should analyze all the data together, not separately for each genotype.

I think genotype is a hard to change factor because it was not randomized. It was replicated 2-4 times. These groups of runs (a tray) are whole plots, not blocks. You varied the two easy to change factors within each whole plot as a replicated full factorial design.

I imagined that you have five genotypes and always performed four replicates (trays). You can delete groups of rows if you used less than four trays. Genotype is a fixed effect and whole plot is a random effect. I attached a design so you can see what might work and how it might be used. I didn't include an interaction term in the model but the data support estimating this effect nonetheless.

[#3]

Thanks for this response, which is very helpful. To clarify the terminology, then: would you call the design a split plot with genotype acting at the whole-plot level (the tray) and the two plant size factors acting at the sub-plot level (the cells within trays)? And finally, as for the interaction term, I am interested in the focal plant size x neighbour plant size interaction, but is it also necessary to consider interactions with genotype (e.g., genotype x focal plant size x neighbour plant size). In other words, what would the "full model" be for this design?

[#4]

(Note that I modified my answer because I missed full meaning about your question regarding the interaction effects.)

Yes, this is a split-plot design as you described and as I understand it.

Yes, the full factorial design supports the interaction model so include the cross terms as follows: seed number = intercept + focal plant + neighbor plant + genotype + (focal plant)*(neighbor plant) + (focal plant)*(genotype) + (neighbor plant)*(genotype) + (focal plant)*(neighbor plant)*(genotype) + error.

[#5]

I updated the design example for your case.

I sorted the columns so that the split-plot nature is apparent. (I assumed that you randomized within the whole plots.) The sorting should also make it easier for you to transfer results if you completed the experiment already. I added a Distribution table script so tht you can interactively explore the design to convince yourself that it is RCBD.

Remember that I always used 4 trays per genotype and I included 5 genotypes but I didn't know what you actually included. I changed the default Whole Plots data column name to Trays to reflect how the whole plots were manifested.

You can use the DOE Dialog table script to return to the Custom Design as I specified and modify it as you see fit. Also, you can use Cols > Recode to easily replace my label (e.g., G1) with the actual genotype designation. I specified the model to include all two-factor interactions and the three-factor interaction as requested.

I hope that this case is a good learning experience (JMP, DOE) so please ask anything if you get stuck.