Would it be incorrect to generalize the analysis to a one-factor rbd despite having an incomplete factorial with two factors, inoculum and fungicide?
It would fine. In fact, it would be an appropriate way to deal with the incomplete factorial if you also included appropriate contrasts to test (1) the effects of fungicide (Edit: including control when combined with inoculum), and (2) the effect of inoculum for the control. Edit: These contrasts could be a subset of pairwise comparisons among the 5 means.
If I'm understanding this right, by turning this into a one-factor analysis, my significant indicator (p-value) is the interaction term or close to an interaction term.
Your interpretation is not correct. The p-value for treatment in the one-factor analysis addresses the alternative hypothesis that at least one of the five means is different than at least one other of the five means; it is not a test of interaction. With the 5 treatments in this study, you are not actually able to address "interaction" if we are defining interaction as an effect of inoculum that differs among fungicides (or vice versa: effects of fungicide that differ depending on presence or absence of inoculum). You can compare fungicides given inoculum; you can compare inoculum given control. I don't see that you have a pertinent experimental setup for any sort of statistical interaction of fungicide and inoculum.
Would it be improper to use the one-factor analysis for my interaction and the two-factor analysis for my main effects?
Yes, it would be incorrect.
To answer your other questions about the bonus question. I did read that paper which I found to be very insightful, but got the impression that pseudoreplication was always a bad or improper form of replication. In my case I don't find this to be true. I would suggest that pseudoreplication of a binary measurement provides greater precision of the treatment. I'm sure there is some reason I am unaware of for why this shouldn't be included in my analysis, which I would be OK with because there are work arounds. I could simply average the binary response variable of the pseudoreplicates and use this average as the response variable. For example 10 of my 18 seedlings emerged in one of the treatments so I'd use 0.55 and ditch the "rep" as one my variables and simplify the analysis whilst providing myself a precise treatment response.
Pseudoreplication is always bad. Subsamples are not bad, it's all in how you incorporate them into the statistical model; pseudoreplication means that you are using subsamples as if they are true replications. For your study, you have two options. (1) You can analyze binary data measured at the subsample-level and include a random effect for plot in addition to block, to accommodate the clustering of 18 measurements within a plot; the plot effect can be specified as Block*Fungicide*Inoculum as in the code I suggested previously. Or (2) you can composite the 18 binary responses for a plot into a binomial response (i.e., number emerged out of 18) which is now a plot-level response and so you no longer include a random effect for plot in the statistical model (although you do now need to consider the distinct possibility of overdispersion). This is like your suggestion of an "average" (which is a proportion), but the binomial distribution is theoretically preferable to analyzing proportion data as a continuous variable (e.g., transforming and then assuming normality).
See
The arcsine is asinine: the analysis of proportions in ecology
Rethinking the Analysis of Non‐Normal Data in Plant and Soil Science
In response to some of your other questions, my replication would be metareplication where I am simply showing repetition and repeatability. These are greenhouse trials, same location and same year, so more of an exercise of repeatability across time than space perhaps. I understand ideally this would be done in another growing season and another location but I have some limitation. Is the suggestion here that I consider this block 9, 10...to 18 and just add this into the analysis? I was under the impression I would need to include this as another variable to show the "contrast" for lack of better words between the experimental replicate 1 versus 2. Just adding blocks sure would make the analysis easier, just want to make sure I'm following modern/conventional statistical procedures the best I can.
I would not recommend treating the second experiment as additional blocks in the first experiment. That approach is not consistent with the experimental design, and it ignores the possibility that the effects of fungicide or inoculum may differ between experiments. (In randomized block designs, we assume that there is no block by treatment interaction.) With only two experiments, in a greenhouse, the first in early season and the second in later season (and where season may matter and is unreplicated), I probably would analyze each experiment separately and subjectively assess whether results are qualitatively similar.
One other question I had was in regard to my error terms. Is there a helpful source on determining these error terms dependent on my design and which factors I choose to run. I know if I do a one-factor rbd or a two factor rbd its just block. But for the binary study as a two-factor with replicates and the binary study if I just do one-factor or average the replicates is where I have some confusion.
Generalized linear mixed models are complex. Start with the Stroup paper linked above, and then move to his text Generalized Linear Mixed Models: Modern Concepts, Methods and Applications . This text Analysis of Generalized Linear Mixed Models in the Agricultural and Natural Resources Sciences goes into much less detail than the Stroup text and may be more accessible initially.
I hope this helps move you forward.
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