Hello Matteo,

For me, the simplest approach would be to apply a three-factor anova:

(a) Mowing regimen (between- factor, 3 levels)

(b) Slope of plot (between- factor, unknown number of levels)

(c) Measurement occasion (within- or repeated-measures factor, 72 levels)

Or, alternatively referred to as a "two-between, one-within" design. How many replications (plots) would be in each mowing x slope combination? If only 1, then several interaction terms will be unavailable (though all main effects are available). If 2 or more, then all interactions could be estimated.

I hope I haven't misunderstood your query.

Good luck with your work!

Maybe it is really easier to think about your design in the mixed model framework.

Thank you Holger Heuer David Morse Jan Pablo Burgard

Indeed, I guess that I might use a Three way-mixed ANOVA model considering:

• two between-subjects factors (Mowing regimes and Slope)
• one within-subjects factor (Dates of measurements)

Please find attached a sketch of the experimental design

Sounds good.

Hello to everyone, I just to share an update on this issue.

Running a mixed model proved to be the best approach so far, but I've faced some issues concerning the block.

The "problem" here is that I have 3 blocks but for each block only one CO2 measurement (dependent variable) per time

Whenever I try to run the mixed model (on SPSS 13) the program gives me an alert reporting" insufficient degrees of freedom for the treatment"

Searching on the web I've found some examples of CRBD RMANOVA but with about 20 repetitions of the same treatment within each block. Indeed running the mixed model with more replicates (within each block) has returned no alerts.

So, I gave myself two possible explanations:

- Block is not really a within-subject factor (might it be a "between-within" subject factor?)

- RMANOVA in RCDB needs to have more than 2 replicates for each treatment within each block

I'm very curious and I would be very happy if you can share your opinion about these hypotheses, thanks!

dear Matteo Francioni , in field experiments where you measured a trait e.g. CO2 emission) more than one and likely more than 3-5 times in each plot, plot is indeed a subject even if per time you have just one measurement per plot. I'm arguing that you actually imposed the plot as a subject (e.g. in SAS, you can do it by the subject statement in GLM and mixed models and by the random statement in Glimmix models).

So block is neither a within-subject, nor a between-subject factor. They are the subject (and are at the same time random), not a factor, nor the levels of a factors

The amount of degrees of freedom needed does not actually depend on the model, but on the amount of total treatments and interactions (and random factors, too). Indeed, if your experiment is quite complex (many treatments with many levels, each and few replicates and pseudoreplicates) you may inadvertently drain all the degrees of freedom for both the fixed factor and interactions and the the random factors, if any, and this would leave you with no DF for the residual error.

This can occur in complex experiments, especially with time-treatments. In such a conditions, you need to collapse some random factor in the total error. By doing so, you will lose power of the analysis, but indeed will be able to estimate the chance your outcomes are somehow depending on your treatments.

Also, the more complex the experiments, the most needed an estimation of the DFs rather than they computations. Usually we use Kenward-Roger approximations for the DF for datasets with n2000.

Lastly, please take into account that if your dataset is unbalanced or its distribution (and distributions by treatment) are not close to the gaussian, separation of means in treatments needs a correction for the distribution. In such a case, I suggest you to use a Tukey-Kramer correction for qualitative data and a Malahanobis distance for multivariate analyses.

I recently published a paper including the SAS pipeline used in the supplementary material. Feel free of catching a look at it and contact me if you need further assistance. The pipeline should also work on JMP but I guess it can be re-computed for SPSS, too.

Please mind that that pipeline has a treatment that is at the one time fixed and random.

Cheers.

Sergio

Saia et al. 2020 Growth-promoting bacteria and arbuscular mycorrhizal fungi differentially benefit tomato and corn depending upon the supplied form of phosphorus. Mycorrhiza DOI: 10.1007/s00572-019-00927-w

Article Growth-promoting bacteria and arbuscular mycorrhizal fungi d...

Sergio Saia Thank you very much for your active participation in this discussion and thank you for sharing your interesting article with everyone.

To the best of my knowledge, a two-way mixed ANOVA model has one within factor (such as time, this might be pre-post-after treatment, years, months, weeks and so on.) and one between factor (such different doses or fertilizer on a crop, a different mowing regime on grassland, etc.).

A two way mixed ANOVA model would work perfectly in the case I had a complete randomized design. However, I have to deal also with blocks (slope) as a source of variation.

So, I tried as suggested also by David Morse and Jan Pablo Burgard , to use a within (time) and between-between (mowing regime-slope) factor analysis, a.k.a. Three-way (WBB) mixed model ANOVA.

As David Morse pointed out in his first reply, since I have only 3 levels for slope (i.e., 3 blocks) and one level of different mowing regimes (i.e., treatment) in each block, some interactions are not detectable and probably that's why SPSS gives me the alert "post-hoc was not carried out because you do not have enough degrees of freedom for the mowing" (or something like this).

So, now the question is: should I remove the block from the three-way mixed model and include blocks again when I want to detect differences between treatments within each date of measurements?

Or maybe I can "bypass" the block design ad assume that this was a complete randomized design (since I do not have more than 1 level of mowing per each block).

In this context, as suggested by @Sergio Saia, (in agronomy trials at least9 blocks are commonly not considered as within/between factors, so I'm wondering if a BBW RMANOVA is the correct model, might it be only a simple BW RMANOVA to be carried out considering the block?

Thank you

So, it seems I've found the problem within the mixed model in SPSS.

Briefly, by default SPSS consider a full factorial design, thus it calculates ALL the interactions between factors. This includes also the interaction between Blocks and Treatments which is a violation of ANOVA assumptions. In other words, a BBW mixed model applied to my example produces the following results:

• Year x Treatment
• Year x Block
• Treatment x Block (not OK)
• Treatment x Year x Block (not OK)

By attempting to calculate the interaction between Treatment and Block, SPSS does not have degrees of freedom for the error (please see the explanation above provided by Sergio Saia). This error produced the alert I mentioned.

This problem can be solved by using a custom model (with no interactions between block and treatment).

I thank everyone who joined this open question and provided help to solve this

Matteo Francioni , just add as a random factor only Year and Block in year [Block IN year is not the interaction of bloc x year), do not compute any interaction between Treatment, from the one side, and block or year from the other side.

Interactions between a fixed factor (the treatment) and a random factor (year or block) are wrong. It would imply that you can control a random factor, whereas this is not your case.

Sure you have to add interaction year and block especilly when your trial was over 3 year or more

Some advantages of blocking pointed out include increased precision with fewer runs, less interruption of operators' time, estimation of variation over time (between blocks), and the easy addition of additional blocks for even higher precision. Results using blocking are compared to analysis without blocking.

Replication is somewhat built in to the block experiment. While the full experiment is not repeated (the usual idea of replication), the pattern of runs within a block can be used to replicate various factors and learn about the factor's effect across blocks (over time). Replication adds generality and accuracy to the experiment described.

So you should do blocking to remove the error and get high precission

I am new in this and I also want to perform a repeated ANOVA with complete block design. The problem here seems to be lackness of degree of freedom. Can you run a split-plot design? Take either the slope or the mow as the main factor, and the other as the sub-factor. That will give you enough degree of freedom. However, I don't know how the time factor will behave in this design.

Hi Zheng Cao

There was no reason why I should consider slope as a main factor, in my case. "Slope" was considered simply because the experimental area was placed on a slope and this could have produced some issues with variability. That is why I've included it as "blocking variable". Thus, there is no reason the "change" the experimental design from CRBD to a split-plot.

Anyway, If you share with us the Exp design and the variables you included I'm sure you can get some useful advice