Hello researchers, I have a question about the type of experimental design to describe my data and details of how to implement it using R or SPSS.
I have three types of roses, three types of soil, two types of graft (soft and hard), and two (or three) types of growth hormones.
I have tried different possible combinations of the above factors each with 3 repeated measurements, and some variables such as the number of roots, length of roots, etc were measured.
I want to compare the univariate mean of each variable for different combinations.
Which kind of experiment is useful to achieve the goal? For example, I want to know which combination performs better on the number of roots.
How to implement it in R.
Finally, I think it is also meaningful to anaylse the above data using multivariate DOE, for example testing the eqaulity of the mean of the vector of, let's say. root length and number of roots for each combination
by the way, which kind of statistical assumption do I need to check, in univariate and multivariate cases.
The sample R codes are needed, please.
Many thanks in advance,
Hamid
Basically, you have a factorial experiment with the repeated measures representing replications. Unfortunately, a full factorial experiment would have 36 (or 54) cells with many levels of interactions. A full analysis would require four simple effect analyses at the two-way levels, many simple, simple effects at the three-way levels, etc. This is not feasible and use up most of the degrees of freedom for interactions that otherwise would improve (reduce) the MS error. One approach to avoid these problems would be to treat the 36 cells as main effects using the released degrees of freedom to compute the MS error and then run multiple comparisons among the 36 cell means. This would tease out possible interaction effects including all possible simple, simple-simple, etc effects. The choice of multiple comparison method you employ would determine the overall error rate. For example, if you chose the Tukey Honestly Significant Difference Procedure, you would control the experiment-wise error-rate, but it would be very conservative. If you chose the Least Significant Difference Procedure, it would be the most powerful, but would not control the experiment-wise error-rate. There are many other options in between. Good luck with your analyses. Jim McLean
I think it's reasonable to assess each dependent variable separately. There's of course some danger when you are recording multiple dependent variables, that you increase the chances of finding some effect or trend simply by chance alone.
To me, it's not clear from the question if you have already collected the data or not. If not, you want to look at a source for different "design of experiments". As James E. McLean notes, you are essentially constructing a large factorial experiment. But there may be designs like split-plot design, strip-plot design, or incomplete block design, that are conceptually more complex, but can save you on the number of experimental units (plants) you need, or allow you to arrange plants in blocks for convenience (like making a block all one soil type). "Design and Analysis of Experiments" by D.C. Montgomery is book you might find.
In terms of analysis, the exact codes may depend upon the specific experimental design you have. In any case, you will using some kind of "general linear model", or perhaps a "mixed effects model". The assumptions will be similar for e.g. all general linear models.
You will probably want to construct one model with all the factors (soil, graft, rose type). And then conduct an analysis of variance. And then use "estimated marginal means" (emmeans) from the model to compare the levels of significant factors or interactions.
With the caveat that I wrote it, there's an example of using a general linear model for a two-way factorial analysis of variance here: https://rcompanion.org/handbook/I_08.html . A few spots in the code need to be updated. IN particular, the lsmeans package can be updated to emmeans, and the code for a compact letter display (cld) needs to be changed.
Overall, a factorial design is ideal. However, to find the best combination, there is no need for complicated models. Conducting all possible combinations and using a clustering or weighting method suffices.
3 (Roses) x 3 (soil) x 2 (grafts) x 3 (hormones) factorial. MANOVA or Covariance should be employed to determine how best to move forward with analysis. If not comparing across rose, then rose can be eliminated from model; with each being independent. This is still a complex model in which simple effects will have to be employed or contrast statements to tease out the a priori hypotheses.
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