I would analyze such designs with one-way ANOVA, though I’m wondering whether factorial-ANOVA or nested-ANOVA can be used.
The research question is the following: is the behavior of a parasitoid affected by the presence of entomopathogenic fungi on the host-insect?
The variables are the following:
X1 = presence of host-insect individuals
X2 = application of entomopathogenic fungi (treatment)
X3 = exposure to parasitoid-insect (days post-treatment)
X4 = entomopathogenic fungi
Y = residence time (seconds), viz. time spent by the parasitoid on the leaf to search, handle and accept the host-insects
The experimental design is attached
A one-way Anova with a factor containg all the possible level-combinations of the original factors is mathematically identical to a full factorial design including all interactions, since the model will fit a mean for each and every factor combination. The problem may be that the one-way Anova does not readily give you the anwers to the possibly relevant questions about the size and/or significance of interactions.
An unbalanced design is not automatically a problem. It can be problematic when some other assumptions (symmetry and variance homogeneity of the residuals) are strongly violated.
I take it that it is some factor levels that are missing? You can analyze what you have ty relabeling the levels, or collect more data to fill them in. However, I am not sure about your data. Perhaps you can do a literature review to find out how others have handled this problem.
If you remove the levels of the factor(s) that are missing, you could then explain that in the text of the article.It is not cheating when you explain what happened.
Marcie Zinn
The "incomplete" character of the design is structural. If the level is untreated, you cannot have any level of the treatment, of course. So the problem does not rely on unbalancing. The design is nested.
In the part of the design where treatments X3 and X4 are relevant, treatments X3 and X4 are fully factorial so, in this part of the design, you can estimate and test the main effects of X3, of X4, and of their interactions.
I suggest you to act in two steps :
1st step: give to the empty cases of your table a level name at your convenience (NONE, or NULL, or whatever you like)
Analyse the model Y~X1+X1:X2, for instance with the Cox Proportional Hazard model, as residence times are not supposed at all to be normally distributed. X1:X2 stands for the interaction of X1 and X2. Without X2 as main factor, this is equivalent to the nesting of X2 within X1.
You'll then estimate the overall effects of three combined levels :
- No host:untreated
- With host:untreated
- With host: treated
2nd step: drop out the untreated cases and analyse the model
Y~X3*X4. You'll have the effects of each exposition time (trhree levels) , of each fungus (2 levels) and of their interaction (8 levels)
If you are reluctant to drop out the untreated cases in the second step, you can study the global model
Y~X1+X1:X2 + X3*X4 but I'm not quite sure that this model be estimable. I'll try it on simulated data and tell you. If the model is not estimable, you'll get an error message saying something like "singular matrix. If it works, you should get directly the results of step1 and step 2.
Regards
Dear Jean-Sébastien,
many thanks for your comment; it is very interesting!
Your proposal about a 2-step approach is a good option. However, I would prefer a one-step approach like Y~X1+X1:X2+X3*X4. On the other hand, it would be useful to test the main effects X3 and X4, therefore, what do you think about Y~X1:X2+X3+X4+X3*X4 (if estimable)?
In my experiment, the (residues of) residence time was normally distributed after a log-transformation, and the framework was homoscedastic. Thus, altough not conceptually correct, could I would use ANOVA?
Another question, which is just my curiosity: would it be correct to analyze this design with one-way ANOVA, or would it be incorrect due to the dependence of some factors? In other words, is one-way ANOVA always applicable?
I am sorry, but the experimental design is not clear to me.
What is the meaning of the No-host-insect / untreated treatment?
What is the objective of comparing this treatment with all the other treatments?
To my view, the No-host-insect / untreated could only be compared with the host-insect-untreated treatment.
Sorry if I have misunderstood the experimental design.
Regards,
Juan
Thanks Juan,
the ojective of the study is to evaluate the activity of a parasitoid (Y) against on leaf discs in the presence/absence of host-insects (X1) when treated (or not; X2) with entomopatogenic fungi (X4) at different time (X3).
Two more questions:
- is there any literature citation documenting that one-way ANOVA is always possible, even when dependancy between factors occur like in factorial designs?
- In the 1st step proposed by Jean-Sébastien Pierre, could the unbalancing be a problem ('treated' has more observations than 'untreated')?
A one-way Anova with a factor containg all the possible level-combinations of the original factors is mathematically identical to a full factorial design including all interactions, since the model will fit a mean for each and every factor combination. The problem may be that the one-way Anova does not readily give you the anwers to the possibly relevant questions about the size and/or significance of interactions.
An unbalanced design is not automatically a problem. It can be problematic when some other assumptions (symmetry and variance homogeneity of the residuals) are strongly violated.
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