Anik Dutta, can you describe and provide some summary information of trait? Is this a continuous number or interval? If interval, does this represent nominal traits or ordered ones (does ordering of traits make sense)? Know more about the response variable will help in suggesting transform or do not transform vs. other approaches entirely.

Hi, Caleb, one is a continuous trait (percentage of leaf area covered by lesion) and the other is also a continuous trait (density of spore in a square cm of leaf).

The way I'm understanding your design "Replication" is being used, essentially, as a block because there are differing environmental conditions between the chambers. I don't think it makes sense to for this to be used as a random variable in this experiment. What I'm not clear about is if there are replicates within each chamber? Without replications I'm not aware of how to separate the effect of chamber from the experimental effect... If only one replicate was done in each chamber, I'm afraid you'll just have to accept the variation that might be caused by the different chambers (maybe, hopefully someone else knows more/better).

With that said and assuming there's only one replicate in each chamber, I think you can simplify your analysis to: trait ~ Iso * Host (the asterisk is short hand for main and interacting effects).

As far as transformations, I tend to use the Box-Cox power transformation. You can transform using the lambda value or use lambda as a guide to other more common transformations (e.g., log). You might refer to this SE thread for more info: https://stats.stackexchange.com/questions/213011/understanding-output-of-powertransform. The YouTube video here might be helpful as well: https://www.youtube.com/watch?v=vGOpEpjz2Ks.

Let me know if you need more information or assistance. Be sure to clear up any misunderstandings I have about your design. Cheers.

Hi Caleb A. Aldridge thank you for your response. Yes, I had three chambers, basically three blocks or you can treat them as replicate. Because I repeated the same experiment three times and put the plants in each of the chamber. For each plant genotype, I have three individual plants in one pot in each chamber. So, it make 9 plants in total when combined. I included individual plant in my model but it did not improve the model and it only explained negligible amount of variation.

So, what do you think now? Can't I decompose the variation using replication as a random effect (although only 3 level) with interactions?

Regarding the transformation, I wanted to know if I used a transformed value for ANOVA, do I have to stick with transformed value for further analyses e.g. correlation or PCA? or I can use original data for other analyses and transformed data only for ANOVA? This question might be silly, but still I am a bit confused.

Anik Dutta, you seem to be confusing some terminology; no worries I can explain.

The experimental design are the conditions (i.e., variables and combinations of variables) placed on subjects. The experimental design you describe is factorial, i.e., does the response vary among conditions produced by categorical variables? Take the example on page 5 in Montgomery's Design and Analysis of Experiments, Fourth Edition (1997): does type of ball and type of driver influence golfers' scores? (See fig. attached). Each point in the square represents a combination of factors.

This experimental design can be replicated so that an estimate/understanding of the "noise" can be gained. If some of the noise is due to a nuisance factor, it can be compensated (i.e., analysis of covariance) or eliminated (i.e., blocking). A complete block design includes full replications of the experimental design and labeling of block. (Measurement of the covariate (e.g., continuous variable) is analogous in analysis of covariance). You can also have replicates within blocks.

From what I read you have 9 plants total, 3 of each genotype. The 3 plants of each genotype were placed together in the same chamber. Each of the 3 plants within a chamber received a different treatment (?). It seems like genotype and chamber are confounding... Not sure if/how you can separate the effects of chamber from genotype without, at least, additional experimentation.

If no confounding then you could include chamber as fixed block. In R it'd look something like:

response ~ block + factor_1 * factor_2

And, I believe using the Box-Cox transformation, as I mention above, would help reduce deviation from assumptions. I'd approach further analyses you mention (PCA and correlation) separately and transform as necessary for that analysis.

I encourage a dive into some chapters on experimental design... Montgomery I mention above, Scheiner and Gurevitvh 2001, or Gotelli and Ellison 2004 (or a mainstay text in your field).

I don't really follow the science here but the reason you do transformations is to get a more correct sampling distribution for your tests and confidence intervals so that you know whether or not it is likely that you can believe your results. As it stands your probabilities are likely correct only for the transformed case. See Montgomery, Design of Experiments or similar books. Best, David Booth

Thanks David Eugene Booth for your suggestions. No, the three plants from each genotype received the same treatment. I just repeated the experiment three times.

Sounds like a completely randomized design( CRD) with replication. You have an interaction term in your model if no no plants got more than 1 treatment then you have the wrong model(a factorial). See the attached handout and my original suggestion of a completely randomized design holds. All is in Montgomery..If I understand you correctly this just is a 1-WAY ANOVA (i.e. CRD). Best, David