What's your total sample size? You probably don't have enough individuals per group.

I second Daniel. You surely don't have replicates for all the combinations of all the levels of all the factors. You will either need more data or use a simpler model (is the 4-way interaction and the 4 3-way interactions really important? Will you be able to interpret these interactions? Will they help to understand the connections of these variables to y?)

I agree with Daniel and Jochen, especially about the interpretability. I have always had difficulty with the interpretation and visualization of four-way interactions (but this is just me). Rarely do I use three-way interactions.

Sample size is also a really important consideration, as others have noted. If you have a within-group sample size of even 5 and two levels per factors, that's 2^6 (64) unique treatment combination, and a required n=360. If the number of levels per group increases to 3, n=3645, or 5*(3^6). And that's for minimum sample size.

I think others will agree that it's probably a good idea to reevaluate the treatments for which you need parameter estimates and if the study design might benefit from inclusion of a hierarchical random effect structure. It seems to me this might be the case.

Hello Shantanu

If your factors are given leveles of quantative variables you better use regression analysis (lm(y~T,conc,ca,gam)) and pick interaction terms based on your professinal knowleddg about your research. Otherwise you have many source of variation in your model.

Also, (if you have qauntative variables as factor) it is better to design your experiment using response surface. 

If it is not the case, hierarchical analysis can help you to handel your analysis more friendly :). You can dived your experiment based on 1 or 2 facrors. So you'll have f1*f2 sections in the results of your thesis. Finally you can do targeted ANOVA for conclusions.

Regards,

Hormoz

thanks all for your answers.