Splitting groups for statistical comparisons I have a question that I cannot answer based in my biological background.
I conducted one experiment with 5 replicates and 4 treatments. 
There were two goals: the first goal was to elucidate if our new method during fermentation could give us better results in relation to a standard. The second goal was to investigate if flask size affects the fermentation.
Well, when I performed one-way ANOVA with all treatments, I got the following with Tukey's test (letters mean the statistical grouping - 95% confidence):


*

*Standard in small flasks (A) 

*Old method in small flasks (AB)

*New method in small flasks (B)

*Old method in big flasks (B)


When I removed the treatment with the big flasks and ran one-way ANOVA, I got the following results in Tukey's test (letters mean the statistical grouping - 95% confidence):


*

*Standard in small flasks (A)

*Old Method in small flasks (B)

*New Method in small flasks (B)


The post hoc test without one treatment (Old Method in Big Flask) makes sense, but my question is if it is statistically fair to analyze as two different experiments:


*

*Old method in small flask VERSUS Old method in big flask

*Standard in small flask VERSUS Old method in small flask VERSUS New method in small flask


In short, I'm asking if I could run two one-way anova with post hoc tests as if I were analyzing two different experiments (subsets of the single experiment). I know that it may sound dumb, but I can't figure out it alone.
 A: Thank you for your edits. Just as I suspected, you have a single experiment with two factors: Method and Flask Size. Your experiment includes 5 replicates for each of these combinations of factors: 
Method      Flask Size
------      ----------
Standard    Small
Old         Small
New         Small
Old         Big

I find it easier to think of your problem in a linear regression setting, so my answer will reflect that. 
Because you haven't included all possible combinations in your study, you can only consider a linear regression model with main effects for Method and Flask Size. (Alternatively, you could consider a two-way ANOVA model with main effects for these factors.) 
Your linear regression model would quantify the main effect of Method via two dummy variables and the effect of Flask Size via a single dummy variable. The model would look like this:
Outcome = beta0 + beta1*DummyOld + beta2*DummyNew + beta3*DummyBig + error

where 
DummyOld = 1 if a replicate comes from Old method and 0 if that replicate comes from either of the remaining two methods

DummyNew = 1 if a replicate comes from New method and 0 if that replicate comes from either of the remaining two methods

and 
DummyBig = 1 if a replicate comes from a big flask and 0 if that replicate comes from a small flask

Note that this model assumes that the effect of Method on the outcome is the same for each Flask Size. It also assumes that the effect of Flask Size on the outcome is the same for each Method. (You can't assume different effects because your study does not include all possible combinations of Method and Flask Size.)
Your study was set up to answer the following research questions (translated in statistical language): 


*

*For a particular flask size, is the mean value of the outcome smaller/larger for the new method compared to the standard method? (Pick smaller or larger, depending on what it means for the new method to be "better" than the standard method.)

*For a particular method, is there a difference in the mean value of the outcome between big flasks and small flasks?
These are the only hypotheses you need to carry out and because they are planned, you can test them directly, without conducting an omnibus F-test for testing the overall significance of the model. 
You can address the first research question by testing the null and alternative planned hypotheses: 
Ho: beta2 = 0 versus Ha: beta2 < 0 (or beta2 > 0, depending on how you define better)
You can address the second research question by testing the null and alternative planned hypotheses: 
Ho: beta3 = 0 versus Ha: beta3 != 0.
So all you need to do is perform these two tests of hypotheses based on the suggested linear regression model. Some people don't adjust the p-values for such tests for multiplicity, arguing that the hypotheses being tested were planned at the study design stage. Personally, I prefer to adjust them for multiplicity and report both adjusted and unadjusted p-values. (Multiplicity arises from performing more than one test from the same data - in your case, two tests.)
You can also report a one-sided confidence interval for beta2 and a two-sided confidence interval for beta3. 
The beauty with the approach suggested here is that:


*

*It acknowledges you have a single study;

*It uses a model which can be used to test two sets of pre-planned hypotheses, inspired by your two research questions;

*It obviates the need for performing an omnibus test of significance of the full model.
I do not use Minitab but I provided enough detail here for you to be able to perform the suggested analyses in Minitab.
