Yes it's good

You can also use bonferroni.

Many people are reluctant to use Fisher's LSD because they believe it does not control the family-wise (FW) alpha adequately. What many of those people do not know is that when there are exactly k=3 groups, it does in fact ensure that the FW alpha = the per contrast alpha. So if you mean you want to use Fisher's LSD to make all pair-wise comparisons among the 3 means for each main effect, that should be just fine. If you need references to convince someone of this, see the following. HTH.

Article A note on the power of Fisher's least significant difference procedure

Baguley, T. (2012). Serious stats: A guide to advanced statistics for the behavioral sciences. Macmillan International Higher Education. (https://www.macmillanihe.com/page/detail/Serious-Stat/?K=9780230577183)

Howell, David C. (multiple years & editions). Statistical Methods for Psychology.

PS- For those who may not have heard, Dr. Howell died just over a year ago. You can read an obituary here:

https://www.legacy.com/obituaries/burlingtonfreepress/obituary.aspx?n=david-charles-howell&pid=190521104&fhid=4514

There are 9 treatment-combinations. If they should all be tested against each other, 32 (!) tests will be performed, and LSD will NOT control the FWER across these tests. So: no, LSD is not ok (if you want to control the FWER). You should better use Tukey's HSD for all-pairwise comparisons. This does control the FWER. You can also use Bonferroni's method, but that will be a bit too conservative.

My concern would be more why you should want to compare the level-combinations. Usually, more-factorial designs are planned to investigate the interaction between the experimnntal factors. With 2 3-level factors, there are (3-1)*(3-1) = 4 interaction terms in the (2-factorial) model. So maybe the relevant number of tests to perform is not 32 but just 4, what will considerably reduce the need for multiplicity correction - and adress the possibly more relevant scientific questions!

Bruce Weaver, in the first link (to the 2006 paper in Pharmaceutical Statistics), the author writes that in a study with 3 groups (placebo, dose1, dose2), both doses can be compared to placebo but not against each other when using Fisher's LSD. I don't understand this. The FWER is kept for all pairwise comparisons among these groups, including doese2 vs dose1. Where is my mistake?

And further, phase-II studies are not just to show the effectiveness, but also the savety and to monitor adverse effects, what is not achieved by jusing small sample sized based on hypothesized effects against placebo. I am further a bit distrubed by the example that drug doses are compared to placebo. In phase-III studies, effectiveness must be shown in comparison to the best known treatment options (what is usually not a placebo treatment). See e.g. https://www.healthline.com/health/clinical-trial-phases#phase-iii: "...investigators need to demonstrate that the medication is at least as safe and effective as existing treatment options." or https://www.fda.gov/patients/clinical-trials-what-patients-need-know/what-are-different-types-clinical-research: "Researchers confirm its effectiveness, monitor side effects, compare it to commonly used treatments, and collect information that will allow the experimental drug or treatment to be used safely" or https://www.australianclinicaltrials.gov.au/what-clinical-trial/phases-clinical-trials: "Phase III studies [...] by comparing the intervention to other standard or experimental interventions (or to non-interventional standard care).  Phase III studies are also used to monitor adverse effects and to collect information that will allow the intervention to be used safely" (emphasis mine).

Good morning Jochen Wilhelm. If one was going to make all pair-wise comparisons among the 9 cells, I think it's even worse than you said. I get 9-choose-2 = 36. ;-)

. display comb(9,2)

36

I was not endorsing all pair-wise comparisons among the 9 cell means. Here's what I said, with emphasis added:

"So if you mean you want to use Fisher's LSD to make all pair-wise comparisons among the 3 means for each main effect, that should be just fine."

In other words, for main effect of A: A1 v A2, A1 v A3, A2 v A3. And for the main effect of B: B1 v B2, B1 v B3, and B2 v B3. According to the usual logic or Fisher's LSD, these pair-wise contrasts would only be carried out if the F-tests for the corresponding main effects are significant.

I hope this clarifies things.

Sudarshan Kharal , what software are you using? It may have features that allow you to generate various kinds of interaction contrasts. Here are some examples for Stata:

https://stats.idre.ucla.edu/stata/faq/how-can-i-explore-interactions-using-the-contrast-command-stata-12/

If you say what software you use, someone may be able to point to similar examples for that program.

Bruce Weaver, you are right... take my 32 as a first approximation :) I don't know what I calculated there :/

Your point is clear and valid to compare the 3 main effects. I thought this is not what is usually intended, and it often is not very meaningful, since the point of the two-factorial design is (usually) that comparisons between the levels of A depend on the level of B (and vice versa), so interactions do play a role. I think it's now clear, and Sudarshan can decide what way suits his demands.

No worries, James Byron Nelson. I never met Dave in person. But I did have some e-mail conversations with him, and always found him to be a true gentleman.

Thank you Bruce Weaver and Jochen Wilhelm for your insightful comments. I am using GenStat for the statistical analysis and will use DMRT to compare the 9 mean values from the interaction effect. For individual effect, I will go with your suggestions.