I have conducted a multi-way ANOVA test, multiple independent variables were significant, but I need to see how the levels of these significant variables differ from each other? Since it is not a one-way ANOVA, I can't use Tukey's HSD test or even the pairwise t-test ? What is your suggestion?
First you want to find out if there are interactive effects between your independent variables on the response variables. Then choose your comparison with or without interactive effects. For example, if two of your variables are interactive then choose to conduct post-hoc on those two together. Tukey's hsd is still valid on these analyses on a pairwise basis. If your treatment groups are large the power of analyses will go down. Other possible options are Bonferroni analysis. There is no specific guidelines for these situations. But these could help.
Dr. Arumugam Thiagarajan
Thank you, actually, I have conducted the ANOVA regardless of variables interaction, because first I am interested only in each's variable individual effect on my response when all of them are included in ANOVA, I just wanna see the contribution of each variable on my response, and second I have 11 categorical variables, imagine the number of interaction terms that would generate in the ANOVA table, and it won't be feasible since I have only 1000 data points.
5 out of 11 categorical independent variables were significant to my response (Hydraulic Conductivity) at alpha = 0.05, type of rock is one of the significant variables, it has 3 levels, I want to know which levels that differ from each other ?
The best approach is probably to use the estimated marginal means or least square means (emmeans, or lsmeans) procedure that can be called from the software you are using. (Available for example in R, SAS, SPSS...). This is usually the most flexible approach, and most tolerant of a wide variety of models.
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