Your DV cannot be Likert data, because this is not numeric. Presumably you converted (coded) the Likert data by numerical values which you then use in the analysis. You must show or argue that this makes sense and that the numbers measure something interpretable, and that sums and differences of such values do have any useful meaning. The next thing you have to do is to show that or argue why the normal error model is appropriate for this (kind of) data. Given all that, the analysis is ok.

The distinction between "significant" and "non-significant" has nothing to do with the fact whether or not there is an effect (or an interaction). You calculate a p-value, what is a kind of a measure of a "statistical signal-to-noise ratio". The lower the p-value, the larger is this ratio. Now you judge if your data gives a a high enough signal-to-noise ratio to say that you can clearly see a "signal" beyond the "noise" (in your data) and that you find it useful to start interpreting (at least the direction) of the signal you see. If the signal-to-noise ratio is too bad, you may find your data unsufficient to start a further interpretation. That's all.

As you already claimed that the interaction is "non.significant" you stated that you find your data insufficient to start any interpretation about the interaction. Full stop. This is NOT equal to the conclusion that "there is no interaction". Whatever the interaction may be: your data is considered insufficient to show clearly enough whether it is positive or negative.

A two-way ANOVA doesn't directly tell you which categories of X1 are significantly different from which other categories; only the follow-up tests tell you that. The two-way ANOVA only tells you three things:

1) The main effect of X1: whether there are differences in the DV associated with the different categories of X1 (technically, whether the variance between these groups is substantially greater than the variance within them)

2) The main effect of X2: whether there are differences in the DV associated with the two categories of X2

3) The interaction: whether the effect of X2 is different at different levels of X1, or vice versa

Stephen, the test on the interaction in this case (3x2 design) is really obtained by an ANOVA that compares the model  Y ~ X1*X2 to the (restricted) model Y ~ X1+X2.

You are quite lucky to have both Jochen and Stephen helping you with this question.

But, if I can add a question:

Can you clarify the nature of your dependent variable.  Is it a) a Likert item, like on a scale of 1 to 5, where the outcome is either 1, 2, 3, 4, 5; or b) a combined Likert scale of e.g. 20 items where the outcome is e.g. 20 - 100, and you have a variety of outcomes?

The dependent variable (= DV?) is categorical or at best "ordered.categorical", so forcing it into standard normal theory ANOVA is very questionable. I would use contingency tables or loglinear models. For ordered categories see the book by McCullagh and Nelder.

Dear Nila

There is no problem in your results, you have significant effects in the main effects but not significant effects in the interaction component.

That reflects the absence of variance interaction or cross interaction relationship between the two variables in all combinations.

Good Luck

@Khalid:

"That reflects the absence of variance interaction or cross interaction relationship between the two variables in all combinations."

No, you can not say this. This would be the interpretation of a "non-significant" result.

@Jochen

Yes, that what I mean for non-significant interaction only.

Thank you for very helpful answers.

It is a between-subject experiment, while I am also curious whether the absence of the interaction is because of my design is not good enough or the 2 IVs are cannot be interacted in nature (since both main effects shows differences even after post hoc)?

Or can I keep my further analysis into the differences between IVs (ignoring the insignificant interactions)?

Dear Nila

The non-significant result also represents a result and will tell the other researchers some things, so it is good to included in the scientific paper.

Good Luck