Correction.
Taguchi is not the best approach for analysing interactions.
ANOVA is the right approach to the analysis.
Taguchi designs really need to rely on an outer array of noise variables as well, which I cannot see that you have used. Seeing as stability to noise or the environment is what Taguchi is all about, by not testing at two environment levels (ie two columns of response) sort of goes against what you would be using a taguchi design for.
You would be better off with a standard factorial design here, with 2 levels per factor I think. Unless you have a specific reason why you need 3 levels per factor.
If you are fitting only linearity and you have numeric inputs you do not need 3 levels on each factor. You can use a centre points instead to check for curvature.
If you have categorical factors with 3 fixed settings then the 3 level factorial designs can be useful.
There is no point creating a factorial design with 3 levels per factor if your only going for a linear fit on numeric inputs. The standard 2 level factorials give you the simplest and least runs for a design.
If you absolutely have to have more than 2 levels on a factor you can use the choice to select optimal design to tell it which interactions you need and get it to reduce the design from there.
I probably haven't explained this enough, but I am worried that you are diving into Taguchi designs without an appreciation of how you are supposed to use them, and trying to use them for something they are not very good at, eg: finding interactions.
My advice on doe is start with the factorial designs as a base of understanding, then move to look at why and where you might use the others.
I get worried by the buzzword that is Taguchi.