The basic problem is that you try to interpret a somewhat complex model by means of statistical significance. I'd recommend that you focus on the estimated parameters of your model. Extract them from the coefficient table and make an interaction plot and you will see what's going on. From the same table, confidence limits can tell you how certain your estimates are.

You may want to have a look at some draft chapters of the book I am writing, see https://www.researchgate.net/project/Book-New-statistics-for-the-design-researcher. The linear models chapter gives a detailed account on contrasts and interaction effects.

Thank you for your replies, however I don't have such coefficient table in my results output, I have rather least-squares mean table and their differences table.

I1 and I2 are categorical variables, the coding of their modalities is (1,2,3) for I1 and (1,2) for I2.

Sorry I am not able to provide the mathematical structure, however I can show the SAS procedure.

The model on SAS is as follow:

Proc MIXED DATA=table covtest;

CLASS Rand1 Rand2 I1 I2;

MODEL D= I1 | I2;

RANDOM intercept /sub=Rand1;

RANDOM intercept /sub = Rand2;

contrast 'A vs C' I1 1 0 -1;

contrast 'A vs C when I2 = 1' I1 1 0 -1

I1*I2 1 0 0 0 -1 0;

contrast 'A vs C when I2 = 2' I1 1 0 -1

I1*I2 0 1 0 0 0 -1;

LSMEANS I1 | I2 /cl diff;

run;

By "I1 is significant" I mean that F and p values are significants (4.32, 0.01). In sum, D is influenced by the three modalities of I1.

Excuse me, can you send the mathematical model?

Sorry but I am not able to provide the mathematical model. I am using directly the model on sas as presented above (to analyse a psychological experiment). My knowledge in statistics is limited.

Now I understand that the contrasts' significativity and the interaction's one could be different because contrasts test only two modalities of I1 while the interaction deals with all three modalities.

Ok.

You have pointed out that one contrast is significantly different from zero and another contrast is not. But an interaction does not test that; an interaction tests whether they are significantly different from EACH OTHER. These are not the same things; see Gelman http://www.stat.columbia.edu/~gelman/research/published/signif4.pdf. For example, if one contrast is p=.049, and another is p=.051 in the same direction, it is very likely that they will not be significantly different from each other, even though one is significant and the other is not.

Thank you Stephen, I understand better the difference between contrasts and interactions.

I also have read that the right way to test hypotheses is to test the targeted contrast but also the other orthogonal contrasts. The hypothesis is confirmed if only the targeted contrast is significant but not the others.