R - lmer() for single factors and interaction give different results?

I was thinking that the difference between

lmer(fit~ a+b +randomF)
lmer(fit~ a*b +randomF)
lmer(fit~ a:b +randomF)


was that the first says "search for fixed effects only", the second "search for fixed effects and interactions" and the third "search for interaction only". Am I wrong?

Particularly, in my case, if I write my model without interactions like:

f.e.model.composers = lmer(Score ~ Relation_PenultimateLast + (1|TrajectoryType) + (1|StimulusType), data=datasheet.complete.composers)


I get a fixed effect like here:

Random effects:
Groups         Name        Variance Std.Dev.
TrajectoryType (Intercept) 0.008906 0.09437
StimulusType   (Intercept) 0.018655 0.13658
Residual                   1.349897 1.16185
Number of obs: 2200, groups:  TrajectoryType, 25; StimulusType, 4

Fixed effects:
Estimate Std. Error       df t value Pr(>|t|)
(Intercept)               2.92128    0.10093  9.78000  28.944 8.35e-11 ***
Relation_PenultimateLast  0.09968    0.02595 23.00000   3.841 0.000834 ***


while, if I define my model including an interaction with a different factor:

f.e.model.composers = lmer(Score ~ Relation_PenultimateLast*Test + (1|TrajectoryType) + (1|StimulusType), data=datasheet.complete.composers)
summary(f.e.model.composers)


the fixed effect that was previously detected (Relation_PenultimateLast), disappears:

Random effects:
Groups         Name        Variance Std.Dev.
TrajectoryType (Intercept) 0.00892  0.09444
StimulusType   (Intercept) 0.01866  0.13659
Residual                   1.34865  1.16131
Number of obs: 2200, groups:  TrajectoryType, 25; StimulusType, 4

Fixed effects:
Estimate Std. Error        df t value Pr(>|t|)
(Intercept)                  2.839e+00  2.040e-01 1.567e+02  13.918   <2e-16
Relation_PenultimateLast     7.525e-02  6.712e-02 7.662e+02   1.121    0.263
Set                          5.492e-02  1.182e-01 2.170e+03   0.465    0.642
Relation_PenultimateLast:Set 1.629e-02  4.127e-02 2.170e+03   0.395    0.693

(Intercept)                  ***
Relation_PenultimateLast
Set
Relation_PenultimateLast:Set


Is this behavior theoretically correct, and I am doing something wrong myself, or is this a weird behavior?

• Yes, when you add new terms, new structure, and new information to your model, it will change. You should be surprised if you add new terms to a model and the old coefficients don't change. – Gregor - reinstate Monica Jan 11 '18 at 0:07
• Thank you @Gregor, two questions: 1) how to move it to stats.stackexchange? 2) How to interpret the diversity between the two results? Have you got any link to help me learn the difference and how to choose the best model for my goal? – Luca Danieli Jan 11 '18 at 0:11
• Also, it's not fair to say the "fixed effect that was previously detected disappears". It changes from 0.0997 to 0.0752, so it decreases by about 25% when Set is 0. I have no idea what your range of Set is, but when Set is 2 the Relation_Penultimate effect is about the same as the original - not to mention the effect of Set itself. – Gregor - reinstate Monica Jan 11 '18 at 0:14
• To your questions 1) I don't know if you can migrate yourself - you could delete the question here and open a new one on the stats site, or wait for automatic migration (takes 5 votes, you have 2 so far). 2) I have no idea what your goal is. Personally, from years of reading Andrew Gelman's blog and book, I think statistical significance is overrated. Using domain knowledge, if the interaction makes sense keep the interaction. As for resources, I like this book. – Gregor - reinstate Monica Jan 11 '18 at 0:17
• @Gregor, thank you! I have read some discussions that suggest p_values to be not precise in representing effects for mixed-effect analysis. Is this an example? Also, my range of Set is 2 indeed. How did you read the information that when Set is 2, the Relation_Penultimate effect is about the same? I would like to learn how to read these kind of information – Luca Danieli Jan 11 '18 at 0:25