interaction term in multilevel analysis using lmer() function - r

Question

level 1 variable:
income - continuous
level 2 variable:
state's general whether: three leveled categorical variable: hot/moderate/cool
used effect coded, and generate two variables because it has three levels.
(weather_ef1, weather_ef2)
enrolled in university - binary : yes/no ( effect coded. yes = -1, no =1)
DV:
math score
grouping variable: household
model 1: (fixed slope)
Dv is predicted by income, enrollment, and the interaction between enrollment and income.
in this case,
lmer(y~ 1 + income + enrollment +income*enrollment+ (1|householdID), data=data)
lmer(y~ 1 + income + enrollment +income:enrollment+ (1|householdID), data=data)
: is it for interaction? or * is it for interaction?
further, do I have to do factor(enrollment)?
or is it okay because it is already effect coded?
model 2: (fixed slope)
DV is predicted by income, weather, and interaction between income and weather
lmer( y ~ 1 + income + weather_ef1 + weather_ef2 + weather_ef1*income
+ weather_ef2*income +(1|houshold_id), data)
lmer ( y ~ l + income + weather_ef1+ weather_ef2 + weather_ef1:income
+ weather_ef2:income + (1|houshold_id), data)
Still confusing * is right or: is right.
I think the effect code variables are already effect coded, so I don't have to
do use the factor(weather_ef1) things.

Answer 1

From the documentation (use ?formula):
The * operator denotes factor crossing: a*b interpreted as a+b+a:b.
In other words a*b adds the main effects of a and b and their interaction. So in your model when you use income*enrollment this is the same as income + enrollment +income:enrollment. The two versions you described for each model should give identical results. You could just have used:
lmer(y~ 1 + income*enrollment+ (1|householdID), data=data)
which also describes the same model.
If your variables are effect coded then you don't need to use factor but be careful about the interpretation of the effects.