Consider the following lme4 model:
mod <- Y ~ X*Condition + (X*Condition|subject)
# Y = logit variable
# X = continuous variable
# Condition = values A and B, dummy coded; the design is repeated
# so all participants go through both Conditions
# subject = random effects for different subjects
summary(model)
Random effects:
Groups Name Variance Std.Dev. Corr
subject (Intercept) 0.85052 0.9222
X 0.08427 0.2903 -1.00
ConditionB 0.54367 0.7373 -0.37 0.37
X:ConditionB 0.14812 0.3849 0.26 -0.26 -0.56
Number of obs: 39401, groups: subject, 219
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.49686 0.06909 36.14 < 2e-16 ***
X -1.03854 0.03812 -27.24 < 2e-16 ***
ConditionB -0.19707 0.06382 -3.09 0.00202 **
X:ConditionB 0.22809 0.05356 4.26 2.06e-05 ***
In Different variance-covariance matrices of random effects per fixed-effect group in lme4 we were discussing the usage of the dummy() function to model random effects.
I was researching the function, and was trying to mimic te results that I got with classic modelling notation. I tested the following models which, according to my logic, should yield the same results as the mod:
mod1 <- glmer(Y ~ X*Condition + (0 + dummy(Condition, "A") + X:dummy(Condition, "A") + dummy(Condition, "B") + X:dummy(Condition, "B")|subject)mod2 <- glmer(Y ~ X*Condition + (1 + X + dummy(Condition, "B") + X:dummy(Condition, "B")|subject)
mod1 was not identical to mod
summary(mod1)
Random effects:
Groups Name Variance Std.Dev. Corr
subject dummy(Condition, "A") 0.8508 0.9224
dummy(Condition, "B") 0.8854 0.9409 0.69
dummy(Condition, "A"):X 0.1120 0.3347 -0.83 -0.73
X:dummy(Condition, "B") 0.1777 0.4215 -0.46 -0.64 0.75
Number of obs: 39401, groups: subject, 219
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.49297 0.06911 36.07 < 2e-16 ***
X -1.04807 0.04032 -25.99 < 2e-16 ***
ConditionB -0.19657 0.06299 -3.12 0.00181 **
X:ConditionB 0.23894 0.05060 4.72 2.33e-06 ***
The random effects results are different. Why is
Intercept != dummy(Condition, "A")
X != X:dummy(Condition, "A")
ConditionB != dummy(Condition, "B")
X:ConditionB != X: dummy(Condition, "B") ? What do the dummy variables mean?
mod2 was equal to mod
Random effects:
Groups Name Variance Std.Dev. Corr
subject (Intercept) 0.85052 0.9222
X 0.08427 0.2903 -1.00
dummy(Condition, "B") 0.54367 0.7373 -0.37 0.37
X:dummy(Condition, "B") 0.14812 0.3849 0.26 -0.26 -0.56
Number of obs: 39401, groups: subject, 219
Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 2.49686 0.06909 36.14 < 2e-16 ***
X -1.03854 0.03812 -27.24 < 2e-16 ***
ConditionB -0.19707 0.06382 -3.09 0.00202 **
X:ConditionB 0.22809 0.05356 4.26 2.06e-05 ***
This means that dummy(Condition, "B") = ConditionB in mod. But what does then dummy(Condition, "A") in mod1 mean?
Update 1, according to @amoeba's answer
So, if I got this right, the imaginary data matrix would look like this:
Intercept(X) Intercept ConditionBX ConditionB dummyA(X) dummyA dummyB(X) dummyB
S1 A 1 5 0 -2 1 5 0 0
S1 B 1 5 1 -2 0 0 1 3
S2 A 1 2 0 -1 1 2 0 0
S2 B 1 2 1 -1 0 0 1 1
In the first case it stands: $Y_1=Intercept*InterceptX+ConditionB*ConditionX$.
In the second case it stands: $Y_2=dummy_AX*dummy_A + dummy_BX*dummy_B$
Is this right?
