I am working on a GLMM model with crossed random effects and I would like to write an equation from the output where the outcome is the probability rather than a log of the odds.
model <- glmer(y ~ x1 + x2 + x3 (1|R1) + (1|R2), data = df, family=binomial)
My question is, does the model above have the same equation as a linear mixed model?
where Yij = the outcome (in log of the odds)
γ00 = intercept under the fixed effects output
u0j = intercept for the first random effect (R1)
u1j = intercept for the second random effect (R2)
y10 = coefficient for the first fixed effect X1
y20 = coefficient for the second fixed effect X2
y30 = coefficient for the third fixed effect X3
e = not quite sure where to find the level 1 error in the output. Is this the standard deviation for the random effects? If I have two crossed as in the glmer formula above do you use both SD's (add them together)?
Question 1 - am I correct to include both the random effects' intercept in the equation at the beginning?
Question 2 - where is the error term in the output for glmer models? Is this the standard deviation for the random effects? If this is the case do you include both standard deviations for the two random effects terms?
Question 3 - How do you transform the equation so that the outcome is probability rather than log of the odds? Specifically I would like the outcome to be a probability of .5 where I can solve for one of the values of the fixed effects.