I have a question on how the random effects are estimated in Lmer
I have a dataset with families with women who have breastcancer. The age
of cancer diagnosis is highly variable between the families. We want to
predict the age of cancer diagnosis (for screening purposes) for 'new'
family members. I have fitted a linear random effects model with lmer (model
1) with age at diagnosis as outcome, a random intercept for family, and
cancer type (three subtypes, on family level) in the fixed part.
I now want to make a regression formula (for a predction rule) which is of
course easy for the fixed part of the model (intercept + coefs for type) but
more difficult for the random effect part.
My practical solution was the following: I fitted a new model (model 2) on
the familiy level aggregated dataset with the estimated random effect as an
outcome, and number of fam members and the distance between the expected age
for the family based on type (the fixed part of model) and the observed mean
age, as predictors.
Model 2 has an R2 of 0.85 and the predicted ages (fixed part + the random
part as 'estimated' in model 2) for some families are 4 years away from the
posterior estimate (from model 1). So we want to do better. I tried some
polynomials in model 2, since we do not expect the effect of the parameters
to be linear, but it did not help a lot. I also included the within family
variance (the sd) in model 2 since I noticed that the predicted ages got
worse in families with high within family variance, also without much
succes.
So basically my question is: is there a (better) way to approximate the
random effects estimated in model 1, to formulate a 'random' part of the
regression formula/a predction rule?
Thank you very much in advance.
Hester Lingsma