A GLMER Question

Hello All,
I am running a GLMER with Gamma distribution and identity link. I have to
add a small constant (1 or 0.001) to my outcome to be able to run my models.

I know in a linear regression with Normality assumption, this only effects
my intercept but my other coefficient estimates will stay the same.

However, in my GLMER models I am noticing the choice of constant effects
all my parameter estimates. I thought since I am using an identity link
this should not happen.

1- Is my understanding wrong?

2- Why are my parameter estimates changing in my Gamma distributed GLMER
with identity link models, depending on the choice of added constant to my
outcome?

Thank you in advance!

Best,

Jamie

I don't necessarily see why you should expect the slopes to be 
invariant; GLMs don't have the same properties as LMs, even with an 
identity link. "Small" will be context-dependent.  If this makes much of 
a difference to your results, or if you have more than a few zero 
values, you might need to think about a more principled approach such as 
a hurdle model ...

Example:

library(lme4)
dd <- data.frame(x = runif(100),
                  f = factor(rep(1:10, 10)))
set.seed(101)
dd$y <- simulate(~ x + (1|f),
                  family = Gamma(link = "identity"),
                  newdata = dd,
                  newparams = list(beta = rep(1,2),
                                   theta = 1,
                                   sigma = 1))[[1]]
dd$y[sample(nrow(dd), size = 10)] <- 0

fitf <- function(off) {
     g1 <- glmer(y + off ~ x + (1|f),
                 family = Gamma(link = "identity"),
                 data = dd)
     return(fixef(g1))
}

try(fitf(0))
offvec <- 10^c((-5):(-2))
cbind(offvec, t(sapply(offvec, fitf)))


      offvec (Intercept)           x
[1,]  1e-05    1.408488 -0.08994451
[2,]  1e-04    1.408578 -0.08994445
[3,]  1e-03    1.409508 -0.09000147
[4,]  1e-02    1.085238  0.45049626
 >

try(fitf(0))
offvec <- 10^c((-5):(-2))
cbind(offvec, t(sapply(offvec, fitf)))