predicted mean from GLMM lower than mean from GAM

Thank you for your replies, Mr. Bolker and Mr. Hadfield.

Yes, I am using LME4 and mgcv (glmer and gam, respectively). I also fit 
a GLM, and I'm getting a similar intercept in the GLM and GAM, but again 
they are both higher than that from the GLMM.

Here is my code. PV_500 is the number of clinical episodes of P. vivax 
malaria during an interval. There were 264 children from 11 villages. 
Children were not considered at risk of acquiring malaria parasites 
after a certain period following treatment, so we fit an offset for 
their time at risk.

GLMM: Pv500 <- glmer(PV_500 ~ 1 + (1|village) + (1|child), family = 
poisson, data = dat, offset=log(YEARATRISK))
GLM: glm(PV_500 ~ 1, family=poisson, data=dat, offset=log(YEARATRISK))
GAM: gam(PV_500 ~ 1, family=poisson, data=dat, offset=log(YEARATRISK))

GLMM intercept = 0.13
GLMM mean = exp(0.13) = 1.14

GLM intercept = 0.47
GLM mean = exp(0.47) = 1.60

GAM intercept = 0.47
GAM mean = exp(0.47) = 1.60

If I do as Mr. Hadfield suggested, and add 0.5*(v_child + v_village), 
then I get the following for the GLMM estimates:

exp(.13 + .5*(.68 +.12)) = 1.70

It is now slightly higher.

Does this make the problem more clear? Thanks again for any more 
suggestions.

Best,

Katie Benton

Re: predicted mean from GLMM lower than mean from GAM