Some of the IELN parameterizations have a "no intercept term" for example p=list(formula=~-1+session). This is an alternative parameterization that has no effect on the real parameter estimates, AIC values, etc. By contrast, the same model including an intercept term would be p=list(formula=~1+session) or simply p=list(formula=~session). In either case, we are specifying that p (the real parameter) values from session to session, so should take on 3 unique values
To clarify, take the example (attached) IELN with 3 sessions. The explicit notation for the model with the intercept is
logit(p_1)=Beta0
logit(p_i)=Beta0+Beta_i-1 , i=2,3
For the no intercept model the equation is
logit(p_i)=Beta'_i, i = 1,2,3 (I am using different terms for Beta and Beta' because they represent different parameterizations)
Because the p_i values are the same under each model , this establishes that there must be a relationship between the betas. It is simply:
Beta'_1=Beta0
Beta'_2=Beta0+Beta_1
Beta'_3=Beta0+Beta_2
That is, the first session is a baseline (Beta0) and the second and third sessions add to that . The no-intercept model simply absorbs the intercept into the 3 Beta' parameters. The attached shows that the real parameter values are identical under the 2 models and shows how the predictions on the logit scale relate to the beta terms under each model.