Dealing with very high autocorrelations in logistic MCMCglmm

Dear List,

I'm using MCMCglmm to estimate logistic mixed models for social network 
data. Estimation works fine until I add a third random effect that 
causes MCMC autocorrelations on all parameters to become extremely high. 
I'm trying to figure out if there is any way to lower autocorrelation 
values while retaining the same specification for the fixed and random 
part of the model.

Here is some information about the model:
* Units of analysis are social ties between individuals, i.e. each row 
in the data frame is one unique social tie.
* Each tie exists between 2 individuals, an Ego and and Alter. Each Ego 
belongs to a Family.
* The binary response is whether the tie is "supportive", i.e. if Alter 
provides support to Ego (1=Yes, 0=No).
* The model includes 3 random effects (aka classifications): Egos, 
Alters, Families. Egos are nested within Families (each Ego belongs to 
one and one only Family). Egos and Alters are cross-classified: the same 
Ego has ties with multiple Alters and the same Alter has ties with 
multiple Egos. Each tie "belongs" to one Ego, one Family (Ego's Family), 
and one Alter, so no multiple membership occurs.
* The data include 1200 ties (1200 dataset rows). There are about 20 
ties for each Ego; 40 ties for each Family; and 2 ties for each Alter.
* This is a fairly simple random-intercept logistic model: The intercept 
has a random component for Egos, Families, and Alters. So I'm interested 
in ego-level, family-level, alter-level variance in the intercept. No 
random slope is involved.
* The model includes covariates at the tie, ego, and alter level (i.e. 
characteristics of ties, egos, and alters).

This is how I'm estimating the model:
```
# Priors
priors.m4b <- list(R= list(V=1,fix=1),
                    G= list(G1=list(V=1, nu=0.002), G2=list(V=1, 
n=0.002), G3=list(V=1, n=0.002)))

# MCMCglmm
MCMCglmm(tie.supportive ~ 1 + tie.family + tie.trust + tie.communication 
+ tie.same.sex + tie.same.age + ego.sex + ego.age + alter.sex + 
alter.age, random= ~ egoID + ego.familyID + alterID, 
family="categorical", data= df, prior= priors.m4b, slice=TRUE, 
nitt=710000, thin=700, burnin= 10000)
```

The estimation results have autocorrelation values of over .50, 
sometimes as high as .90 for the fixed effects; and over .70 for the 
variance components, resulting in uselessly low effective sample sizes.
However, the problem only occurs when adding the Alter random effect 
(alterID). The same model with just the Ego and Family random effects is 
estimated well, with low autocorrelation values. Unfortunately, the main 
aim of the analysis is precisely to examine the model with the Alter 
random effect.
Also, the problem seems to be reduced when keeping the Alter random 
effect while removing some tie-level (i.e. level-1) covariates, in 
particular tie.trust and tie.communication. These are tie-level measures 
of how much trust and communication there is on that tie (i.e. between 
that Ego and that Alter) on a 1-to-5 scale (I'm treating these as 
continuous and centered variables).

The questions on which I wanted to ask your thoughts are the following:
* Am I specifying the model in MCMCglmm in the right way? I have read 
Jarrod Hadfield's Course Notes and a number of other tutorials, but 
obviously I'm using MCMCglmm on a different type of model and data than 
presented in most tutorials, so I'm still not 100% sure.
* What might be the problem with the Alter random effect, that is 
causing autocorrelations to suddenly rocket in this way?
* What might be the problem with tie.trust and tie.communication, that 
causes autocorrelations to be way lower when these two level-1 
covariates are removed?
* Do you have any suggestions on how to deal with such high 
autocorrelations? Is changing the model the only solution? I've tried to 
increase the nitt and thin (up to nitt=1210000, thin=1200, 
burnin=10000), but autocorrelations seem to stay above .50 or .60.

Thank you very much for any thought, comment or suggestion.

Raffaele Vacca
Department of Sociology and Criminology & Law
University of Florida