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Hi,

I am having trouble choosing the best estimation method, and because different methods lead to different conclusions in my case, I wonder what solution to put more trust in.

I have a 3-level binary random intercepts logit regression, and am under the impression the data is underdispersed. I think the PLQ2 allowing for extra-binomial distribution would be the best option. Certainly because I have very small cluster sizes and many singletons. However, this does not converge, even for the simplest model, even when first estimating PLQ1 and then going 'more' on PLQ2, or first binomial than 'more' on extra-binomial, or when allowing negative values everywhere and changing the convergence tolerance, and changing IGLS to RIGL and all these changes together.

So the dilemma is to now choose between:
- PQL2 and not allow for extra-binomial. Note: the variation on level one goes to 0.365 when estimating in PLQ1, so I feel like have severe underdispersion.
- PLQ1 and allow for extra-binomial distribution. However the second level estimation is preferred by Rodríguez G. and Goldman N. (2001; Improved estimation procedures for multilevel models with binary response: a case-study. Journal of the Royal Statistical Society, Series A, 164, 339-355.)
- MCMC however this takes a very long time, and does not seem to allow for extra-binomial distribution.

What would be the 'least harm' option, and based on which principles. Or are there no such principles yet. My personal preference goes to PQL1 extra-binomial, but more for subjective reasons.

If you are analysing binary data then it is now accepted that you cannot validly allow for under/overdispersion in a Bernouilli model. (You can if you are analysing proportions and are thefore fitting a binomial model.) See Redundant Overdispersion Parameters in Multilevel Models for Categorical Responses by Anders Skrondal and Sophia Rabe-Hesketh JOURNAL OF EDUCATIONAL AND BEHAVIORAL STATISTICS December 2007 vol. 32 no. 4,419-430. So start by not allowing extra -binomial variation. Pql 2nd will order will generally be quick and give reasonabaly good estimates except when the higher-level variance is large and when the number of lower levels units in a higher level is small - eg individuals in households is often problematic. When that is the case or you cannot get Pql 2nd oder to converge, use MLq 1st order to get things going and then switch to MCMC. The relatively recently introduced MCMC options can speed things up by giving more independent chains and therefore not having to run for so long. I typically tick on the use ortoghonal parameterization ( this wil never be worse) and the hierachical centering (which is often better) at the highest level -see the final chapters of Bill Browne's MCMC manual.

Thank you,

I will definitely take this advice into consideration.