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I'm fitting a longitudinal mixed effect Poisson model with 9332 occasions nested within 85 health care units). My data exhibit overdispersion. I have added overdispersed parameter to accout for overdispersion as suggested in this forum. (see below for results of the null model)
I learnt that in discrete models such Poisson regression, the ICC and VPC are not constant across the data but depend on the fixed part of the model.
In my case I have a between variance for ID et within variance for within-Health centers. Is it possible to compute ICC as follow?
ICC= .6990216/ (.6990216+.6443424) = .52035159

Regression :


runmlwin consultants cons, ///
level3(ID: cons) level2(Wave:cons) level1(Wave:) ///
discrete(distribution(poisson) offset(lnpop)) igls irr sd nopause

runmlwin consultants cons, ///
level3(ID: cons) level2(Wave:cons) level1(Wave:) ///
discrete(distribution(poisson) offset(lnpop)) mcmc(on) irr initsprevious nopause


Results :
MLwiN 2.35 multilevel model Number of obs = 9332
Poisson response model
Estimation algorithm: MCMC

-----------------------------------------------------------
| No. of Observations per Group
Level Variable| Groups Minimum Average Maximum
----------------+------------------------------------------
ID | 85 12 109.8 132
Wave | 9332 1 1.0 1
-----------------------------------------------------------

Burnin = 500
Chain = 5000
Thinning = 1
Run time (seconds) = 36.4
Deviance (dbar) = 71431.84
Deviance (thetabar) = 62314.25
Effective no. of pars (pd) = 9117.60
Bayesian DIC = 80549.44
------------------------------------------------------------------------------
consultants | IRR Std. Dev. ESS P [95% Cred. Interval]
-------------+----------------------------------------------------------------
cons | 48.00869 .8004014 3 0.000 46.48676 49.05223
------------------------------------------------------------------------------

------------------------------------------------------------------------------
Random-effects Parameters | Mean Std. Dev. ESS [95% Cred. Int]
-----------------------------+------------------------------------------------
Level 3: ID |
var(cons) | .6990216 .1101347 4588 .5159748 .9426054
-----------------------------+------------------------------------------------
Level 2: Wave |
var(cons) | .6443424 .0098829 3171 .6253806 .6639223
------------------------------------------------------------------------------

Thank you for your help.

Hi davydzombre,

These days people almost exclusively use the latent response approach to calculating ICC/VPC coefficients. However, there isn't a latent response formulation of the Poisson model so you can't do this.

What you propose is a simple statistic to summarize the proportion of higher-level (level-2 + level-3) variation which lies at level-3. In general this is fine as long as you state that it is only based on the higher levels. However, in your case your level-2 is not a level in the conventional sense. Rather it is a psuedo-level used as a device to allow for overdispersion. If you have no overdispersion then the level-2 variance would be zero and your proposed statistic would equal 1, but you wouldn't want to infer that there is necessarily severe clustering as the level-3 variance may well itself not be large. I'm afraid there isn't really a simple solution as far as I am aware.

One alternative you could explore, though I haven't myself, is to calculate the so-called median incidence rate ratio.

Best wishes

George