[R-sig-ME] How to report and quantify the random effect in a logistic model?

[Pi]

Hello.

I'm fitting a logistic regression model with mixed effects using the
package glmmTMB. (Because the dataset is quite large and lme4 produces out
of memory errors, even increasing memory.limit).

I need help to interpret and report the output.

    Family: binomial  ( logit )
    Formula:      OUTPUT ~ SEX + YEAR + OTHER +  (1|CITY/ID)
    Data: mydata

          AIC       BIC    logLik  deviance  df.resid
     890000  891000 -450000  889000    1000000

    Random effects:

    Conditional model:
     Groups Name        Variance Std.Dev.
     ID:CITY (Intercept) 10.0    3.1
     CITY    (Intercept) 1.5    1.2
    Number of obs: 1000009, groups:  ID:CITY, 200000; CITY, 20

    Conditional model:
                 Estimate Std.Error z value Pr(>|z|)
    (Intercept)  0.79   0.28    2.9   0.005
    SEX1       -0.21   0.017 -12   <2e-16
    YEAR        0.48   0.0048 100   <2e-16
    OTHER       -0.70   0.005 -130   <2e-16

Output from sjPlot:

    Random Effects
    σ2 3.29
    τ00 ID:CITY 10.0
    τ00 CITY 1.5
    ICC 0.78


How can I report the effect of CITY and its confidence interval?
I think most people would report the CI of the odds of the Intercept using
the variance to calculate

     { exp(Intercept-1.96*1.2) , exp(Intercept+1.96*1.2) }

but this doesn't take into account the standard error of the Intercept from
the conditional model, 0.28. How should I combine them?
(I think this is the estimated deviation for the Intercept excluding the
random effects).

Is it acceptable to ignore the Intercept value and just say...?
"The 95% CI for the odds ratio for the CITY is
{exp(-1.96\*1.2),exp(+1.96\*1.2)}"

or

"The odds of OUTPUT is multiplied by a number between exp(-1.96\*1.2) and
exp(+1.96\*1.2) due to the variability of CITY". I don't know how to
include here the 95%IC argument.

I have also tried to calculate the residuals with residuals(model) but it
produces an out of memory error.
Is there any way to do it approximately? or is there already some useful
information in the output?

I just need to say what proportion of the total variance (or of the
residuals) is explained by the variable CITY.

The σ2 value reported by sjPlot is the variance of the residuals.

What about the variance of ID:CITY (Intercept) 10.0? How should I report
it? How can I split it into variance due to ID and variance due to the
interaction of ID and CITY?

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