# Interpret Negative-Binomial mixed-model output in R

I am modelling the number of jobs in a year, with a random intercept to account for regional variation. How do I interpret the model output when using the model to predict/ for inference?

For example, using the log-link function I would do the following to work out the expected number of jobs in year 2 in London:

exp(Intercept + Year*2 + London_intercept) = exp(0.23290 + -0.13369*2 + 0.42820729) = 1.482496


But this value is way too low, I expect way more jobs than 1.5 in a year. Do I instead interpret it as 70% increase over the mean? If so what is the mean? Could someone clarify what to do please?

summary output:

> summary(m8)
Generalized linear mixed model fit by maximum likelihood (Laplace Approximation) ['glmerMod']
Family: Negative Binomial(0.9321)  ( log )
Formula: Jobs ~ 1 + Year + (1 | Region)
Data: df_jobs

AIC      BIC   logLik deviance df.resid
3554.5   3575.9  -1773.3   3546.5     1564

Scaled residuals:
Min      1Q  Median      3Q     Max
-0.7836 -0.6187 -0.5081  0.3613  7.2595

Random effects:
Groups Name        Variance Std.Dev.
Region (Intercept) 0.1597   0.3996
Number of obs: 1568, groups:  Region, 14

Fixed effects:
Estimate Std. Error z value Pr(>|z|)
(Intercept)  0.23290    0.13610   1.711    0.087 .
Year        -0.13369    0.01572  -8.502   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Correlation of Fixed Effects:
(Intr)
Year -0.536


Random intercepts for region:

\$Region
(Intercept)
East Midlands         -0.02413869
East of England        0.18654921
Kent, Surrey & Sussex  0.14610941
London                 0.42820729
North East            -0.42892509
North West             0.48852281
Northern Ireland      -0.26330415
Scotland               0.41420383
South West             0.15415043
Thames Valley         -0.80072297
Wales                 -0.05975189
Wessex                -0.53037342
West Midlands          0.15126500
Yorkshire & Humber     0.21339728


This model has the following interpretations:

• The estimate fo Year is the negative binomial regression estimate for a one unit increase in year. Assuming that each "unit" is 1 year then for each additional year, the difference in the logs of expected number of jobs is expected to decrease by -0.13369, exponentiating this gives 0.875 which means that overall there is a 12.5% decreasee in jobs for every 1 year increase

• You can add in the intercept and random effects to obtain individual estimates for each region - as you did - however these will be shrunken due to partial pooling. Also, note that your formula exp(Intercept + Year*2 + London_intercept) = exp(0.23290 + -0.13369*2 + 0.42820729) = 1.482496 means that there is a 48% increase in jobs associated with a 2 year increase in Year

• If you want unshrunken estimates then you should fit fixed effects for Year

• It is common in models such as these to have a nonlinear association with time, so you might want to consider adding an additional nonlinear term (eg quadratic) or using splines. If using the former it may be best to centre the Year variable first.

• Another possibility is to allow for the "effect"of Year to vary by region, by fitting random slopes:

    Jobs ~ 1 + Year + (Year | Region)