Interpretation of Dispersion Parameter and Intercept in ZINB Model I am doing my first own data analysis and I'm new to more complicated mixed models. I fitted a ZINB model with the glmmTMB package with the following code:
model <- glmmTMB(Y ~ 1 + day + NAw*RSSw*gender + NAb*RSSw*gender +
               add + (1 + day + NAw||ID), data = data, 
               ziformula=~1, family="nbinom2")

Dispersion parameter for nbinom2 family (): 10.5

My question is: how do I interpret the Dispersion parameter?
I did the testDispersion() function from the DHARMa package and it says the model is underdispersed. But I'm confused because the Dispersion parameter is 10.5. I read somewhere that sometimes this is not really relevant when using a negative binomial distribution?
testDispersion(model, alternative="less") #underdispersion?
DHARMa nonparametric dispersion test via sd of residuals fitted vs. simulated
data:  simulationOutput
dispersion = 1.9795e-06, p-value < 2.2e-16
alternative hypothesis: less

Also: how do I interpret this significant Intercept in de zero-inflated part?
Zero-inflation model:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  -1.2626     0.1117   -11.3   <2e-16 ***


In the nbinom2 family, conditional on the covariates, the mean number of counts is $\mu$ and the variance is $\mu+\frac{\mu^2}{\lambda}$ where $\lambda$ is the dispersion parameter. So, you can see when the estimated dispersion parameter is 10.5 and the mean is small, the variance is approximately $\mu$. This is close to what the variance would be in the Poisson regression.
For the intercept coefficient, without the zero-inflation, that tells you that when the other covariates are 0, the estimated population mean counts would be $e^{-1.2626}$.
