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 ***

 A: 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 $e^{-1.2626}$.
A: First, regarding that dispersion test: from the doc for testDispersion: "The test statistics is biased to lower values under quite general conditions, and will therefore tend
to test significant for underdispersion. It is recommended to use this test only for overdispersion,
i.e. use alternative == "greater"".
Underdispersion is rare. The Negative Binomial model can only handle overdispersion wrt the Poisson.
Regarding how to interpret the overdispersion parameter, it depends on how exactly that package has parameterized it, but generally the idea is that like a Poisson, the variance is a function of the mean, but now it's been inflated by the overdispersion parameter $\frac{1}{\alpha}$: $\mathbb{V}[x]=E[x] + \frac{1}{\alpha} E[x]^2$.
Finally, the significant intercept for the zero inflation suggests that there is a zero-inflation component, with probability $e^{-1.26}\approx 0.28$ of getting a zero based on the zero inflation.
