Beta regression: how to interpret the p-value of the phi coefficient

Question

I am applying a beta regression to my proportion data (breeding success). My Phi coefficients show a p value of 0.043, which seems considerably higher to most examples I have looked at. I fail to find an explanation (that I understand) of what the phi coefficient shows, and what its p-value signifies.

Call: betareg(formula = BS ~ log(Var1), data = df1)

Standardized weighted residuals 2:
    Min      1Q  Median      3Q     Max 
-0.9986 -0.9617 -0.4764  0.4077  2.3898 

Coefficients (mean model with logit link):
             Estimate Std. Error z value Pr(>|z|)
(Intercept)    4.7546     0.9926   4.790 1.67e-06 ***
log(Var1)     -1.1192     0.2554  -4.383 1.17e-05 ***

Phi coefficients (precision model with identity link):
      Estimate Std. Error z value Pr(>|z|)
(phi)    44.24      21.90    2.02   0.0433 *
 --- 
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Type of estimator: ML (maximum likelihood) 
Log-likelihood: 9.896 on 3 Df 
Pseudo R-squared: 0.6939 
Number of iterations: 57 (BFGS) + 4 (Fisher scoring)

Answer 1

In beta regression you assume that the dependent variable is beta-distributed with expectation $\mu$ and variance $\mu \cdot (1 - \mu)/ (1 + \phi)$. Thus, $\phi$ is a precision parameter: the higher $\phi$ the lower the variance for given mean $\mu$.

In principle, the precision parameter $\phi$ can depend on regressors - just like the mean $\mu$ as well. In the model you have estimated, you have used a constant $\phi$, i.e., this is like an intercept-only model. Therefore, the $p$-value does not correspond to a very interesting null hypothesis, it tests whether $\phi = 0$. This is similar to other regression models where the $p$-value of the intercept is also not of much interest. But in the case where $\phi$ can depend on regressors you would often be interested to test whether this is actually the case or not (i.e., whether the corresponding coefficient differs from zero or not).