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

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)

• Have you read the betareg vignette (pdf)? May 19, 2021 at 14:08
• Yes.. but I couldn't find an answer to my question. May 19, 2021 at 18:44

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).
• In your particular model it tells you that $\phi$ is not significantly different from $0$. But this non-significance is practically irrelevant, I would say, because you will keep the parameter in the model anyway. Also, it does not convey any information about how well the beta distribution fits your data. May 26, 2021 at 13:47