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I am looking for advice/input on how to report results from a beta regression output. My data is bound between 0 and 1, as ratios, and I am looking at a simple relationship between the response variable (D_Ratio), and predictor (body length, or BL) variable which is continuous. I used the betareg function from the betareg package in R.

For example, here is my R output:

Call:
    betareg(formula = D_Ratio ~ BL, data = wild, link = c("cloglog"))

    Standardized weighted residuals 2:
        Min      1Q  Median      3Q     Max 
     -1.4137 -0.6463 -0.1782  0.3970  2.6160 

Coefficients (mean model with cloglog link):
            Estimate Std. Error z value Pr(>|z|)    
(Intercept) -2.14147    0.51930  -4.124 3.73e-05 ***
     BL      0.05252    0.01673   3.139  0.00169 ** 

Phi coefficients (precision model with identity link): 

    Estimate Std. Error z value Pr(>|z|)
(phi)1.9522     0.2969   6.576 4.82e-11 ***
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 

Type of estimator: ML (maximum likelihood)
Log-likelihood: 8.766 on 3 Df
Pseudo R-squared: 0.2058
Number of iterations: 13 (BFGS) + 1 (Fisher scoring) 

Firstly, I noticed that there are two tables to consider; the coefficients from the mean model link and the coefficients from the precision model. Which coefficients do I report? I am finding different answers in other threads... Right now I am thinking it should be the pseudo R squared, Z value, P value from the mean model...Or does the "Estimate" coefficient term mean something significant, like slope? I ask because I am under the impression that this relationship is not a straight line.

Unfortunately, I am a relatively new R user so if there is a coding issue here, please let me know.

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  • $\begingroup$ Is this specific form of a generalized linear model? $\endgroup$ Jan 13 '17 at 6:25
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    $\begingroup$ @MichaelChernick Yes, see this paper for details. $\endgroup$
    – Roland
    Jan 13 '17 at 7:47
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The beta regression model can have two submodels: (1) a regression model for the mean - similar to a linear regression model or a binary regression model; (2) a regression model for the precision parameter - similar to the inverse of a variance in a linear regression model or the dispersion in a GLM.

So far you have just used regressors in (1) but just a constant in (2). I would encourage you to check whether the model D_Ratio ~ BL | BL with the regressor BL in both parts leads to an improved fit.

If not, then you can probably best report the coefficients from the mean equation as you would for a binary regression model. And then you can add the precision parameter estimate (as you would in a linear regression), the pseudo-R-squared and/or log-likelihood and/or AIC/BIC.

If the regressor plays a role in both parts of the model, then probably report both sets of coefficients.

You can also use the function mtable(betareg_object,...) from the memisc package to generate such a table. Export to LaTeX is also available. Furthermore, you might consider a scatterplot of D_RATIO ~ BL with the fitted mean regression line plus possibly some quantiles (e.g., 5% and 95%). The vignette("betareg", package = "betareg") has some examples like that.

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    $\begingroup$ Thank you for your helpful answer. I am trying out the mtable function via memisc package, and I am wondering if it gives the slope of the regression line? or is it the log odds ratio? Also, it gives two values per cell in the output table, one is in parentheses, do you know what they represent? $\endgroup$
    – Kat Y
    Jan 23 '17 at 3:38
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    $\begingroup$ The mtable(x) reports the coefficient and standard error - just like the first two columns in summary (x). $\endgroup$ Jan 23 '17 at 15:20
  • $\begingroup$ I was wondering if you can clarify what it means when the precision model is significant but the mean model is not, and vice versa? $\endgroup$
    – Kat Y
    Jun 17 '17 at 21:41
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    $\begingroup$ If a regressor is significant in the mean model, the mean changes when the regressor changes (ceteris paribus). See Figure 1 in vignette("betareg", package = "betareg") what this can look like. Similarly, if a regressor is significant in the precision model, then the precision changes. Note that the variance depends on both parameters (see Equation 1). And, of course, it is possible that a certain regressor only influences one or the other parameter. $\endgroup$ Jun 17 '17 at 21:49

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