title: Why is the standard error different in these two fitting methods (R Logistic Regression and Beta Regression) for a common dataset?
body: <p>I am trying to understand the difference between two different fitting methods for a data set with a bounded response variable.  The response variable is a fraction and therefore has a range of [0,1].  I have uncovered through my Google searching that there are a lot of different methods out there as this is a common operation.  I am currently interested in the difference between the stock R GLM fit and the Beta regression offered in the betareg package.  I am using the GasolineYield data set from the "betareg" package as my sample data set.  Before I post the code and the results my two questions are the following:</p>
  
  <ol>
  <li><p>Am I performing the Logistic Regression fit in R using the builtin R GLM correctly?</p></li>
  <li><p>Why are the standard errors reported in the Beta regression so much smaller than the standard errors for the R logistic regression?</p></li>
  </ol>
  
  <p><strong>R Setup Code</strong></p>
  
  <pre><code>library(betareg)
  data("GasolineYield", package = "betareg")
  </code></pre>
  
  <p><strong>Beta Regression code from the "betareg" package</strong></p>
  
  <pre><code>gy = betareg(yield ~ batch + temp, data = GasolineYield)
  summary(gy)
  </code></pre>
  
  <p><strong>Beta Regression summary output</strong></p>
  
  <pre><code>Call:
  betareg(formula = yield ~ batch + temp, data = GasolineYield)
  
  Standardized weighted residuals 2:
      Min      1Q  Median      3Q     Max 
  -2.8750 -0.8149  0.1601  0.8384  2.0483 
  
  Coefficients (mean model with logit link):
                Estimate Std. Error z value Pr(&gt;|z|)    
  (Intercept) -6.1595710  0.1823247 -33.784  &lt; 2e-16 ***
  batch1       1.7277289  0.1012294  17.067  &lt; 2e-16 ***
  batch2       1.3225969  0.1179020  11.218  &lt; 2e-16 ***
  batch3       1.5723099  0.1161045  13.542  &lt; 2e-16 ***
  batch4       1.0597141  0.1023598  10.353  &lt; 2e-16 ***
  batch5       1.1337518  0.1035232  10.952  &lt; 2e-16 ***
  batch6       1.0401618  0.1060365   9.809  &lt; 2e-16 ***
  batch7       0.5436922  0.1091275   4.982 6.29e-07 ***
  batch8       0.4959007  0.1089257   4.553 5.30e-06 ***
  batch9       0.3857930  0.1185933   3.253  0.00114 ** 
  temp         0.0109669  0.0004126  26.577  &lt; 2e-16 ***
  
  Phi coefficients (precision model with identity link):
        Estimate Std. Error z value Pr(&gt;|z|)    
  (phi)    440.3      110.0   4.002 6.29e-05 ***
  ---
  Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
  
  Type of estimator: ML (maximum likelihood)
  Log-likelihood:  84.8 on 12 Df
  Pseudo R-squared: 0.9617
  Number of iterations: 51 (BFGS) + 3 (Fisher scoring) 
  </code></pre>
  
  <p><strong>R GLM Logistic Regression code from stock R</strong></p>
  
  <pre><code>glmfit = glm(yield ~ batch + temp, data = GasolineYield, family = "binomial")
  summary(glmfit)
  </code></pre>
  
  <p><strong>R GLM Logistic Regression summary output</strong></p>
  
  <pre><code>Call:
  glm(formula = yield ~ batch + temp, family = "binomial", data = GasolineYield)
  
  Deviance Residuals: 
        Min         1Q     Median         3Q        Max  
  -0.100459  -0.025272   0.004217   0.032879   0.082113  
  
  Coefficients:
               Estimate Std. Error z value Pr(&gt;|z|)
  (Intercept) -6.130227   3.831798  -1.600    0.110
  batch1       1.720311   2.127205   0.809    0.419
  batch2       1.305746   2.481266   0.526    0.599
  batch3       1.562343   2.440712   0.640    0.522
  batch4       1.048928   2.152385   0.487    0.626
  batch5       1.125075   2.176242   0.517    0.605
  batch6       1.029601   2.229773   0.462    0.644
  batch7       0.540401   2.294474   0.236    0.814
  batch8       0.497355   2.288564   0.217    0.828
  batch9       0.378315   2.494881   0.152    0.879
  temp         0.010906   0.008676   1.257    0.209
  
  (Dispersion parameter for binomial family taken to be 1)
  
      Null deviance: 2.34184  on 31  degrees of freedom
  Residual deviance: 0.07046  on 21  degrees of freedom
  AIC: 36.631
  
  Number of Fisher Scoring iterations: 5
  </code></pre>