Judging the quality of a statistical model for a percentage

Question

I have a data set with multiple predictors, and a single response variable which is a percentage, thus is bounded between 0 and 100. I cannot share the dataset unfortunately. I would like to build a simple model for the response. I then tried, perhaps mindlessly, to use logistic regression using glm. R throws the following error:

    logistic_regression <- glm(y ~ x1 + x5 + x11 , data = df_red, family = binomial(link="logit"))
Error in eval(expr, envir, enclos) : 
  the values of y must be 0 <= y <= 1

The error message might not be exactly that because I had to translate it from my language (I had never seen R throw errors in my language before!). The concept is there, however, and it's right: my $y\in[0,100]$. I must rescale it so that $y^*\in[0,1]$. I then get

    logistic_regression <- glm(y/100 ~ x1 + x5 + x11 , data = df_red, family = binomial(link="logit"))
Warning message:
In eval(expr, envir, enclos) :
  #successes not integers in the glm binomial model!

This time, I don't get an error but a warning. This makes sense: after all, the response variable for logistic regression should be a binary variable, not a continuous one. On the other hand, the model ran. My questions:

1. How do I judge the quality of this model? Does it make sense to look at residuals vs fitted, distribution of residuals, etc.? I am mainly interested in prediction, thus point estimates and prediction intervals for unseen data. A secondary goal would be interpretation of coefficients: if I increase $x_1$ by 1 ceteris paribus, does $y$ increase by a fixed amount? A fixed ratio? Neither of those? The third goal is inference on the coefficients: I care more about uncertainty estimates for $\hat{y}$, but if I can have confidence intervals for the coefficients of the model, that would be good, too.
2. Does the model make sense at all? Should I do something completely different, such as for example beta regression, or can I use something simpler/more similar to what I did?

Answer 1

If your data (to be predicted) is pure fractions/percentages, then logistic regression is not really appropriate.

The closest continuous analogue would probably be to assume the data is logit-normal. I have done this before in similar situations and gotten reasonable results (however my "ad hoc" approach was not really statistically rigorous).

In the more statistically rigorous GLM framework, you could still use a logit link function as in logistic regression, but the (conditional) data distribution will no longer be Bernoulli distributed. As you hint in your last point, beta regression is probably the most common approach for predicting a fraction/percentage (i.e. you assume a Beta distribution rather than a logit-normal). This would certainly be more appropriate than logistic regression, and I imagine it should be straightforward in R (which I do not use, so YMMV).

The second factor to consider though, as mentioned in the comments, is the possibility of using the raw counts rather than the percentages. This will affect the relative uncertainty of your data, e.g. $0.5=\frac{1}{2}=\frac{500}{1000}$ but the second ratio has much less uncertainty than the first. So if you can use the raw counts that approach should be strongly preferred!

For integer count data, possible distributions would be Binomial or Poisson. For continuous "count" (i.e. non-negative weight) data, a Gamma distribution could be appropriate. (Note that the Beta distribution can be interpreted as the mixing-fraction for a binary mixture of two Gamma-distributed components.)

In the count case you can model the two "component-counts" as your primary variables, i.e. if $y=\frac{A}{A+B}$ then you can model $A[x]$ and $B[x]$. Or you could model $A$ and the "total mass" $C=A+B$, depending on the relative correlations. For example if $C$ is relatively constant, $B$ will be strongly correlated to $A$, so $C$ may be a better 2nd variable to target, i.e. more independent.