modeling response variable that is proportion I am attempting to model data where the response variable y is a proportion (ratio of counts=successes /(successes+failures) in the range (0,1) the predictors are also proportions (proportion of total) in  (0,1). I have 24 observations with each representing the monthly measure of the variables.  I am currently fitting a log-log linear model as the interpretation of the coefficients is exactly what I am after i.e. a 1% increase in the predictor => a beta% change in the response. 
Here is the log-log linear model using sample data:
 library(tidyverse)
set.seed(1)
dat=data.frame(month=1:12,success=sample(85:99,size=12,replace=TRUE),total=rep(100,12),
               var1=runif(12),var2=runif(12,.3,1)) %>% 
               mutate(failures=total-success, prop=(success/total)*100) %>% 
               mutate_at(c("prop","var1","var2"),list(ln=log))

log.log.lr <- lm(prop_ln ~ var1_ln+var2_ln, dat)
summary(log.log.lr)     

 Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  4.55155    0.02844 160.025   <2e-16 ***
var1_ln      0.05286    0.02018   2.619   0.0278 *  
var2_ln      0.01307    0.04132   0.316   0.7589    

Assuming model diagnostics check out the interpretation of the results would be:


*

*one percent change in var1 yields a .05% change in prop_ln

*one percent change in var2 yields a .01% change in prop_ln


I am not sure if the log-log linear model approach is valid  given that the proportion is discrete which is why I am not able to use beta regression, so I performed a binomial regression.
Binomial regression:
Trials = cbind(dat$success, dat$failures)

model.log = glm(Trials ~ var1+var2,
                data = dat,
                family = binomial(link="logit"))
summary(model.log)
Coefficients:
            Estimate Std. Error z value Pr(>|z|)    
(Intercept)  1.43279    0.44069   3.251  0.00115 ** 
var1         1.48684    0.38192   3.893  9.9e-05 ***
var2         0.09792    0.61626   0.159  0.87375     

The interpretation of the var2 being an (exp(.09)-1)% increase in the odds of a success is not quite same kind of interpretation I am looking for.
My two questions  are:


*

*Is the approach using the log-log linear model valid?

*If the log-log method is flawed how do I use the binomial regression results to arrive at a similar interpretation? 


I am not able to provide any real data, TIA.
 A: When dealing with count data, it is generally preferable to model the raw counts rather than converting them to a proportion prior to modelling.  For data of the kind you have generated here, I would suggest using a count-based GLM (e.g., binomial, Poisson, negative binomial, etc.) with a logarithmic link function, and enter your explanatory variables into the regression equation through their logarithms.  (Note that you can do this directly using functions in the regression equation, so you don't need to create new logarithmic explanatory variables.)  This will automatically connect the response variable to the raw (untransformed) explanatory variables via a log-linear relationship, which means that you can interpret the coefficients as rates of change relative to the size of the initial variable.
For example, if we were to fit a Poisson regression to this data, we would do the following.  (Observe here that we do not need to create the mutated variables for the proportion, or for the logarithms of any of the variables.  The response variable is entered directly as a count, with an appropriate offset variable, and the explanatory variables are entered into the regression equation via their logarithms.)
#Generate simulated data
set.seed(1);
DATA <- data.frame(month   = 1:12,
                   success = sample(85:99, size = 12, replace = TRUE),
                   total   = rep(100, 12),
                   var1    = runif(12),
                   var2    = runif(12,.3,1));

#Fit a Poisson model using your data
MODEL <- glm(success ~ log(var1) + log(var2), offset = total, 
             family = poisson (link = 'log'), data = DATA);

Under this model, we obtain the following summary output:
#See summary of the model
summary(MODEL);

Call:
glm(formula = success ~ log(var1) + log(var2), family = poisson(link = "log"), 
    data = DATA, offset = total)

Deviance Residuals: 
     Min        1Q    Median        3Q       Max  
-0.53226  -0.42997  -0.03965   0.18130   0.85817  

Coefficients:
             Estimate Std. Error   z value Pr(>|z|)    
(Intercept) -95.46439    0.07064 -1351.424   <2e-16 ***
log(var1)     0.03645    0.05088     0.716    0.474    
log(var2)    -0.01615    0.10253    -0.158    0.875    
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for poisson family taken to be 1)

    Null deviance: 2.7760  on 11  degrees of freedom
Residual deviance: 2.2211  on  9  degrees of freedom
AIC: 84.522

Number of Fisher Scoring iterations: 3

As you can see, unsurprisingly, there is no evidence of a relationship between var1 or var2 and the response variable success.  That is good, because we generated those variables independently, so the model is making a correct inference here.  Nevertheless, setting that aside, if we were to use the coefficient estimates as estimates of the relationship with the response, then we would estimate that a small $\Delta$% change in var1 is associated with a small $0.03645 \times \Delta$% change in success, and similarly, a small $\Delta$% change in var2 is associated with a small $-0.01615 \times \Delta$% change in success.$^\dagger$

$^\dagger$ We refer here to "small" percentage changes, since the log-linear relationship measures rates of change relative to the existing size of the variable.  For non-small changes, this leads to a distinction between exact percentage changes relative to an initial base, versus the logarithmic changes (see e.g., here).
A: It seems you want to estimate the marginal effect of var1 and var2 on the proportion of success. Let's call this variable 'success rate'. 


*

*Because your dependent variable is already a proportion, taking the logarithm makes the interpretation less straight forward. suggest you run OLS with logarithmic transformations of the right-hand side (RHS) variables. This way, the interpretation is: "a 1% increase in RHS variable $x_j$ leads to a success rate  increase of  $100 \times \beta_j$%" (e.g. $\beta_1=.06$ means a 1% increase in $x_1$ leads to a 6% increase in success rate). 

*An improvement to 1 would be to estimate censored regression with censoring bounds of [0,1]. You should avoid approach 1 if you plan to do any forecasting or parametric bootstrap, since approach 1 can lead to predicted values less than 0 and greater than 1. Be aware, however, that calculation of the marginal effects is a bit different with censored regression.  

*If you want to take the logistic regression approach, your dependent variable needs to be binary, not a proportion. You should avoid this approach, unless you have a dataset where the observational units are individual trails (with 1 or 0 for the dependent variabls), you should avoid this approach. 
