Is there any possible statistical test for this data? I have a data set of the daily dosage of a drug that participants take against some signs and symptoms. The data are highly skewed to the left because there are a lot of patients with 0mg per day of drug intake. My goal is to compare the dosage intake between those patients that have a specific sign and those who do not have this sign.
Here are the quantiles and histogram of the data for the whole population(n=56):
   0%  25%  50%  75% 100% 
   0    0    0   15   30 


My goal is to compare this variable between two categories of this population:
1(those with a specific clinical sign)(n=10): [0  0  0  0 10  0  0 10  0  0]


2(those without that clinical sign)(n=46): [0.0 20.0 10.0  0.0  0.0  0.0  0.0 15.0  0.0 20.0 10.0  0.0 10.0 10.0  0.0 15.0  0.0  0.0  0.0  2.5  0.0 15.0  0.0  0.0 15.0  0.0  2.5 15.0  0.0 30.0 20.0 10.0 15.0 30.0 20.0  0.0  2.5  2.5 15.0  0.0  0.0  0.0 20.0  0.0 12.5 30.0]


But as you see, the data are highly skewed with high number of zeros in the whole population and also in each group.
My question is what is the most suitable test for comparing these two groups?

*

*Due to non-normality, I suppose t-tests can not be used.


*Due to high number of ties, I suppose I can't use a Wilcoxon test.
(I tried jitter as low as (-.1, .1), but each time results are different.)


*I read that permutation and bootstrap tests are suitable for comparing customized indices, and for mean and median its better to stick with student and Wilocxon tests. Since the latter are more common and well-known in the scientific medical literature.
In fact, can I use a t-test or Wilcoxon test for this variable in this population? If no, is there any transformation that I can use to make the data suitable for t-test (or Wilcoxon test) ?
 A: I've been thinking about this further and I wanted to capture some of my thoughts in an answer rather than lengthy comments.  I'll openly admit that I'm not an expert in this area and I don't know what people typically do for this type of analysis.  (I'll restate some of my comments below for context.)
Two group analysis
Given your data, I can see how it could be structured as a two-group analysis.  As stated in the comments, "H0 is that the drug intake doesn't affect the prevalence of the clinical sign."  In this case the two groups are patients with the clinical sign and patients without.  For a t-test or Wilcox test, this would give a comparison of the mean/mean rank of the observed dosing in each group.  As presented in the OP, it would appear that the drug suppresses the clinical sign except perhaps at the highest dose.  This is complicated by the fact that there are so few observations (n=10) with the clinical sign.
Comparing the means or ranks of dosing between the two groups seems unnatural to me as does using logistic regression (with or without Firth's adjustment), given the data.  It could very well be that I'm just biased in the way I'm thinking about the data, and those are perfectly valid approaches.  If that was the approach taken, I would probably opt for the Wilcox-test because of the small sample size.  Using R that gives a p-value of 0.03641, given your data.
Dose-response
My first instinct is to look for a dose-response for the clinical sign, but there doesn't seem to be enough data to really do that.  Again, the data would suggest that dosing suppresses the clinical sign, except at the highest does, which seems odd.  Given the relative umber of individuals that show the clinical sign in the absence of dosing, my guess is that the drug is intended to suppress it, but this seems counter to the observations with the clinical sign at the highest dose.
And all of these counter-intuitive observations could be due to the very small sample size.
Binomial Distribution
One thought I had was to assume that the drug is extremely effective at suppressing the clinical sign and turn this into a binomial problem.  This binarization of the data does lead to data loss and may not be appropriate, however.
If we combine all the data, n=56, and then draw 10 observations at random (simulating a random draw of the clinical sign group), what is the probability of drawing only 2 (or fewer) that have taken the drug at any dose?  This is a simple Binomial distribution problem, with the probability of "success" (drawing a dosed subject) is 27/56 is ~0.0684.
Interpretation
From this, I would say that the data is suggestive that the drug suppresses the clinical sign, but the dataset is very small (only n=10 in the clinical sign group) and there are some inconsistencies (in particular, presence of the clinical sign in at the highest dose, no obvious dose-response).
I hope this is helpful.
