Should I use t-test on highly skewed and discrete data? I have samples from a highly skewed dataset about users' participation (e.g.: number of posts), that have different sizes (but not less than 200) and I want to compare their mean. For that, I'm using two-sample unpaired t-tests(and t-tests with the Welch’s factor, when the samples had different variances). As I have heard that, for really large samples, it doesn't matter that the sample are not normal distributed.
My metrics are discrete, they are counts of the number of each user's participation. Of course we have those users who participate much more than the others, but I'm not considering them as outliers. Here are the data description: https://docs.google.com/spreadsheets/d/1WhSKgYIuP35eRsukHVoUFUlITNwO_RRcYoOoR9EmXHg/edit?usp=sharing
My problem: someone, reviewing what I've done, said that the tests I am using were not suitable for my data. They suggested to log-transform my samples before using the t-tests.
I do know that I can't log-transform these, because all of them have zero-values on the samples. My guess is, if I can't use t-test, I should use the Mann Whitney U test.
Are they wrong? Am I wrong? If they are wrong, is there a book or scientific paper which I could cite/show them? If I am wrong, which test should I use? 
 A: You should not use the t-test or even Welch's modified t-test on very skewed data, because these tests tend to be conservative (e.g., alpha and power of these tests can be reduced; Zimmerman and Zumbo, 1993).
Then which test should you use? Your response variable is discrete count data with many 0's, and you want to compare means of two independent groups. I suggest use zero-inflated negative binomial regression. This page has a great tutorial on this technique using R.
Reference:
D.W. Zimmerman & B.D. (1993). Rank Transformations and the Power of the Student t Test and Welch t' Test for Non-Normal Populations With Unequal Variances, Canadian Journal of Experimental Psychology, 1993, 47:3, 523-539
A: To $T$ or not to $T$ -- is that the question?
I would suggest backing off for a moment and asking yourself, "What IS the question?"  Is the question, "Are the means of populations 1 and 2 the same?", or is the question, "Is the usage distribution the same in populations 1 and 2?", or is the question, "Are the medians of populations 1 and 2 the same?", or is the question something else yet?
At $\nu > 350$ degrees of freedom the difference between using sample variances vs population variances is a minor issue.  Questions of data provenance are much more important.  These are questions like how did these data come to be?  Was any sort of random sampling mechanism involved?  Also critical are questions related to the analysis, like those asked above.
If you answer those questions, your choice of test statistic will be clearer.  Of course, this answer precedes your question.
Now, supposing that the question really is about the means, we have to ask if $N(0, 1)$ is a reasonable approximation to the distribution of the test statistic.  The heavily skewed distributions you are dealing with cause me to doubt this.  I'd recommend using an Edgeworth expansion and compare that answer with the answer given by the standard Normal.  Note that Edgeworth expansions are not free of problems themselves, but if the two methods are giving radically different answers I would tend to trust the Edgeworth expansion answer more than the the $N(0, 1)$ answer.
A: While it will come with its own set of limitations, propensity scoring may be a way to ensure sample equality (Connelly et al., 2013). 
