Comparing two heavily skewed, overdispersed counts I have, as the title suggests, two heavily skewed, overdispersed histograms. The data ranges from 0 minutes to 85334 minutes. 90% of the data is below 15 minutes, and takes the form of a positive-skewed exponential/power distribution. Then, there's just a huge tail. There are two groups with similar data structures—one for Conversation A and Conversations B. 
I'm solid enough with basic statistics to know that comparing the means, STD, p-values, etc. is pretty useless, but I'm not good enough to know how I can compare these two, or what metrics I can compare with one another to see if being in A or B has any significant effect on the data. I've done some research, and it looks like negative binomial regression fittings will suit my purposes best.
I'm using the MASS package in R, w/ the calls
glm.nb(conversation$A_times ~ 1):
Coefficients:
(Intercept)
    5.624

Degrees of Freedom: 1674 (i.e. Null); 1674 Residual
Null Deviance:      1850
Residual Deviance:  1850    AIC: 17130

and glm.nb(conversation$B_times ~ 1):
Coefficients:
(Intercept)
    4.768

Degrees of Freedom: 1072 (i.e. Null); 1072 Residual
Null Deviance:      1234
Residual Deviance:  1234    AIC: 12390

Now, I imagine that the goal here is to compare two coefficients (or sets thereof) for significant differences, but I'm not actually sure what to do with this info. What are some directions I can take to learn more and really figure out what I'm doing? 
 A: There are actually many ways you can analyse the data. The key question is --- which one makes the most sense? For example, as a start you can compare the proportion of zero counts, if that's a sensible thing to compare. 
The approach you were considering, namely, negative binomial (NB) regression is one of many methods to deal with overdispersion in count models. I would not recommend it, however, as it does have distributional assumptions, namely, that the data does actually follow the negative binomial distribution. You can, however, use NB regression, or even Poisson regression for the data, if you use the Huber/sandwich standard errors. You can even consider the class of zero-inflated Poisson/NB regression, which basically fits a mixture model of 0 together with a Poisson/NB regression. Again you should use Huber/sandwich SE to guard against departures against distributional assumptions. 
All of the above actually model the mean of the data. You can also model the median using quantile regression, if that makes more sense. Finally, there's also the option of transforming the data first. Using square-root transform is easy, but is not so interpretable. A log transform would fail on your 0 counts, but you can use $\log(x+1)$ as an approximate log transform, although again the interpretability is still a problem. You can however, still fit a zero-inflated model on the transformed data, again using Huber/sandwich estimators.  
