What test should I choose if I want to see if two groups are different from each other in many categories? I am currently analysing data of a big webshop with over 31,000 vendors selling goods, i.e. each vendor is selling items over the webshop.
The cool thing is that we know the gender of these vendors (=individuals) and the categories of their products. Now, we want to see if there are categorical gender differences, i.e. we would like to test, if males sell more/less items than females of a certain category and if these differences are significant. Here is a graph of the data:
 
The question I have now is which statistical test is appropriate for this kind of analysis?  
So far I was using a chi-squared test to see if the distributions over all categories are significantly different between males and females. Now I would like to know: which statistical test is valid to use to show me for instance that the two distributions between males, females of the category "Apparel" is also significant?
 A: I think you can compare them by individual categories using a t test or Wilcoxon rank sum whichever is more appropriate based on distributions.  The tests wll take account of the differences in sample sizes.  Most importantly though since you would be doing 20 or more test a p-value adjustment for multiplicity should be applied.
A: Another approach could be logistic regression.  Let $p_i$ be the probability that an iten of category $i$ is sold by a male, and $n_i$ be the total number of sales of category $i$.  Then you can make a logistic regression model for $y_i$ the number of sales of products in category $i$ by males (and $n_i - y_i$ being the number of sales by females). You should probably remember the possibility of overdispersion.  One advantahe of this approach is that you can first try a fixed effects model, where the probabilities $p_i$ are constant (which will probably be a bad fit). Then you can expand the model with random effects for the categories, possibly using other covariables, either covariables of the product categories or seller covariables. 
