What method to use


I am having trouble figuring how to analyze a set of data I have at my hand. Basically, let say I have a N boxes and in the i-th box there is n(i,j) balls colored with j-th color. Distributions of balls are very skewed and dispersed - that is to say they aren't "nice" i.e. Poisson distributed. Then, I add m(i) balls of new color (let say blue) in i-th box. For a given level of significance (such as p-value), how could I analyze the situation so I could deduce statements such as:
  • Number of added blue balls is strongly correlated with number of red balls when also yellow are present, but is poorly correlated when there are no yellows.
  • It is unlikely that number of added blue balls depend only on total number of balls in the boxed.
  • Number of added blue balls is uncorrelated with number of present green balls
I understand how can I calculate correlation between any pair of colors but what I'm after is how this may change in the presence of other colors. Any kick in the right direction is appreciated.