How do I calculate clustering of symptoms? I am trying to assess whether a risk factor (A) has an impact on clustering of 3 types of symptoms (B, C, D).
I know that A has a significant association with B, C, and D. 
So first I did a multinomial logistic regression, with outcomes 0 (no B, C, or D), 1 (either B or C or D), 2 (any combination of 2 or 3 symptoms). This gave me significant results, so exposure to A gives you an increased chance of having any combination of B, C, and D.
But here's my problem:
Let's say the chances of someone developing symptoms (B, C, or D), without exposure to A, is 0.1
That means that purely on chance level, the chances of developing 2 symptoms is 0.1*0.1=0.01
Now lets say that the chances of developing symptoms with exposure to A is 0.2.
Therefore getting 2 symptoms would be 0.2*0.2=0.04.
The problem now is that with my multinomial regression, I think that the fact that the outcome is significant, is merely based on the fact that 0.04 is bigger than 0.01 - because there are significant associations between A, and BCD, you will always get a significant association with combinations of B, C, and D.
But what I want to know is whether the chance of clustering of symptoms is actually higher after exposure to A, compared with the chance of clustering that can happen naturally, independent of exposure to A.
Does anyone have experience with such an analysis, or have any idea how to go about this?
 A: It's not clear what you mean by "clustering" but it seems like you're interested in the number of co-occurring symptoms.
In that case, one simple model would be to just count the number of symptoms, and model this count as a function of A. Then you end up with four classes of patients that can be analyzed with multinomial logistic regression with one input, A.
In your question you implicitly assume that symptoms appear independently. I don't think that's a good assumption, but if you think it's plausible you can also use binomial regression with 3 "trials" (or "size" of 4 or just k=4 depending on who you ask) since you have one trial representing each symptom.
A: SSDecontrol makes a good suggestion which is that you not treat these as independent events. Probit analysis might provide a more intuitive model than multinomial logit since it allows for dependence among outcomes. The challenge will be in running it using classic, closed form solutions since probit models with more than 3 levels are known to be computationally intensive, even prohibitive depending on the machine used. Hierarchical bayes approaches would provide a good approximating workaround to this barrier. 
In addition, modeling with and without risk factor A relative to the symptoms may not make sense if these conditions are pre-existing or baked into the data. Why not collapse the information into classes or groups in a tree-like structure as a function of the combinatorics of observed behaviors? So, the presence of absence of risk factor A would produce 2 groups, and on down through all of the possible combinations. This would generate many more classes than the 4 proposed by SSDecontrol but would be more informative from an analytic point of view. Of course, you could fold in additional regressors as appropriate, for a conditional answer. 
If the classes produced by the combinatorics are treated as one big experimental factor, as in ANOVA, then the t-values associated with each level or class within that factor could act as effect sizes or heuristics of relative importance. Finally, if you really want to nail the idea of relative importance for each class, use Ulrike Gromping's RELAMPO model for a rigorous multivariate approach to estimating this.
