Co-occurrence of properties in a population I have 150 properties that may occur in a population of 10000 people. Individual people may have none, one or a couple of these properties. The properties are not mutually exclusive and have different frequencies in the population.
I would like to answer two questions here:


*

*Are particular sets of these properties associated, that is do they occur together in a person more often than can be expected by chance?
The method should not only analyse for pair-wise co-occurrence of two properties, but also identify sets of properties that occur together.
Ideally I'd like to have a probability measure like a p-value for a given set of properties telling me the likelihood of observing this combination of properties just by chance.

*Given a sample of e.g. 50 out of the 10000 people I'd like to know whether the co-occurrence of properties observed in the sample is significantly different from a random sample of people from the population.
How would I best address this in statistical / mathematical terms? Are there any tools you could recommend for the calculation?
 A: For a much smaller number of properties, consider a log-linear model, or perhaps some other generalized linear model depending on the underlying process generating your data. Specifically, each of the "properties" of interest should be considered a binary variable (presence vs absence of property). Note that this approach can flexibly handle variables with any number of categories. The gist of this approach is that you are modeling log cell counts in the P-way table based on the P variables used. Your question number (1) involves testing for interactions between the variables. Your question (2) would involve creating a new categorical variable based on group membership, and testing whether or not this new variable is significantly related to log cell count.
Depending upon your objectives, you should look into latent class models. This is similar to PCA, but adapted to categorical variables. If you would indeed rather use a large number of variables as stated in your question, this will combine the initial variables in an optimal way, thereby reducing them down to a smaller number of varibles that captures as large a proportion of the total variance as possible. This may help achieving your objective (1), as you can see which variables get grouped together.
