Suppose that I have three populations with four, mutually exclusive characteristics. I take random samples from each population and construct a crosstab or frequency table for the characteristics that I am measuring. Am I correct in saying that:
1) If I wanted to test whether there is any relationship between the populations and the characteristics (e.g. whether one population has a higher frequency of one of the characteristics), I should run a chi-squared test and see whether the result is significant.
2) If the chi-squared test is significant, it only shows me that there is some relationship between the populations and characteristics, but not how they are related.
3) Furthermore, not all of the characteristics need to be related to the population. For example, if the different populations have significantly different distributions of characteristics A and B, but not of C and D, then the chi-squared test may still come back as significant.
4) If I wanted to measure whether or not a specific characteristic is affected by the population, then I can run a test for equal proportions (I have seen this called a z-test, or as prop.test() in R) on just that characteristic.
In other words, is it appropriate to use the prop.test() to more accurately determine the nature of a relationship between two sets of categories when the chi-squared test says that there is a significant relationship?