I am looking to correlate an ordinal variable (four categories) with a continuous variable (not normal). The dataset is large, around n=700. I am currently using Kendall's Tau-b because Goodman-Kruskal does not handle continuous data very well. Is there an appropriate correlation coefficient for this case?
Alternatively, should I apply multilevel models to this problem? As described here: How to correctly assess the correlation between ordinal and a continuous variable?
Lastly, is this better conceived of as a matter of inference, i.e. asking whether the means of the four categories (say they are called 1,2,3, and 4) roughly follow the following pattern: 4>3>2>1? If this is the case, what statistic should I use to measure how closely the data follows that pattern, and how should I simulate the null distribution?
