Use chi square test of independence, with Ho, the null hypothesis = independence, and look for evidence that the two or more variables are connected. Do not look for just linear relationships. Way too limiting.
So for example, a large number of random samples of urine in which the following quantities are determined:
Xi/creatinine, and percentile ranked. X1..Xi…Xn are the first, ith and nth component of the urine detected by HPLC/MS. Shortest retention time to longest, lowest detectable mass peak to the highest detectable mass peak at each retention time to form an ordered series of molecules. Create categorical variables from the percentile ranks, say tertiles or quartiles, then create contingency tables for the Chi Squared Test of Independence. If Chi Squared is positive, look further at dose-response for quantification. If present, the detected association is even stronger, as noted by Bradford Hill.
Starting with pairwise comparisons and then proceeding to three way and so on, areX1 and X2 independent? Is a tenuous relationship possible? Are X1 and X3 independent? And so on through all of the data generated by this powerful technique.
Until all comparisons have been made and possible dependent relationships flagged at a given level of statistical significance. Although chi square is not a robust statistic, it still may be able to aid in mapping less direct relationships (where there are degrees of separation between strongly interacting markers), in roughly ranking the degrees of separation between any two strongly interacting markers.
So for example, we would expect to see ethyl sulfate and ethyl glucuronide highest in urine samples with high amounts of alcohol, trailing off as the amount of alcohol in urine trails off, and as the alcohol consumption was deeper in the past, and next to non-existent in those who never drink, with the exception of those with serious yeast overgrowth. Are not all of these things potentially valuable pieces of medical information?
And would we be surprised to find that the percentile score of vitamin C in urine correlates to the percentile score of the amount of hydroxyproline and hydroxylysine in urine? Would we be surprised if low B12 in urine correlates to high methylmalonic acid? Would we be surprised to find that the latter correlation does not work so well if a person is on long term antibiotics? That treatment will reduce propionic acid (a screen of propionic, butyric, acetic acids in urine vs antibiotic use will turn up a possible relationship here), a significant source of methylmalonic acid. In those who are B12 sufficient, the bulk of methylmalonic acid is converted as the CoA derivative into succinyl CoA, which is converted to succinate, and thence to carbon dioxide and water.
Imagine applying this technique to psych patients, for example depressed or paranoid individuals, looking at percentile ranked components in their urine vs so called normal individuals. All kinds of interesting relationships could be found, including compounds produced by colonic flora that influence (not cause) moods and fears. The concentrations/biological activities of these compounds circulating in the bloodstream can be medically manipulated for good or ill, and can thus affect a person’s fears or his moods.