Finding the variation of multivariate variable explained by univariate variable I wanted to hear your inputs on the following topic-
I have a dataset of microbiome profile and nutrient content from individual plants. I want to understand whether the variation in microbiome profile is explained by nutrient content of the plants. However, the microbiome profile is a multivariate data (with many features).
Details:
Microbiome data: 500 features (count data)
nutrient content: Continuous variable
######Microbiome data looks like this (Samples in columns and features in rows###

####Nutrient data looks like this
Samples
AMLEAF100Y  34.56uM/mg
AMLEAF113Y 33.56 uM/mg
AMLEAF10.  32.56uM/mg
....
Could you please give me a lead on how do I find whether the nutrient content significantly associates with microbiome data? Thanks in advance
 A: IIUC, you want to know if at least one microbiome feature is associated somehow with the nutrient content.
You can think of your problem as having 500 times the problem of testing for independence. So, you might want to proceed by doing 500 independence hypothesis tests and claiming relevance of the nutrient content if at least one of those tests is positive, i.e. has found a microbiome feature that is not independent of the nutrient contents.
The problem of this approach is that hypothesis tests can raise a false alarm with some low probability given by the significance level: if you use an independence test with alpha equal to 0.05 (5%), then, this test, when applied to 100 cases where you have independence, will nevertheless wrongly claim a dependency in 5 cases.
So, if you have 500 tests for your 500 microbiome features, and you chose for your hypothesis tests a significance level of e.g. alpha equal to 0.05, you will find about 25 dependent features even in the case where the nutrient content has absolutely no influence.
What you should do is mitigate this effect of considering lots of tests at the same time. Possible approaches are e.g. the Bonferroni correction or the Benjamini–Hochberg procedure.
In a nutshell, the Bonferroni correction just makes it more difficult for an individual test to be positive, while the Benjamini–Hochberg procedure gives you the possibility of choosing an upper bound for the percentage of how many of your positive tests are to be expected false.
