# 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

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.

• Hi frank, Thank you for the input. I understand your approach. I shall definitely look into it. However, I wanted to see if microbiome community (not a single feature) can explain this variation. I was wondering if the multivariate microbiome data can be decomposed into univariate and then we could see the relationships? Any leads/idea ? Commented Mar 3, 2022 at 7:53
• You wrote: "I want to understand whether the variation in microbiome profile is explained by nutrient content". I.e., whether variation in the microbiome features could be explained by variation in the nutrient content. I.e. whether there is a dependence. That's what I described. I don't understand your comment. Do you want to do regression of each of the microbiome features on the nutrient? Commented Mar 3, 2022 at 8:08
• Thanks for the input. Sorry I was not clear maybe. I want to ask if by any means the multivariate microbiome data be transformed to univariate space and then the relationship be investigated Commented Mar 3, 2022 at 9:00
• There are ways to transform your microbiome data to $\mathbb{R}^1$, e.g. one-dimensional PCA or one-dimensional autoencoders. But unless you have strong evidence that your data is indeed located near a one-dimensional submanifold in your 500 dimensional space, this is not the approach I would recommend. Commented Mar 3, 2022 at 10:06