I want to relate how similar people are on their metabolite levels, to their similarity on personal attributes.
Let's say I have measured level for 500 metabolites in 1000 subjects. For each metabolite I generate z scores by subtracting the metabolite mean from the levels. I then generate the 1000 by 1000 correlation matrix for the metabolite z-scores (each element of the matrix is a sample correlation, sample size = 500). Plotting the matrix (ordered by subjects' personal attributes (e.g. age, height, BMI, sex)) as a heatmap, it is clear that similarity in some personal attributes predicts correlation in metabolite levels.
I'd like to model what predicts inter-subject correlation in metabolite levels. One can imagine modelling it with the pairwise subject similarity in various personal attributes, e.g. a multiple regression
metabCorr = b0 + b1 * ageDiff + b2 * heightDiff + ... + e
where
- metabCorr = lower triangle of the metabolite levels correlation matrix, as a vector
- ageDiff = corresponding pairwise differences in age between subjects
- heightDiff = corresponding pairwise differences in height between subjects
Standard multiple regression software won't yield correct regression coefficient p-values. That is because it assumes the rows of the dataset to which it is fit, are independent. Whereas here each row relates to a pair of individuals, making each row related to many other rows. Perhaps p-values would be correct if degrees of freedom were adjusted, e.g. to number of subjects (1000) minus number of explanatory variables.
Cannonical correlation analysis seems to be the answer. However I don't think I have enough data to accurately measure linear combos for both the metabolites and for the personal attributes. Therefore I thought to combine the metabolite level similarities into the one summary statistic, namely correlation.
Furthermore, how can I do this modelling in R?
Any thoughts on this would be greatly appreciated.
You can use something like ridge regression to solve the inference problem. See the first link below. lasso, ridge and elastic net are implemented in R. See the second link below for an R program for an elastic net program which does ridge regression and lasso as well. The difficulty is that the inference step can be a bit difficult but is certainly not impossible. The first link gives some advice. See also the first attached file.
Perhaps a better way to go is to run the regression as a standard lasso variable selection problem on the original independent variables . If you do that all of the links and attachments are still useful references since they give examples. The inferences would still be Bayesian. If p>n, lasso is still a good choice. See the first attached file.
Best, David Booth
https://www.google.com/search?q=statistical+inference+in+ridge+regression&ie=utf-8&oe=utf-8&client=firefox-b-1
https://www.amazon.com/Everyone-Advanced-Analytics-Graphics-Addison-Wesley/dp/013454692X/ref=sr_1_1?s=books&ie=UTF8&qid=1528311527&sr=1-1&keywords=Jared+lander
Thanks for your answer, and the references especially the Shmueli one. I am familiar with Elasticnet. I think you are suggesting a multivariate multiple regression, i.e. multivariate Dependent Variables - the metabolites, with independent variables - personal attributes. This would obviously need to be penalised as there are more variables than subjects. That might be possible, I have only ever seen univariate penalised regression before, but will invesetigate. It is similar to what I was saying about Cannonical Correlation as a possibler solution, except in this situation multivariate penalised regression would be a better match for the scenario, as (I think) I can assume the personal attributes are causal on the metabolites.
It does sound a bit like overkill to me. I think in the first instance I will do an ordinary multiple regression as I suggested in my question, then do permutations to find the correct p-values. I would do that by permuting the person id attached to the rows of the metabolite dataset, thus creating a dataset following the null where metabolites were unrelated to the personal attributes. Etc...
An additional comment. Since you are using difference's as IVs it seems like you might want to consider whether or not your models are actually errors in variables models and are also complicated from that perspective. Wish you the best, David
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