Reddit mentions of Methods of Multivariate Analysis

Linear mixed effects models (also called mixed models or random effect models) are exactly what you want to base everything off of. Unfortunately, they are a tad bit advanced if you ever need to customize it. The basic idea is that in a linear model, you can treat some effects as random instead of fixed and it allows you to account for noise at multiple measurement levels. Let me give you some book recommendations that can teach you about this stuff while simultaneously building your stats background. (listed in order of approachability)

A Modern Approach to Regression with R is a fantastic book that starts with basic linear models and ends with a nice basic look at mixed models. Its also decently cheap for a textbook.

A similar book that covers more topics but assumes a slightly higher starting level of familiarity is An Introduction to Statistical Learning, which is probably the best text book I have ever used, plus its ultra cheap. Its coverage of mixed models is minimal, but you will learn a ton about modeling and how things like machine learning algorithms can be cast into a statistical framework.

There is an R package called lme4 that does mixed modeling. It isn't my favorite one to use (check out nlme) but the author wrote a pdf book about mixed models. It reads like an academic paper, but it is pretty comprehensive.

Profile analysis is related to mixed models, but it generally appears like a separate topic in most books. Now that I am thinking about it, it probably isn't the best idea for your application because it really only helps with one area of the problem whereas a mixed model can address the entire thing. If you are curious, look up books about multivariate analysis (not multiple analysis. very different). An example is Methods of Multivariate Analysis, but its pretty expensive and starts at a graduate level of understanding (that being said it is actually a lovely read if you are at that level).

There are other resources online that talk about this kind of stuff. Mixed models are especially useful in medical studies and pharmaceutical research, so searching for mixed models with those kinds of key words might bring up different perspectives based on different fields. Find whatever suits you best and be patient when learning this stuff. It takes time to learn and to settle in your mind.