If multiple variables add up to 1, are they independent of each other? I am trying to test for association between continuous fractions of cell types in a sample (e.g. immune cells, cancer cells, fibroblasts...) and tumour grade (categorical/binary/ordinal, grade 1 or 2). The cell fractions always add up to 1, because altogether the cell fractions make up 100% of the sample.
Many statistical tests assume independence but I cannot find a helpful definition of independence in this context. As an example, say samples 1 and 2 each have fractions A, B, and C... an increase in fraction A in sample 1 would have to result in a decrease in fractions B or C or both, also in sample 1. This suggests non-independence. However, fractions in sample 1 have no impact on sample 2... are the fraction variables therefore independent?
I'm also at a loss regarding how to test for association between multiple non-normally distributed, (maybe non-independent), continuous variables on one binary outcome variable. There are tests which meet some of these assumptions but I can't find one which meets all.
Any help is much appreciated, thank you!
 A: If you have $k$ different cell types then you have $k-1$ linearly independent cell-type fractions within each sample. This is a standard situation when you have multiple mutually exclusive categories for a variable. This is a type of compositional data, which might be a better choice for one of your tags. Whether observations among samples are independent depends on the study design.
If you try to include all cell-type proportions as predictors, that set of predictors will be linearly dependent and software will either refuse to fit the model or will (perhaps silently) remove one anyway. So it's best to choose one to omit yourself, perhaps the one that is least likely to differ among samples. (There's no need for predictors to have any particular distribution, normal or otherwise.)
Although you are thinking of the cell types as predictors and the tumor grade as the outcome, you could equivalently turn this around with tumor grade as predictor and cell-type proportions as outcomes. To my mind, that might make even more sense biologically.
With cell-type fractions as outcomes, you could evaluate the log-odds of cell types versus a reference cell type with multinomial logistic regression. You model the actual numbers in each cell-type class, as illustrated in the above link, as the variance of a proportion estimate depends on the number of observations.
