# 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!

• Update: I conducted a variance inflation factor (VIF) test between all cell fractions and VIF was low for each fraction, suggesting low multicollinearity. I'm taking this as evidence for independence, but would like to know if I'm wrong... Commented Mar 28, 2022 at 13:13

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.

• Great answer, thank you. This also explains why when analysing a previous dataset the model only returned p-values for 3 out of 4 fractions. I will look into the possibility of reversing predictor/outcomes since as you say it may be biologically more valid. Commented Mar 28, 2022 at 13:36
• @Bethan unless the total number of cells in each tumor was so large that there was little sampling error in the cell-type fractions, with cell-type fractions as predictors you should use case weights for the tumors related to the number of cells analyzed for each tumor. The more cells you analyze in a tumor the more confidence you have in the fractions and thus the more weight that tumor should get. That takes care of itself with a properly performed multinomial logistic regression based on cell counts in the categories.
– EdM
Commented Mar 28, 2022 at 14:35
• Thank you, this is a valid point, however I don't actually know the number of cells in each tumour. These cell fractions were inferred from bulk DNA methylation data. I used R to determine methylation profiles unique to each cell type, and then used the profiles to predict cell fractions based on methylation levels in each sample. In the future it might be possible to predict cell numbers in a similar way, but probably not before this research project is due! Commented Mar 28, 2022 at 15:22
• @Bethan my answer implicitly assumed that you had cell counts available. If all you have is proportions, you should look more closely into what's available here or online under compositional-data. As predictors, log transformations of the proportions might be preferable. This thread and this thread might be of particular interest.
– EdM
Commented Mar 28, 2022 at 15:49
• I hadn't previously encountered compositional data prior to this analysis & it's helpful to know that the methods are different. Thanks for the pointers, I will look into it more. The second thread looks particularly useful Commented Mar 28, 2022 at 15:55