Analyze proportions

Question

I have a dataset containing multiple proportions that add up to 1. I am interested in the change of these proportions along a gradient (see below for example data).

gradient <- 1:99
A1 <- gradient * 0.005
A2 <- gradient * 0.004
A3 <- 1 - (A1 + A2)

df <- data.frame(gradient = gradient,
                 A1 = A1,
                 A2 = A2,
                 A3 = A3)

require(ggplot2)
require(reshape2)
dfm <- melt(df, id = "gradient")
ggplot(dfm, aes(x = gradient, y = value, fill = variable)) +
  geom_area()

Additional information: It need not be necessarily linear, I did this just for easiness of the example. The original counts from which these proportions are calculated are also available. The real dataset contains more variable adding up to 1 (eg. B1, B2 & B3, C1 to C4, etc) - so a hint for a multivariate solution is would be also helpful... But for now I'll stick on the univariate side of statistics.

Question: How can one analyze such kind of data? I´ve read a little bit around, and perhaps a multinomial model or a glm is suited? - If I run 3 (or 2) glms, how can I incorporate the constraint that the predicted values sum up to 1? I dont want to only plot such kind of data, I also want to do a deeper regression like analysis. I preferably want to use R - how can i do this in R?

Answer 1

In one dimension, this sounds like a job for beta regression (with or without variable dispersion). This is a regression model with beta-distributed dependent variable, naturally 0-1 constrained. An R package is betareg and a paper describing its use is here.
For more than two proportions the usual extension of the Beta distribution leads to Dirichlet regression. An R package DirichletReg is available, described e.g. here.

There are some reasons not to use logit links and multinomial logistic regression for true compositional data, mostly to do with what strong assumptions they imply for the variance. However, if your data are all actually normalised counts (abundances?), those assumptions may be correct and Peter's suggestion would probably be the way to go.

Answer 2

I am not sure exactly what you are trying to find out, but what about a multinomial logistic regression with gradient as the independent variable?

In R, one way to do this is the mlogit function in the mlogit library. See this vignette