# Appendix S1. R code to perform nonlinear beta regression.

# Adair, EC, Laliberté, E and SE Hobbie. Estimating litter decomposition
# rate in single-pool models using nonlinear beta regression. In review.

# September 9, 2011
# Tested under R 2.13.1


# New  function to generate random data from a beta distribution
rbeta2 <- function(n, mu, phi, ...) {
	# Smithson and Verkuilen (2006) parameterization
	shape1 <- mu * phi
	shape2 <- (1 - mu) * phi
	rbeta(n, shape1, shape2, ...)
}

# New probability density function for beta distribution
dbeta2 <- function(x,mu,phi,...) {
  # Smithson and Verkuilen (2006) parameterization
  shape1 <- mu*phi
	shape2 <- (1-mu)*phi
	dbeta(x,shape1,shape2,...)
}

# Generate decomposition data
# time (days)
time <- c(30, 60, 100, 150, 200, 250, 300, 350, 400, 500, 600, 700, 1000, 1500)

# number of replicates
reps <- 5

# full time vector with reps
time <- rep(time, reps)

# k = decomposition rate, with phi = 5
k <- 0.002

# Proportion of litter mass remaining
propmass <- rbeta2(n = length(time), mu = exp(-k * time), phi = 5)

# Plot data
plot(propmass ~ time)

# Suppose some values were = 0 and some were greater or equal to 1
# as sometimes occur in real decomposition data
propmass2 <- propmass
propmass2[sample(1:length(time), 5)] <- 0
propmass2[sample(1:length(time), 5)] <- c(1, 1.02, 1.06, 1.01, 1.11)

# For beta regression, need to transform data to ]0, 1[ interval
# Smithson and and Verkuilen (2006) transformation could be used
sv2006 <- function(x, s = 0.5) {
      n <- length(x)
      x.max <- max(x)
      # compress between 0 and 1
      # we need to divide by range
      # using theoretical minimum = 0
      if (x.max > 1) x <- x / x.max
      # then convert data to 
      if (any(x >= 1) | any(x <= 0) ) x <- (x * (n - 1) + s ) / n
      return(x)
}

# Apply SV transformation to propmass2
propmass.SV <- sv2006(propmass2)

# Merge vectors into data frame (using the original propmass)
xydat <- data.frame(x = time, y = propmass)

# Get parameter estimates using maximul likelihood estimation
# load bbmle library
library(bbmle) # tested with version 1.0.3
# Refer to Bolker (2008) Ecological Models and Data in R
# Chapter 7 for more details
fit <- mle2(y ~ dbeta2(mu = exp(-k * x), phi = phi), data = xydat,
                start = list(k = 0.002, phi = 5),
                control = list(parscale = c(k = 0.001, phi = 1) ),
                method = "L-BFGS-B",
                lower = c(k = 0.00001, phi = 0.00001))

# Summary, including tests of coefficients
summary(fit)
                  
# Extract parameters
# k is the litter decomposition rate
# phi is the precision parameter of the beta distribution
params <- coef(fit)