Beta Distribution

TODO:

1. Figure out how to fit this model in glmmTMB, which can be extended to longitudinal repeated measures.
2. Reference for the betareg R package: https://cran.r-project.org/web/packages/betareg/vignettes/betareg.pdf
3. Discuss page 5 quotation

Motivation

https://cran.r-project.org/web/packages/betareg/vignettes/betareg.pdf

“Our interest in what follows will be more closely related to the recent literature, i.e., modeling continous random variables that assume values in (0, 1), such as rates, proportions, and concentration or inequality indices (e.g., Gini).”

Application of Approach

Page 5: “It is noteworthy that the beta regression model described above was developed to allow practitioners to model continuous variates that assume values in the unit interval such as rates, proportions, and concentration or inequality indices (e.g., Gini). However, the data types that can be modeled using beta regressions also encompass proportions of “successes” from a number of trials, if the number of trials is large enough to justify a continuous model. In this case, beta regression is similar to a binomial generalized linear model (GLM) but provides some more flexibility – in particular when the trials are not independent and the standard binomial model might be too strict. In such a situation, the fixed dispersion beta regression is similar to the quasi-binomial model (McCullagh and Nelder 1989) but fully parametric. Furthermore, it can be naturally extended to variable dispersions."

Generate Data

N = 300 #this should be divisible by however many groups you use!
number.groups <- 2
number.timepoints <- 1
set.seed(6282021)

    dat <- data.frame(
                      'USUBJID' = rep(paste0('Subject_', formatC(1:N, width = 4, flag = '0')), length.out= N*number.timepoints),
                     'Group' = c(rep('Group_1', 0.75*N), rep('Group_2', 0.25*N)),
                      'Time' = rep(paste0('Time_', 1:number.timepoints), each = N),
                      stringsAsFactors=F)

    # Design Matrix
    X <- model.matrix( ~ Group , data = dat)
    Beta <- matrix(0, nrow = ncol(X), dimnames=list(colnames(X), 'param'))
    Beta[] <- c(-.5, 1)

    # Parameters:
    mu <- X %*% Beta
    mu <- as.vector(mu)
    mu <- exp(mu)/(1 + exp(mu))
    phi <- 10

    # Re-parameterize, see pg 3 of PDF
    shape1.i <- mu*phi
    shape2.i <- phi - mu*phi

    # Confirm that parmeterization is equivalent:
    unique(shape1.i/(shape1.i + shape2.i))
    unique(mu)

    # Generate Data:
    dat$Y <- stats::rbeta(n = N, shape1 = shape1.i, shape2 = shape2.i)

    # check
    hist(dat$Y, xlim = c(0,1))
    hist(dat$Y[dat$Group == 'Group_1'], xlim = c(0,1))
    hist(dat$Y[dat$Group == 'Group_2'], xlim = c(0,1))
    aggregate(Y ~ 1, FUN = mean, data = dat)
    aggregate(Y ~ Group, FUN = mean, data = dat)
    aggregate(Y ~ Group, FUN = var, data = dat)

Fit Models

library(betareg)
mod <- betareg::betareg(Y ~ Group, data = dat)
summary(mod)

# Parameter recovery:
unique(shape1.i/(shape1.i + shape2.i)) # generating parameters
aggregate(Y ~ Group, FUN = mean, data = dat) # observed

Custom Estimation Code

#----
Beta_loglike <- function(vP, dat){

  # the 'par' (parameters to estimate) are only passed as a vector
  Beta.hat <- vP[c('b0', 'b1')]
  Beta.hat <- matrix(Beta.hat, ncol = 1)
  phi <- vP['phi']

  X <- model.matrix( ~ Group , data = dat)
  mu <- X %*% Beta.hat
  mu <- as.vector(mu)
  mu <- exp(mu)/(1 + exp(mu))
  Y <- dat$Y

  loglike <- lgamma(phi) - lgamma(mu*phi) - lgamma((1-mu)*phi) +
    (mu*phi - 1)*log(Y) +
    ( (1-mu)*phi - 1)*log(1-Y)

  loglike <- -1*sum(loglike, na.rm = T)
  return(loglike)

}

#---------
# optimize
vP <- c('b0' = 0, 'b1' = 1, 'phi' = 2)
out <- optim(par = vP,
             fn = Beta_loglike,
             method = 'BFGS',
             hessian = T,
             dat = dat)
SE <- sqrt(diag(solve(out$hessian)))
cbind('Custom code Beta' = out$par,
      'Custom code SE' = SE,
      'Model Beta' = coef(mod),
      'Model SE' = sqrt(diag(vcov(mod))))