I am trying to estimate parameters in a beta regression using optim in R. I can do so if I parameterize the model with mu and phi as is done in the package betareg. That optim code is shown below. However, I cannot estimate alpha and beta directly with optim. I have tried maybe a half-doze variations of the code at the bottom. I suspect this is more of a coding issue than a statistical issue, but can move this to Cross Validated if requested.
First I create simulated data with 1000 samples of the proportion my.survival in each of 2 years. I model my.survival as a function of year:
set.seed(1234)
# create data
year <- 1:2
years <- length(year) # number of years
n.trials <- 1000 # samples per year
# define parameters in linear predictor
B0 <- 0.7
B1 <- -0.05
# obtain mu and phi
my.means <- exp(B0 + B1 * year) / (1 + exp(B0 + B1 * year))
my.means
my.phi <- 100
# convert mu and phi into annual alpha and beta
ab.fun <- function(mu, phi)
{
a <- mu * phi
b <- phi - mu * phi
ab <- data.frame(a, b)
return(ab)
}
alphabeta <- ab.fun(my.means, my.phi)
alphabeta
# a b
# 1 65.70105 34.29895
# 2 64.56563 35.43437
my.alpha <- alphabeta$a
my.beta <- alphabeta$b
# obtain random samples of annual proportion surviving
my.survival <- rep(NA, (years*n.trials))
my.year <- rep(NA, (years*n.trials))
iter <- 1
for(i in 1:years) {
for(j in 1:n.trials) {
my.survival[iter] <- rbeta(1, my.alpha[i], my.beta[i])
my.year[iter] <- i
iter <- iter + 1
}
}
head(my.survival)
head(my.year)
Then I estimate the parameters with the betareg package:
library(betareg)
gy <- betareg(my.survival ~ my.year)
summary(gy)
# Call:
# betareg(formula = my.survival ~ my.year)
#
# Standardized weighted residuals 2:
# Min 1Q Median 3Q Max
# -3.4235 -0.6520 -0.0099 0.6561 3.5155
#
# Coefficients (mean model with logit link):
# Estimate Std. Error z value Pr(>|z|)
# (Intercept) 0.721712 0.014738 48.97 < 2e-16 ***
# my.year -0.067093 0.009293 -7.22 5.19e-13 ***
#
# Phi coefficients (precision model with identity link):
# Estimate Std. Error z value Pr(>|z|)
# (phi) 100.69 3.17 31.77 <2e-16 ***
# ---
# Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#
# Type of estimator: ML (maximum likelihood)
# Log-likelihood: 3268 on 3 Df
# Pseudo R-squared: 0.02544
# Number of iterations: 8 (BFGS) + 2 (Fisher scoring)
# Warning message:
# In deparse(x$call, width.cutoff = floor(getOption("width") * 0.85)) :
# invalid 'cutoff' value for 'deparse', using default
I obtain virtually the same estimates of B0, B1 and phi with optim:
# estimate parameters with optim
beta.reg = function(betas, my.survival, my.year){
phi = betas[1]
B0 = betas[2]
B1 = betas[3]
n <- length(my.survival)
llh <- rep(0, n)
for(i in 1:n){
mu <- exp(B0 + B1 * my.year[i]) / (1 + exp(B0 + B1 * my.year[i]))
alpha <- mu * phi
beta <- phi - mu * phi
llh[i] = llh[i] + dbeta(my.survival[i], alpha, beta, log = TRUE)
}
-sum(llh)
}
alpha.init <- 2
beta.init <- 2
phi.init <- 25
B0.init <- 0
B1.init <- 0
my.model <- optim(c(phi.init, B0.init, B1.init),
beta.reg, my.survival = my.survival, my.year = my.year, method = "BFGS", hessian=TRUE)
my.model$par
#[1] 100.71561736 0.72166040 -0.06705041
my.model$value
#[1] -3268.381
covmat <- solve(my.model$hessian)
mySE <- diag(covmat)^0.5
mySE
#[1] 3.170301989 0.014736516 0.009291484
However, I cannot estimate the annual alpha's and beta's with optim. I have tried several variations of the code below. In the version of the code below I simplied the formulae of mu by hardcoding the values of year. I know I can derive alpha and beta from the estimates of mu and phi from the optim code above and use the delta method to obtain SE's. However, I thought it would be more convenient and ultimately useful to estimate alpha and beta directly with optim if possible.
I wonder whether the issue is optim does not allow derived parameters and below alpha and beta are derived from phi, B0 and B1?
# estimate parameters with optim
beta.reg = function(betas, my.survival, my.year){
B0 = betas[1]
B1 = betas[2]
alpha1 = betas[3]
alpha2 = betas[4]
beta1 = betas[5]
beta2 = betas[6]
phi = betas[7]
n <- length(my.survival)
llh <- rep(0, n)
# here I try to simplify by hardcoding the values of year to 1:2
mu1 <- exp(B0 + B1 * 1) / (1 + exp(B0 + B1 * 1))
mu2 <- exp(B0 + B1 * 2) / (1 + exp(B0 + B1 * 2))
alpha1 = mu1 * phi
beta1 = phi - mu1 * phi
alpha2 = mu2 * phi
beta2 = phi - mu2 * phi
iter <- 1
for(i in 1:years){
for(i in 1:n.trials){
if(i == 1) llh[iter] = llh[iter] + dbeta(my.survival[iter], alpha1, beta1, log = TRUE)
if(i == 2) llh[iter] = llh[iter] + dbeta(my.survival[iter], alpha2, beta2, log = TRUE)
iter <- iter + 1
}
}
-sum(llh)
}
B0.init <- 0
B1.init <- 0
alpha.init <- rep(2, years)
beta.init <- rep(2, years)
phi.iter <- 100
my.model <- optim(c(B0.init, B1.init, alpha.init, beta.init, phi.iter),
beta.reg, my.survival = my.survival, my.year = my.year, method = "BFGS", hessian=TRUE)
my.model$par
#[1] 1.0013198 -0.2096154 2.0000000 2.0000000 2.0000000 2.0000000 338.7167214
my.model$value
#[1] -9.090913
covmat <- solve(my.model$hessian)
# Error in solve.default(my.model$hessian) :
# Lapack routine dgesv: system is exactly singular: U[3,3] = 0
mySE <- diag(covmat)^0.5
mySE
I have not been able to estimate
alphaandbetafrom abeta regressionwithoptim. So, I resort here to using a Bayesian model withJAGS. In this answer I return to using 10 years of data instead of just two. I first usebetaregthenJAGS. At the bottom of this post I work through thedelta methodto estimate the95% CIofalphaandbetawithbetareg.The Bayesian point estimates of
alphaandbetaare pretty close to the true values. The estimated95% CIfrombetaregalso are close to the estimated95% credible intervalsreturned byJAGSfor the two years I checked: Years 2 and 10.I remain interested in how to get estimates of
alphaandbetafromoptim. Note that in this post I obtain the95%CIofalphaandbetaseparately. I do not know whether it is better to somehow estimate their95%CItogether.First I create some fake data:
Then I use
betaregto getpoint estimatesof annualalphaandbeta. Their95%CIare obtained at the very bottom of this post:Here is the
JAGScode. Extracting the estimates ofalphaandbetafor each year is a little cumbersome. My answer is based on the two posts below:https://stats.stackexchange.com/questions/230106/error-in-jags-beta-regression-model-r-value-out-of-range-in-gammafn
https://stats.stackexchange.com/questions/41536/how-can-i-model-a-proportion-with-bugs-jags-stan
EDIT - September 8, 2023
Below I work through the
delta methodto obtain the95%CIofalphaandbetafrombetareg. Then I compare those estimates with the95% credible intervalestimates returned by myJAGScode. Both approaches return similar estimates. I only show the results forYear 2. Results for any year from 1 - 10 can be obtained by changingyear = 2to the desiredyear. I have only checked estimates for years 2 and 10.