betareg.stan {phylopairs}

betareg.stan

Description

Fits one of two beta regression models to a dataset in a Bayesian framework using Stan software via the rstan package. Response variable must be bounded between 0 and 1. Users can fit either (1) a basic beta regression model or a (2) beta mixed-effects model in which there are covarying random intercepts in the linear predictor. For the latter, users must supply a covariance matrix. In both models, users can choose one of four link functions. Users can alter parameters for model-parameter prior distributions and Bayesian sampling settings. See details.

Usage

betareg.stan(des, y, model="beta.ols", link="logit", covmat=NULL, 
  iter=2000, chains=4, coef.u=0, coef.sd=10, phi.shape=0.01, phi.rate=0.01,
  physig2.u=-1, physig2.sd=1, cores=4, ...)

Arguments

des

A vector of predictor variable observations OR, in the case of multiple predictors, a matrix in which each column is a vector of observations of a given predictor. betareg.stan() adds a column of 1s to make this a design matrix whose first column corresponds to the model intercept (unless such a column already exists).

y

One of "beta.ols" or "beta.mm" for a basic beta regression model or beta mixed model, respectively; defaults to "beta.ols".

model

A vector of response variable observations.

link

The link function to be used. Default is "logit". Other possible choices are "probit", "cloglog" (complementary log-log), and "loglog".

covmat

Covariance matrix for model residuals (a lineage-pair covariance matrix if analyzing lineage-pair data or a phylogenetic vcv matrix if analyzing bounded species data).

iter

Number of iterations to run on each chain; defaults to 2000 (more are often necessary).

chains

Number of Markov chains to run; defaults to 4.

coef.u

Mean of the Gaussian prior for each preditor variable coefficient; defaults to 0.

coef.sd

SD of the Gaussian prior for each preditor variable coefficient; defaults to 10.

phi.shape

Shape parameter for gamma prior of beta distribution's precision parameter (phi); defaults to 0.01.

phi.rate

Rate parameter for gamma prior of beta distribution's precision parameter (phi); defaults to 0.01.

physig2.u

Mean of the lognormal prior for the scale of the residual covariance; defaults to -1.

physig2.sd

SD of the lognormal prior for the scale of the residual covariance; defaults to 1.

cores

Number of cores to be used; defaults to 4 (one chain per core).

...

additional arguments passed to rstan::sampling, including control parmeters (see rstan::sampling documentation)

Details

The two models that can be chosen differ in terms of their linear predictor on the scale of the link function. These predictors are

1. Simple beta regression: g(u) = XB

2. Beta mixed-effects regression: g(u) = XB + u, u~N(0, physig2*C), where C is a lineage-pair covariance matrix (if analyzing lineage-pair data) or a phylogenetic vcv matrix (if analyzing species data) and physig2 is the scaling parameter for C.

NOTE: For model 2, the user must supply a covariance matrix.

Prior Distributions for Model Parameters: The underlying stan models assume the following prior distributions for model parameters.

1. Regression coefficients: Gaussian prior (users can set prior mean and sd).

2. Precision parameter phi: gamma prior (users can set prior shape and rate of gamma).

3. physig: lognormal prior (users can set prior mean and sd).

Value

A list containing two elements: (1) the posterior distribution of model parameters, and (2) the log-likelihood of the posteriors for use in downstream analyses (e.g. the calculation of model fitting metrics like loo or waic)

References

Anderson, S. A. S., et al. In review. The comparative analysis of lineage-pair data.

Examples

## Example 1: Fit beta regression models with different link functions to independent data
# Load a dataset of independent response observations simulated with a logit link function
data(data5)
# Note: data were simulated with Coef[1]=1 (intercept), Coef[2]=0.8 (slope), phi=5

# Run the betareg function
result1 = betareg.stan(des=data5[,3], y=data5[,4], iter=1000, cores=2)

# Fit the model again but this time use a probit link function
result2 = betareg.stan(des=data5[,3], y=data5[,4], link="probit", iter=1000, cores=2)

# Compare posterior parameter estimates
result1[[1]]
result2[[1]]

## Example 2: Fit beta regression models to a dataset with simulated non-independence
# Load a dataset of non-independent response observations simulated with a logit link function
data(data7)
# Note: data were simulated with Coef[1]=1 (intercept), Coef[2]=0.8 (slope), phi=5
# Load the lineage-pair covariance matrix that arose from those simulations
data(sim.cov.pairs)

# Run the betareg function
result1 = betareg.stan(des=data7[,3], y=data7[,4], model="beta.mm", cov=sim.cov.pairs, 
 iter=1000, cores=2)

# Fit the model again but this time without the covariance matrix
result2 = betareg.stan(des=data7[,3], y=data7[,4], iter=1000, cores=2)

# Fit the model a third time with the cov. matrix but now with a probit link function
result3 = betareg.stan(des=data7[,3], y=data7[,4], model="beta.mm", 
  cov=sim.cov.pairs, link="probit", iter=1000, cores=2)

# Compare posterior parameter estimates
result1[[1]]
result2[[1]]
result3[[1]]

# Compare the fit of the three models via leave-on-out (loo) cross validation.
loo1 = suppressWarnings(loo::loo(result1[[2]]))
loo2 = suppressWarnings(loo::loo(result2[[2]]))
loo3 = suppressWarnings(loo::loo(result3[[2]]))
loo::loo_compare(loo1, loo2, loo3)