flexreg_binom: Flexible Regression Models for Bounded Discrete Responses

Description

The function fits some flexible regression models for bounded discrete responses via a Bayesian approach to inference based on Hamiltonian Monte Carlo algorithm. Available regression models are the flexible beta-binomial (type = "FBB", default), the beta-binomial (type = "BetaBin"), and the binomial one (type = "Bin").

Usage

flexreg_binom(
  formula,
  data,
  type = "FBB",
  n,
  link.mu = "logit",
  prior.beta = "normal",
  hyperparam.beta = 100,
  hyper.theta.a = NULL,
  hyper.theta.b = NULL,
  link.theta = NULL,
  prior.psi = NULL,
  hyperparam.psi = NULL,
  n.chain = 1,
  n.iter = 5000,
  warmup.perc = 0.5,
  thin = 1,
  verbose = TRUE,
  ...
)

Value

The flexreg_binom function returns an object of class `flexreg`, i.e. a list with the following elements:

call: the function call.
type: the type of regression model.
formula: the original formula.
link.mu: a character specifing the link function in the mean model.
link.theta: a character specifing the link function in the overdispersion model.
model: an object of class `stanfit` containing the fitted model.
response: the response variable, assuming values in (0, 1).
design.X: the design matrix for the mean model.
design.Z: the design matrix for the overdispersion model (if defined).

Arguments

formula: an object of class "formula": a symbolic description of the model to be fitted (y ~ x or y ~ x | z, see Details).
data: an optional data.frame, list, or object that is coercible to a data.frame through as.data.frame containing the variables in the model. If not found in data, the variables in formula are taken from the environment from which the function flexreg_binom is called.
type: a character specifying the type of regression model. Current options are "FBB" (flexible beta-binomial, default), "BetaBin" (beta-binomial), and "Bin" (binomial).
n: a character specifying the name of the variable containing the total number of trials.
link.mu: a character specifying the link function for the mean model. Currently, "logit" (default), "probit", "cloglog", and "loglog" are supported.
prior.beta: a character specifying the prior distribution for the regression coefficients of the mean model, beta. Currently, "normal" (default) and "cauchy" are supported.
hyperparam.beta: a positive numeric (vector of length 1) specifying the hyperprior scale parameter for the prior distribution of beta regression coefficients. The default is 100 if the prior is "normal", 2.5 if it is "cauchy".
hyper.theta.a: a numeric (vector of length 1) specifying the first shape parameter for the beta prior distribution of theta.
hyper.theta.b: a numeric (vector of length 1) specifying the second shape parameter for the beta prior distribution of theta.
link.theta: a character specifying the link function for the overdispersion model. Currently, "identity" (default), "logit", "probit", "cloglog", and "loglog" are supported. If link.theta = "identity", the prior distribution for theta is a beta.
prior.psi: a character specifying the prior distribution for the regression coefficients of the overdispersion model,psi. Not supported if link.theta="identity". Currently, "normal" (default) and "cauchy" are supported.
hyperparam.psi: a positive numeric (vector of length 1) specifying the hyperprior scale parameter for the prior distribution of psi regression coefficients. The default is 100 if the prior is "normal", 2.5 if it is "cauchy".
n.chain: a positive integer specifying the number of Markov chains. The default is 1.
n.iter: a positive integer specifying the number of iterations for each chain (including warm-up). The default is 5000.
warmup.perc: the percentage of iterations per chain to discard.
thin: a positive integer specifying the period for saving samples. The default is 1.
verbose: a logical (with default TRUE) indicating whether to print intermediate output.
...: additional arguments from sampling.

Details

Let Y be a random variable whose distribution can be specified in the type argument and $\mu$ be the mean of Y/n. The flexreg_binom function links the parameter $\mu$ to a linear predictor through a function $g_1(\cdot)$ specified in link.mu: $$g_1(\mu) = x^t \bold{\beta},$$ where $\bold{\beta}$ is the vector of regression coefficients for the mean model. The prior distribution and the related hyperparameter of $\bold{\beta}$ can be specified in prior.beta and hyperparam.beta. By default, link.theta="identity", meaning that the overdispersion parameter $\theta$ is assumed to be constant. In that case, the prior distribution for $\theta$ is a beta with shape hyperparameters $a$ and $b$ that can be specified in hyper.theta.a and hyper.theta.b. If not specified, $a=b=1$, otherwise if only one hyperparameter is specified, the other is set equal. It is possible to extend the model by linking $\theta$ to an additional (possibly overlapping) set of covariates through a proper link function $g_2(\cdot)$ specified in the link.theta argument: $$g_2(\theta) = z^t \bold{\psi},$$ where $\bold{\psi}$ is the vector of regression coefficients for the overdispersion model. The prior distribution and the related hyperparameter of $\bold{\psi}$ can be specified in prior.psi and hyperparam.psi. In flexreg_binom, the regression model for the mean and, where appropriate, for the overdispersion parameter can be specified in the formula argument with a formula of type y ~ x1 + x2 | z1 + z2 where covariates on the left of "|" are included in the regression model for the mean, whereas covariates on the right of "|" are included in the regression model for the overdispersion.

If the second part is omitted, i.e., y ~ x1 + x2, the overdispersion is assumed constant for each observation.

References

Ascari, R., Migliorati, S. (2021). A new regression model for overdispersed binomial data accounting for outliers and an excess of zeros. Statistics in Medicine, 40(17), 3895--3914. doi:10.1002/sim.9005

Examples

Run this code

if (FALSE) {
data(Bacteria)
fbb <- flexreg_binom(y ~ females,  n = "n", data = Bacteria, type = "FBB")
}

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