# flexreg_binom: Flexible Regression Models for Binomial data In FlexReg: Regression Models for Bounded Responses

## Description

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

## Usage

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 flexreg_binom( formula, data, type = "FBB", n = NULL, 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.iter = 5000, burnin.perc = 0.5, n.chain = 1, thin = 1, verbose = TRUE, ... ) 

## Arguments

 formula an object of class formula: a symbolic description of the model to be fitted (of type y ~ x or y ~ x | z). data an optional data frame, list, or object that is coercible to a data frame through base::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 is called. type a character specifying the type of regression model. Current options are the flexible beta-binomial "FBB" (default), the beta-binomial "BetaBin", and the binomial one "Bin". n the total number of trials. link.mu a character specifying the link function for the mean model (mu). Currently, "logit" (default), "probit", "cloglog", and "loglog" are supported. prior.beta a character specifying the prior distribution for the beta regression coefficients of the mean model. Currently, "normal" (default) and "cauchy" are supported. hyperparam.beta a positive numeric (vector of length 1) specifying the hyperprior standard deviation parameter for the prior distribution of beta regression coefficients. A value of 100 is suggested if the prior is "normal", 2.5 if "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 (theta). 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 psi regression coefficients of the overdispersion model (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 standard deviation parameter for the prior distribution of psi regression coefficients. A value of 100 is suggested if the prior is "normal", 2.5 if "cauchy". n.iter a positive integer specifying the number of iterations for each chain (including warmup). The default is 5000. burnin.perc the percentage of iterations per chain to discard. n.chain a positive integer specifying the number of Markov chains. The default is 1. thin a positive integer specifying the period for saving samples. The default is 1. verbose TRUE (default) or FALSE: flag indicating whether to print intermediate output. ... additional arguments for rstan::sampling.

## Details

Let Y be a random variable whose distribution can be specified in the type argument and μ be the mean of Y/n. The flexreg_binom function links the parameter μ to a linear predictor through a function g(\cdot) specified in link.mu:

g(μ_i) = x_i^t \bold{β},

where \bold{β} is the vector of regression coefficients for the mean model. By default, link.theta="identity", meaning that the overdispersion parameter θ is assumed to be constant. It is possible to extend the model by linking θ to an additional (possibly overlapping) set of covariates through a proper link function q(\cdot) specified in the link.theta argument:

q(θ_i) = z_i^t \bold{ψ},

where \bold{ψ} is the vector of regression coefficients for the overdispersion model. 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 \sim x_1 + x_2 | z_1 + z_2 where covariates on the left of ("|") are included in the regression model for the mean and covariates on the right of ("|") are included in the regression model for the overdispersion.

If the second part is omitted, i.e., y \sim x_1 + x_2, the overdispersion is assumed constant for each observation.

## Value

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

 call the function call. 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).

## References

Ascari, R., and 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

 1 2 3 4 5 ## Not run: data(Bacteria) fbb <- flexreg_binom(y~females, n=n, data=Bacteria, type="FBB") ## End(Not run) 

FlexReg documentation built on Jan. 17, 2022, 5:06 p.m.