HRQoL (version 1.0)

BBreg:

Description

BBreg function fits a beta-binomial logistic regression model, i.e., it links the probability parameter of a beta-binomial distribution with the given covariates by means of a logistic link function. The estimation of the parameters in the model is done via maximum likelihood estimation.

Usage

BBreg(formula,m,data,maxiter=100)

Arguments

formula

an object of class "formula" (or one that can be coerced to that class): a symbolic description of the model to be fitted.

m

maximum score number in each beta-binomial observation.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula).

maxiter

the maximum number of iterations in the estimation process. Default 100.

Value

BBreg returns an object of class "BBreg". The function summary (i.e., summary.BBreg) can be used to obtain or print a summary of the results.

coefficients

the estimated value of the regression coefficients.

vcov

the variance-covariance matrix of the estimated coefficients of the regression.

phi

the estimation of the dispersion parameter of the beta-binomial distribution.

psi

the estimation of the logarithm of the dispersion parameter of the beta-binomial distribution.

psi.var

the variance of the estimated logarithm of the dispersion parameter of the beta-binomial distribution.

conv

convergence of the methodology. If the method has converged it returns "yes", otherwise "no".

fitted.values

the fitted mean values of the model.

deviance

the deviance of the model.

df

degrees of freedom of the model.

null.deviance

null-deviance, the deviance for the null model. The null model will only include an intercept as the estimation of the probability parameter.

null.df

the degrees of freedom for the null model.

iter

number of iterations in the estimation process.

X

the model matrix.

y

the dependent response variable in the model.

m

maximum score number in each beta-binomial observation.

balanced

if the response beta-binomial variable is balanced it returns "yes", otherwise "no".

nObs

number of observations.

call

the matched call.

formula

the formula supplied.

Details

There are two different ways of defining a regression model based on the beta-binomial distribution: (i) the marginal regression approach, (ii) hierarchical generalized linear model approach. Najera-Zuloaga et al. (2017) proved that the first approach is more adequate when the interest consists of the interpretation of the regression coefficients. Consequenlty, this function is based on the first approach, i.e., the marginal regression approach. Once the marginal density function of the beta-binomial distribution is explicity calculated, we connect the probability parameter with the given covariates by means of a logistic link function: $$logit(p)=log(p/(1-p))=X*beta$$ where \(X\) a model matrix composed by the given covariates and \(beta\) are the regression coefficients of the model. Remplacing the previous linear predictor in the marginal density function of the beta-binomial distribution, we can derive maximum likelihood estimations of both regression and dispersion parameters. Forcina and Franconi (1988) presented an estimation algorithm based on the Newton-Raphson approach. This function performs the estimation of the parameters following the presented methodology.

References

Forcina A. & Franconi L. (1988): Regression analysis with Beta-Binomial distribution, Revista di Statistica Applicata, 21, 7-12 Najera-Zuloaga J., Lee D.-J. & Arostegui I. (2017): Comparison of beta-binomial regression model approaches to analyze health related quality of life data, Statistical Methods in Medical Research, DOI: 10.1177/0962280217690413

Examples

# We simulate a covariate, fix the paramters of the beta-binomial 
# distribution and simulate a response variable.

# Then we apply the model, and try to get the same values.
set.seed(18)
k <- 1000
m <- 10
x <- rnorm(k,5,3)

beta <- c(-10,2)
p <- 1/(1+exp(-(beta[1]+beta[2]*x)))
phi <- 1.2

y <- rBB(k,m,p,phi)

model <- BBreg(y~x,m)
model