HRQoL (version 1.0)

BIreg:

Description

BIreg function fits a binomial logistic regression model, i.e., it links the probability parameter of a binomial distribution with the given covariates by means of a logistic link function. There is the option to include a dispersion parameter in the binomial distribution, which will be estimated by the bias corrected method of moments.

Usage

BIreg(formula,m,data,disp=FALSE,maxiter=20)

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

number of trials in each 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).

disp

if TRUE a dispersion parameter will be estimated. Default FALSE.

maxiter

the maximum number of iterations in IWLS method. Default 20.

Value

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

coefficients

the estimated value of the regression coefficients.

vcov

the variance and covariance matrix of the estimated regression coefficients.

phi

if disp TRUE, it returns the estimated value of the dispersion parameter. If disp FALSE, then the estimated value is 1. Default FALSE.

fitted.values

the fitted mean values of the model.

residuals

working residuals, i.e. the residuals in the final iteration of the IWLS method.

deviance

deviance of the model.

df

degrees of freedom of the model.

null.deviance

null-deviance, deviance for the null model. The null model will include only an intercept.

df.null

degrees of freedom for the null model.

iter

number of iterations in the IWLS method.

conv

if the algorithm has converged it returns "yes", otherwise "no".

X

model matrix.

y

dependent variable in the model.

balanced

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

m

number of trials in each binomial observation.

nObs

number of observations.

call

the matched call.

formula

the formula supplied.

Details

BIreg function performs a regression model linking by a logistic function the probability paramater of a binomial distribution with a linear predictor that consists of the given covariates. Following the exponential family theory, the binomial distribution with dispersion parameter has the following log-likelihood function: $$l=[y*log(p/(1-p))+m*log(1-p)]/phi+c(y,phi)$$ where \(c()\) is a known function. If we any dispersion parameter is not considered the usual density function of the binomial distribution will be used, $$l=y*log(p)+(m-y)*log((1-p)).$$ As explained before we link the probablity parameter with the given covariates by $$logit(p)=log(p/(1-p))=x_i'*beta$$ where \(beta\) are the regression coefficients and \(x_i\) is the \(i\)th row of a full rank design matrix \(X\) composed by the given covariables. The estimation of the regression parameters \(beta\) is done via maximum likelihood approach, where the iterative weighted least square (IWLS) method is applied. If disp is TRUE, a dispersion parameter will be added in the binomial distribution and, consequently, the method will deal with the general definition of the log-likelihood formula, otherwise the usual and simpler one will be used. In case the dispersion parameter is included, the estimation will be done with a bias-corrected method of moments: $$phi=Var[y]/[(m-q)*p*(1-p)]$$ where \(q\) is the number of estimated regression paramters, and \(p\) is the estimated probability parameter. The deviance of the model is defined by the ratio between the log-likelihood of the estimated model and saturated or null model. If the dispersion paramter is included the scaled deviance is obtained dividing the deviance by the dispersion parameter.

References

Pawitan Y. (2001): In All Likelihood: Statistical Modelling and Inference Using Likelihood, Oxford University Press Williams D. A. (1982): Extra-Binomial Variation in Logistic Linear Regression, Journal of the Royal Statistical Society. Series C, 31, 144-148

See Also

Iterative weighted least squares method function BIiwls in R-package HRQoL.

Examples

  set.seed(1234)
  # We simulate a covariable and construct the outcome variable applying
  # an inverse logit link function on it.
  
  m <- 10                       
  k <-100                       
  covariate <- rnorm(k,2,0.5)  
  
  beta <- c(-6,4)
  p <- 1/(1+exp(-(beta[1]+beta[2]*covariate)))
  
  # without dispersion parameter
  outcome <- rBI(k,m,p)
  model <- BIreg(outcome~covariate,m,disp=FALSE)
  model
  
  # with dispersion parameter
  phi <- 2
  outcome.disp <- rBI(k,m,p,phi)
  model.disp <- BIreg(outcome.disp~covariate,m,disp=TRUE)
  model.disp