HRQoL (version 1.0)

BImm:

Description

BImm function performs binomial logistic mixed-effects models, i.e., it allows the inclusion of gaussian random effects in the linear predictor of a logistic binomial regression model. The structure of the random part of the model can be expecified by two different ways: (i) determining the random.formula argument, or (ii) especifying the model matrix of the random effects, Z, and determining the number of random effects in each random component, nRandComp.

Usage

BImm(fixed.formula,random.formula,Z=NULL,nRandComp=NULL,m,data,maxiter=100)

Arguments

fixed.formula

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

random.formula

an object of class "formula" (or one that can be coerced to that class): a symbolic description of the random part of the model to be fitted. It must be specified in cases where the model matrix of the random effects Z is not determined.

Z

the design matrix of the random effects. If the random.formula argument is specified this argument should not be specified, as we will be specifying twice the random structure of the model.

nRandComp

the number of random components/levels in each random effect of the model. It must be especified as a vector, where the 'i'th value must correspond with the number of random components of the 'i'th random effect. It must be only determined when we specify the random structure of the model by the model matrix of the random effects, Z.

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).

maxiter

the maximum number of iterations in the parameters estimation algorithm. Default 100.

Value

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

fixed.coef

estimated value of the fixed coefficients in the regression.

fixed.vcov

variance and covariance matrix of the estimated fixed coefficients in the regression.

random.coef

predicted random effects of the regression.

sigma.coef

estimated value of the random effects variance parameters.

sigma.var

variance of the estimated value of the random effects variance parameters.

fitted.values

the fitted mean values of the probability parameter of the conditional beta-binomial distribution.

conv

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

deviance

deviance of the model.

df

degrees of freedom of the model.

nRand

number of random effects.

nComp

number of random components.

nRandComp

number of random effects in each random component of the model.

namesRand

names of the random components.

iter

number of iterations in the estimation method.

nObs

number of observations in the data.

y

dependent response variable in the model.

X

model matrix of the fixed effects.

Z

model matrix of the random effects.

balanced

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

m

maximum score number in each binomial observation.

call

the matched call.

formula

the formula supplied.

Details

The model that is performed by this function is a especial case of generalized linear mixed models (GLMMs), in which conditioned on some random components the response variable has a binomial distribution. As in the binomial (logistic) regression a logit link function is applied to the probability parameter of the conditioned binomial distribution, allowing the inclusion of random effects in the linear predictor, $$logit(p)=X*beta+Z*u,$$ where \(p\) is the probability parameter, \(X\) a full rank matrix composed by the covariables, \(beta\) the fixed effects, \(Z\) the design matrix for the random effects ang \(u\) are the random effects. These random effects are independent and have a normal distribution with the same variance and mean 0. The model estimates the fixed effects, predicts the random effects, and gets the estimation of the random effects variance parameters. The estimation procedure is done by likelihood approximation, via iteartive weighted least squares method. The process is performed in two steps: (i) fixed and random parameters are estimated for some given values of the random effects variance parameters, and (ii) random effects variance parameters are estimated for some given regression and random coefficients using a penalized profile likelihood. The estimation approach iterates between (i) and (ii) until convergence is obtained.

References

Breslow N. E. & Calyton D. G. (1993): Approximate Inference in Generalized Linear Mixed Models, Journal of the American Statistical Association, 88, 9-25 McCulloch C. E. & Searle S. R. (2001): Generalized, Linear, and Mixed Models, Jhon Wiley & Sons Pawitan Y. (2001): In All Likelihood: Statistical Modelling and Inference Using Likelihood, Oxford University Press

Examples

set.seed(5)
# Fixing parameters for the simulation:
nObs <- 1000
m <- 10
beta <- c(1.5,-1.1)
sigma <- 0.8

# Simulating the covariate:
x <- runif(nObs,-5,5)

# Simulating the random effects:
z <- as.factor(rBI(nObs,5,0.5,2))
u <- rnorm(6,0,sigma)

# Getting the linear predictor and probability parameter.
X <- model.matrix(~x)
Z <- model.matrix(~z-1)
eta <- beta[1]+beta[2]*x+crossprod(t(Z),u)
p <- 1/(1+exp(-eta))

# Simulating the response variable
y <- rBI(nObs,m,p)

# Apply the model
model <- BImm(fixed.formula = y~x,random.formula = ~z,m=m)
model