BBmm: Beta-binomial mixed-effects model

In PROreg: Patient Reported Outcomes Regression Analysis

Beta-binomial mixed-effects model

Description

BBmm function performs beta-binomial mixed-effects models, i.e., it allows the inclusion of gaussian random effects in the linear predictor of a logistic beta-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.

For the estimation of the parameters the joint log-likelihood can be optimized by means of two different approaches: (i) BB-Delta, specific Delta algorithm developed for beta-binomial mixed-effect regression models, and (ii) usual Newton-Raphson method.

Usage

BBmm(fixed.formula,X,y,random.formula,Z=NULL,nRandComp=NULL,m,data,
      method="BB-Delta",maxiter=50,show=FALSE,nDim=1,tolerance=10^(-6))

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. It must be specified in cases where the model matrix of the fixed effects X and the outcome variable y are not specified.
X	a matrix class object containing the covariate structure of the fixed part of the model to be fitted. If the fixed.formula argument is specified this argument should not be defined, as we will be specifying twice the fixed structure of the model.
y	a vector containing the outcome variable(s). If joint analysis is expected, the outcome variables must be listed one after another in a vector.
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 effects in each random component of the model. It must be especified as a vector, where the 'i'th value must correspond with the number of random effects of the 'i'th random component. It must be only determined when we specify the random structure of the model by the model matrix of the random effects, Z.
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).
method	the methodology for performing the estimation of the parameters. Options "BB-Delta" or "NR". Default "BB-Delta".
maxiter	the maximum number of iterations in the parameters estimation algorithm. Default 50.
show	logical parameter. If TRUE, the step by step optimization process will be shown in the screen. FALSE by default.
nDim	number of response outcome variables involved in the multidimensional analysis. nDim=1 by default.
tolerance	tolerance of the estimated linear predictor in the estimation process. tolerance=10^(-6) by default.

Details

BBmm function performs beta-binomial mixed-effects regression models. It extends the beta-binomial regression model including gaussian random effects in the linear predictor. The model is defined as, conditional on some gaussian random effects u, the response variable y follows a beta-binomial distribution with parameters m, p and \phi,

y|u \sim BB(m,p,\phi), \text{and } u \sim N(0,D)

where the regression model is defined as,

\log(p/(1-p))=X*\beta+Z*u

and D is determined by some dispersion parameters icluded in the parameter vector \theta.

The estimation of the regression parameters \beta and the prediction of the random effects u is done via the extended likelihood, where the marginal likelihood is approximated to the h-likelihood by a first order Laplace approximation,

h=f(y|\beta,u,\theta)+f(u|\theta).

The previous formula do not have a closed form and numerical methods are needed for the estimation precedure. Two approches are available in the function in order to perform the fixed and random effects estimation: (i) A special case of the Delta algorithm developed for beta-binomial mixed-effects model estimations, and (ii) the general Newton-Raphson algorithm available in R-package rootSolve.

On the other hand, the estimation of dispersion parameters by the h-likelihood can be bias due to the previous estimation of the regression parameters. Consequenlty, a penalization of the h-likelihood must be performed to get unbiased estimations of the dispersion parameters. Lee and Nelder (1996) proposed the adjusted profile h-likelihood, where the following penalization is applied,

h(\theta)=h+(1/2)*\log[det(2\pi H^{-1})],

being H the Hessian matrix of the model, replacing the terms \beta and u involved in the previous formula with their estimates.

The method iterates between both estimation processes until convergence is reached based on the pre-established tolerance of the linear predictor.

Value

BBmm returns an object of class "BBmm".

The function summary (i.e., summary.BBmm) can be used to obtain or print a summary of the results..

fixed.coef	estimated value of the fixed coefficients of the regression.
fixed.vcov	variance and covariance matrix of the estimated fixed coefficients of 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.
phi.coef	estimated value of the dispersion parameter of the conditional beta-binomial distribution.
psi.coef	estimated value of the logrithm of the dispersion parameter of the conditional beta-binomial distribution.
psi.var	variance of the estimation of the logarithm of the conditional beta-binomial distribution.
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.
null.deviance	null-deviance, deviance of the null model. The null model will only include an intercept as the estimation of the probability parameter.
null.df	degrees of freedom of the null 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.
D	variance and covariance matrix of the random effects.
balanced	if the response dependent variable is balanced it returns "yes", otherwise "no".
m	maximum score number in each beta-binomial observation.
call	the matched call.
formula	the formula supplied.

Author(s)

J. Najera-Zuloaga

D.-J. Lee

I. Arostegui

References

Najera-Zuloaga, J.; Lee, D.-J.; Esteban, C. & Arostegui, I. (2023): Multidimensional beta-binomial regression model: a joint analysis of patient-reported outcomes. Statistical Modelling, 0(0).

Najera-Zuloaga J., Lee D.-J. & Arostegui, I. (2019): A beta-binomial mixed effects model approach for analysing longitudinal discrete and bounded outcomes. Biometrical Journal, 61(3), 600-615.

Najera-Zuloaga J., Lee D.-J. & Arostegui I. (2018): Comparison of beta-binomial regression model approaches to analyze health related quality of life data. Statistical Methods in Medical Research, 27(10), 2989-3009.

Lee Y. & Nelder J. A. (1996): Hierarchical generalized linear models, Journal of the Royal Statistical Society. Series B, 58, 619-678

See Also

The multiroot and uniroot functions of the R-package rootSolve for the general Newton-Raphson algorithm.

Examples

set.seed(7)

# Defining the parameters
k <- 100
m <- 10
phi <- 0.5
beta <- c(1.5,-1.1)
sigma <- 0.5

# Simulating the covariate and random effects
x <- runif(k,0,10)
X <- model.matrix(~x)
z <- as.factor(rBI(k,4,0.5,2))
Z <- model.matrix(~z-1)
u <- rnorm(5,0,sigma)


# The linear predictor and simulated response variable
eta <- beta[1]+beta[2]*x+crossprod(t(Z),u)
p <- 1/(1+exp(-eta))
y <- rBB(k,m,p,phi)
dat <- data.frame(cbind(y,x,z))
dat$z <- as.factor(dat$z)

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

#---------#

# Multidimensional regression with 2 dimensions

set.seed(5)
nId <- 25

# Simulation

m <- 10
beta1 <- c(1,-0.5)
beta2 <- c(-1,0.5)
beta <- cbind(beta1,beta2)

x1 <- rnorm(nId, 1,2)
X1 <- model.matrix(~x1)
x2 <- rnorm(nId, -1,1)
X2 <- model.matrix(~x2)

sigma <- 0.6
u <- rnorm(nId,0,sigma)

eta1 <- beta1[1]+x1*beta1[2]+u
p1 <- 1/(1+exp(-eta1))
eta2 <- beta2[1]+x2*beta2[2]+u
p2 <- 1/(1+exp(-eta2))

phi1 <- 0.3
phi2 <- 1
phi <- c(phi1,phi2)

y1 <- rBB(nId,m,p1,phi1)
y2 <- rBB(nId,m,p2,phi2)
y <- c(y1,y2)

# Define matrices

X <- Matrix::bdiag(X1,X2)
X <- as.matrix(X)
Z. <- diag(nId)
Z <- kronecker(rbind(1,1),Z.)

# Fit the model

Model.multi <- BBmm(X=X,y=y,Z=Z,nRandComp = nId,m=m,nDim=2)
Model.multi

PROreg documentation built on May 29, 2024, 11:29 a.m.