Bayesian Inference for Zero/One Inflated Beta Regression

Description

Description: zoib fits a zero/one inflated beta regression model and obtains the Bayesian Inference for the model via the Markov Chain Monte Carlo approach implemented in JAGS.

Usage

zoib(model, data, n.response=1, joint = TRUE,
     zero.inflation = TRUE, one.inflation = TRUE, 
     random = 0, EUID, link.mu = "logit", link.x0 = "logit",
     link.x1 = "logit", prec.int = matrix(1e-3, n.response, 4), 
     prior.beta = matrix("DN", n.response, 4), 
     prec.DN =  matrix(1e-3, n.response, 4),
     lambda.L2 =  matrix(1e-3, n.response, 4), 
     lambda.L1 =  matrix(1e-3, n.response, 4),
     lambda.ARD =  matrix(1e-3, n.response, 4), prior.Sigma = "VC.unif",  
     scale.unif = 20,scale.halfcauchy = 20,     
     n.chain = 2, n.iter = 5000, n.burn =200, n.thin = 2, inits=NULL, seeds=NULL)

Arguments

Details

************** model **************
When there are multiple responses y, each y should be separated by "|", such as "y1 | y2 | y3" on the left hand side (LHS) of the formula. On the right side of the formula, it could include up to 5 parts in the following order: xb | xd | x0 | x1 | z, where xb represents the fixed-effects covariates/factors in the link function of the mean of the beta distribution, xd represents the fixed-effects covariates/factors in the link function of the sum of the two shape parameters of the beta distribution, x0 represents the fixed-effect covariates/factors in the link function of Pr(y=0), x1 represents the fixed-effect covariates/factors in the link function of Pr(y=1|y>0), and z represents the random-effect covariates/factors. The current version of the package only accomodates z being the same across all the link function who have a random component. xb and xd should always be specified, even if xd contains only an intercept. If there is no zero inflation in any of the y's, then the x0 part can be omitted, similarly for the x1 part and the random effects part z.

For example, if there are 3 response variables and 2 independent variables (xx1, xx2), and none of the y's have zero inflation, and all have one inflation, then specification of y1 | y2 | y3 ~ xx1 + xx2 | 1 | xx1 | xx2, together with zero.inflation = c(F,F,F) and one.inflation = c(T,T,T), implies xb = (1, xx1, xx2), xd = 1 (intercept), x0 = NULL, x1 = (1, xx1) for all y's, z = (1, xx2). If y1 has inflation at 0, and y3 has inflation at 1, and there is no random effect, model y1 | y2 | y3 ~ xx1 + xx2 | xx1 | xx1 | xx2, together with zero.inflation = c(T,F,F) and one.inflation=c(F,F,T), implies xb = (1, xx1, xx2), xd = (1, xx1), x0 = (1, xx1) for y1, x1 = (1, xx2) for y3. Refer to Formula in package Formula for more details on the specification of terms in formula such as interaction terms and categorical x's, etc.

************* random *************
Denote the four link functions by

Eqn 1. g(mu) = xb*b, where g is the link function (logit, probit or cloglog), and mu is the mean of the Beta distribution
Eqn 2. log(eta) = xd*d, where eta is the sum of the two shape parameters of the Beta distribution
Eqn 3. g(p0) = x0*b0, where g is the link function (logit, probit or cloglog), and p0 = Pr(y=0)
Eqn 4. g(p1) = x1*b1, where g is the link function (logit, probit or cloglog), and p1 = Pr(y=1|y>0)
then random =
0: no random effect (default);
1: only the linear predictor in Eqn 1 has a random component; that is, g(mu) = xb*b +z*gamma
2: only the linear predictor in Eqn 2 has a random component; that is, log(eta) = xd*d+z*gamma
3: only the linear predictor in Eqn 3 has a random component; that is, g(p0) = x0*b0 +z*gamma
4: only the linear predictor in Eqn 4 has a random component; that is, g(p1) = x1*b1+z*gamma
12: the linear predictors in Eqns 1 and 2 contain random component z*gamma
13: the linear predictors in Eqns 1 and 3 contain random component z*gamma
14: the linear predictors in Eqns 1 and 4 contain random component z*gamma
23: the linear predictors in Eqns 2 and 3 contain random component z*gamma
24: the linear predictors in Eqns 2 and 4 contain random component z*gamma
34: the linear predictors in Eqns 3 and 4 contain random component z*gamma
123: the linear predictors in Eqns 1, 2 and 3 contain random component z*gamma
134: the linear predictors in Eqns 1, 3 and 4 contain random component z*gamma
124: the linear predictors in Eqn 1, 2 and 4 contain random component z*gamma
1234: the linear predictors in all Eqns contain random component z*gamma

*************** prior.beta ***************
1. "DN": Diffuse Normal
2. "L1": L1-like shrinkage prior
3. "L2": L2-like shrinkage prior
4. "ARD": ARD-like shrinkage prior

************** prior.Sigma **************
1. "VC.unif": Sigma is diagonal, the prior for the standard deviation of each random variable follows a uniform distribution;
2. "VC.halfcauchy": Sigma is diagonal, the prior for the standard deviation of each random variable follows a half-Cauchy distribution with a large scale parameter;
3. "UN.unif": Sigma is fully parameterized, the prior for the standard deviation of each random variable follows a uniform distribution; each element in the correlation matrix follows an independent unif(0,1) distribution with necessary constraints to ensure positive definiteness.
4. "UN.halfcauchy". Sigma is fully parameterized, the prior for the standard deviation of each random variable follows an half-Cauchy distribution; each element in the correlation matrix follows an independent unif(0,1) distribution with necessary constraints to ensure positive definiteness.

************** inits **************
If supplied, should be in the following parameter order:
inits = list(list(b=, d=, b0=, b1=, sigma=, R=),list(b=, d=, b0=, b1=, sigma=, R=), ...)
The notations b, d, b0, b1 are the same as in the "random" section above, with each specified in a matrix format of dimension (nx+1)*n.response, where nx is number of regression coefficients corresponding to the respective x's (+1 because of the intercept). sigma is a vector containing the standard deviation of each random component, and R contains the lowe.r triangle of the correlation matrix. For example, in a 3x3 correlation matrix, R is specified as c(1,r21,1,r31,r32,1). Each inner list contains the starting values for one MCMC chain. If initial values are specified only for a subset of the parameters in a model, NULL should be used for the rest of the unspecified parameters (whose initial values will be generated automatically by the function).

Value

Author(s)

Fang Liu (fang.liu.131@nd.edu)

References

Liu, F. and Kong, Y. (2015). ZOIB: an R Package for Bayesian Inferences in Beta and Zero One Inflated Beta Regression Models, The R Journal, 7(2):34-51

Liu, F. and Li, Q. (2014) A Bayesian Model for Joint Analysis of Multivariate Repeated Measures and Time to Event Data in Crossover Trials, Statistical Methods in Medical Research, doi: 10.1177/0962280213519594

Examples

  #dontrun{
    # refer to data sets GasolineYield, BiRepeated, and AlcoholUse in package zoib
    # for examples on fixed effect models, mixed effects models, joint modeling 
    # of bivariate beta variables with repeated measures, and modelling clustered 
    # beta variables with inflation at 0 using zoib
  #}

Results