R: Bayesian estimation of (zero-inflated) Discrete Weibull...

bdw.reg {BDgraph}

R Documentation

Bayesian estimation of (zero-inflated) Discrete Weibull regression

Description

Bayesian estimation of the parameters for Discrete Weibull (DW) regression. The conditional distribution of the response given the predictors is assumed to be DW with parameters q and beta, dependent on the predictors, and, with an additional parameter pi under zero inflation.

Usage

bdw.reg( data, formula = NA, iter = 5000, burnin = NULL, 
         dist.q = dnorm, dist.beta = dnorm, 
         par.q = c( 0, 1 ), par.beta = c( 0, 1 ), par.pi = c( 1, 1 ), 
         initial.q = NULL, initial.beta = NULL, initial.pi = NULL, 
         ZI = FALSE, scale.proposal = NULL, adapt = TRUE, print = TRUE )

Arguments

`data`	`data.frame` or `matrix` corresponding to the data, containing the variables in the model.
`formula`	object of class formula as a symbolic description of the model to be fitted. For the case of `data.frame`, it is taken as the model frame (see `model.frame)`.
`iter`	number of iterations for the sampling algorithm.
`burnin`	number of burn-in iterations for the sampling algorithm.
`dist.q`	Prior density for the regression coefficients associated to the parameter `q`. The default is a Normal distribution (`dnorm`). Any density function which has two parameters and can support the `log = TRUE` flag can be used, e.g. `dnorm`, `dlnorm`, `dunif` etc.
`dist.beta`	Prior density for the regression coefficients associated to the parameter `beta`. The default is a Normal distribution (`dnorm`). Any density function which has two parameters and can support the `log = TRUE` flag can be used, e.g. `dnorm`, `dlnorm`, `dunif` etc.
`par.q`	vector of length two corresponding to the parameters of `dist.q`.
`par.beta`	vector of length two corresponding to the parameters of `dist.beta`.
`par.pi`	vector of length two corresponding to the parameters of the `beta` prior density on `pi`.
`initial.q, initial.beta, initial.pi`	vector of initial values for the regression coefficients and for `pi` (if `ZI` = `TRUE`).
`ZI`	logical: if FALSE (default), the conditional distribution of the response given the predictors is assumed to be DW with parameters `q` and `beta`. If TRUE, a zero-inflated DW distribution will be applied.
`scale.proposal`	scale of the proposal function. Setting to lower values results in an increase in the acceptance rate of the sampler.
`adapt`	logical: if TRUE (default), the proposals will be adapted. If FALSE, no adapting will be applied.
`print`	logical: if TRUE (default), tracing information is printed.

Details

The regression model uses a logit link function on q and a log link function on beta, the two parameters of a DW distribution, with probability mass function given by

DW(y) = q^y^β - q^(y+1)^β, y = 0, 1, 2, …

For the case of zero inflation (ZI = TRUE), a zero-inflated DW is considered:

f(y) = ( 1 - pi ) I(y = 0 ) + pi DW(y)

where 0 ≤q pi ≤q 1 and I(y = 0 ) is an indicator for the point mass at zero for the response y.

Value

`sample`	MCMC samples
`q.est`	posterior estimates of `q`
`beta.est`	posterior estimates of `beta`
`pi.est`	posterior estimates of `pi`
`accept.rate`	acceptance rate of the MCMC algorithm

Author(s)

Veronica Vinciotti, Reza Mohammadi a.mohammadi@uva.nl, and Pariya Behrouzi

References

Vinciotti, V., Behrouzi, P., and Mohammadi, R. (2022) Bayesian structural learning of microbiota systems from count metagenomic data, arXiv preprint, doi: 10.48550/arXiv.2203.10118

Peluso, A., Vinciotti, V., and Yu, K. (2018) Discrete Weibull generalized additive model: an application to count fertility, Journal of the Royal Statistical Society: Series C, 68(3):565-583, doi: 10.1111/rssc.12311

Haselimashhadi, H., Vinciotti, V. and Yu, K. (2018) A novel Bayesian regression model for counts with an application to health data, Journal of Applied Statistics, 45(6):1085-1105, doi: 10.1080/02664763.2017.1342782

Mohammadi, R. and Wit, E. C. (2019). BDgraph: An R Package for Bayesian Structure Learning in Graphical Models, Journal of Statistical Software, 89(3):1-30, doi: 10.18637/jss.v089.i03

Examples

## Not run: 
# - - Example 1

q    = 0.6
beta = 1.1
n    = 500

y = BDgraph::rdweibull( n = n, q = q, beta = beta )

output = bdw.reg( data = y, y ~ ., iter = 5000 )

output $ q.est
output $ beta.est

traceplot( output $ sample[ , 1 ], acf = T, pacf = T )
traceplot( output $ sample[ , 2 ], acf = T, pacf = T )

# - - Example 2

q    = 0.6
beta = 1.1
pii  = 0.8
n    = 500

y_dw = BDgraph::rdweibull( n = n, q = q, beta = beta )
z = rbinom( n = n, size = 1, prob = pii ) 
y = z * y_dw

output = bdw.reg( data = y, iter = 5000, ZI = TRUE )

output $ q.est
output $ beta.est
output $ pi.est

traceplot( output $ sample[ , 1 ], acf = T, pacf = T )
traceplot( output $ sample[ , 2 ], acf = T, pacf = T )
traceplot( output $ sample[ , 3 ], acf = T, pacf = T )

# - - Example 3

theta.q    = c( 0.1, -0.1, 0.34 )  # true parameter
theta.beta = c( 0.1, -.15, 0.5  )  # true parameter

n  = 500

x1 = runif( n = n, min = 0, max = 1.5 )
x2 = runif( n = n, min = 0, max = 1.5 )

reg_q = theta.q[ 1 ] + x1 * theta.q[ 2 ] + x2 * theta.q[ 3 ]
q     = 1 / ( 1 + exp( - reg_q ) )

reg_beta = theta.beta[ 1 ] + x1 * theta.beta[ 2 ] + x2 * theta.beta[ 3 ]
beta     = exp( reg_beta )

y = BDgraph::rdweibull( n = n, q = q, beta = beta )

data = data.frame( x1, x2, y ) 

output = bdw.reg( data, y ~. , iter = 5000 )

# - - Example 4

theta.q    = c( 1, -1, 0.8 )  # true parameter
theta.beta = c( 1, -1, 0.3 )  # true parameter
pii = 0.8

n  = 500

x1 = runif( n = n, min = 0, max = 1.5 )
x2 = runif( n = n, min = 0, max = 1.5 )

reg_q = theta.q[ 1 ] + x1 * theta.q[ 2 ] + x2 * theta.q[ 3 ]
q     = 1 / ( 1 + exp( - reg_q ) )

reg_beta = theta.beta[ 1 ] + x1 * theta.beta[ 2 ] + x2 * theta.beta[ 3 ]
beta     = exp( reg_beta )

y_dw = BDgraph::rdweibull( n = n, q = q, beta = beta )
z    = rbinom( n = n, size = 1, prob = pii ) 
y    = z * y_dw

data = data.frame( x1, x2, y ) 

output = bdw.reg( data, y ~. , iter = 5000 )

## End(Not run)

[Package BDgraph version 2.65 Index]