Version: 0.1
Title: Zero-Inflated and Hurdle Modelling Using Bayesian Inference
Description: When considering count data, it is often the case that many more zero counts than would be expected of some given distribution are observed. It is well established that data such as this can be reliably modelled using zero-inflated or hurdle distributions, both of which may be applied using the functions in this package. Bayesian analysis methods are used to best model problematic count data that cannot be fit to any typical distribution. The package functions are flexible and versatile, and can be applied to varying count distributions, parameter estimation with or without explanatory variable information, and are able to allow for multiple hurdles as it is also not uncommon that count data have an abundance of large-number observations which would be considered outliers of the typical distribution. In lieu of throwing out data or misspecifying the typical distribution, these extreme observations can be applied to a second, extreme distribution. With the given functions of this package, such a two-hurdle model may be easily specified in order to best manage data that is both zero-inflated and over-dispersed.
Date: 2017-07-01
Depends: R (≥ 3.3.0)
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding: UTF-8
LazyData: true
RoxygenNote: 6.0.1
NeedsCompilation: no
Packaged: 2017-07-01 19:29:37 UTC; Earvin
Author: Earvin Balderama [aut, cre], Taylor Trippe [aut]
Maintainer: Earvin Balderama <ebalderama@luc.edu>
Repository: CRAN
Date/Publication: 2017-07-02 00:04:08 UTC

The Generalized Pareto Distribution

Description

Density, distribution function, quantile function and random generation for the Generalized Pareto distribution with parameters mu, sigma, and xi.

Usage

dgpd(x, mu = 0, sigma = 1, xi = 1, log = F)

mgpd(x, mu = 0, sigma = 1, xi = 1, log = F)

pgpd(q, mu = 0, sigma = 1, xi = 1, lower.tail = T)

qgpd(p, mu = 0, sigma = 1, xi = 1, lower.tail = T)

rgpd(n, mu = 0, sigma = 1, xi = 1)

Arguments

x, q

vector of quantiles.

mu

location parameter.

sigma

(non-negative) scale parameter.

xi

shape parameter.

log

logical; if TRUE, probabilities p are given as log(p).

lower.tail

logical; if TRUE, probabilities are P[X \le x], otherwise, P[X > x].

p

numeric predictor matrix.

n

number of random values to return.

Details

The generalized pareto distribution has density

f(x) = \frac{\sigma^{\frac{1}{\xi}}}{(\sigma + \xi(x-\mu))^{\frac{1}{\xi}+1}}

Value

dgpd gives the continuous density, pgpd gives the distribution function, qgpd gives the quantile function, and rgpd generates random deviates.

mgpd gives a probability mass function for a discretized version of GPD.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

Examples

dexp(1,rate=.5) #Exp(rate) equivalent to gpd with mu=0 AND xi=0, and sigma=1/rate.
dgpd(1,mu=0,sigma=2,xi=0) #cannot take xi=0.
dgpd(1,mu=0,sigma=2,xi=0.0000001) #but can get close.

##"mass" function of GPD
mgpd(8) == pgpd(8.5) - pgpd(7.5)

Extreme Count Probability Likelihood

Description

PE is used to calculate the likelihood of a user-defined 'extreme' value count observation in a double-hurdle regression model.

Usage

PE(p, q, log = T)

Arguments

p

vector of zero-count probabilities.

q

vector of 'extreme' count probabilities.

log

logical operator. If TRUE, probabilities p and q are given as log(p), log(q).

Value

A vector of probabilities.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle

Typical Count Probability Likelihood

Description

PT is used to calculate the likelihood of a user-defined 'typical' value count observation in a double-hurdle regression model.

Usage

PT(p, q, log = T)

Arguments

p

vector of zero-count probabilities.

q

vector of 'typical' count probabilities.

log

logical operator. If TRUE, probabilities p and q are given as log(p), log(q).

Value

A vector of probabilities.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle

Zero Count Probability Likelihood

Description

PZ is used to calculate the likelihood of a zero-value count observation in a single or double-hurdle regression model.

Usage

PZ(p, log = T)

Arguments

p

vector of zero-count probabilities.

log

logical operator. If TRUE, probabilities p and q are given as log(p), log(q).

Value

A vector of probabilities.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle

Distributional Likelihood for Hurdle Model Count Data Regression

Description

dist_ll is the data likelihood fuction for hurdle model regression using hurdle.

Usage

dist_ll(y, hurd = Inf, lam = NULL, size = 1, mu = NULL, xi = NULL,
  sigma = NULL, dist = c("poisson", "nb", "lognormal", "gpd"), g.x = F,
  log = T)

Arguments

y

numeric response vector.

hurd

numeric threshold for 'extreme' observations of two-hurdle models. Inf for one-hurdle models.

lam

current value for the poisson likelihood lambda parameter.

size

size parameter for negative binomial likelihood distributions.

mu

current value for the negative binomial or log normal likelihood mu parameter.

xi

current value for the generalized pareto likelihood xi parameter.

sigma

current value for the generalized pareto likelihood sigma parameter.

dist

character specification of response distribution.

g.x

logical operator. TRUE if operating within the third component of the likelihood function (the likelihood of 'extreme' observations).

log

logical operator. if TRUE, probabilities p are given as log(p).

Details

Currently, Poisson, Negative Binomial, log-Normal, and Generalized Pareto distributions are available.

Value

The log-likelihood of the zero-inflated Poisson fit for the current iteration of the MCMC algorithm.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle

Hurdle Model Count Data Regression

Description

hurdle is used to fit single or double-hurdle regression models to count data via Bayesian inference.

Usage

hurdle(y, x = NULL, hurd = Inf, dist = c("poisson", "nb", "lognormal"),
  dist.2 = c("gpd", "poisson", "lognormal", "nb"),
  control = hurdle_control(), iters = 1000, burn = 500, nthin = 1,
  plots = FALSE, progress.bar = TRUE)

Arguments

y

numeric response vector.

x

numeric predictor matrix.

hurd

numeric threshold for 'extreme' observations of two-hurdle models. Inf for one-hurdle models.

dist

character specification of response distribution.

dist.2

character specification of response distribution for 'extreme' observations of two-hurdle models.

control

list of parameters for controlling the fitting process, specified by hurdle_control.

iters

number of iterations for the Markov chain to run.

burn

numeric burn-in length.

nthin

numeric thinning rate.

plots

logical operator. TRUE to output plots.

progress.bar

logical operator. TRUE to print progress bar.

Details

Setting dist and dist.2 to be the same distribution creates a single dist-hurdle model, not a double-hurdle model. However, this is being considered in future package updates.

Value

hurdle returns a list which includes the items

pD

measure of model dimensionality p_D where p_D = \bar{D} - D(\bar{\theta}) is the "mean posterior deviance - deviance of posterior means"

DIC

Deviance Information Criterion where DIC = \bar{D} - p_D

PPO

Posterior Predictive Ordinate (PPO) measure of fit

CPO

Conditional Predictive Ordinate (CPO) measure of fit

pars.means

posterior mean(s) of third-component parameter(s) if hurd != Inf

ll.means

posterior means of the log-likelihood distributions of all model components

beta.means

posterior means regression coefficients

dev

posterior deviation where D = -2LogL

beta

posterior distributions of regression coefficients

pars

posterior distribution(s) of third-component parameter(s) if hurd != Inf

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

Examples

#Generate some data:
p=0.5; q=0.25; lam=3;
mu=10; sigma=7; xi=0.75;
n=200

set.seed(2016)
y <- rbinom(n,1,p)
nz <- sum(1-y)
extremes <- rbinom(sum(y),1,q)
ne <- sum(extremes)
nt <- n-nz-ne
yt <- sample(mu-1,nt,replace=TRUE,prob=dpois(1:(mu-1),3)/(ppois(mu-1,lam)-ppois(0,lam)))
yz <- round(rgpd(nz,mu,sigma,xi))
y[y==1] <- c(yt,yz)
g <- hurdle(y)

Control Parameters for Hurdle Model Count Data Regression

Description

Various parameters for fitting control of hurdle model regression using hurdle.

Usage

hurdle_control(a = 1, b = 1, size = 1, beta.prior.mean = 0,
  beta.prior.sd = 1000, beta.tune = 1, pars.tune = 0.2, lam.start = 1,
  mu.start = 1, sigma.start = 1, xi.start = 1)

Arguments

a

shape parameter for gamma prior distributions.

b

rate parameter for gamma prior distributions.

size

size parameter for negative binomial likelihood distributions.

beta.prior.mean

mu parameter for normal prior distributions.

beta.prior.sd

standard deviation for normal prior distributions.

beta.tune

Markov-chain tuning for regression coefficient estimation.

pars.tune

Markov chain tuning for parameter estimation of 'extreme' observations distribution.

lam.start

initial value for the poisson likelihood lambda parameter.

mu.start

initial value for the negative binomial or log normal likelihood mu parameter.

sigma.start

initial value for the generalized pareto likelihood sigma parameter.

xi.start

initial value for the generalized pareto likelihood xi parameter.

Value

A list of all input values.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle

Zero-inflated Negative Binomial Data Likelihood

Description

Data likelihood fuction for zero-inflated negative binomial model regression using zero_nb.

Usage

loglik_zinb(y, z, mu, size, p)

Arguments

y

numeric response vector.

z

vector of binary operators. z == 0 for observations considered belnging to the negative binomial distribution, z == 1 for observations considered to be 'extra' zeros.

mu

current value for the negative binomial likelihood mu parameter.

size

size parameter for negative binomial distribution.

p

vector of 'extra' zero-count probabilities.

Value

The log-likelihood of the zero-inflated negative binomial fit for the current iteration of the MCMC algorithm.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

zero_nb

Zero-inflated Poisson Data Likelihood

Description

Data likelihood fuction for zero-inflated Poisson model regression using zero_poisson.

Usage

loglik_zip(y, z, lam, p)

Arguments

y

numeric response vector.

z

vector of binary operators. z == 0 for observations considered belnging to the negative binomial distribution, z == 1 for observations considered to be 'extra' zeros.

lam

current value for the Poisson likelihood lambda parameter.

p

vector of 'extra' zero-count probabilities.

Value

The log-likelihood of the zero-inflated Poisson fit for the current iteration of the MCMC algorithm.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

zero_poisson

Density Function for Discrete Log Normal Distribution

Description

Density function of the discrete log normal distribution whose logarithm has mean equal to meanlog and standard deviation equal to sdlog.

Usage

mlnorm(x, meanlog = 0, sdlog = 1, log = T)

Arguments

x

vector of quantiles.

meanlog

mean of the distribution on the log scale.

sdlog

standard deviation of the distribution on the log scale.

log

logical; if TRUE, probabilities p are given as log(p).

Value

Discrete log-normal distributional density.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

MCMC Second-Component Parameter Update Function for Hurdle Model Count Data Regression

Description

MCMC algorithm for updating the second-component likelihood parameters in hurdle model regression using hurdle.

Usage

update_beta(y, x, hurd, dist, like.part, beta.prior.mean, beta.prior.sd, beta,
  XB, beta.acc, beta.tune, g.x = F)

Arguments

y

numeric response vector of observations within the bounds of the second component of the likelihood function, y[0 < y \& y < hurd]

x

optional numeric predictor matrix for response observations within the bounds of the second component of the likelihood function, y[0 < y \& y < hurd].

hurd

numeric threshold for 'extreme' observations of two-hurdle models.

dist

character specification of response distribution for the third component of the likelihood function.

like.part

numeric vector of the current third-component likelihood values.

beta.prior.mean

mu parameter for normal prior distributions.

beta.prior.sd

standard deviation for normal prior distributions.

beta

numeric matrix of current regression coefficient parameter values.

XB

x*beta[,1] product matrix for response observations within the bounds of the second component of the likelihood function, y[0 < y \& y < hurd].

beta.acc

numeric matrix of current MCMC acceptance rates for regression coefficient parameters.

beta.tune

numeric matrix of current MCMC tuning values for regression coefficient estimation.

g.x

logical operator. TRUE if operating within the third component of the likelihood function (the likelihood of 'extreme' observations).

Value

A list of MCMC-updated regression coefficients for the estimation of the second-component likelihood parameter as well as each coefficient's MCMC acceptance ratio.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle
dist_ll

MCMC Third-Component Parameter Update Function for Hurdle Model Count Data Regression

Description

MCMC algorithm for updating the third-component likelihood parameters in hurdle model regression using hurdle.

Usage

update_pars(y, hurd, dist, like.part, a, b, size, lam, mu, xi, sigma, lam.acc,
  mu.acc, xi.acc, sigma.acc, lam.tune, mu.tune, xi.tune, sigma.tune, g.x = F)

Arguments

y

numeric response vector of observations within the bounds of the third component of the likelihood function, y[y \ge hurd].

hurd

numeric threshold for 'extreme' observations of two-hurdle models.

dist

character specification of response distribution for the third component of the likelihood function.

like.part

numeric vector of the current third-component likelihood values.

a

shape parameter for gamma prior distributions.

b

rate parameter for gamma prior distributions.

size

size parameter for negative binomial likelihood distributions.

lam

current value for the poisson likelihood lambda parameter.

mu

current value for the negative binomial or log normal likelihood mu parameter.

xi

current value for the generalized pareto likelihood xi parameter.

sigma

current value for the generalized pareto likelihood sigma parameter.

lam.acc, mu.acc, xi.acc, sigma.acc

current MCMC values for third-component parameter acceptance rates.

lam.tune, mu.tune, xi.tune, sigma.tune

current MCMC tuning values for each third-component parameter.

g.x

logical operator. TRUE if operating within the third component of the likelihood function (the likelihood of 'extreme' observations).

Value

A list of MCMC-updated likelihood estimator(s) for the third-component parameter(s) and each parameter's MCMC acceptance ratio.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle
dist_ll

MCMC Probability Update Function for Hurdle Model Count Data Regression

Description

MCMC algorithm for updating the likelihood probabilities in hurdle model regression using hurdle.

Usage

update_probs(y, x, hurd, p, q, beta.prior.mean, beta.prior.sd, pZ, pT, pE, beta,
  XB2, XB3, beta.acc, beta.tune)

Arguments

y

numeric response vector.

x

optional numeric predictor matrix.

hurd

numeric threshold for 'extreme' observations of two-hurdle models.

p

numeric vector of current 'p' probability parameter values for zero-value observations.

q

numeric vector of current 'q' probability parameter values for 'extreme' observations.

beta.prior.mean

mu parameter for normal prior distributions.

beta.prior.sd

standard deviation for normal prior distributions.

pZ

numeric vector of current 'zero probability' likelihood values.

pT

numeric vector of current 'typical probability' likelihood values.

pE

numeric vector of current 'extreme probability' likelihood values.

beta

numeric matrix of current regression coefficient parameter values.

XB2

x*beta[,2] product matrix.

XB3

x*beta[,3] product matrix.

beta.acc

numeric matrix of current MCMC acceptance rates for regression coefficient parameters.

beta.tune

numeric matrix of current MCMC tuning values for regression coefficient estimation.

Value

A list of MCMC-updated regression coefficients for the estimation of the parameters 'p' (the probability of a zero-value observation) and 'q' (the probability of an 'extreme' observation) as well as each coefficient's MCMC acceptance ratio.

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

See Also

hurdle
dist_ll

Zero-Inflated Negative Binomial Regression Model

Description

zero_nb is used to fit zero-inflated negative binomial regression models to count data via Bayesian inference.

Usage

zero_nb(y, x, size, a = 1, b = 1, mu.start = 1, beta.prior.mean = 0,
  beta.prior.sd = 1, iters = 1000, burn = 500, nthin = 1, plots = T,
  progress.bar = T)

Arguments

y

numeric response vector.

x

numeric predictor matrix.

size

size parameter for negative binomial likelihood distributions.

a

shape parameter for gamma prior distributions.

b

rate parameter for gamma prior distributions.

mu.start

initial value for mu parameter.

beta.prior.mean

mu parameter for normal prior distributions.

beta.prior.sd

standard deviation for normal prior distributions.

iters

number of iterations for the Markov chain to run.

burn

numeric burn-in length.

nthin

numeric thinning rate.

plots

logical operator. TRUE to output plots.

progress.bar

logical operator. TRUE to print progress bar.

Details

Fits a zero-inflated negative binomial (ZINB) model.

Value

zero_nb returns a list which includes the items

mu

numeric vector; posterior distribution of mu parameter

beta

numeric matrix; posterior distributions of regression coefficients

p

numeric vector; posterior distribution of parameter 'p', the probability of a given zero observation belonging to the model's zero component

ll

numeric vector; posterior log-likelihood

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>

Zero-Inflated Poisson Regression Model

Description

zero_poisson is used to fit zero-inflated poisson regression models to count data via Bayesian inference.

Usage

zero_poisson(y, x, a = 1, b = 1, lam.start = 1, beta.prior.mean = 0,
  beta.prior.sd = 1, iters = 1000, burn = 500, nthin = 1, plots = T,
  progress.bar = T)

Arguments

y

numeric response vector.

x

numeric predictor matrix.

a

shape parameter for gamma prior distributions.

b

rate parameter for gamma prior distributions.

lam.start

initial value for lambda parameter.

beta.prior.mean

mu parameter for normal prior distributions.

beta.prior.sd

standard deviation for normal prior distributions.

iters

number of iterations for the Markov chain to run.

burn

numeric burn-in length.

nthin

numeric thinning rate.

plots

logical operator. TRUE to output plots.

progress.bar

logical operator. TRUE to print progress bar.

Details

Fits a zero-inflated Poisson (ZIP) model.

Value

zero_poisson returns a list which includes the items

lam

numeric vector; posterior distribution of lambda parameter

beta

numeric matrix; posterior distributions of regression coefficients

p

numeric vector; posterior distribution of parameter 'p', the probability of a given zero observation belonging to the model's zero component

ll

numeric vector; posterior log-likelihood

Author(s)

Taylor Trippe <ttrippe@luc.edu>
Earvin Balderama <ebalderama@luc.edu>