Package 'abms'

Chilean National Health Survey (2016-2017)

Description

ens is a data-base that reo collected by the Chilean National Health Survey in order to study the health status of the population and policy making. This survey was performed between 2016 and 2017

Usage

ens

Format

A Data frame with 3238 rows and 15 variables (columns):

pas

Blood systolic pressure

pad

Blood diastolic pressure

age

In years

waist

diameter in centimeters

bmi

body mass index

weight

In centimeters

height

In centimeters

hypertension

Presence of hypertension (1: Yes, 0: No)

sedentary

sedentary person (1: Yes, 0: No)

smoker

1: Yes, 0: No

diabetes

1: Yes, 0: No

depression

1: Yes, 0: No

male

1: Yes, 0: female

scholar

Years of formal education

Source

Chilean Health Ministry (https://epi.minsal.cl/encuesta-ens-anteriores/)

Examples

data(ens)

---

Logistic Regression Data generator

Description

It generates N observations of the Binomial distribution with parameters ni (the i-th's individual sizes) and p (the success probability), where the coefficients are indexed on p via the logistic function.

Usage

gen_base_binomial_reg(N, beta, Covariates, ni = rep(1, N))

Arguments

N	The number of observations that will be generated. It must be a positive integer.
beta	A vector of coefficients including the intercept. It can be a matrix.
Covariates	A data.frame object with the predictors (without intercept) for which we want to test if they are relevant to the response variable. It can also be a ($n x p$) matrix.
ni	A vector of size $n$ that represent the i-th individual size (the size parameter of the binomial distribution). It can also be a ($n x 1$) matrix. For default, all individual size are fixed at 1.

Value

The function return a table with the sample of size N from the Binomial distribution indexed with the predictors indicated in the Covariates argument, the ni, the number of failures (ni - y), and the predictors for each individual .

Examples

N<-200    #Number of extractions
beta<-c(1, 0, 2, 0, 3, 2)    #Coefficient vector
p<-length(beta)
aux_cov<-rnorm((p-1)*N, 0,1)
Covariates<-data.frame(matrix(aux_cov, ncol=p-1, nrow=N))   #Generating the Covariates data.frame
colnames(Covariates)<-c("X1", "X2", "X3", "X4", "X5")
base<-gen_base_binomial_reg(N, beta, Covariates, ni=rep(1, N))    #Generating the data
base

---

Negative Binomial Regression Data generator

Description

It generates N observations of the Negative binomial distribution with parameters r (number of success) and $p$ (success probability), where the coefficients are indexed on $p$ via the logistic function.

Usage

gen_base_NegBinomial_reg(N, beta, r, Covariates)

Arguments

N	The number of observations that will be generated. It must be a positive integer.
beta	A vector of coefficients including the intercept. It can be a matrix.
r	The number of success parameter. It must be a positive integer.
Covariates	A data.frame object with the predictors (without intercept) for which we want to test if they are relevant to the response variable. It can also be a ($n x p$) matrix.

Value

The function return a sample of size N from the Negative binomial distribution indexed with the predictors indicated in the Covariates argument, and the predictors for each individual.

Examples

N<-10   #Number of extractions
beta<-c(0.5, -0.8,  1.0,  0,  0.4, -0.7)  #Coefficient vector
p<-length(beta)
r<-2    #Number of success parameter
aux_cov<-rnorm((p-1)*N, 0,1)
Covariates<-data.frame(matrix(aux_cov, ncol=p-1, nrow=N))   #Generating the Covariates data.frame
colnames(Covariates)<-c("X1", "X2", "X3", "X4", "X5")
base<-gen_base_NegBinomial_reg(N, beta, r, Covariates)    #Generating the data
base

---

Bayesian variable selection models via a spike-and-slab methodology.

Description

A Bayesian model selection methodology based on the spike-and-slab strategy and an augmentation technique for Linear, Logistic, Negative Binomial, Quantile, and Skew Normal Regression. The model considers a response vector $y$ of size $n$ and $p$ predictors to perform coefficient estimation and asses which ones are relevant to explain the response distribution. Other parameters related to the family selected are also estimated. Summary results can be provided using the summary_gibbs() R function.

Usage

gibbs_abms(
  y,
  Covariates,
  family = "LiR",
  first_excluded = 0,
  nchain = 10000,
  burnin = 2000,
  tau2 = 1000,
  rho = 1,
  ni = rep(1, length(y)),
  alpha = 0.5,
  a0 = 1,
  b0 = 1,
  d = 2,
  b2 = 1/2,
  count.iteration = TRUE
)

Arguments

y	A vector of size $n$ with observed responses. It can also be a ($n x 1$) matrix.
Covariates	A data.frame object with the predictors (without the intercept) for which we want to test if they are relevant to the response variable. It can also be a ($n x p$) matrix.
family	A character object that describes the hierarchical regression model that will be used. If family="LiR", then a Linear regresion model will be fitted (gaussian errors). If family="LoR", then a Logistic regresion model will be fitted (binomial distribution). If family="NBR", then a Negative Binomial regresion model will be fitted (mean $r(1-p)/p$). If family="QR", then a Quantile regresion model will be fitted (Asymmetric Laplace errors). If family="SNR", then a Skew normal regresion model will be fitted (Skew-Normal errors). The argument is fixed at family="LiR" by default.
first_excluded	A non-negative integer that indicates which first columns will not be tested. For example, if first_excluded=2, the two first columns of Covariates will not be tested. Intercept is always excluded for the selection process.
nchain	The Gibbs sampler's chain size, it must be a non-negative integer. The default value is 10,000
burnin	The burn-in period of the Gibbs sampler, it must be a non-negative integer and greater than nchain. The default value is 2,000
tau2	The variance prior of each coefficient, it must be a positive real number. Fixed at 1 by deafault
rho	The parameter of the Womack prior, it must be a positive real number. Fixed at 1 by deafault
ni	For Logistic regression only. A vector of size $n$ that represent the i-th individual size (the size parameter of the Binomial distribution) that it must be a positive integer. It can also be a ($n x 1$) matrix. For default, all individual size are fixed at 1.
alpha	For Quantile regression only. The desired quantile for which we want to perform Quantile regression. alpha must be between ($0,1$). By default, alpha=0.5, that is, median regression.
a0	This argument depends on the family choosen. For family="LiR", is the shape hyper-parameter of the $Gamma$ prior to the variance parameter ($\sigma^2$) of the Gaussian distribution. For family="NBR" is the shape hyper-parameter of the $Gamma$ prior to the parameter $r$ the Negative Binomial distribution (the number of successes until the experiment is stopped). For family="QR" is the shape hyper-parameter of the $Gamma$ prior to thevariance parameter ($\sigma^2$) of the Asymmetric Laplace distribution. Note thas this argument do not exist for family=LoR and family=SNR. For all hierarchical regression models, it must be a positive real number and its fixed at 1 by deafault.
b0	This argument depends on the family choosen. For family="LiR" is the scale hyper-parameter of the $Gamma$ prior to the variance parameter ($\sigma^2$) of the Gaussian distribution. For family="NBR" is the scale hyper-parameter of the $Gamma$ prior to the parameter $r$ the Negative Binomial distribution (the number of successes until the experiment is stopped). For family="QR" is the scale hyper-parameter of the $Gamma$ prior to the variance parameter ($\sigma^2$) of the Asymmetric Laplace distribution. Note thas this argument do not exist for family=LoR and family=SNR. For all hierarchical regression models, it must be a positive real number and its fixed at 1 by deafault.
d	For the Skew-Normal regression only. It is the location hyper-parameter of the t-student prior to the parameter $lambda$ (asymmetric parameter of the Skew-Normal distribution). By default is fixed at 2, which is recommended.
b2	For the Skew-Normal regression only. It is the scale hyper-parameter of the t-student prior to the parameter lambda (asymmetric parameter of the Skew-Normal distribution). By default is fixed at 1/2, which is recommended.
count.iteration	A logical argument. If TRUE, a counter for the Gibbs sampler iterations will be displayed. Fixed at TRUE by deafult.

Value

A abms object with the following variables:

family	This character object prints the name of the fitted hierarchical regression model. It needs to be extracted from the list 'Default'.
prednames	A character object that prints the predictors names, using the columns names of the Covariates argument. It needs to be extracted from the list 'Default'.
Seconds	How many seconds the method took. It needs to be extracted from the list 'Default'.
tau2	The tau2 that was used as argument.
y	The y response vector that was used as argument.
Covariates	The Covariates data frame or matrix that was used as argument.
beta_chain	The coefficients sample for each Gibbs sampler iteration. A (nchain x $p$) matrix
sigma2_chain	For the Linear, Quantile and Skew-Normal regression only. The variance parameter ($\sigma^2$) sample for each Gibbs sampler iteration. A (nchain x 1) matrix
r_chain	For the Negative-Binomial regression only. The number of failure parameter ($r$) sample for each Gibbs sampler iteration. A (nchain x 1) matrix
lambda_chain	For the Skew-Normal regression only. The asymmetric parameter ($\lambda$) sample for each Gibbs sampler iteration. A (nchain x 1) matrix
model_chain	The model selected at each Gibbs sampler iteration. A (nchain x $p$) matrix.
Z_chain	For internal use.
t_chain	For internal use.

References

Azzalini (1985). A class of distributions which includes the normal ones, Scandinavian Journal of Statistics 12(2): 171:178.

Bayes, C. and Branco, M. (2007). Bayesian inference for the skewness parameter of the scalar skew-normal distribution. Brazilian Journal of Probability and Statistics. 21: 141:163.

Kotz, S., Kozubowski, T. and Podgorski, K. (2001). The Laplace Distribution and Generalization, first edn, Birkhauser Basel.

Polson, N., Scott, J., and Windle, J. (2013). Bayesian Inference for Logistic Models Using Polya Gamma Latent Variables. Journal of the American Statistical Association, 108: 1339:1349.

Zhou, W. and Carin, L. (2013). Negative Binomial Process Count and Mixture Modeling. arXiv:1405.0506v1.

Examples

##################################################
## 	    	Gibbs for Linear Regression 		##
##################################################

## Simulating data
set.seed(31415)
N<-200
r_beta<-as.matrix(c(1, 0, 2, 0))
r_p<-length(r_beta)
r_sigma2<-1.5
X<-matrix( c(rep(1, N), rnorm((r_p -1)*N)), ncol=r_p )
Xbeta<-X%*%r_beta
y<-rnorm(N, mean=Xbeta , sd=sqrt(r_sigma2))
Covariates<-X[,2:(length(r_beta))];
colnames(Covariates)<-c("X1", "X2", "X3")

## Fitting the model
fit<- gibbs_abms(y, Covariates, family="LiR", first_excluded=0, nchain=1000, burnin=20,
 a0=1, b0=1)

summary_gibbs(fit, BF=FALSE)	#Summary results

## For more examples, see "Model Ilustrations.R" file in
## https://github.com/SirCornflake/BMS

---

Title

Description

Sampling from the Chines Restaurant distribution.

Usage

rCRT(n, b, c)

Arguments

n	Number of observations. It must be a positive integer.
b	Parameter distribution, a non-negative integer. The number of Bernoulli independent variables that are added. It can be a vector
c	Parameter distribution, a positive real number. Used calculate the success probability the j-th Bernoulli independent variable, that is, c/(c +j -1). It can be a vector

Value

This function generates n random variables from the $CRT$($b,c$) distribution

References

Pitman, Jim (1995). "Exchangeable and Partially Exchangeable Random Partitions". Probability Theory and Related Fields. 102 (2): 145:158

Examples

#Generating 4 random variables with parameters b=2 and c=1
rCRT(4,2,1)

---

Summary function for abms objects

Description

For abms objects, it returns the posterior mean, standard deviation, and 95% centered credible interval for each parameter. Additionally, it provides all explored models alongside the conditional Bayes factors and marginal Bayes factors estimator between the most probable model and the others that have arisen.

Usage

summary_gibbs(fit, BF = FALSE)

Arguments

fit	An abms object. Such object is obtained by fitting a regression model with the gibbs_abms() function.
BF	A logical object. if TRUE, then the Bayes factor comparison is shown.BF=FALSE by default.

Value

A summary of the inference performed by the Bayesian model obtained by the gibbs_abms() function. The variables are:

Mean_IC	A table with the posterior mean, standard deviation, and 95% centered credible interval for each parameter
Explored_Models	A table with all explored models. If BF=TRUE, the conditional Bayes factors and marginal Bayes factors estimator between the most probable model and the others that have arisen are displayed.

Examples

## See \code{gibbs_abms()} help page function

---

For internal use Womack prior

Description

Womack probability mass function with 'K' predictors and parameter 'rho'.

Usage

womack(K, rho)

Arguments

K	Number of predictors. It must be a positive integer.
rho	Value for the "rho" parameter. It must be positive real number

Value

Given that all models of the same hierarchy has the same prior probability, this function returns one value for each hierarchy. including the null model (size=0)

References

Womack, A., Fuentes, C., and Rodriguez-Taylor, D. (2015). "Model Space Priors for Objective Sparse Bayesian Regression." arXiv:1511.04745. 8:24

Examples

#Fixing rho=1 and 3 predictors
womack(K=3, rho=1)
#it returns Womack prior for all models of size 0 (the null model), 1,2 and 3 (the full model)

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