Empirical Bayes estimation of hyperparameters

Description

Compute emprirical Bayes estimates of the error variance and distribution of the regression coefficients.

Usage

ebayes_est(y, X, emu = FALSE, dof = min(50, round(0.5 * (dim(X)[1] -
  dim(X)[2]))))

Arguments

Details

This function computes the adaptive FAB confidence interval for each coefficient in a linear regression model.

Value

A list (s,sigma2,tau2,mu) where

1. s an estimate of the error standard deviation

2. sigma2 an estimate of the error variance, independent of s

3. tau2 an estimate of the coefficient variance, independent of s

4. mu an estimate of the coefficient mean, independent of s

Author(s)

Peter Hoff

FAB regression coefficient intervals

Description

Compute the adaptive FAB t-intervals for the coefficients of a regression model.

Usage

fabregCI(y, X, alpha = 0.05, dof = min(50, round(0.5 * (dim(X)[1] -
  dim(X)[2]))), verbose = TRUE)

Arguments

Details

This function computes the adaptive FAB confidence interval for each coefficient in a linear regression model.

Value

A matrix where each row corresponds to the interval and OLS estimate of a coefficient.

Author(s)

Peter Hoff

FAB t-interval

Description

Computation of a 1-alpha FAB t-interval

Usage

fabtCI(y, psi = c(0, 100, 1, 2), alpha = 0.05)

Arguments

Details

A FAB interval is the "frequentist" interval procedure that is Bayes optimal: It minimizes the prior expected interval width among all interval procedures with exact 1-alpha frequentist coverage. This function computes the FAB t-interval for the mean of a normal population with an unknown variance, given a user-specified prior distribution determined by psi. The prior is that the population mean and variance are independently distributed as normal and inverse-gamma random variables. Referring to the elements of psi as mu, t2, s20, nu0, the prior is determined as follows:

1. mu is the prior expectation of the mean

2. t2 is the prior variance of the mean

3. the population variance is inverse-gamma(nu0/2,nu0 s20/2)

Author(s)

Peter Hoff

Examples

y<-rnorm(10)
fabtCI(y,c(0,10,1,5)) 
fabtCI(y,c(0,1/10,1,5))
fabtCI(y,c(2,10,1,5))
fabtCI(y,c(0,1/10,1,5)) 

FAB t-interval with z-optimal w-function

Description

Computation of a 1-alpha FAB t-interval using the w-function of the optimal FAB z-interval

Usage

fabtwzCI(y, s2, df, muw, t2w, s2w, alpha = 0.05)

Arguments

Details

Internal function for multifabCIhom. It's similar to fabtCI, the difference being that fabtwzCI uses the w-function that leads to the optimal CI if the sampling variance were known.

Author(s)

Chaoyu Yu

Examples

y<-rnorm(10)
fabtwzCI(y,10,9,1,5,10)
fabtwzCI(y,1/10,9,1,5,1/10)
fabtwzCI(y,10,1,9,5,10)
fabtwzCI(y,1/10,9,1,5,1/10)

z-optimal FAB t-interval

Description

Computation of a 1-alpha FAB t-interval using z-optimal spending function

Usage

fabtzCI(y, s, dof, alpha = 0.05, psi = list(mu = 0, tau2 = 1e+05, sigma2 =
  1))

Arguments

Examples

n<-10 
y<-rnorm(n) 
fabtzCI(mean(y),sqrt(var(y)/n),n-1)  
t.test(y)$conf.int 

FAB z-interval

Description

Computation of a 1-alpha FAB z-interval

Usage

fabzCI(y, mu, t2, s2, alpha = 0.05)

Arguments

Details

A FAB interval is the "frequentist" interval procedure that is Bayes optimal: It minimizes the prior expected interval width among all interval procedures with exact 1-alpha frequentist coverage. This function computes the FAB z-interval for the mean of a normal population with an known variance, given a user-specified prior distribution determined by psi. The prior is that the population mean is normally distributed. Referring to the elements of psi as mu, t2, s2, the prior and population variance are determined as follows:

1. mu is the prior expectation of the mean

2. t2 is the prior variance of the mean

3. s2 is the population variance

Author(s)

Peter Hoff

Examples

y<-0
fabzCI(y,0,10,1)
fabzCI(y,0,1/10,1)
fabzCI(y,2,10,1)
fabzCI(y,0,1/10,1) 

Hierarchical heteroscedastic model estimates

Description

Estimate across-group heterogeneity of means and variances

Usage

hhetmodel(y, g)

Arguments

Details

This function estimates parameters in a hierarchical model for normally distributed groups, where the across-group model for means is normal and the across group model for variances is inverse-gamma.

Value

A vector (mu,t2,s20,nu0), where

1. mu is the mean of the group means

2. t2 is the variance of the group means

3. the the distribution of group variances is inverse-gamma(nu0/2,nu0 s20/2)

Author(s)

Peter Hoff

Hierarchical homoscedastic model estimates

Description

Estimate across-group heterogeneity of means

Usage

hhommodel(y, g, group, p1)

Arguments

Details

This function estimates parameters in a hierarchical model for normally distributed groups, where the across-group model for means is normal and the variance is the same across groups.

Value

A vector (s2,df,muw,t2w,s2w), where

1. s2 is the pooled variance

2. df is the degree of freedom of the t-quantiles

3. muw is the estimate mean of the group means

4. t2w is the estimate variance of the group means

5. s2w is the estimate within-group variance

Author(s)

Chaoyu Yu

Multigroup FAB t-intervals

Description

Computation of 1-alpha FAB t-intervals for heteroscedastic multigroup data.

Usage

multifabCI(y, g, alpha = 0.05)

Arguments

Details

For each group j, this function computes an estimate of the parameters in a hierarchical model for means and variances from data other than group j, and uses this information to construct a FAB t-interval for group j. These intervals have 1-alpha frequentist coverage, assuming within-group normality.

Author(s)

Peter Hoff

Examples

## -- simulated data
p<-10 ; n<-10
y<-rnorm(n*p) ; g<-rep(1:p,n) 

## -- more interesting data takes longer 
# data(radon) ; y<-radon[,2] ; g<-radon[,1] 

## -- FAB t-intervals
FCI<-multifabCI(y,g) 

## -- UMAU t-intervals 
ybar<-tapply(y,g,mean) ; ssd<-tapply(y,g,sd) ; n<-table(g) 
qtn<-cbind( qt(.025,n-1),  qt(.975,n-1) ) 
UCI<-sweep(sweep(qtn,1,ssd/sqrt(n),"*"),1,ybar,"+") 

mean( (UCI[,2]-UCI[,1])/(FCI[,2]-FCI[,1]) , na.rm=TRUE)

Multigroup FAB t-intervals for the homoscedastic model

Description

Computation of 1-alpha FAB t-intervals for homoscedastic multigroup data.

Usage

multifabCIhom(y, g, alpha = 0.05, prop = 0.5)

Arguments

Details

For each group j, this function computes an estimate of the parameters in a hierarchical model for means using data from other groups, and uses this information to construct a FAB t-interval for group j. These intervals have 1-alpha frequentist coverage, assuming within-group normality and that the within group variance is the same across groups.

Author(s)

Chaoyu Yu

Examples

## -- simulate the data
mu = 0; sigma2 = 10; tau2 = 1; p =100; 
theta = rnorm(p,mu,sqrt(tau2))
ns = round(runif(p,2,18))
Y=c()
for(i in 1:p){
 d2 = rnorm(ns[i],theta[i],sqrt(sigma2))
 d1 = rep(i,ns[i])
 d = cbind(d1,d2)
 Y = rbind(Y,d)}
y = Y[,2]
g = Y[,1]

## -- FAB t-intervals
FCI = multifabCIhom(y,g)  

## -- UMAU t-intervals 
ybar<-tapply(y,g,mean) ; ssd<-tapply(y,g,sd) ; n<-table(g) 
qtn<-cbind( qt(.025,n-1),  qt(.975,n-1) ) 
UCI<-sweep(sweep(qtn,1,ssd/sqrt(n),"*"),1,ybar,"+") 

mean( (UCI[,2]-UCI[,1])/(FCI[,2]-FCI[,1]) , na.rm=TRUE)

Prior predictive acceptance probability

Description

Compute prior probability of falling into an acceptance region

Usage

pppaccept(w, theta, mu, t2, s20, nu0, n, alpha)

Arguments

Details

Internal function for fabtCI. This function computes the prior probability that the value theta will fall in the acceptance region of a biased level-alpha test, under a specified prior predictive distribution.

Author(s)

Peter Hoff

Minnesota Radon Dataset

Description

Radon levels in 919 homes from 85 Minnesota counties

Usage

data(radon)

Format

A numeric matrix

Source

http://www.stat.columbia.edu/~gelman/arm/software/

Bayes-optimal spending function

Description

Compute Bayes optimal spending function

Usage

sfabz(theta, psi, alpha = 0.05)

Arguments

Details

This function computes the value of s that minimizes the acceptance probability of a biased level-alpha test for a normal population with known variance, under a specified prior predictive distribution.

Author(s)

Peter Hoff

UMAU regression coefficient intervals

Description

Compute the usual t-intervals for the coefficients of a regression model

Usage

umauregCI(y, X, alpha = 0.05)

Arguments

Details

This function computes the 'usual' uniformly most accurate unbiased confidence interval for each coefficient in a linear regression model.

Value

A matrix where each row corresponds to the interval and OLS estimate of a coefficient.

Author(s)

Peter Hoff

Bayes-optimal w-function

Description

Compute Bayes optimal w-function

Usage

wfabt(theta, mu, t2, s20, nu0, n, alpha)

Arguments

Details

Internal function for fabtCI. This function computes the value of w that minimizes the acceptance probability of a biased level-alpha test under a specified prior predictive distribution.

Author(s)

Peter Hoff