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

Value

a two-dimensional vector of the left and right endpoints of the interval

Author(s)

Peter Hoff

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

Value

a two-dimensional vector of the left and right endpoints of the interval

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) 

FAB inference for generalized linear models

Description

asymptotic FAB p-values and confidence intervals for parameters in generalized linear regression models

Usage

glmFAB(cformula, FABvars, lformula = NULL, alpha = 0.05,
  silent = FALSE, ...)

Arguments

Value

an object of the class glmFAB which inherits from glm

Author(s)

Peter Hoff

Examples

# n observations, p FAB variables, q=2 control variables 

n<-100 ; p<-25 

# X is design matrix for params of interest
# beta is vector of true parameter values 
# v a variable in the linking model - used to share info across betas

v<-rnorm(p) ; beta<-(2 - 2*v + rnorm(p))/3 ; X<-matrix(rnorm(n*p),n,p)/8

# control coefficients and variables  
alpha1<-.5 ; alpha2<- -.5
w1<-rnorm(n)/8
w2<-rnorm(n)/8

# simulate data 
lp<-1 + alpha1*w1 + alpha2*w2 + X%*%beta 
y<-rpois(n,exp(lp))

# fit model
fit<-glmFAB(y~w1+w2,X,~v,family=poisson)

fit$FABpv
fit$FABci 
summary(fit) # look at p-value column 

FAB inference for linear models

Description

FAB p-values and confidence intervals for parameters in linear regression models

Usage

lmFAB(cformula, FABvars, lformula = NULL, alpha = 0.05,
  rssSplit = TRUE, silent = FALSE)

Arguments

Value

an object of the class lmFAB which inherits from lm

Author(s)

Peter Hoff

Examples

# n observations, p FAB variables, q=2 control variables 

n<-100 ; p<-25 

# X is design matrix for params of interest
# beta is vector of true parameter values 
# v a variable in the linking model - used to share info across betas

v<-rnorm(p) ; beta<-(2 - 2*v + rnorm(p))/3 ; X<-matrix(rnorm(n*p),n,p)/8

# control coefficients and variables  
alpha1<-.5 ; alpha2<- -.5
w1<-rnorm(n)/8
w2<-rnorm(n)/8

# simulate data 
lp<-1 + alpha1*w1 + alpha2*w2 + X%*%beta 
y<-rnorm(n,lp) 

# fit model
fit<-lmFAB(y~w1+w2,X,~v)

fit$FABpv
fit$FABci 
summary(fit) # look at p-value column 

Marginal MLEs for the Fay-Herriot model

Description

Marginal MLEs for the Fay-Herriot random effects model where the covariance matrix for the sampling model is known to scale.

Usage

mmleFH(y, X, V, ss0 = 0, df0 = 0)

Arguments

Value

a list of parameter estimates including

1. beta, the estimated regression coefficients

2. t2, the estimate of \tau^2

3. s2, the estimate of \sigma^2

Author(s)

Peter Hoff

Examples

 
n<-30 ; p<-3 
X<-matrix(rnorm(n*p),n,p)  
beta<-rnorm(p) 
theta<-X%*%beta + rnorm(n)  
V<-diag(n) 
y<-theta+rnorm(n) 
mmleFH(y,X,V) 

Marginal MLEs for the Fay-Herriot model with known covariance

Description

Marginal MLEs for the Fay-Herriot random effects model where the covariance matrix for the sampling model is known

Usage

mmleFHP(y, X, Sigma)

Arguments

Value

a list of parameter estimates including

1. beta, the estimated regression coefficients

2. t2, the estimate of \tau^2

Author(s)

Peter Hoff

Examples

n<-30 ; p<-3 
X<-matrix(rnorm(n*p),n,p)  
beta<-rnorm(p) 
theta<-X%*%beta + rnorm(n)  
Sigma<-diag(n) 
y<-theta+rnorm(n) 
mmleFHP(y,X,Sigma) 

QR decomposition

Description

QR decomposition for lmFAB objects

Usage

## S3 method for class 'lmFAB'
qr(x, ...)

Arguments

Value

qr decomposition for a design matrix

Residual sum of squares split

Description

Split residual sum of squares from normal linear regression

Usage

rssSplit(fit, df0 = max(1, floor(fit$df/10)), seed = -71407)

Arguments

Value

a two-dimensional vector of independent sums of squares

Author(s)

Peter Hoff

Examples

n<-30 ; p<-6 ; sigma2<-1.5 
X<-matrix(rnorm(n*p),n,p) 
y<-X%*%rnorm(6) + sqrt(sigma2)*rnorm(n) 
ss<-rssSplit(lm(y~ -1+X)) 
df<-as.numeric( substring(names(ss),first=3)) 
ss/df

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.

Value

a scalar value giving the optimal tail-area probability

Author(s)

Peter Hoff

Examples

thetas<-seq(-1,1,length=100)  
s<-NULL 
for(theta in thetas){ s<-c(s,sfabz(theta,list(mu=0,tau2=1,sigma2=1)) ) }  
plot(thetas,s,type="l") 

Summarizing Generalized Linear Model Fits with FAB Inference

Description

summary method for class glmFAB

Usage

## S3 method for class 'glmFAB'
summary(object, dispersion = NULL,
  correlation = FALSE, symbolic.cor = FALSE, ...)

Arguments

Details

A mod of summary.glm that shows FAB p-values in table

Value

A list of summary statistics of the fitted generalized linear model

Examples

# n observations, p FAB variables, q=2 control variables 

n<-100 ; p<-25 

# X is design matrix for params of interest
# beta is vector of true parameter values 
# v a variable in the linking model - used to share info across betas

v<-rnorm(p) ; beta<-(2 - 2*v + rnorm(p))/3 ; X<-matrix(rnorm(n*p),n,p)/8

# control coefficients and variables  
alpha1<-.5 ; alpha2<- -.5
w1<-rnorm(n)/8
w2<-rnorm(n)/8

# simulate data 
lp<-1 + alpha1*w1 + alpha2*w2 + X%*%beta 
y<-rpois(n,exp(lp))

# fit model
fit<-glmFAB(y~w1+w2,X,~v,family=poisson)

fit$FABpv
fit$FABci 
summary(fit) # look at p-value column 

Summarizing Linear Model Fits with FAB Inference

Description

summary method for class lmFAB

Usage

## S3 method for class 'lmFAB'
summary(object, correlation = FALSE,
  symbolic.cor = FALSE, ...)

Arguments

Details

A mod of summary.lm that shows FAB p-values in table

Value

A list of summary statistics of the fitted linear model

Examples

# n observations, p FAB variables, q=2 control variables 

n<-100 ; p<-25 

# X is design matrix for params of interest
# beta is vector of true parameter values 
# v a variable in the linking model - used to share info across betas

v<-rnorm(p) ; beta<-(2 - 2*v + rnorm(p))/3 ; X<-matrix(rnorm(n*p),n,p)/8

# control coefficients and variables  
alpha1<-.5 ; alpha2<- -.5
w1<-rnorm(n)/8
w2<-rnorm(n)/8

# simulate data 
lp<-1 + alpha1*w1 + alpha2*w2 + X%*%beta 
y<-rnorm(n,lp) 

# fit model
fit<-lmFAB(y~w1+w2,X,~v)

fit$FABpv
fit$FABci 
summary(fit) # look at p-value column