Package 'FABInference' reference manual

Title:	FAB p-Values and Confidence Intervals
Description:	Frequentist assisted by Bayes (FAB) p-values and confidence interval construction. See Hoff (2019) <arXiv:1907.12589> "Smaller p-values via indirect information", Hoff and Yu (2019) <doi:10.1214/18-EJS1517> "Exact adaptive confidence intervals for linear regression coefficients", and Yu and Hoff (2018) <doi:10.1093/biomet/asy009> "Adaptive multigroup confidence intervals with constant coverage".
Authors:	Peter Hoff
Maintainer:	Peter Hoff <[email protected]>
License:	GPL-3
Version:	0.1
Built:	2025-03-04 02:45:50 UTC
Source:	https://github.com/pdhoff/fabinference

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

Arguments

`y`	a numeric scalar, a normally distributed statistic
`s`	a numeric scalar, the standard error of y
`dof`	positive integer, degrees of freedom for s
`alpha`	the type I error rate, so 1-alpha is the coverage rate
`psi`	a list of parameters for the spending function, including mu, the prior expectation of E[y] tau2, the prior variance of E[y] sigma2 the variance of y

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

FAB z-interval

Description

Computation of a 1-alpha FAB z-interval

Usage

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

Arguments

`y`	a numeric scalar
`mu`	a numeric scalar
`t2`	a positive numeric scalar
`s2`	a positive numeric scalar
`alpha`	the type I error rate, so 1-alpha is the coverage rate

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:

mu is the prior expectation of the mean
t2 is the prior variance of the mean
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) 

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

Arguments

`cformula`	formula for the control variables
`FABvars`	matrix of regressors for which to make FAB p-values and CIs
`lformula`	formula for the linking model (just specify right-hand side)
`alpha`	error rate for CIs (1-alpha CIs will be constructed)
`silent`	show progress (TRUE) or not (FALSE)
`...`	additional arguments to be passed to `glm`

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 

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

Arguments

`cformula`	formula for the control variables
`FABvars`	matrix of regressors for which to make FAB p-values and CIs
`lformula`	formula for the linking model (just specify right-hand side)
`alpha`	error rate for CIs (1-alpha CIs will be constructed)
`rssSplit`	use some residual degrees of freedom to help fit linking model (TRUE/FALSE)
`silent`	show progress (TRUE) or not (FALSE)

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 

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

Arguments

`y`	direct data following normal model $y\sim N(\theta,V\sigma^2)$
`X`	linking model predictors $\theta\sim N(X\beta,\tau^2 I)$
`V`	covariance matrix to scale
`ss0`	prior sum of squares for estimate of $\sigma^2$
`df0`	prior degrees of freedom for estimate of $\sigma^2$

Value

a list of parameter estimates including

beta, the estimated regression coefficients
t2, the estimate of $\tau^2$
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) 

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)
mmleFHP(y, X, Sigma)

Arguments

`y`	direct data following normal model $y\sim N(\theta,\Sigma)$
`X`	linking model predictors $\theta\sim N(X\beta,\tau^2 I)$
`Sigma`	covariance matrix in sampling model

Value

a list of parameter estimates including

beta, the estimated regression coefficients
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) 

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

Arguments

`x`	`lmFAB` object
`...`	see `qr.lm`, if you can find it

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

Arguments

`fit`	lm object
`df0`	degrees of freedom for the smaller of the two residual sums of squares
`seed`	random seed for constructing the basis vectors of the split

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

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)
sfabz(theta, psi, alpha = 0.05)

Arguments

theta

value of theta being tested

psi

a list of parameters for the spending function, including

mu, the prior expectation of E[y]
tau2, the prior variance of E[y]
sigma2 the variance of y

alpha

level of test

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") 

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

Arguments

`object`	an object of class `glmFAB`
`dispersion`	see `summary.glm`
`correlation`	see `summary.glm`
`symbolic.cor`	see `summary.glm`
`...`	see `summary.glm`

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 

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

Arguments

`object`	an object of class `lmFAB`
`correlation`	see `summary.lm`
`symbolic.cor`	see `summary.lm`
`...`	see `summary.lm`

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 


# 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

Package 'FABInference'

Help Index

z-optimal FAB t-interval

Description

Usage

Arguments

Value

Author(s)

Examples

FAB z-interval

Description

Usage

Arguments

Details

Value

Author(s)

Examples

FAB inference for generalized linear models

Description

Usage

Arguments

Value

Author(s)

Examples

FAB inference for linear models

Description

Usage

Arguments

Value

Author(s)

Examples

Marginal MLEs for the Fay-Herriot model

Description

Usage

Arguments

Value

Author(s)

Examples

Marginal MLEs for the Fay-Herriot model with known covariance

Description

Usage

Arguments

Value

Author(s)

Examples

QR decomposition

Description

Usage

Arguments

Value

Residual sum of squares split

Description

Usage

Arguments

Value

Author(s)

Examples

Bayes-optimal spending function

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Summarizing Generalized Linear Model Fits with FAB Inference

Description

Usage

Arguments

Details

Value

Examples

Summarizing Linear Model Fits with FAB Inference

Description

Usage

Arguments

Details

Value

Examples