Package 'ashr' reference manual

Title:	Methods for Adaptive Shrinkage, using Empirical Bayes
Description:	The R package 'ashr' implements an Empirical Bayes approach for large-scale hypothesis testing and false discovery rate (FDR) estimation based on the methods proposed in M. Stephens, 2016, "False discovery rates: a new deal", <DOI:10.1093/biostatistics/kxw041>. These methods can be applied whenever two sets of summary statistics---estimated effects and standard errors---are available, just as 'qvalue' can be applied to previously computed p-values. Two main interfaces are provided: ash(), which is more user-friendly; and ash.workhorse(), which has more options and is geared toward advanced users. The ash() and ash.workhorse() also provides a flexible modeling interface that can accommodate a variety of likelihoods (e.g., normal, Poisson) and mixture priors (e.g., uniform, normal).
Authors:	Matthew Stephens [aut], Peter Carbonetto [aut, cre], Chaoxing Dai [ctb], David Gerard [aut], Mengyin Lu [aut], Lei Sun [aut], Jason Willwerscheid [aut], Nan Xiao [aut], Mazon Zeng [ctb]
Maintainer:	Peter Carbonetto <[email protected]>
License:	GPL (>=3)
Version:	2.2-66
Built:	2024-09-12 06:08:01 UTC
Source:	https://github.com/stephens999/ashr

Adaptive Shrinkage

Description

Implements Empirical Bayes shrinkage and false discovery rate methods based on unimodal prior distributions.

Usage

ash(
  betahat,
  sebetahat,
  mixcompdist = c("uniform", "halfuniform", "normal", "+uniform", "-uniform",
    "halfnormal"),
  df = NULL,
  ...
)

ash.workhorse(
  betahat,
  sebetahat,
  method = c("fdr", "shrink"),
  mixcompdist = c("uniform", "halfuniform", "normal", "+uniform", "-uniform",
    "halfnormal"),
  optmethod = c("mixSQP", "mixIP", "cxxMixSquarem", "mixEM", "mixVBEM", "w_mixEM"),
  df = NULL,
  nullweight = 10,
  pointmass = TRUE,
  prior = c("nullbiased", "uniform", "unit"),
  mixsd = NULL,
  gridmult = sqrt(2),
  outputlevel = 2,
  g = NULL,
  fixg = FALSE,
  mode = 0,
  alpha = 0,
  grange = c(-Inf, Inf),
  control = list(),
  lik = NULL,
  weights = NULL,
  pi_thresh = 1e-10
)
ash(
  betahat,
  sebetahat,
  mixcompdist = c("uniform", "halfuniform", "normal", "+uniform", "-uniform",
    "halfnormal"),
  df = NULL,
  ...
)

ash.workhorse(
  betahat,
  sebetahat,
  method = c("fdr", "shrink"),
  mixcompdist = c("uniform", "halfuniform", "normal", "+uniform", "-uniform",
    "halfnormal"),
  optmethod = c("mixSQP", "mixIP", "cxxMixSquarem", "mixEM", "mixVBEM", "w_mixEM"),
  df = NULL,
  nullweight = 10,
  pointmass = TRUE,
  prior = c("nullbiased", "uniform", "unit"),
  mixsd = NULL,
  gridmult = sqrt(2),
  outputlevel = 2,
  g = NULL,
  fixg = FALSE,
  mode = 0,
  alpha = 0,
  grange = c(-Inf, Inf),
  control = list(),
  lik = NULL,
  weights = NULL,
  pi_thresh = 1e-10
)

Arguments

`betahat`	a p vector of estimates
`sebetahat`	a p vector of corresponding standard errors
`mixcompdist`	distribution of components in mixture used to represent the family G. Depending on the choice of mixture component, the family G becomes more or less flexible. Options are: uniform G is (approximately) any symmetric unimodal distribution normal G is (approximately) any scale mixture of normals halfuniform G is (approximately) any unimodal distribution +uniform G is (approximately) any unimodal distribution with support constrained to be greater than the mode. -uniform G is (approximately) any unimodal distribution with support constrained to be less than the mode. halfnormal G is (approximately) any scale mixture of truncated normals where the normals are truncated at the mode If you are happy to assume a symmetric distribution for effects, you can use "uniform" or "normal". If you believe your effects may be asymmetric, use "halfuniform" or "halfnormal". If you want to allow only positive/negative effects use "+uniform"/"-uniform". The use of "normal" and "halfnormal" is permitted only if df=NULL.
`df`	appropriate degrees of freedom for (t) distribution of (betahat-beta)/sebetahat; default is NULL which is actually treated as infinity (Gaussian)
`...`	Further arguments of function `ash` to be passed to `ash.workhorse`.
`method`	specifies how ash is to be run. Can be "shrinkage" (if main aim is shrinkage) or "fdr" (if main aim is to assess false discovery rate or false sign rate (fsr)). This is simply a convenient way to specify certain combinations of parameters: "shrinkage" sets pointmass=FALSE and prior="uniform"; "fdr" sets pointmass=TRUE and prior="nullbiased".
`optmethod`	The method used to compute maximum-likelihood estimates of the mixture weights. The default setting, “mixSQP”, uses the fast sequential quadratric programming (SQP) method implemented in the mixsqp package. Alternative methods include the interior-point method implemented in the REBayes package (`optmethod = "mixIP"`), and a simple Expectation Maximization (EM) algorithm (`optmethod = "mixEM"`). For more details on the different options, see the help for functions `estimate_mixprop`, `mixSQP`, `mixIP`, `mixEM`, `w_mixEM` and `mixVBEM`.
`nullweight`	scalar, the weight put on the prior under "nullbiased" specification, see `prior`
`pointmass`	Logical, indicating whether to use a point mass at zero as one of components for a mixture distribution.
`prior`	string, or numeric vector indicating Dirichlet prior on mixture proportions: “nullbiased”, `c(nullweight,1,...,1)`, puts more weight on first component; “uniform” is `c(1,1...,1)`; “unit” is (1/K,...,1/K), for `optmethod = mixVBEM` version only.
`mixsd`	Vector of standard deviations for underlying mixture components.
`gridmult`	the multiplier by which the default grid values for mixsd differ by one another. (Smaller values produce finer grids.)
`outputlevel`	Determines amount of output. There are several numeric options: 0 = just fitted g; 1 = also PosteriorMean and PosteriorSD; 2 = everything usually needed; 3 = also include results of mixture fitting procedure (including matrix of log-likelihoods used to fit mixture). 4 and 5 are reserved for outputting additional data required by the (in-development) flashr package. The user can also specify the output they require in detail (see Examples).
`g`	The prior distribution for beta. Usually this is unspecified (NULL) and estimated from the data. However, it can be used in conjuction with fixg=TRUE to specify the g to use (e.g. useful in simulations to do computations with the "true" g). Or, if g is specified but fixg=FALSE, the g specifies the initial value of g used before optimization, (which also implicitly specifies mixcompdist).
`fixg`	If TRUE, don't estimate g but use the specified g - useful for computations under the "true" g in simulations.
`mode`	either numeric (indicating mode of g) or string "estimate", to indicate mode should be estimated, or a two dimension numeric vector to indicate the interval to be searched for the mode.
`alpha`	Numeric value of alpha parameter in the model.
`grange`	Two dimension numeric vector indicating the left and right limit of g. Default is c(-Inf, Inf).
`control`	A list of control parameters passed to optmethod.
`lik`	Contains details of the likelihood used; for general ash. Currently, the following choices are allowed: normal (see function lik_normal(); binomial likelihood (see function lik_binom); likelihood based on logF error distribution (see function lik_logF); mixture of normals likelihood (see function lik_normalmix); and Poisson likelihood (see function lik_pois).
`weights`	a vector of weights for observations; use with optmethod = "w_mixEM"; this is currently beta-functionality.
`pi_thresh`	a threshold below which to prune out mixture components before computing summaries (speeds up computation since empirically many components are usually assigned negligible weight). The current implementation still returns the full fitted distribution; this only affects the posterior summaries.

Details

The ash function provides a number of ways to perform Empirical Bayes shrinkage estimation and false discovery rate estimation. The main assumption is that the underlying distribution of effects is unimodal. Novice users are recommended to start with the examples provided below.

In the simplest case the inputs to ash are a vector of estimates (betahat) and their corresponding standard errors (sebetahat), and degrees of freedom (df). The method assumes that for some (unknown) "true" vector of effects beta, the statistic (betahat[j]-beta[j])/sebetahat[j] has a $t$ distribution on $df$ degrees of freedom. (The default of df=NULL assumes a normal distribution instead of a t.)

By default the method estimates the vector beta under the assumption that beta ~ g for a distribution g in G, where G is some unimodal family of distributions to be specified (see parameter mixcompdist). By default is to assume the mode is 0, and this is suitable for settings where you are interested in testing which beta[j] are non-zero. To estimate the mode see parameter mode.

As is standard in empirical Bayes methods, the fitting proceeds in two stages: i) estimate g by maximizing a (possibly penalized) likelihood; ii) compute the posterior distribution for each beta[j] | betahat[j],sebetahat[j] using the estimated g as the prior distribution.

A more general case allows that beta[j]/sebetahat[j]^alpha | sebetahat[j] ~ g.

Value

ash returns an object of class "ash", a list with some or all of the following elements (determined by outputlevel)

`fitted_g`	fitted mixture
`loglik`	log P(D\|fitted_g)
`logLR`	log[P(D\|fitted_g)/P(D\|beta==0)]
`result`	A dataframe whose columns are:

NegativeProb: A vector of posterior probability that beta is negative.
PositiveProb: A vector of posterior probability that beta is positive.
lfsr: A vector of estimated local false sign rate.
lfdr: A vector of estimated local false discovery rate.
qvalue: A vector of q values.
svalue: A vector of s values.
PosteriorMean: A vector consisting the posterior mean of beta from the mixture.
PosteriorSD: A vector consisting the corresponding posterior standard deviation.

`call`	a call in which all of the specified arguments are specified by their full names
`data`	a list containing details of the data and models used (mostly for internal use)
`fit_details`	a list containing results of mixture optimization, and matrix of component log-likelihoods used in this optimization

Functions

ash.workhorse(): Adaptive Shrinkage with full set of options.

Examples


beta = c(rep(0,100),rnorm(100))
sebetahat = abs(rnorm(200,0,1))
betahat = rnorm(200,beta,sebetahat)
beta.ash = ash(betahat, sebetahat)
names(beta.ash)
head(beta.ash$result) # the main dataframe of results
head(get_pm(beta.ash)) # get_pm returns posterior mean
head(get_lfsr(beta.ash)) # get_lfsr returns the local false sign rate
graphics::plot(betahat,get_pm(beta.ash),xlim=c(-4,4),ylim=c(-4,4))

## Not run: 
# Why is this example included here? -Peter
CIMatrix=ashci(beta.ash,level=0.95)
print(CIMatrix)

## End(Not run)

# Illustrating the non-zero mode feature.
betahat=betahat+5
beta.ash = ash(betahat, sebetahat)
graphics::plot(betahat,get_pm(beta.ash))
betan.ash=ash(betahat, sebetahat,mode=5)
graphics::plot(betahat,get_pm(betan.ash))
summary(betan.ash)

# Running ash with different error models
beta.ash1 = ash(betahat, sebetahat, lik = lik_normal())
beta.ash2 = ash(betahat, sebetahat, lik = lik_t(df=4))

e = rnorm(100)+log(rf(100,df1=10,df2=10)) # simulated data with log(F) error
e.ash = ash(e,1,lik=lik_logF(df1=10,df2=10))

# Specifying the output
beta.ash = ash(betahat, sebetahat, output = c("fitted_g","logLR","lfsr"))

#Running ash with a pre-specified g, rather than estimating it
beta = c(rep(0,100),rnorm(100))
sebetahat = abs(rnorm(200,0,1))
betahat = rnorm(200,beta,sebetahat)
true_g = normalmix(c(0.5,0.5),c(0,0),c(0,1)) # define true g
## Passing this g into ash causes it to i) take the sd and the means
## for each component from this g, and ii) initialize pi to the value
## from this g.
beta.ash = ash(betahat, sebetahat,g=true_g,fixg=TRUE)

# running with weights
beta.ash = ash(betahat, sebetahat, optmethod="w_mixEM",
               weights = c(rep(0.5,100),rep(1,100)))

# Different algorithms can be used to compute maximum-likelihood
# estimates of the mixture weights. Here, we illustrate use of the
# EM algorithm and the (default) SQP algorithm.
set.seed(1)
betahat  <- c(8.115,9.027,9.289,10.097,9.463)
sebeta   <- c(0.6157,0.4129,0.3197,0.3920,0.5496)
fit.em   <- ash(betahat,sebeta,mixcompdist = "normal",optmethod = "mixEM")
fit.sqp  <- ash(betahat,sebeta,mixcompdist = "normal",optmethod = "mixSQP")
range(fit.em$fitted$pi - fit.sqp$fitted$pi)

beta = c(rep(0,100),rnorm(100))
sebetahat = abs(rnorm(200,0,1))
betahat = rnorm(200,beta,sebetahat)
beta.ash = ash(betahat, sebetahat)
names(beta.ash)
head(beta.ash$result) # the main dataframe of results
head(get_pm(beta.ash)) # get_pm returns posterior mean
head(get_lfsr(beta.ash)) # get_lfsr returns the local false sign rate
graphics::plot(betahat,get_pm(beta.ash),xlim=c(-4,4),ylim=c(-4,4))

## Not run: 
# Why is this example included here? -Peter
CIMatrix=ashci(beta.ash,level=0.95)
print(CIMatrix)

## End(Not run)

# Illustrating the non-zero mode feature.
betahat=betahat+5
beta.ash = ash(betahat, sebetahat)
graphics::plot(betahat,get_pm(beta.ash))
betan.ash=ash(betahat, sebetahat,mode=5)
graphics::plot(betahat,get_pm(betan.ash))
summary(betan.ash)

# Running ash with different error models
beta.ash1 = ash(betahat, sebetahat, lik = lik_normal())
beta.ash2 = ash(betahat, sebetahat, lik = lik_t(df=4))

e = rnorm(100)+log(rf(100,df1=10,df2=10)) # simulated data with log(F) error
e.ash = ash(e,1,lik=lik_logF(df1=10,df2=10))

# Specifying the output
beta.ash = ash(betahat, sebetahat, output = c("fitted_g","logLR","lfsr"))

#Running ash with a pre-specified g, rather than estimating it
beta = c(rep(0,100),rnorm(100))
sebetahat = abs(rnorm(200,0,1))
betahat = rnorm(200,beta,sebetahat)
true_g = normalmix(c(0.5,0.5),c(0,0),c(0,1)) # define true g
## Passing this g into ash causes it to i) take the sd and the means
## for each component from this g, and ii) initialize pi to the value
## from this g.
beta.ash = ash(betahat, sebetahat,g=true_g,fixg=TRUE)

# running with weights
beta.ash = ash(betahat, sebetahat, optmethod="w_mixEM",
               weights = c(rep(0.5,100),rep(1,100)))

# Different algorithms can be used to compute maximum-likelihood
# estimates of the mixture weights. Here, we illustrate use of the
# EM algorithm and the (default) SQP algorithm.
set.seed(1)
betahat  <- c(8.115,9.027,9.289,10.097,9.463)
sebeta   <- c(0.6157,0.4129,0.3197,0.3920,0.5496)
fit.em   <- ash(betahat,sebeta,mixcompdist = "normal",optmethod = "mixEM")
fit.sqp  <- ash(betahat,sebeta,mixcompdist = "normal",optmethod = "mixSQP")
range(fit.em$fitted$pi - fit.sqp$fitted$pi)

Performs adaptive shrinkage on Poisson data

Description

Uses Empirical Bayes to fit the model

$y_j | \lambda_j ~ Poi(c_j \lambda_j)$

with

$h(lambda_j) ~ g()$

where $h$ is a specified link function (either "identity" or "log" are permitted).

Usage

ash_pois(y, scale = 1, link = c("identity", "log"), ...)
ash_pois(y, scale = 1, link = c("identity", "log"), ...)

Arguments

`y`	vector of Poisson observations.
`scale`	vector of scale factors for Poisson observations: the model is $y[j]~Pois(scale[j]*lambda[j])$ .
`link`	string, either "identity" or "log", indicating the link function.
`...`	other parameters to be passed to ash

Details

The model is fit in two stages: i) estimate $g$ by maximum likelihood (over the set of symmetric unimodal distributions) to give estimate $\hat{g}$ ; ii) Compute posterior distributions for $\lambda_j$ given $y_j,\hat{g}$ . Note that the link function $h$ affects the prior assumptions (because, e.g., assuming a unimodal prior on $\lambda$ is different from assuming unimodal on $\log\lambda$ ), but posterior quantities are always computed for the for $\lambda$ and *not* $h(\lambda)$ .

Examples

   beta = c(rep(0,50),rexp(50))
   y = rpois(100,beta) # simulate Poisson observations
   y.ash = ash_pois(y,scale=1)
beta = c(rep(0,50),rexp(50))
   y = rpois(100,beta) # simulate Poisson observations
   y.ash = ash_pois(y,scale=1)

Credible Interval Computation for the ash object

Description

Given the ash object returned by the main function ash, this function computes a posterior credible interval (CI) for each observation. The ash object must include a data component to use this function (which it does by default).

Usage

ashci(
  a,
  level = 0.95,
  betaindex,
  lfsr_threshold = 1,
  tol = 0.001,
  trace = FALSE
)
ashci(
  a,
  level = 0.95,
  betaindex,
  lfsr_threshold = 1,
  tol = 0.001,
  trace = FALSE
)

Arguments

`a`	the fitted ash object
`level`	the level for the credible interval, (default=0.95)
`betaindex`	a vector consisting of locations of betahat where you would like to compute the credible interval
`lfsr_threshold`	a scalar, if specified then computes CIs only for observations more significant than that threshold.
`tol`	passed to uniroot; indicates desired accuracy.
`trace`	a logical variable denoting whether some of the intermediate results of iterations should be displayed to the user. Default is FALSE.

Details

Uses uniroot to find credible interval, one at a time for each observation. The computation cost is linear in number of observations.

Value

A matrix, with 2 columns, ith row giving CI for ith observation

Examples

beta = c(rep(0,20),rnorm(20))
sebetahat = abs(rnorm(40,0,1))
betahat = rnorm(40,beta,sebetahat)
beta.ash = ash(betahat, sebetahat)

CImatrix=ashci(beta.ash,level=0.95)

CImatrix1=ashci(beta.ash,level=0.95,betaindex=c(1,2,5))
CImatrix2=ashci(beta.ash,level=0.95,lfsr_threshold=0.1)
beta = c(rep(0,20),rnorm(20))
sebetahat = abs(rnorm(40,0,1))
betahat = rnorm(40,beta,sebetahat)
beta.ash = ash(betahat, sebetahat)

CImatrix=ashci(beta.ash,level=0.95)

CImatrix1=ashci(beta.ash,level=0.95,betaindex=c(1,2,5))
CImatrix2=ashci(beta.ash,level=0.95,lfsr_threshold=0.1)

ashr

Description

The main function in the ashr package is ash, which should be examined for more details. For simplicity only the most commonly-used options are documented under ash. For expert or interested users the documentation for function ash.workhorse provides documentation on all implemented options.

Author(s)

Maintainer: Peter Carbonetto [email protected]

Authors:

Matthew Stephens [email protected]
David Gerard
Mengyin Lu
Lei Sun
Jason Willwerscheid
Nan Xiao

Other contributors:

Chaoxing Dai [contributor]
Mazon Zeng [contributor]

Compute loglikelihood for data from ash fit

Description

Return the log-likelihood of the data for a given g() prior

Usage

calc_loglik(g, data)
calc_loglik(g, data)

Arguments

`g`	the fitted g, or an ash object containing g
`data`	a data object, see set_data

Compute loglikelihood ratio for data from ash fit

Description

Return the log-likelihood ratio of the data for a given g() prior

Usage

calc_logLR(g, data)
calc_logLR(g, data)

Arguments

`g`	the fitted g, or an ash object containing g
`data`	a data object, see set_data

Generic function of calculating the overall mean of the mixture

Description

Generic function of calculating the overall mean of the mixture

Usage

calc_mixmean(m)
calc_mixmean(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns scalar, the mean of the mixture distribution.

Generic function of calculating the overall standard deviation of the mixture

Description

Generic function of calculating the overall standard deviation of the mixture

Usage

calc_mixsd(m)
calc_mixsd(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns scalar

Compute loglikelihood for data under null that all beta are 0

Description

Return the log-likelihood of the data betahat, with standard errors betahatsd, under the null that beta==0

Usage

calc_null_loglik(data)
calc_null_loglik(data)

Arguments

data

a data object; see set_data

Compute vector of loglikelihood for data under null that all beta are 0

Description

Return the vector of log-likelihoods of the data points under the null

Usage

calc_null_vloglik(data)
calc_null_vloglik(data)

Arguments

data

a data object; see set_data

Compute vector of loglikelihood for data from ash fit

Description

Return the vector of log-likelihoods of the data betahat, with standard errors betahatsd, for a given g() prior on beta, or an ash object containing that

Usage

calc_vloglik(g, data)
calc_vloglik(g, data)

Arguments

`g`	the fitted g, or an ash object containing g
`data`	a data object, see set_data

Compute vector of loglikelihood ratio for data from ash fit

Description

Return the vector of log-likelihood ratios of the data betahat, with standard errors betahatsd, for a given g() prior on beta, or an ash object containing that, vs the null that g() is point mass on 0

Usage

calc_vlogLR(g, data)
calc_vlogLR(g, data)

Arguments

`g`	the fitted g, or an ash object containing g
`data`	a data object, see set_data

cdf_conv

Description

compute cdf of mixture m convoluted with error distribution either normal of sd (s) or student t with df v at locations x

Usage

cdf_conv(m, data)
cdf_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

cdf_post

Description

evaluate cdf of posterior distribution of beta at c. m is the prior on beta, a mixture; c is location of evaluation assumption is betahat | beta ~ t_v(beta,sebetahat)

Usage

cdf_post(m, c, data)
cdf_post(m, c, data)

Arguments

`m`	mixture distribution with k components
`c`	a scalar
`data`	details depend on model

Value

an n vector containing the cdf for beta_i at c

Examples

beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
cdf0 = cdf_post(ash.beta$fitted_g,0,set_data(betahat,sebetahat))
graphics::plot(cdf0,1-get_pp(ash.beta))
beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
cdf0 = cdf_post(ash.beta$fitted_g,0,set_data(betahat,sebetahat))
graphics::plot(cdf0,1-get_pp(ash.beta))

cdf method for ash object

Description

Computed the cdf of the underlying fitted distribution

Usage

cdf.ash(a, x, lower.tail = TRUE)
cdf.ash(a, x, lower.tail = TRUE)

Arguments

`a`	the fitted ash object
`x`	the vector of locations at which cdf is to be computed
`lower.tail`	(default=TRUE) whether to compute the lower or upper tail

Details

None

Generic function of computing the cdf for each component

Description

Generic function of computing the cdf for each component

Usage

comp_cdf(m, y, lower.tail = TRUE)
comp_cdf(m, y, lower.tail = TRUE)

Arguments

`m`	a mixture (eg of type normalmix or unimix)
`y`	locations at which cdf to be computed
`lower.tail`	boolean indicating whether to report lower tail

Value

it returns a vector of probabilities, with length equals to number of components in m

comp_cdf_conv

Description

compute the cdf of data for each component of mixture when convolved with error distribution

Usage

comp_cdf_conv(m, data)
comp_cdf_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

Value

a k by n matrix of cdfs

comp_cdf_conv.normalmix

Description

returns cdf of convolution of each component of a normal mixture with N(0,s^2) at x. Note that convolution of two normals is normal, so it works that way

Usage

## S3 method for class 'normalmix'
comp_cdf_conv(m, data)
## S3 method for class 'normalmix'
comp_cdf_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	a list with components x and s to be interpreted as a normally-distributed observation and its standard error

Value

a k by n matrix

cdf of convolution of each component of a unif mixture

Description

cdf of convolution of each component of a unif mixture

Usage

## S3 method for class 'unimix'
comp_cdf_conv(m, data)
## S3 method for class 'unimix'
comp_cdf_conv(m, data)

Arguments

`m`	a mixture of class unimix
`data`	see set_data()

Value

a k by n matrix

comp_cdf_post

Description

evaluate cdf of posterior distribution of beta at c. m is the prior on beta, a mixture; c is location of evaluation assumption is betahat | beta ~ t_v(beta,sebetahat)

Usage

comp_cdf_post(m, c, data)
comp_cdf_post(m, c, data)

Arguments

`m`	mixture distribution with k components
`c`	a scalar
`data`	details depend on model

Value

a k by n matrix

Examples

beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
comp_cdf_post(get_fitted_g(ash.beta),0,data=set_data(beta,sebetahat))
beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
comp_cdf_post(get_fitted_g(ash.beta),0,data=set_data(beta,sebetahat))

Generic function of calculating the component densities of the mixture

Description

Generic function of calculating the component densities of the mixture

Usage

comp_dens(m, y, log = FALSE)
comp_dens(m, y, log = FALSE)

Arguments

`m`	mixture of k components generated by normalmix() or unimix() or igmix()
`y`	is an n-vector of location
`log`	whether to use log-scale on densities

Value

A k by n matrix of densities

comp_dens_conv

Description

compute the density of data for each component of mixture when convolved with error distribution

Usage

comp_dens_conv(m, data, ...)
comp_dens_conv(m, data, ...)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model
`...`	other arguments

Value

a k by n matrix of densities

comp_dens_conv.normalmix

Description

returns density of convolution of each component of a normal mixture with N(0,s^2) at x. Note that convolution of two normals is normal, so it works that way

Usage

## S3 method for class 'normalmix'
comp_dens_conv(m, data, ...)
## S3 method for class 'normalmix'
comp_dens_conv(m, data, ...)

Arguments

`m`	mixture distribution with k components
`data`	a list with components x and s to be interpreted as a normally-distributed observation and its standard error
`...`	other arguments (unused)

Value

a k by n matrix

density of convolution of each component of a unif mixture

Description

density of convolution of each component of a unif mixture

Usage

## S3 method for class 'unimix'
comp_dens_conv(m, data, ...)
## S3 method for class 'unimix'
comp_dens_conv(m, data, ...)

Arguments

`m`	a mixture of class unimix
`data`	see set_data()
`...`	other arguments (unused)

Value

a k by n matrix

Generic function of calculating the first moment of components of the mixture

Description

Generic function of calculating the first moment of components of the mixture

Usage

comp_mean(m)
comp_mean(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns a vector of means.

comp_mean.normalmix

Description

returns mean of the normal mixture

Usage

## S3 method for class 'normalmix'
comp_mean(m)
## S3 method for class 'normalmix'
comp_mean(m)

Arguments

`m`	a normal mixture distribution with k components

Value

a vector of length k

comp_mean.tnormalmix

Description

Returns mean of the truncated-normal mixture.

Usage

## S3 method for class 'tnormalmix'
comp_mean(m)
## S3 method for class 'tnormalmix'
comp_mean(m)

Arguments

`m`	A truncated normal mixture distribution with k components.

Value

A vector of length k.

Generic function of calculating the second moment of components of the mixture

Description

Generic function of calculating the second moment of components of the mixture

Usage

comp_mean2(m)
comp_mean2(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns a vector of second moments.

comp_postmean

Description

output posterior mean for beta for each component of prior mixture m,given data

Usage

comp_postmean(m, data)
comp_postmean(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

comp_postmean2

Description

output posterior mean-squared value given prior mixture m and data

Usage

comp_postmean2(m, data)
comp_postmean2(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

comp_postprob

Description

compute the posterior prob that each observation came from each component of the mixture m,output a k by n vector of probabilities computed by weighting the component densities by pi and then normalizing

Usage

comp_postprob(m, data)
comp_postprob(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

comp_postsd

Description

output posterior sd for beta for each component of prior mixture m,given data

Usage

comp_postsd(m, data)
comp_postsd(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

Examples

beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
ash.beta = ash(betahat,1,mixcompdist="normal")
data= set_data(betahat,rep(1,100))
comp_postmean(get_fitted_g(ash.beta),data)
comp_postsd(get_fitted_g(ash.beta),data)
comp_postprob(get_fitted_g(ash.beta),data)
beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
ash.beta = ash(betahat,1,mixcompdist="normal")
data= set_data(betahat,rep(1,100))
comp_postmean(get_fitted_g(ash.beta),data)
comp_postsd(get_fitted_g(ash.beta),data)
comp_postprob(get_fitted_g(ash.beta),data)

Generic function to extract the standard deviations of components of the mixture

Description

Generic function to extract the standard deviations of components of the mixture

Usage

comp_sd(m)
comp_sd(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns a vector of standard deviations

comp_sd.normalmix

Description

returns sds of the normal mixture

Usage

## S3 method for class 'normalmix'
comp_sd(m)
## S3 method for class 'normalmix'
comp_sd(m)

Arguments

`m`	a normal mixture distribution with k components

Value

a vector of length k

comp_sd.normalmix

Description

Returns standard deviations of the truncated normal mixture.

Usage

## S3 method for class 'tnormalmix'
comp_sd(m)
## S3 method for class 'tnormalmix'
comp_sd(m)

Arguments

`m`	A truncated normal mixture distribution with k components.

Value

A vector of length k.

Function to compute the local false sign rate

Description

Function to compute the local false sign rate

Usage

compute_lfsr(NegativeProb, ZeroProb)
compute_lfsr(NegativeProb, ZeroProb)

Arguments

`NegativeProb`	A vector of posterior probability that beta is negative.
`ZeroProb`	A vector of posterior probability that beta is zero.

Value

The local false sign rate.

Brief description of function.

Description

Explain here what this function does.

Usage

cxxMixSquarem(matrix_lik, prior, pi_init, control)
cxxMixSquarem(matrix_lik, prior, pi_init, control)

Arguments

`matrix_lik`	Description of argument goes here.
`prior`	Description of argument goes here.
`pi_init`	Description of argument goes shere.
`control`	Description of argument goes here.

Find density at y, a generic function

Description

Find density at y, a generic function

Usage

dens(x, y)
dens(x, y)

Arguments

`x`	A mixture of k components generated by `normalmix` or `unimix`.
`y`	An n-vector of the location.

dens_conv

Description

compute density of mixture m convoluted with normal of sd (s) or student t with df v at locations x

Usage

dens_conv(m, data)
dens_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

The log-F distribution

Description

Density function for the log-F distribution with df1 and df2 degrees of freedom (and optional non-centrality parameter ncp).

Usage

dlogf(x, df1, df2, ncp, log = FALSE)
dlogf(x, df1, df2, ncp, log = FALSE)

Arguments

`x`	vector of quantiles
`df1`, `df2`	degrees of freedom
`ncp`	non-centrality parameter. If omitted the central F is assumed.
`log`	logical; if TRUE, probabilities p are given as log(p).

Value

The density function.

Estimate mixture proportions of a mixture g given noisy (error-prone) data from that mixture.

Description

Estimate mixture proportions of a mixture g given noisy (error-prone) data from that mixture.

Usage

estimate_mixprop(
  data,
  g,
  prior,
  optmethod = c("mixSQP", "mixEM", "mixVBEM", "cxxMixSquarem", "mixIP", "w_mixEM"),
  control,
  weights = NULL
)
estimate_mixprop(
  data,
  g,
  prior,
  optmethod = c("mixSQP", "mixEM", "mixVBEM", "cxxMixSquarem", "mixIP", "w_mixEM"),
  control,
  weights = NULL
)

Arguments

`data`	list to be passed to log_comp_dens_conv; details depend on model
`g`	an object representing a mixture distribution (eg normalmix for mixture of normals; unimix for mixture of uniforms). The component parameters of g (eg the means and variances) specify the components whose mixture proportions are to be estimated. The mixture proportions of g are the parameters to be estimated; the values passed in may be used to initialize the optimization (depending on the optmethod used)
`prior`	numeric vector indicating parameters of "Dirichlet prior" on mixture proportions
`optmethod`	name of function to use to do optimization
`control`	list of control parameters to be passed to optmethod, typically affecting things like convergence tolerance
`weights`	vector of weights (for use with w_mixEM; in beta)

Details

This is used by the ash function. Most users won't need to call this directly, but is exported for use by some other related packages.

Value

list, including the final loglikelihood, the null loglikelihood, an n by k likelihood matrix with (j,k)th element equal to $f_k(x_j)$ , the fit and results of optmethod

gen_etruncFUN

Description

Produce function to compute expectation of truncated error distribution from log cdf and log pdf (using numerical integration)

Usage

gen_etruncFUN(lcdfFUN, lpdfFUN)
gen_etruncFUN(lcdfFUN, lpdfFUN)

Arguments

`lcdfFUN`	the log cdfFUN of the error distribution
`lpdfFUN`	the log pdfFUN of the error distribution

Density method for ash object

Description

Return the density of the underlying fitted distribution

Usage

get_density(a, x)
get_density(a, x)

Arguments

`a`	the fitted ash object
`x`	the vector of locations at which density is to be computed

Details

None

Return lfsr from an ash object

Description

These functions simply return elements of an ash object, generally without doing any calculations. (So if the value was not computed during the original call to ash, eg because of how outputlevel was set in the call, then NULL will be returned.) Accessing elements in this way rather than directly from the ash object will help ensure compatability moving forward (e.g. if the internal structure of the ash object changes during software development.)

Usage

get_lfsr(x)

get_lfdr(a)

get_svalue(a)

get_qvalue(a)

get_pm(a)

get_psd(a)

get_pp(a)

get_np(a)

get_loglik(a)

get_logLR(a)

get_fitted_g(a)

get_pi0(a)
get_lfsr(x)

get_lfdr(a)

get_svalue(a)

get_qvalue(a)

get_pm(a)

get_psd(a)

get_pp(a)

get_np(a)

get_loglik(a)

get_logLR(a)

get_fitted_g(a)

get_pi0(a)

Arguments

`x`	an ash fit (e.g. from running ash)
`a`	an ash fit (e.g. from running ash)

Value

a vector (ash) of local false sign rates

Functions

get_lfsr(): local false sign rate
get_lfdr(): local false discovery rate
get_svalue(): svalue
get_qvalue(): qvalue
get_pm(): posterior mean
get_psd(): posterior standard deviation
get_pp(): positive probability
get_np(): negative probability
get_loglik(): log-likelihood
get_logLR(): log-likelihood ratio
get_fitted_g(): fitted g mixture
get_pi0(): pi0, the proportion of nulls

Sample from posterior

Description

Returns random samples from the posterior distribution for each observation in an ash object. A matrix is returned, with columns corresponding to observations and rows corresponding to samples.

Usage

get_post_sample(a, nsamp)
get_post_sample(a, nsamp)

Arguments

`a`	the fitted ash object
`nsamp`	number of samples to return (for each observation)

Examples

beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
ash.beta = ash(betahat,1,mixcompdist="normal")
post.beta = get_post_sample(ash.beta,1000)
beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
ash.beta = ash(betahat,1,mixcompdist="normal")
post.beta = get_post_sample(ash.beta,1000)

Constructor for igmix class

Description

Creates an object of class igmix (finite mixture of univariate inverse-gammas)

Usage

igmix(pi, alpha, beta)
igmix(pi, alpha, beta)

Arguments

`pi`	vector of mixture proportions
`alpha`	vector of shape parameters
`beta`	vector of rate parameters

Details

None

Value

an object of class igmix

Examples

igmix(c(0.5,0.5),c(1,1),c(1,2))

igmix(c(0.5,0.5),c(1,1),c(1,2))

Likelihood object for Binomial error distribution

Description

Creates a likelihood object for ash for use with Binomial error distribution

Usage

lik_binom(y, n, link = c("identity", "logit"))
lik_binom(y, n, link = c("identity", "logit"))

Arguments

`y`	Binomial observations
`n`	Binomial number of trials
`link`	Link function. The "identity" link directly puts unimodal prior on Binomial success probabilities p, and "logit" link puts unimodal prior on logit(p).

Details

Suppose we have Binomial observations y where $y_i\sim Bin(n_i,p_i)$ . We either put an unimodal prior g on the success probabilities $p_i\sim g$ (by specifying link="identity") or on the logit success probabilities $logit(p_i)\sim g$ (by specifying link="logit"). Either way, ASH with this Binomial likelihood function will compute the posterior mean of the success probabilities $p_i$ .

Examples

   p = rbeta(100,2,2) # prior mode: 0.5
   n = rpois(100,10)
   y = rbinom(100,n,p) # simulate Binomial observations
   ash(rep(0,length(y)),1,lik=lik_binom(y,n))
p = rbeta(100,2,2) # prior mode: 0.5
   n = rpois(100,10)
   y = rbinom(100,n,p) # simulate Binomial observations
   ash(rep(0,length(y)),1,lik=lik_binom(y,n))

Likelihood object for logF error distribution

Description

Creates a likelihood object for ash for use with logF error distribution

Usage

lik_logF(df1, df2)
lik_logF(df1, df2)

Arguments

`df1`	first degree of freedom parameter of F distribution
`df2`	second degree of freedom parameter of F distribution

Examples

   e = rnorm(100) + log(rf(100,df1=10,df2=10)) # simulate some data with log(F) error
   ash(e,1,lik=lik_logF(df1=10,df2=10))
e = rnorm(100) + log(rf(100,df1=10,df2=10)) # simulate some data with log(F) error
   ash(e,1,lik=lik_logF(df1=10,df2=10))

Likelihood object for normal error distribution

Description

Creates a likelihood object for ash for use with normal error distribution

Usage

lik_normal()
lik_normal()

Examples

   z = rnorm(100) + rnorm(100) # simulate some data with normal error
   ash(z,1,lik=lik_normal())
z = rnorm(100) + rnorm(100) # simulate some data with normal error
   ash(z,1,lik=lik_normal())

Likelihood object for normal mixture error distribution

Description

Creates a likelihood object for ash for use with normal mixture error distribution

Usage

lik_normalmix(pilik, sdlik)
lik_normalmix(pilik, sdlik)

Arguments

`pilik`	a k vector of mixture proportions (k is the number of mixture components), or an n*k matrix that the j'th row the is mixture proportions for betahat_j
`sdlik`	a k vector of component-wise standard deviations, or an n*k matrix that the j'th row the is component-wise standard deviations for betahat_j

Examples

   e = rnorm(100,0,0.8) 
   e[seq(1,100,by=2)] = rnorm(50,0,1.5) # generate e~0.5*N(0,0.8^2)+0.5*N(0,1.5^2)
   betahat = rnorm(100)+e
   ash(betahat, 1, lik=lik_normalmix(c(0.5,0.5),c(0.8,1.5)))
e = rnorm(100,0,0.8) 
   e[seq(1,100,by=2)] = rnorm(50,0,1.5) # generate e~0.5*N(0,0.8^2)+0.5*N(0,1.5^2)
   betahat = rnorm(100)+e
   ash(betahat, 1, lik=lik_normalmix(c(0.5,0.5),c(0.8,1.5)))

Likelihood object for Poisson error distribution

Description

Creates a likelihood object for ash for use with Poisson error distribution

Usage

lik_pois(y, scale = 1, link = c("identity", "log"))
lik_pois(y, scale = 1, link = c("identity", "log"))

Arguments

`y`	Poisson observations.
`scale`	Scale factor for Poisson observations: y~Pois(scale*lambda).
`link`	Link function. The "identity" link directly puts unimodal prior on Poisson intensities lambda, and "log" link puts unimodal prior on log(lambda).

Details

Suppose we have Poisson observations y where $y_i\sim Poisson(c_i\lambda_i)$ . We either put an unimodal prior g on the (scaled) intensities $\lambda_i\sim g$ (by specifying link="identity") or on the log intensities $log(\lambda_i)\sim g$ (by specifying link="log"). Either way, ASH with this Poisson likelihood function will compute the posterior mean of the intensities $\lambda_i$ .

Examples

   beta = c(rnorm(100,50,5)) # prior mode: 50
   y = rpois(100,beta) # simulate Poisson observations
   ash(rep(0,length(y)),1,lik=lik_pois(y))

beta = c(rnorm(100,50,5)) # prior mode: 50
   y = rpois(100,beta) # simulate Poisson observations
   ash(rep(0,length(y)),1,lik=lik_pois(y))

Likelihood object for t error distribution

Description

Creates a likelihood object for ash for use with t error distribution

Usage

lik_t(df)
lik_t(df)

Arguments

`df`	degree of freedom parameter of t distribution

Examples

   z = rnorm(100) + rt(100,df=4) # simulate some data with t error
   ash(z,1,lik=lik_t(df=4))
z = rnorm(100) + rt(100,df=4) # simulate some data with t error
   ash(z,1,lik=lik_t(df=4))

log_comp_dens_conv

Description

compute the log density of the components of the mixture m when convoluted with a normal with standard deviation s or a scaled (se) student.t with df v, the density is evaluated at x

Usage

log_comp_dens_conv(m, data)
log_comp_dens_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

Value

a k by n matrix of log densities

log_comp_dens_conv.normalmix

Description

returns log-density of convolution of each component of a normal mixture with N(0,s^2) or s*t(v) at x. Note that convolution of two normals is normal, so it works that way

Usage

## S3 method for class 'normalmix'
log_comp_dens_conv(m, data)
## S3 method for class 'normalmix'
log_comp_dens_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	a list with components x and s to be interpreted as a normally-distributed observation and its standard error

Value

a k by n matrix

log density of convolution of each component of a unif mixture

Description

log density of convolution of each component of a unif mixture

Usage

## S3 method for class 'unimix'
log_comp_dens_conv(m, data)
## S3 method for class 'unimix'
log_comp_dens_conv(m, data)

Arguments

`m`	a mixture of class unimix
`data`	see set_data()

Value

a k by n matrix of densities

loglik_conv

Description

find log likelihood of data using convolution of mixture with error distribution

Usage

loglik_conv(m, data)
loglik_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

loglik_conv.default

Description

The default version of loglik_conv.

Usage

## Default S3 method:
loglik_conv(m, data)
## Default S3 method:
loglik_conv(m, data)

Arguments

`m`	mixture distribution with k components
`data`	data whose details depend on model

mixcdf

Description

Returns cdf for a mixture (generic function)

Usage

mixcdf(x, y, lower.tail = TRUE)
mixcdf(x, y, lower.tail = TRUE)

Arguments

`x`	a mixture (eg of type normalmix or unimix)
`y`	locations at which cdf to be computed
`lower.tail`	boolean indicating whether to report lower tail

Details

None

Value

an object of class normalmix

Examples

mixcdf(normalmix(c(0.5,0.5),c(0,0),c(1,2)),seq(-4,4,length=100))

mixcdf(normalmix(c(0.5,0.5),c(0,0),c(1,2)),seq(-4,4,length=100))

mixcdf.default

Description

The default version of mixcdf.

Usage

## Default S3 method:
mixcdf(x, y, lower.tail = TRUE)
## Default S3 method:
mixcdf(x, y, lower.tail = TRUE)

Arguments

`x`	a mixture (eg of type normalmix or unimix)
`y`	locations at which cdf to be computed
`lower.tail`	boolean indicating whether to report lower tail

Estimate mixture proportions of a mixture model by EM algorithm

Description

Given the individual component likelihoods for a mixture model, estimates the mixture proportions by an EM algorithm.

Usage

mixEM(matrix_lik, prior, pi_init = NULL, control = list())
mixEM(matrix_lik, prior, pi_init = NULL, control = list())

Arguments

`matrix_lik`	a n by k matrix with (j,k)th element equal to $f_k(x_j)$ .
`prior`	a k vector of the parameters of the Dirichlet prior on $\pi$ . Recommended to be rep(1,k)
`pi_init`	the initial value of $\pi$ to use. If not specified defaults to (1/k,...,1/k).
`control`	A list of control parameters for the SQUAREM algorithm, default value is set to be control.default=list(K = 1, method=3, square=TRUE, step.min0=1, step.max0=1, mstep=4, kr=1, objfn.inc=1,tol=1.e-07, maxiter=5000, trace=FALSE).

Details

Fits a k component mixture model

$f(x|\pi)= \sum_k \pi_k f_k(x)$

to independent and identically distributed data $x_1,\dots,x_n$ . Estimates mixture proportions $\pi$ by maximum likelihood, or by maximum a posteriori (MAP) estimation for a Dirichlet prior on $\pi$ (if a prior is specified). Uses the SQUAREM package to accelerate convergence of EM. Used by the ash main function; there is no need for a user to call this function separately, but it is exported for convenience.

Value

A list, including the estimates (pihat), the log likelihood for each interation (B) and a flag to indicate convergence

Estimate mixture proportions of a mixture model by Interior Point method

Description

Given the individual component likelihoods for a mixture model, estimates the mixture proportions.

Usage

mixIP(matrix_lik, prior, pi_init = NULL, control = list(), weights = NULL)
mixIP(matrix_lik, prior, pi_init = NULL, control = list(), weights = NULL)

Arguments

`matrix_lik`	a n by k matrix with (j,k)th element equal to $f_k(x_j)$ .
`prior`	a k vector of the parameters of the Dirichlet prior on $\pi$ . Recommended to be rep(1,k)
`pi_init`	the initial value of $\pi$ to use. If not specified defaults to (1/k,...,1/k).
`control`	A list of control parameters to be passed to REBayes::KWDual
`weights`	weights to be assigned to the observations (an n vector)

Details

Optimizes

$L(pi)= sum_j w_j log(sum_k pi_k f_{jk}) + h(pi)$

subject to pi_k non-negative and sum_k pi_k = 1. Here

$h(pi)$

is a penalty function h(pi) = sum_k (prior_k-1) log pi_k. Calls REBayes::KWDual in the REBayes package, which is in turn a wrapper to the mosek convex optimization software. So REBayes must be installed to use this. Used by the ash main function; there is no need for a user to call this function separately, but it is exported for convenience.

Value

A list, including the estimates (pihat), the log likelihood for each interation (B) and a flag to indicate convergence

Generic function of calculating the overall second moment of the mixture

Description

Generic function of calculating the overall second moment of the mixture

Usage

mixmean2(m)
mixmean2(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns scalar

Generic function of extracting the mixture proportions

Description

Generic function of extracting the mixture proportions

Usage

mixprop(m)
mixprop(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

it returns a vector of component probabilities, summing up to 1.

Estimate mixture proportions of a mixture model using mix-SQP algorithm.

Description

Estimate mixture proportions of a mixture model using mix-SQP algorithm.

Usage

mixSQP(matrix_lik, prior, pi_init = NULL, control = list(), weights = NULL)
mixSQP(matrix_lik, prior, pi_init = NULL, control = list(), weights = NULL)

Arguments

`matrix_lik`	A matrix containing the conditional likelihood values, possibly normalized.
`prior`	A vector of the parameters of the Dirichlet prior on the mixture weights.
`pi_init`	The initial estimate of the mixture weights.
`control`	A list of settings for the mix-SQP optimization algorithm; see `mixsqp` for details.
`weights`	The weights to be assigned to the observations. Must be a vector of length equal the number of rows of `matrix_lik`. If `weights = NULL`, all observations are assigned the same weight.

Value

A list object including the estimates (pihat) and a flag (control) indicating convergence success or failure.

Estimate posterior distribution on mixture proportions of a mixture model by a Variational Bayes EM algorithm

Description

Given the individual component likelihoods for a mixture model, estimates the posterior on the mixture proportions by an VBEM algorithm. Used by the ash main function; there is no need for a user to call this function separately, but it is exported for convenience.

Usage

mixVBEM(matrix_lik, prior, pi_init = NULL, control = list())
mixVBEM(matrix_lik, prior, pi_init = NULL, control = list())

Arguments

`matrix_lik`	a n by k matrix with (j,k)th element equal to $f_k(x_j)$ .
`prior`	a k vector of the parameters of the Dirichlet prior on $\pi$ . Recommended to be rep(1,k)
`pi_init`	the initial value of the posterior parameters. If not specified defaults to the prior parameters.
`control`	A list of control parameters for the SQUAREM algorithm, default value is set to be control.default=list(K = 1, method=3, square=TRUE, step.min0=1, step.max0=1, mstep=4, kr=1, objfn.inc=1,tol=1.e-07, maxiter=5000, trace=FALSE).

Details

Fits a k component mixture model

$f(x|\pi) = \sum_k \pi_k f_k(x)$

to independent and identically distributed data $x_1,\dots,x_n$ . Estimates posterior on mixture proportions $\pi$ by Variational Bayes, with a Dirichlet prior on $\pi$ . Algorithm adapted from Bishop (2009), Pattern Recognition and Machine Learning, Chapter 10.

Value

A list, whose components include point estimates (pihat), the parameters of the fitted posterior on $\pi$ (pipost), the bound on the log likelihood for each iteration (B) and a flag to indicate convergence (converged).

second moment of truncated Beta distribution

Description

Compute second moment of the truncated Beta.

Usage

my_e2truncbeta(a, b, alpha, beta)
my_e2truncbeta(a, b, alpha, beta)

Arguments

`a`	left limit of distribution
`b`	right limit of distribution
`alpha`, `beta`	shape parameters of Beta distribution

second moment of truncated gamma distribution

Description

Compute second moment of the truncated gamma.

Usage

my_e2truncgamma(a, b, shape, rate)
my_e2truncgamma(a, b, shape, rate)

Arguments

`a`	left limit of distribution
`b`	right limit of distribution
`shape`	shape of gamma distribution
`rate`	rate of gamma distribution

Expected Squared Value of Truncated Normal

Description

Computes the expected squared values of truncated normal distributions with parameters a, b, mean, and sd. Arguments can be scalars, vectors, or matrices. Arguments of shorter length will be recycled according to the usual recycling rules, but a and b must have the same length. Missing values are accepted for all arguments.

Usage

my_e2truncnorm(a, b, mean = 0, sd = 1)
my_e2truncnorm(a, b, mean = 0, sd = 1)

Arguments

`a`	The lower limit for the support of the truncated normal. Can be `-Inf`.
`b`	The upper limit for the support. Can be `Inf`. `a` and `b` must have the same length, and each element of `a` should be less than or equal to the corresponding element of `b`.
`mean`	The mean of the untruncated normal.
`sd`	The standard deviation of the untruncated normal. Standard deviations of zero are interpreted as numerically (rather than exactly) zero, so that the square of the untruncated mean is returned if it lies within `[a, b]` and the square of the nearer of `a` and `b` is returned otherwise.

Value

The expected squared values of truncated normal distributions with parameters a, b, mean, and sd. If any of the arguments is a matrix, then a matrix will be returned.

my_e2trunct

Description

Compute second moment of the truncated t. Uses results from O'Hagan, Biometrika, 1973

Usage

my_e2trunct(a, b, df)
my_e2trunct(a, b, df)

Arguments

`a`	left limit of distribution
`b`	right limit of distribution
`df`	degree of freedom of error distribution

mean of truncated Beta distribution

Description

Compute mean of the truncated Beta.

Usage

my_etruncbeta(a, b, alpha, beta)
my_etruncbeta(a, b, alpha, beta)

Arguments

`a`	left limit of distribution
`b`	right limit of distribution
`alpha`, `beta`	shape parameters of Beta distribution

mean of truncated gamma distribution

Description

Compute mean of the truncated gamma.

Usage

my_etruncgamma(a, b, shape, rate)
my_etruncgamma(a, b, shape, rate)

Arguments

`a`	left limit of distribution
`b`	right limit of distribution
`shape`	shape of gamma distribution
`rate`	rate of gamma distribution

my_etrunclogf

Description

Compute expectation of truncated log-F distribution.

Usage

my_etrunclogf(a, b, df1, df2)
my_etrunclogf(a, b, df1, df2)

Arguments

`a`	Left limit of distribution.
`b`	Right limit of distribution.
`df1`, `df2`	degrees of freedom

Expected Value of Truncated Normal

Description

Computes the means of truncated normal distributions with parameters a, b, mean, and sd. Arguments can be scalars, vectors, or matrices. Arguments of shorter length will be recycled according to the usual recycling rules, but a and b must have the same length. Missing values are accepted for all arguments.

Usage

my_etruncnorm(a, b, mean = 0, sd = 1)
my_etruncnorm(a, b, mean = 0, sd = 1)

Arguments

`a`	The lower limit for the support of the truncated normal. Can be `-Inf`.
`b`	The upper limit for the support. Can be `Inf`. `a` and `b` must have the same length, and each element of `a` should be less than or equal to the corresponding element of `b`.
`mean`	The mean of the untruncated normal.
`sd`	The standard deviation of the untruncated normal. Standard deviations of zero are interpreted as numerically (rather than exactly) zero, so that the untruncated mean is returned if it lies within `[a, b]` and the nearer of `a` and `b` is returned otherwise.

Value

The expected values of truncated normal distributions with parameters a, b, mean, and sd. If any of the arguments is a matrix, then a matrix will be returned.

my_etrunct

Description

Compute second moment of the truncated t. Uses results from O'Hagan, Biometrika, 1973

Usage

my_etrunct(a, b, df)
my_etrunct(a, b, df)

Arguments

`a`	left limit of distribution
`b`	right limit of distribution
`df`	degree of freedom of error distribution

Variance of Truncated Normal

Description

Computes the variance of truncated normal distributions with parameters a, b, mean, and sd. Arguments can be scalars, vectors, or matrices. Arguments of shorter length will be recycled according to the usual recycling rules, but a and b must have the same length. Missing values are accepted for all arguments.

Usage

my_vtruncnorm(a, b, mean = 0, sd = 1)
my_vtruncnorm(a, b, mean = 0, sd = 1)

Arguments

`a`	The lower limit for the support of the truncated normal. Can be `-Inf`.
`b`	The upper limit for the support. Can be `Inf`. `a` and `b` must have the same length, and each element of `a` should be less than or equal to the corresponding element of `b`.
`mean`	The mean of the untruncated normal.
`sd`	The standard deviation of the untruncated normal.

Value

The variance of truncated normal distributions with parameters a, b, mean, and sd. If any of the arguments is a matrix, then a matrix will be returned.

ncomp

Description

ncomp

Usage

ncomp(m)
ncomp(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

ncomp.default

Description

The default version of ncomp.

Usage

## Default S3 method:
ncomp(m)
## Default S3 method:
ncomp(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Constructor for normalmix class

Description

Creates an object of class normalmix (finite mixture of univariate normals)

Usage

normalmix(pi, mean, sd)
normalmix(pi, mean, sd)

Arguments

`pi`	vector of mixture proportions
`mean`	vector of means
`sd`	vector of standard deviations

Details

None

Value

an object of class normalmix

Examples

normalmix(c(0.5,0.5),c(0,0),c(1,2))

normalmix(c(0.5,0.5),c(0,0),c(1,2))

pcdf_post

Description

“parallel" vector version of cdf_post where c is a vector, of same length as betahat and sebetahat

Usage

pcdf_post(m, c, data)
pcdf_post(m, c, data)

Arguments

`m`	mixture distribution with k components
`c`	a numeric vector with n elements
`data`	depends on context

Value

an n vector, whose ith element is the cdf for beta_i at c_i

Examples

beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
c = pcdf_post(get_fitted_g(ash.beta),beta,set_data(betahat,sebetahat))
beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
c = pcdf_post(get_fitted_g(ash.beta),beta,set_data(betahat,sebetahat))

The log-F distribution

Description

Distribution function for the log-F distribution with df1 and df2 degrees of freedom (and optional non-centrality parameter ncp).

Usage

plogf(q, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)
plogf(q, df1, df2, ncp, lower.tail = TRUE, log.p = FALSE)

Arguments

`q`	vector of quantiles
`df1`, `df2`	degrees of freedom
`ncp`	non-centrality parameter. If omitted the central F is assumed.
`lower.tail`	logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].
`log.p`	logical; if TRUE, probabilities p are given as log(p).

Value

The distribution function.

Diagnostic plots for ash object

Description

Generate several plots to diagnose the fitness of ASH on the data

Usage

plot_diagnostic(
  x,
  plot.it = TRUE,
  sebetahat.tol = 0.001,
  plot.hist,
  xmin,
  xmax,
  breaks = "Sturges",
  alpha = 0.01,
  pch = 19,
  cex = 0.25
)
plot_diagnostic(
  x,
  plot.it = TRUE,
  sebetahat.tol = 0.001,
  plot.hist,
  xmin,
  xmax,
  breaks = "Sturges",
  alpha = 0.01,
  pch = 19,
  cex = 0.25
)

Arguments

`x`	the fitted ash object
`plot.it`	logical. whether to plot the diagnostic result
`sebetahat.tol`	tolerance to test the equality of betahat
`plot.hist`	logical. whether to plot the histogram of betahat when sebetahat is not constant
`xmin`, `xmax`	range of the histogram of betahat to be plotted
`breaks`	histograms parameter (see `hist`)
`alpha`	error level for the de-trended diagnostic plot
`pch`, `cex`	plot parameters for dots

Details

None.

Plot method for ash object

Description

Plot the cdf of the underlying fitted distribution

Usage

## S3 method for class 'ash'
plot(x, ..., xmin, xmax)
## S3 method for class 'ash'
plot(x, ..., xmin, xmax)

Arguments

`x`	the fitted ash object
`...`	Arguments to be passed to methods,such as graphical parameters (see `plot`)
`xmin`	xlim lower range, default is the lowest value of betahat
`xmax`	xlim upper range, default is the highest value of betahat

Details

None

Generic function to extract which components of mixture are point mass on 0

Description

Generic function to extract which components of mixture are point mass on 0

Usage

pm_on_zero(m)
pm_on_zero(m)

Arguments

`m`	a mixture of k components generated by normalmix() or unimix() or igmix()

Value

a boolean vector indicating which components are point mass on 0

post_sample

Description

returns random samples from the posterior, given a prior distribution m and n observed datapoints.

Usage

post_sample(m, data, nsamp)
post_sample(m, data, nsamp)

Arguments

`m`	prior distribution (eg of type normalmix)
`data`	a list with components x and s, each vectors of length n, to be interpreted as a normally-distributed observations and corresponding standard errors
`nsamp`	number of random samples to return for each observation

Details

exported, but mostly users will want to use 'get_post_sample'

Value

an nsamp by n matrix

post_sample.normalmix

Description

returns random samples from the posterior, given a prior distribution m and n observed datapoints.

Usage

## S3 method for class 'normalmix'
post_sample(m, data, nsamp)
## S3 method for class 'normalmix'
post_sample(m, data, nsamp)

Arguments

`m`	mixture distribution with k components
`data`	a list with components x and s to be interpreted as a normally-distributed observation and its standard error
`nsamp`	number of samples to return for each observation

Value

a nsamp by n matrix

post_sample.unimix

Description

returns random samples from the posterior, given a prior distribution m and n observed datapoints.

Usage

## S3 method for class 'unimix'
post_sample(m, data, nsamp)
## S3 method for class 'unimix'
post_sample(m, data, nsamp)

Arguments

`m`	mixture distribution with k components
`data`	a list with components x and s to be interpreted as a normally-distributed observation and its standard error
`nsamp`	number of samples to return for each observation

Value

a nsamp by n matrix

Compute Posterior

Description

Return the posterior on beta given a prior (g) that is a mixture of normals (class normalmix) and observation $betahat ~ N(beta,sebetahat)$

Usage

posterior_dist(g, betahat, sebetahat)
posterior_dist(g, betahat, sebetahat)

Arguments

`g`	a normalmix with components indicating the prior; works only if g has means 0
`betahat`	(n vector of observations)
`sebetahat`	(n vector of standard errors/deviations of observations)

Details

This can be used to obt

Value

A list, (pi1,mu1,sigma1) whose components are each k by n matrices where k is number of mixture components in g, n is number of observations in betahat

postmean

Description

postmean

Usage

postmean(m, data)
postmean(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

postmean2

Description

output posterior mean-squared value given prior mixture m and data

Usage

postmean2(m, data)
postmean2(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

postsd

Description

output posterior sd given prior mixture m and data

Usage

postsd(m, data)
postsd(m, data)

Arguments

`m`	mixture distribution with k components
`data`	details depend on the model

Print method for ash object

Description

Print the fitted distribution of beta values in the EB hierarchical model

Usage

## S3 method for class 'ash'
print(x, ...)
## S3 method for class 'ash'
print(x, ...)

Arguments

`x`	the fitted ash object
`...`	not used, included for consistency as an S3 generic/method.

Details

None

prune

Description

prunes out mixture components with low weight

Usage

prune(m, thresh = 1e-10)
prune(m, thresh = 1e-10)

Arguments

`m`	What is this argument?
`thresh`	the threshold below which components are removed

Function to compute q values from local false discovery rates

Description

Computes q values from a vector of local fdr estimates

Usage

qval.from.lfdr(lfdr)
qval.from.lfdr(lfdr)

Arguments

lfdr

a vector of local fdr estimates

Details

The q value for a given lfdr is an estimate of the (tail) False Discovery Rate for all findings with a smaller lfdr, and is found by the average of the lfdr for all more significant findings. See Storey (2003), Annals of Statistics, for definition of q value.

Value

vector of q values

Takes raw data and sets up data object for use by ash

Description

Takes raw data and sets up data object for use by ash

Usage

set_data(betahat, sebetahat, lik = NULL, alpha = 0)
set_data(betahat, sebetahat, lik = NULL, alpha = 0)

Arguments

`betahat`	vector of betahats
`sebetahat`	vector of standard errors
`lik`	a likelihood (see e.g., lik_normal())
`alpha`	specifies value of alpha to use (model is for betahat/sebetahat^alpha \| sebetahat)

Details

The data object stores both the data, and details of the model to be used for the data. For example, in the generalized version of ash the cdf and pdf of the likelihood are stored here.

Value

data object (list)

Summary method for ash object

Description

Print summary of fitted ash object

Usage

## S3 method for class 'ash'
summary(object, ...)
## S3 method for class 'ash'
summary(object, ...)

Arguments

`object`	the fitted ash object
`...`	not used, included for consistency as an S3 generic/method.

Details

summary prints the fitted mixture, the fitted log likelihood with 10 digits and a flag to indicate convergence

Constructor for tnormalmix class

Description

Creates an object of class tnormalmix (finite mixture of truncated univariate normals).

Usage

tnormalmix(pi, mean, sd, a, b)
tnormalmix(pi, mean, sd, a, b)

Arguments

`pi`	Cector of mixture proportions (length k say).
`mean`	Vector of means (length k).
`sd`	Vector of standard deviations (length k).
`a`	Vector of left truncation points of each component (length k).
`b`	Cector of right truncation points of each component (length k).

Value

An object of class “tnormalmix”.

Examples

tnormalmix(c(0.5,0.5),c(0,0),c(1,2),c(-10,0),c(0,10))

tnormalmix(c(0.5,0.5),c(0,0),c(1,2),c(-10,0),c(0,10))

Constructor for unimix class

Description

Creates an object of class unimix (finite mixture of univariate uniforms)

Usage

unimix(pi, a, b)
unimix(pi, a, b)

Arguments

`pi`	vector of mixture proportions
`a`	vector of left hand ends of uniforms
`b`	vector of right hand ends of uniforms

Details

None

Value

an object of class unimix

Examples

unimix(c(0.5,0.5),c(0,0),c(1,2))
unimix(c(0.5,0.5),c(0,0),c(1,2))

vcdf_post

Description

vectorized version of cdf_post

Usage

vcdf_post(m, c, data)
vcdf_post(m, c, data)

Arguments

`m`	mixture distribution with k components
`c`	a numeric vector
`data`	depends on context

Value

an n vector containing the cdf for beta_i at c

Examples

beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
c = vcdf_post(get_fitted_g(ash.beta),seq(-5,5,length=1000),data = set_data(betahat,sebetahat))
beta = rnorm(100,0,1)
betahat= beta+rnorm(100,0,1)
sebetahat=rep(1,100)
ash.beta = ash(betahat,1,mixcompdist="normal")
c = vcdf_post(get_fitted_g(ash.beta),seq(-5,5,length=1000),data = set_data(betahat,sebetahat))

Estimate mixture proportions of a mixture model by EM algorithm (weighted version)

Description

Given the individual component likelihoods for a mixture model, and a set of weights, estimates the mixture proportions by an EM algorithm.

Usage

w_mixEM(matrix_lik, prior, pi_init = NULL, weights = NULL, control = list())
w_mixEM(matrix_lik, prior, pi_init = NULL, weights = NULL, control = list())

Arguments

`matrix_lik`	a n by k matrix with (j,k)th element equal to $f_k(x_j)$ .
`prior`	a k vector of the parameters of the Dirichlet prior on $\pi$ . Recommended to be rep(1,k)
`pi_init`	the initial value of $\pi$ to use. If not specified defaults to (1/k,...,1/k).
`weights`	an n vector of weights
`control`	A list of control parameters for the SQUAREM algorithm, default value is set to be control.default=list(K = 1, method=3, square=TRUE, step.min0=1, step.max0=1, mstep=4, kr=1, objfn.inc=1,tol=1.e-07, maxiter=5000, trace=FALSE).

Details

Fits a k component mixture model

$f(x|\pi)= \sum_k \pi_k f_k(x)$

to independent and identically distributed data $x_1,\dots,x_n$ with weights $w_1,\dots,w_n$ . Estimates mixture proportions $\pi$ by maximum likelihood, or by maximum a posteriori (MAP) estimation for a Dirichlet prior on $\pi$ (if a prior is specified). Here the log-likelihood for the weighted data is defined as $l(\pi) = \sum_j w_j log f(x_j | \pi)$ . Uses the SQUAREM package to accelerate convergence of EM. Used by the ash main function; there is no need for a user to call this function separately, but it is exported for convenience.

Value

A list, including the estimates (pihat), the log likelihood for each interation (B) and a flag to indicate convergence

Package 'ashr'

Help Index

Adaptive Shrinkage

Description

Usage

Arguments

Details

Value

Functions

See Also

Examples

Performs adaptive shrinkage on Poisson data

Description

Usage

Arguments

Details

Examples

Credible Interval Computation for the ash object

Description

Usage

Arguments

Details

Value

Examples

ashr

Description

Author(s)

See Also

Compute loglikelihood for data from ash fit

Description

Usage

Arguments

Compute loglikelihood ratio for data from ash fit

Description

Usage

Arguments

Generic function of calculating the overall mean of the mixture

Description

Usage

Arguments

Value

Generic function of calculating the overall standard deviation of the mixture

Description

Usage

Arguments

Value

Compute loglikelihood for data under null that all beta are 0

Description

Usage

Arguments

Compute vector of loglikelihood for data under null that all beta are 0

Description

Usage

Arguments

Compute vector of loglikelihood for data from ash fit

Description

Usage

Arguments

Compute vector of loglikelihood ratio for data from ash fit

Description

Usage

Arguments

cdf_conv

Description

Usage

Arguments

cdf_post

Description

Usage

Arguments

Value

Examples

cdf method for ash object

Description

Usage

Arguments

Details

Generic function of computing the cdf for each component

Description

Usage