Help for package islasso

Type:

Package

Title:

The Induced Smoothed Lasso

Version:

1.5.2

Date:

2024-01-22

Depends:

glmnet (≥ 4.0), Matrix (≥ 1.0-6), R (≥ 4.0.0)

Imports:

stats, utils, graphics

Suggests:

knitr, lars, xfun, rmarkdown

Description:

An implementation of the induced smoothing (IS) idea to lasso regularization models to allow estimation and inference on the model coefficients (currently hypothesis testing only). Linear, logistic, Poisson and gamma regressions with several link functions are implemented. The algorithm is described in the original paper; see <doi:10.1177/0962280219842890> and discussed in a tutorial <doi:10.13140/RG.2.2.16360.11521>.

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Encoding:

UTF-8

NeedsCompilation:

yes

Packaged:

2024-01-23 09:00:46 UTC; gianlucasottile

Author:

Gianluca Sottile [aut, cre], Giovanna Cilluffo [aut, ctb], Vito MR Muggeo [aut, ctb]

Maintainer:

Gianluca Sottile <gianluca.sottile@unipa.it>

LazyLoad:

yes

Repository:

CRAN

Date/Publication:

2024-01-23 15:20:05 UTC

The Induced Smoothed lasso: A practical framework for hypothesis testing in high dimensional regression

Description

This package implements an induced smoothed approach for hypothesis testing in lasso regression.

Details

Package:	islasso
Type:	Package
Version:	1.5.w
Date:	2024-01-22
License:	GPL-2

islasso is used to fit generalized linear models with a L1-penalty on (some) regression coefficients. Along with point estimates, the main advantage is to return the full covariance matrix of estimate. The resulting standard errors can be used to make inference in the lasso framework. The main function is islasso and the correspoinding fitter function islasso.fit, and many auxiliary functions are implemented to summarize and visualize results: summary.islasso, predict.islasso, logLik.islasso, deviance.islasso, residuals.islasso.

islasso.path is used to fit Fit a generalized linear model via the Induced Smoothed Lasso. The regularization path is computed for the lasso or elasticnet penalty at a grid of values for the regularization parameter lambda. Along with coefficients profile, the main advantage is to return also the standard errors profile. The resulting standard errors can be used to make inference in the lasso framework. The main function is islasso.path and the correspoinding fitter function islasso.path.fit, and many auxiliary functions are implemented to summarize and visualize results: summary.islasso.path, predict.islasso.path, logLik.islasso.path, deviance.islasso.path, residuals.islasso.path, coef.islasso.path, fitted.islasso.path.

Author(s)

Gianluca Sottile based on some preliminary functions by Vito Muggeo.

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

References

Cilluffo, G, Sottile, G, S, La Grutta, S and Muggeo, VMR (2019). The Induced Smoothed lasso: A practical framework for hypothesis testing in high dimensional regression. Statistical Methods in Medical Research, DOI: 10.1177/0962280219842890.

Sottile, G, Cilluffo, G, Muggeo, VMR (2019). The R package islasso: estimation and hypothesis testing in lasso regression. Technical Report on ResearchGate. doi:10.13140/RG.2.2.16360.11521.

Examples


set.seed(1)
n <- 100
p <- 30
p1 <- 10  #number of nonzero coefficients
coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.veri, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
mu <- drop(X%*%coef)
y <- mu + rnorm(n, 0,sigma)

o <- islasso.path(y ~ ., data = data.frame(y = y, X), 
                  family = gaussian())
temp <- GoF.islasso.path(o) 

lambda <- temp$lambda.min["BIC"]
o <- islasso(y ~ ., data = data.frame(y = y, X), 
             family = gaussian(), lambda = lambda)
o
summary(o, pval = .05)

Optimization for the selection of the tuning parameter

Description

This function extracts the value of the tuning parameter which minimizes the AIC/BIC/AICc/eBIC/GCV/GIC criterion in “islasso.path”.

Usage

GoF.islasso.path(object, plot = TRUE, ...)

Arguments

object

a fitted model object of class "islasso.path".

plot

a logical flag indicating if each criterion have to be plotted

...

further arguments passed to or from other methods.

Details

Minimization of the Akaike Information Criterion (AIC), or Bayesian Information Criterion (BIC) or several other criteria are sometimes employed to select the tuning parameter as an alternative to the cross validation. The model degrees of freedom (not necessarly integers as in the plain lasso) used in all methods are computed as trace of the hat matrix at convergence.

Value

A list of

gof

the goodness of fit measures

minimum

the position of the optimal lambda values

lambda.min

the optimal lambda values

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

set.seed(1)
n <- 100
p <- 30
p1 <- 10  #number of nonzero coefficients
coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.veri, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
mu <- drop(X%*%coef)
y <- mu + rnorm(n, 0, sigma)

o <- islasso.path(y ~ ., data = data.frame(y = y, X))
GoF.islasso.path(o)

Prostate Cancer Data

Description

These data come from a study that examined the correlation between the level of prostate specific antigen and a number of clinical measures in men who were about to receive a radical prostatectomy. It is data frame with 97 rows and 9 columns.

Usage

data(Prostate)

Format

The data frame has the following components:

lcavol: log(cancer volume)
lweight: log(prostate weight)
age: age
lbph: log(benign prostatic hyperplasia amount)
svi: seminal vesicle invasion
lcp: log(capsular penetration)
gleason: Gleason score
pgg45: percentage Gleason scores 4 or 5
lpsa: log(prostate specific antigen)

Source

Stamey, T.A., Kabalin, J.N., McNeal, J.E., Johnstone, I.M., Freiha, F., Redwine, E.A. and Yang, N. (1989)
Prostate specific antigen in the diagnosis and treatment of adenocarcinoma of the prostate: II. radical prostatectomy treated patients, Journal of Urology 141(5), 1076–1083.

Optimization for the selection of the tuning parameter

Description

This function performs a minimization of the AIC/BIC criterion for selecting the tuning parameter in “islasso”.

Usage

aic.islasso(object, method = c("AIC", "BIC", "AICc", "GCV", "GIC"), 
  interval, g = 0, y, X, 
  intercept = FALSE, family = gaussian(), alpha = 1, offset, 
  weights, unpenalized, control = is.control(), trace = TRUE)

Arguments

object

a fitted model object of class "islasso".

method

the criterion to optimize, AIC, BIC, AICc, GCV, GIC.

interval

the lower and upper limits of \lambda wherein the AIC/BIC criterion should be optimized. Can be missing, if object has been obtained via cross-validation (and therefore includes the range of lambdas)

g

a value belonging to the interval [0, 1]. Classical BIC is returned by letting g = 0 (default value), whereas extended BIC corresponds to the case g = 0.5.

y

if object is missing, the response vector of length n.

X

if object is missing, the design matrix of dimension n * p.

intercept

if object is missing, if TRUE the intercept is added to the model matrix.

family

if object is missing, a description of the error distribution, family=gaussian, family=binomial and family=poisson are implemented with canonical link.

alpha

The elasticnet mixing parameter, with 0\le\alpha\le 1. The penalty is defined as

(1-\alpha)/2||\beta||_2^2+\alpha||\beta||_1.

alpha=1 is the lasso penalty, and alpha=0 the ridge penalty.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases.

weights

observation weights. Default is 1 for each observation.

unpenalized

a vector used to specify the unpenalized estimators; unpenalized has to be a vector of logicals.

control

a list of parameters for controlling the fitting process (see islasso.control for more details).

trace

Should the iterative procedure be printed? TRUE is the default value.

Details

Value

the optimal lambda value is returned

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

set.seed(1)
n <- 100
p <- 100
p1 <- 20  #number of nonzero coefficients
coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.veri, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
mu <- drop(X%*%coef)
y <- mu + rnorm(n, 0, sigma)

o <- islasso(y ~ ., data = data.frame(y = y, X))

## Not run: 
#use the evaluation interval of the fit
lambda_aic <- aic.islasso(o, method = "AIC") 

#overwrites the evaluation interval for lambda
lambda_bic <- aic.islasso(o, interval = c(.1, 30), method = "BIC") 

#overwrites the evaluation interval for lambda using eBIC criterion
lambda_ebic <- aic.islasso(o, interval = c(.1, 30), method = "BIC", g = .5) 

## End(Not run)

General Linear Combination method for `islasso` objects

Description

General linear hypotheses and confidence intervals estimation for linear combinantions of the regression coefficients in islasso fits

Usage

## S3 method for class 'islasso'
anova(object, A, b = NULL, ci, ...)

Arguments

object

a fitted model object of class "islasso".

A

matrix (or vector) giving linear combinations of coefficients by rows, or a character vector giving the hypothesis in symbolic form (see Details).

b

right-hand-side vector for hypothesis, with as many entries as rows in the hypothesis matrix A; can be omitted, in which case it defaults to a vector of zeroes.

ci

optionally, a two columns matrix of estimated confidence intervals for the estimated coefficients.

...

not used.

Details

For the islasso regression model with coefficients \beta, the null hypothesis is

H_0:A\beta=b

where A and b are known matrix and vector.

The hypothesis matrix A can be supplied as a numeric matrix (or vector), the rows of which specify linear combinations of the model coefficients, which are tested equal to the corresponding entries in the right-hand-side vector b, which defaults to a vector of zeroes.

Alternatively, the hypothesis can be specified symbolically as a character vector with one or more elements, each of which gives either a linear combination of coefficients, or a linear equation in the coefficients (i.e., with both a left and right side separated by an equals sign). Components of a linear expression or linear equation can consist of numeric constants, or numeric constants multiplying coefficient names (in which case the number precedes the coefficient, and may be separated from it by spaces or an asterisk); constants of 1 or -1 may be omitted. Spaces are always optional. Components are separated by plus or minus signs. Newlines or tabs in hypotheses will be treated as spaces. See the examples below.

Value

An object of class "anova.islasso" which contains the estimates, the standard errors, the Wald statistics and corresponding p value of each linear combination and of the restriced model.

Author(s)

The main function of the same name was inspired by the R function previously implemented by Vito MR Muggeo.

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

set.seed(1)
n <- 100
p <- 100
p1 <- 10  #number of nonzero coefficients
coef.true <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.true, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
eta <- drop(X %*% coef)
mu <- eta
y <- mu + rnorm(n, 0, sigma)

o <- islasso(y ~ . - 1, data = data.frame(y = y, X), 
             family = gaussian())
anova(o, A = diag(p), b = coef)
anova(o, A = c("X1 + X2 + X3 + X4 + X5 = -7.5"))
anova(o, A = c("X1 + X2 + X3 + X4 + X5 = 0"))
anova(o, A = c("X6 + X7 + X8 + X9 + X10"), b = 8.75)
anova(o, A = c("X6 + X7 + X8 + X9 + X10"), b = 0)
anova(o, A = c("X1 + X2 + X3 + X4 + X5 = -7.5",
               "X6 + X7 + X8 + X9 + X10 = 8.75"))
anova(o, A = c("X1 + X2 + X3 + X4 + X5",
               "X6 + X7 + X8 + X9 + X10"), b = c(-7.5, 8.75))
anova(o, A = c("X1 + X2 + X3 + X4 + X5",
               "X6 + X7 + X8 + X9 + X10"))

Breast Cancer microarray experiment

Description

This data set details microarray experiment for 52 breast cancer patients. The binary variable status is used to indicate whether or not the patient has died of breast cancer (status = 0 = did not die of breast cancer, status = 1 = died of breast cancer). The other variables contain the amplification or deletion of the considered genes.

Rather than measuring gene expression, this experiment aims to measure gene amplification or deletion, which refers to the number of copies of a particular DNA sequence within the genome. The aim of the experiment is to find out the key genomic factors involved in agressive and non-agressive forms of breast cancer.

The experiment was conducted by the Dr.\ John Bartlett and Dr.\ Caroline Witton in the Division of Cancer Sciences and Molecular Pathology of the University of Glasgow at the city's Royal Infirmary.

Usage

data(breast)

Source

Dr. John Bartlett and Dr. Caroline Witton, Division of Cancer Sciences and Molecular Pathology, University of Glasgow, Glasgow Royal Infirmary.

References

Augugliaro L., Mineo A.M. and Wit E.C. (2013) dgLARS: a differential geometric approach to sparse generalized linear models, Journal of the Royal Statistical Society. Series B., Vol 75(3), 471-498.

Wit E.C. and McClure J. (2004) "Statistics for Microarrays: Design, Analysis and Inference" Chichester: Wiley.

confint method for `islasso` objects

Description

confint method for islasso objects

Usage

## S3 method for class 'islasso'
confint(object, parm, level = 0.95,
  type.ci = "wald", trace = TRUE, ...)

Arguments

object

a fitted model object of class "islasso".

parm

a specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.

level

the confidence level required.

type.ci

Only Wald-type confidence intervals are implemented yet! type.ci = "wald" estimates and standard errors are used to build confidence interval

trace

if TRUE (default) a bar shows the iterations status.

...

additional argument(s) for methods.

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples


set.seed(1)
n <- 100
p <- 100
p1 <- 10  #number of nonzero coefficients
coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.veri, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
eta <- drop(X%*%coef)

##### gaussian ######
mu <- eta
y <- mu + rnorm(n, 0, sigma)

o <- islasso(y ~ ., data = data.frame(y = y, X), 
             family = gaussian())

ci <- confint(o, type.ci = "wald", parm = 1:10)
ci
plot(ci)

Blood and other measurements in diabetics

Description

The diabetes data frame has 442 rows and 3 columns. These are the data used in the Efron et al "Least Angle Regression" paper.

Format

This data frame contains the following columns:

x: a matrix with 10 columns
y: a numeric vector
x2: a matrix with 64 columns

Details

The x matrix has been standardized to have unit L2 norm in each column and zero mean. The matrix x2 consists of x plus certain interactions.

Source

https://web.stanford.edu/~hastie/Papers/LARS/LeastAngle_2002.ps

References

Efron, Hastie, Johnstone and Tibshirani (2003) "Least Angle Regression" (with discussion) Annals of Statistics

Auxiliary for controlling islasso model fitting

Description

Auxiliary function for controlling the islasso model fitting.

Usage

is.control(sigma2 = -1, tol = 1E-05, itmax = 1E+3, stand = TRUE,
  trace = 0, nfolds = 5, seed = NULL, adaptive = FALSE, g = .5,
  b0 = NULL, V0 = NULL, c = .5)

Arguments

sigma2

optional. The fixed value of dispersion parameter. If -1 (default) it is estimated from the data

tol

tollerance value to declare convergence, dafault to 1e-5

itmax

maximum number of iterations, default to 1000

stand

if TRUE (default), the covariates are standardized prior to fitting the model. However the coefficients are always returned on the original scale.

trace

Should the iterative procedure be printed? 0: no printing, 1 = compact printing, 2 = enlarged printing, 3 = compact printing including Fisher scoring information (only used in glm family).

nfolds

if lambda is unspecified in islasso, the number of folds to be used to perform cross valdation. Default to 5, and nfolds>2 is allowed. Although nfolds can be as large as the sample size (leave-one-out CV), it is not recommended for large datasets. nfolds is ignored if lambda is supplied.

seed

optional, the seed to be used to split the dataframe and to perform cross validation. Useful to make reproducible the results.

adaptive

experimental, if TRUE the adaptive LASSO is implemented.

g

a value belonging to the interval [0, 1]. Classical BIC is returned by letting g = 0 (default value), whereas extended BIC corresponds to the case g = 0.5.

b0

optional, starting values for the regression coefficients. If NULL, the point estimates from glmnet are used.

V0

optional, starting value for the estimates covariance matrix, If NULL, the identity matrix is used.

c

the weight of the mixture in the induced smoothed lasso, the default is c = .5. c = -1 means to compute it at each step of the iterative algorithm.

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

The Induced Smoothed lasso

Description

islasso is used to fit lasso regression models wherein the nonsmooth L_1 norm penalty is replaced by a smooth approximation justified under the induced smoothing paradigm. Simple lasso-type or elastic-net penalties are permitted and Linear, Logistic, Poisson and Gamma responses are allowed.

Usage

islasso(formula, family = gaussian, lambda, alpha = 1, data, weights, subset,
        offset, unpenalized, contrasts = NULL, control = is.control())

Arguments

formula

an object of class “formula” (or one that can be coerced to that class): the ‘usual’ symbolic description of the model to be fitted.

family

the assumed response distribution. Gaussian, (quasi) Binomial, (quasi) Poisson, and Gamma are allowed. family=gaussian is implemented with identity link, family=binomial is implemented with logit or probit links, family=poisson is implemented with log link, and family=Gamma is implemented with inverse, log and identity links.

lambda

Value of the tuning parameter in the objective. If missing, the optimal lambda is computed using cv.glmnet.

alpha

The elastic-net mixing parameter, with 0\le\alpha\le 1. The penalty is defined as

(1-\alpha)/2||\beta||_2^2+\alpha||\beta||_1.

alpha=1 is the lasso penalty, and alpha=0 the ridge penalty.

data

an optional data frame, list or environment (or object coercible by as.data.frame to a data frame) containing the variables in the model. If not found in data, the variables are taken from environment(formula), typically the environment from which islasso is called.

weights

observation weights. Default is 1 for each observation.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases.

unpenalized

optional. A vector of integers or characters indicating any covariate (in the formula) with coefficients not to be penalized. The intercept, if included in the model, is always unpenalized.

contrasts

an optional list. See the contrasts.arg of model.matrix.default.

control

a list of parameters for controlling the fitting process (see islasso.control for more details).

Details

islasso estimates regression models by imposing a lasso-type penalty on some or all regression coefficients. However the nonsmooth L_1 norm penalty is replaced by a smooth approximation justified under the induced smoothing paradigm. The advantage is that reliable standard errors are returned as model output and hypothesis testing on linear combinantions of the regression parameters can be carried out straightforwardly via the Wald statistic. Simulation studies provide evidence that the proposed approach controls type-I errors and exhibits good power in different scenarios.

Value

A list of

coefficients

a named vector of coefficients

se

a named vector of standard errors

residuals

the working residuals

fitted.values

the fitted values

rank

the estimated degrees of freedom

family

the family object used

linear.predictors

the linear predictors

deviance

the family deviance

aic

the Akaike Information Criterion

null.deviance

the family null deviance

iter

the number of iterations of IWLS used

weights

the working weights, that is the weights in the final iteration of the IWLS fit

df.residual

the residual degrees of freedom

df.null

the degrees of freedom of a null model

converged

logical. Was the IWLS algorithm judged to have converged?

model

if requested (the default), the model frame used.

call

the matched call

formula

the formula supplied

terms

the terms object used

data

he data argument.

offset

the offset vector used.

control

the value of the control argument used

xlevels

(where relevant) a record of the levels of the factors used in fitting.

lambda

the lambda value used in the islasso algorithm

alpha

the elasticnet mixing parameter

dispersion

the estimated dispersion parameter

internal

internal elements

contrasts

(only where relevant) the contrasts used.

Author(s)

The main function of the same name was inspired by the R function previously implemented by Vito MR Muggeo.

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

References

Sottile, G, Cilluffo, G, Muggeo, VMR (2019). The R package islasso: estimation and hypothesis testing in lasso regression. Technical Report on ResearchGate. doi:10.13140/RG.2.2.16360.11521.

Examples


set.seed(1)
n <- 100
p <- 100
p1 <- 10  #number of nonzero coefficients
coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.veri, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
eta <- drop(X%*%coef)

##### gaussian ######
mu <- eta
y <- mu + rnorm(n, 0, sigma)

o <- islasso(y ~ ., data = data.frame(y = y, X), 
             family = gaussian())
o
summary(o)
coef(o)
fitted(o)
predict(o, type="response")
plot(o)
residuals(o)
deviance(o)
AIC(o)
logLik(o)

## Not run: 
# for the interaction
o <- islasso(y ~ X1 * X2, data = data.frame(y = y, X), 
             family = gaussian())

##### binomial ######
coef <- c(c(1,1,1), rep(0, p-3))
X <- matrix(rnorm(n*p), n, p)
eta <- drop(cbind(1, X)%*%c(-1, coef))
mu <- binomial()$linkinv(eta)
y <- rbinom(n, 100, mu)
y <- cbind(y, 100-y)

o <- islasso(cbind(y1, y2) ~ ., 
             data = data.frame(y1 = y[,1], y2 = y[,2], X), 
             family = binomial())
summary(o, pval = .05)

##### poisson ######
coef <- c(c(1,1,1), rep(0, p-3))
X <- matrix(rnorm(n*p), n, p)
eta <- drop(cbind(1, X)%*%c(1, coef))
mu <- poisson()$linkinv(eta)
y <- rpois(n, mu)

o <- islasso(y ~ ., data = data.frame(y = y, X), 
             family = poisson())
summary(o, pval = .05)

##### Gamma ######
coef <- c(c(1,1,1), rep(0, p-3))
X <- matrix(rnorm(n*p), n, p)
eta <- drop(cbind(1, X)%*%c(-1, coef))
mu <- Gamma(link="log")$linkinv(eta)
shape <- 10
phi <- 1 / shape
y <- rgamma(n, scale = mu / shape, shape = shape)

o <- islasso(y ~ ., data = data.frame(y = y, X), 
             family = Gamma(link = "log"))
summary(o, pval = .05)

## End(Not run)

Internal Functions

Description

Internal islasso functions.

Usage

fmt.perc(probs, digits)
qqNorm(x, probs=seq(.005, .995, l=200), centre=FALSE, scale=FALSE, 
  leg=TRUE, mean=0, sd=1, add=FALSE, p.col=1, dF=FALSE, ylab, ...)
ginv2(X, tol = sqrt(.Machine$double.eps))
modelX(n, p, rho=.5, scale=TRUE)
printCoefmat2(x, digits = max(3L, getOption("digits") - 2L), 
  signif.stars = getOption("show.signif.stars"), 
  signif.legend = signif.stars, dig.tst = max(1L, min(5L, digits - 1L)), 
  cs.ind = 1:k, tst.ind = k + 1, zap.ind = integer(), P.values = NULL, 
  has.Pvalue = nc >= 4L && length(cn <- colnames(x)) && 
    substr(cn[nc], 1L, 3L) %in% c("Pr(", "p-v"), 
  eps.Pvalue = .Machine$double.eps, na.print = "NA", quote = FALSE, 
  right = TRUE, ...)
.checkinput(X, y, family, alpha, intercept, weights, offset, 
  unpenalized, control)
.startpoint(X, y, lambda, alpha, weights, offset, mustart, 
  family, intercept, setting)
.islasso(prep, start, Lambda, fam, link)

checkinput.islasso.path(X, y, family, lambda, nlambda, lambda.min.ratio, 
  alpha, intercept, weights, offset, unpenalized, control)
startpoint.islasso.path(X, y, lambda, alpha, weights, offset, mustart, 
  family, intercept, setting)
islasso.path.fit.glm(prep, start, lambda, fam, link)
islasso.path.fitter2(b, se, c, fix.c, l, alpha, x, y, weights, sigma2, 
  cov.unscaled, n, intercept, tol, unpenalized, offset, family, maxIter)
interpolate(y1, y2, x1, x2, x.new)

makeHyp(cnames, hypothesis, rhs = NULL) 
printHyp(L, b, nms)

cislasso(object, a, ci)
ci.fitted.islasso(object, newx, ci = NULL, type.ci = "wald", 
  conf.level=.95, only.ci = FALSE)
  
grad.islasso(object, ...)
hess.islasso(object, ...)
is.influence(model, do.coef = TRUE)
islasso.diag(glmfit)
islasso.diag.plots(glmfit, glmdiag = islasso.diag(glmfit), 
  subset = NULL, iden = FALSE, labels = NULL, ret = FALSE)
predislasso(object, newdata, type = c("response", "terms"), 
  terms = NULL, na.action = na.pass, ...)

Details

These functions are not intended for users.

Author(s)

Gianluca Sottile (gianluca.sottile@unipa.it)

The Induced Smoothed lasso path

Description

islasso.path is used to fit a generalized linear model via induced smoothed lasso method. The regularization path is computed for the lasso or elasticnet penalty at a grid of values for the regularization parameter lambda. Fits linear, logistic, poisson and gamma regression models.

Usage

islasso.path(formula, family = gaussian(), lambda = NULL, nlambda = 100, 
        lambda.min.ratio = ifelse(nobs < nvars, 1E-2, 1E-03), alpha = 1, data, 
        weights, subset, offset, contrasts = NULL, unpenalized, control = is.control())

Arguments

formula

an object of class “formula” (or one that can be coerced to that class): the ‘usual’ symbolic description of the model to be fitted.

family

lambda

A user supplied lambda sequence. Typical usage is to have the program compute its own lambda sequence based on nlambda and lambda.min.ratio. Supplying a value of lambda overrides this.

nlambda

The number of lambda values - default is 100.

lambda.min.ratio

Smallest value for lambda, as a fraction of lambda.max, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size nobs relative to the number of variables nvars. If nobs > nvars, the default is 0.00001, close to zero. If nobs < nvars, the default is 0.001. A very small value of lambda.min.ratio will lead to a saturated fit in the nobs < nvars case.

alpha

The elastic-net mixing parameter, with 0\le\alpha\le 1. The penalty is defined as

(1-\alpha)/2||\beta||_2^2+\alpha||\beta||_1.

alpha=1 is the lasso penalty, and alpha=0 the ridge penalty.

data

weights

observation weights. Default is 1 for each observation.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

offset

this can be used to specify an a priori known component to be included in the linear predictor during fitting. This should be NULL or a numeric vector of length equal to the number of cases.

contrasts

an optional list. See the contrasts.arg of model.matrix.default.

control

a list of parameters for controlling the fitting process (see islasso.control for more details).

unpenalized

optional. A vector of integers or characters indicating any covariate (in the formula) with coefficients not to be penalized. The intercept, if included in the model, is always unpenalized.

Details

The sequence of models implied by lambda is fit the islasso method. islasso estimates regression models by imposing a lasso-type penalty on some or all regression coefficients. However the nonsmooth L_1 norm penalty is replaced by a smooth approximation justified under the induced smoothing paradigm. The advantage is that reliable standard errors are returned as model output and hypothesis testing on linear combinantions of the regression parameters can be carried out straightforwardly via the Wald statistic. Simulation studies provide evidence that the proposed approach controls type-I errors and exhibits good power in different scenarios.

Value

A list of

call

the matched call.

Info

a named matrix containing information about lambda values, estimated degrees of freedom, estimated dispersion parameters, deviance, loglikelhood, number of iterations and convergence criteria.

GoF

a named matrix containing information criteria, i.e., AIC, BIC, AICc, eBIC, GCV, GIC.

Coef

a length(lambda) x nvars matrix of coefficients.

SE

a length(lambda) x nvars matrix of standard errors.

Weights

a length(lambda) x nvars matrix of the weight of the mixture in the induced smoothed lasso.

Linear.predictors

a length(lambda) x nvars matrix of linear predictors

Fitted.values

a length(lambda) x nvars matrix of fitted values

Residuals

a length(lambda) x nvars matrix of working residuals

Input

a named list containing several input arguments, i.e., the numbers of observations and predictors, if an intercept ha to be estimated, the model matrix and the response vector, the observation weights, the offset, the family object used, The elasticnet mixing parameter and the vector used to specify the unpenalized estimators.

control

the value of the control argument used.

formula

the formula supplied.

model

if requested (the default), the model frame used.

terms

the terms object used.

data

he data argument.

xlevels

(where relevant) a record of the levels of the factors used in fitting.

contrasts

(only where relevant) the contrasts used.

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

References

Sottile, G, Cilluffo, G, Muggeo, VMR (2019). The R package islasso: estimation and hypothesis testing in lasso regression. Technical Report on ResearchGate. doi:10.13140/RG.2.2.16360.11521.

Examples


set.seed(1)
n <- 100
p <- 30
p1 <- 10  #number of nonzero coefficients
coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
sigma <- 1

coef <- c(coef.veri, rep(0, p-p1))

X <- matrix(rnorm(n*p), n, p)
eta <- drop(X%*%coef)

##### gaussian ######
mu <- eta
y <- mu + rnorm(n, 0, sigma)

o <- islasso.path(y ~ ., data = data.frame(y = y, X), 
             family = gaussian(), nlambda = 30L)
o
summary(o, lambda = 10)
coef(o, lambda = 10)
fitted(o, lambda = 10)
predict(o, type="response", lambda = 10)
plot(o, xvar = "coef")
residuals(o, lambda = 10)
deviance(o, lambda = 10)
logLik(o, lambda = 10)
GoF.islasso.path(o)

## Not run: 
##### binomial ######
coef <- c(c(1,1,1), rep(0, p-3))
X <- matrix(rnorm(n*p), n, p)
eta <- drop(cbind(1, X)%*%c(-1, coef))
mu <- binomial()$linkinv(eta)
y <- rbinom(n, 100, mu)
y <- cbind(y, 100-y)

o <- islasso.path(cbind(y1, y2) ~ ., 
             data = data.frame(y1 = y[,1], y2 = y[,2], X), 
             family = binomial(), nlambda = 30L)
temp <- GoF.islasso.path(o)
summary(o, pval = .05, lambda = temp$lambda.min["BIC"])

##### poisson ######
coef <- c(c(1,1,1), rep(0, p-3))
X <- matrix(rnorm(n*p), n, p)
eta <- drop(cbind(1, X)%*%c(1, coef))
mu <- poisson()$linkinv(eta)
y <- rpois(n, mu)

o <- islasso.path(y ~ ., data = data.frame(y = y, X), 
             family = poisson(), nlambda = 30L)
temp <- GoF.islasso.path(o)
summary(o, pval = .05, lambda = temp$lambda.min["BIC"])

##### Gamma ######
coef <- c(c(1,1,1), rep(0, p-3))
X <- matrix(rnorm(n*p), n, p)
eta <- drop(cbind(1, X)%*%c(-1, coef))
mu <- Gamma(link="log")$linkinv(eta)
shape <- 10
phi <- 1 / shape
y <- rgamma(n, scale = mu / shape, shape = shape)

o <- islasso.path(y ~ ., data = data.frame(y = y, X), 
             family = Gamma(link = "log"), nlambda = 30L)
temp <- GoF.islasso.path(o)
summary(o, pval = .05, lambda = temp$lambda.min["BIC"])

## End(Not run)

Diagnostics plots for Induced Smoothing Lasso Model

Description

Diagnostics plots for Induced Smoothing Lasso Model

Usage

## S3 method for class 'islasso'
plot(x, ...)

Arguments

x

an object of class islasso, usually, a result of a call to islasso.

...

other graphical parameters for the plot

Details

The plot on the top left is a plot of the standard deviance residuals against the fitted values. The plot on the top right is a normal QQ plot of the standardized deviance residuals. The red line is the expected line if the standardized residuals are normally distributed, i.e. it is the line with intercept 0 and slope 1. The bottom two panels are plots of link and variance functions. On the left is squared standardized Pearson residuals against the fitted values. On the right working vector against the linear predictor.

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

## Not run: 
  set.seed(1)
  n <- 100
  p <- 100
  p1 <- 20  #number of nonzero coefficients
  coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
  sigma <- 1

  coef <- c(coef.veri, rep(0, p-p1))

  X <- matrix(rnorm(n*p), n, p)
  mu <- drop(X%*%coef)
  y <- mu + rnorm(n, 0,sigma)

  lambda <- 2
  o <- islasso(y ~ ., data = data.frame(y = y, X), 
               family = gaussian(), lambda = lambda)
  plot(o)

## End(Not run)

plot coefficient profile from a fitted "islasso.path" object.

Description

Produces a coefficient profile plot of the coefficient paths for a fitted "islasso.path" object.

Usage

## S3 method for class 'islasso.path'
plot(x, 
  yvar = c("coefficients", "se", "gradient", "weight", "gof"), 
  gof = c("none", "AIC", "BIC", "AICc", "eBIC", "GCV", "GIC"), 
  label = FALSE, legend = FALSE, ...)

Arguments

x

an object of class islasso, usually, a result of a call to islasso.path.

yvar

What is on the Y-axis. "coef" plot the log-lambda sequence against the coefficients; "se" plot the log-lambda sequence against the standard deviations; "gradient" plot the log-lambda sequence against the gradient; "weight" plot the log-lambda sequence against the mixture weight of the islasso method; "gof" plot the log-lambda sequence against the chosen criterion.

gof

the chosen criterion to highlight the active variables. "none" doesn't highlight active variables.

label

a logical flag indicating if some labels have to be added.

legend

a logical flag indicating if legend has to be shown.

...

other graphical parameters for the plot, i.e., main, xlab, ylab, xlim, ylim, lty, col, lwd, cex.axis, cex.lab, cex.main, gof_lty, gof_col and gof_lwd. The last three parameters are used to modify aspects of the legend, and of the goodness of fit measure used.

Details

A coefficient profile plot is produced for Induced Smoothing Lasso Model path.

Examples

## Not run: 
  set.seed(1)
  n <- 100
  p <- 30
  p1 <- 10  #number of nonzero coefficients
  coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
  sigma <- 1

  coef <- c(coef.veri, rep(0, p-p1))

  X <- matrix(rnorm(n*p), n, p)
  mu <- drop(X%*%coef)
  y <- mu + rnorm(n, 0,sigma)

  o <- islasso.path(y ~ ., data = data.frame(y = y, X), 
                    family = gaussian())
  par(mfrow = c(2, 2))
  plot(o, yvar = "coefficients", gof = "AICc", label = TRUE)
  plot(o, yvar = "se", gof = "AICc")
  plot(o, yvar = "gradient", gof = "AICc")
  plot(o, yvar = "gof", gof = "AICc")

## End(Not run)

Prediction method for islasso fitted objects

Description

Prediction method for islasso fitted objects

Usage

## S3 method for class 'islasso'
predict(object, newdata = NULL, 
  type = c("link", "response", "coefficients", "class", "terms"), 
  se.fit = FALSE, ci = NULL, type.ci = "wald", 
  level = .95, terms = NULL, na.action = na.pass, ...)

Arguments

object

a fitted object of class "islasso".

newdata

optionally, a data frame in which to look for variables with which to predict. If omitted, the fitted linear predictors are used.

type

the type of prediction required. The default is on the scale of the linear predictors; the alternative "response" is on the scale of the response variable. Thus for a default binomial model the default predictions are of log-odds (probabilities on logit scale) and type = "response" gives the predicted probabilities. The coefficients option returns coefficients. Type "class" applies only to "binomial" models, and produces the class label. The "terms" option returns a matrix giving the fitted values of each term in the model formula on the linear predictor scale.

se.fit

logical switch indicating if confidence intervals are required.

ci

optionally, a two columns matrix of estimated confidence intervals for the estimated coefficients.

type.ci

Only Wald-type confidence intervals are implemented yet! type.ci = "wald" estimates and standard errors are used to build confidence interval

level

the confidence level required.

terms

with type = "terms" by default all terms are returned. A character vector specifies which terms are to be returned.

na.action

function determining what should be done with missing values in newdata. The default is to predict NA.

...

further arguments passed to or from other methods.

Value

An object depending on the type argument

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

 set.seed(1)
 n <- 100
 p <- 100
 p1 <- 20  #number of nonzero coefficients
 coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
 sigma <- 1

 coef <- c(coef.veri, rep(0, p-p1))

 X <- matrix(rnorm(n*p), n, p)
 mu <- drop(X%*%coef)
 y <- mu + rnorm(n, 0,sigma)
 lambda <- 2
 o <- islasso(y ~ ., data = data.frame(y = y, X), lambda = lambda)
 predict(o, type = "response")

Prediction method for islasso.path fitted objects

Description

Prediction method for islasso fitted objects

Usage

## S3 method for class 'islasso.path'
predict(object, newdata, type = c("link", "response",
  "coefficients", "class"), lambda, ...)

Arguments

object

a fitted object of class "islasso.path".

newdata

optionally, a data frame in which to look for variables with which to predict. If omitted, the fitted linear predictors are used.

type

lambda

Value(s) of the penalty parameter lambda at which predictions are required. Default is the entire sequence used to create the model.

...

further arguments passed to or from other methods.

Value

An object depending on the type argument

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

## Not run: 
 set.seed(1)
 n <- 100
 p <- 30
 p1 <- 10  #number of nonzero coefficients
 coef.veri <- sort(round(c(seq(.5, 3, l=p1/2), seq(-1, -2, l=p1/2)), 2))
 sigma <- 1

 coef <- c(coef.veri, rep(0, p-p1))

 X <- matrix(rnorm(n*p), n, p)
 mu <- drop(X%*%coef)
 y <- mu + rnorm(n, 0,sigma)

 o <- islasso.path(y ~ ., data = data.frame(y = y, X), 
                   family = gaussian())
 temp <- GoF.islasso.path(o)
 predict(o, type = "response", lambda = temp$lambda.min)

## End(Not run)

Simulate model matrix and response

Description

Simulate model matrix and response from a specified distribution.

Usage

simulXy(n, p, interc = 0, beta, family = gaussian(), prop =
  0.1, lim.b = c(-3, 3), sigma = 1, size = 1, rho = 0,
  scale = TRUE, seed, X)

Arguments

n

number of observations.

p

total number of covariates in the model matrix.

interc

the model intercept.

beta

the vector of p coefficients in the linear predictor.

family

a description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. Only gaussian, binomial or poisson are allowed.

prop

if beta is missing, prop represent the quote of non-null coefficients out of p. The default is 0.10 p.

lim.b

if beta is missing, the coefficients come from uniform variates in lim.b. The default is (-3,3).

sigma

if family is 'gaussian', the standard deviation of the response. The default is 1.

size

if family is 'binomial', the number of trials to build the response vector. The default is 1.

rho

correlation value to define the variance covariance matrix to build the model matrix, i.e., rho^|i-j| i,j = 1,...,p and i different from j. The default is 0.

scale

Should the columns of the mdoel matrix be scaled? The default is TRUE

seed

optional, the seed to generate the data.

X

optional, the model matrix.

Examples


n <- 100
p <- 100
beta <- c(runif(10, -3, 3), rep(0, p-10))
dat <- simulXy(n, p,  beta = beta, seed=1234)

summary method for islasso fitted objects

Description

summary method for islasso fitted objects

Usage

## S3 method for class 'islasso'
summary(object, pval = 1, which, use.t = FALSE,
  type.pval = "wald", ...)

Arguments

object

fitted "islasso" object

pval

a threshold p-value value indicating which coefficients should be printed. If pval = 0.10, say, only the variables/coefficients with p-value\le 0.10 are printed. Possible unpenalized coefficients (including the intercept if in the model) are always printed, regardless of their p-value.

which

a specification of which parameters are to be given p-values. If missing, all parameters are considered.

use.t

if TRUE, the p-values are computed using the t-distribution with residual model degrees of freedom

type.pval

Only Wald-type confidence intervals are implemented yet! type.pval = "wald" (default) estimates and standard errors are used to build confidence interval

...

not used

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

## Not run: 
#continues example from ?islasso
summary(o, pval = .1) #print just the "borderline" significant coefficients

## End(Not run)

summary method for islasso.path fitted objects

Description

summary method for islasso.path fitted objects

Usage

## S3 method for class 'islasso.path'
summary(object, pval = 1, use.t = FALSE, lambda, ...)

Arguments

object

fitted "islasso.path" object

pval

use.t

if TRUE, the p-values are computed using the t-distribution with residual model degrees of freedom

lambda

Value of the penalty parameter lambda at which summary are required.

...

not used

Author(s)

Maintainer: Gianluca Sottile <gianluca.sottile@unipa.it>

Examples

## Not run: 
#continues example from ?islasso.path
summary(o, pval = .1, lambda = 5) #print just the "borderline" significant coefficients

## End(Not run)

The Induced Smoothed lasso: A practical framework for hypothesis testing in high dimensional regression

Description

Details

Author(s)

References

Examples

Optimization for the selection of the tuning parameter

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Prostate Cancer Data

Description

Usage

Format

Source

Optimization for the selection of the tuning parameter

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

General Linear Combination method for islasso objects

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Breast Cancer microarray experiment

Description

Usage

Source

References

confint method for islasso objects

Description

Usage

Arguments

Author(s)

See Also

Examples

Blood and other measurements in diabetics

Description

Format

Details

Source

References

Auxiliary for controlling islasso model fitting

Description

Usage

Arguments

Author(s)

The Induced Smoothed lasso

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Internal Functions

Description

Usage

Details

Author(s)

The Induced Smoothed lasso path

Description

Usage

Arguments

Details

General Linear Combination method for `islasso` objects

confint method for `islasso` objects