Help for package islasso

Type:

Package

Title:

The Induced Smoothed Lasso

Version:

1.6.0

Date:

2025-07-30

Depends:

glmnet (≥ 4.0), R (≥ 4.1.0)

Imports:

stats, utils, graphics, cli, gridExtra, ggplot2

Suggests:

knitr, lars, xfun, rmarkdown, roxygen2

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

RoxygenNote:

7.3.2

NeedsCompilation:

yes

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

Packaged:

2025-07-31 12:32:13 UTC; gianlucasottile

Date/Publication:

2025-07-31 12:50:02 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.

Fits regression models with a smoothed L1 penalty under the induced smoothing paradigm. Supports linear, logistic, Poisson, and Gamma responses. Enables reliable standard errors and Wald-based inference.

Usage

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

Arguments

formula

A symbolic formula describing the model.

family

Response distribution. Can be gaussian, binomial, poisson, or Gamma.

lambda

Regularization parameter. If missing, it is estimated via cv.glmnet.

alpha

Elastic-net mixing parameter (0 \le \alpha \le 1).

data

A data frame or environment containing the variables in the model.

weights

Observation weights. Defaults to 1.

subset

Optional vector specifying a subset of rows to include.

offset

Optional numeric vector of offsets in the linear predictor.

unpenalized

Vector indicating variables (by name or index) to exclude from penalization.

contrasts

Optional contrasts specification for factor variables.

control

A list of parameters to control model fitting. See is.control.

Details

Package:	islasso
Type:	Package
Version:	1.6.0
Date:	2025-07-30
License:	GPL-2

islasso fits generalized linear models with an L1 penalty on selected coefficients. It returns both point estimates and full covariance matrices, enabling standard error-based inference. Related methods include: summary.islasso, predict.islasso, logLik.islasso, deviance.islasso, and residuals.islasso.

islasso.path fits regularization paths using the Induced Smoothed Lasso. It computes coefficients and standard errors across a grid of lambda values. Companion methods include: summary.islasso.path, predict.islasso.path, logLik.islasso.path, residuals.islasso.path, coef.islasso.path, and fitted.islasso.path.

The non-smooth L1 penalty is replaced by a smooth approximation, enabling inference through standard errors and Wald tests. The approach controls type-I error and shows strong power in various simulation settings.

Value

A list with components such as:

coefficients

Estimated coefficients

se

Standard errors

fitted.values

Fitted values

deviance, aic, null.deviance

Model diagnostic metrics

residuals, weights

IWLS residuals and weights

df.residual, df.null, rank

Degrees of freedom

converged

Logical; convergence status

model, call, terms, formula, data, offset

Model objects

xlevels, contrasts

Factor handling details

lambda, alpha, dispersion

Model parameters

internal

Other internal values

Author(s)

Gianluca Sottile, based on preliminary work by Vito Muggeo. Maintainer: gianluca.sottile@unipa.it

Gianluca Sottile gianluca.sottile@unipa.it

References

Cilluffo, G., Sottile, G., La Grutta, S., Muggeo, VMR (2019). *The Induced Smoothed lasso: A practical framework for hypothesis testing in high dimensional regression*, Statistical Methods in Medical Research. DOI: 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: doi:10.13140/RG.2.2.16360.11521

Cilluffo G., Sottile G., La Grutta S., Muggeo V.M.R. (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 V.M.R. (2019) The R package islasso: estimation and hypothesis testing in lasso regression. Technical Report. DOI: 10.13140/RG.2.2.16360.11521

Examples

n <- 100; p <- 100

beta <- c(rep(1, 5), rep(0, p - 5))
sim1 <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
o <- islasso(y ~ ., data = sim1$data, family = gaussian())

summary(o, pval = 0.05)
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 = sim1$data, family = gaussian())

##### binomial ######
beta <- c(c(1,1,1), rep(0, p-3))
sim2 <- simulXy(n = n, p = p, beta = beta, interc = 1, seed = 1,
                size = 100, family = binomial())
o2 <- islasso(cbind(y.success, y.failure) ~ .,
              data = sim2$data, family = binomial())
summary(o2, pval = 0.05)

##### poisson ######
beta <- c(c(1,1,1), rep(0, p-3))
sim3 <- simulXy(n = n, p = p, beta = beta, interc = 1, seed = 1,
                family = poisson())
o3 <- islasso(y ~ ., data = sim3$data, family = poisson())
summary(o3, pval = 0.05)

##### Gamma ######
beta <- c(c(1,1,1), rep(0, p-3))
sim4 <- simulXy(n = n, p = p, beta = beta, interc = -1, seed = 1,
                dispersion = 0.1, family = Gamma(link = "log"))
o4 <- islasso(y ~ ., data = sim4$data, family = Gamma(link = "log"))
summary(o4, pval = 0.05)

## End(Not run)

Select Optimal Lambda via Goodness-of-Fit Criteria

Description

Extracts the tuning parameter lambda minimizing multiple information criteria from a fitted islasso.path object. Supported criteria include AIC, BIC, AICc, eBIC, GCV, and GIC.

Usage

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

Arguments

object

A fitted model of class "islasso.path".

plot

Logical. If TRUE (default), displays plots for each criterion over the lambda path.

...

Additional arguments passed to lower-level plotting or diagnostic methods.

Details

This function identifies the optimal regularization parameter lambda by minimizing various information-based selection criteria. Degrees of freedom are computed as the trace of the hat matrix, which may be fractional under induced smoothing. This provides a robust alternative to cross-validation, especially in high-dimensional settings.

Value

A list with components:

gof

Matrix of goodness-of-fit values across lambda values.

minimum

Index positions of the minimum for each criterion.

lambda.min

Optimal lambda values that minimize each criterion.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

set.seed(1)
n <- 100; p <- 30
beta <- c(runif(10, -2, 2), rep(0, p - 10))
sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
fit <- islasso.path(y ~ ., data = sim$data, family = gaussian())
GoF.islasso.path(fit)

Prostate Cancer Data

Description

This dataset originates from a study examining the correlation between prostate-specific antigen levels and various clinical measures in men scheduled for radical prostatectomy. It contains 97 rows and 9 variables.

Format

A data frame with 97 observations and 9 variables:

lcavol: Log of cancer volume
lweight: Log of prostate weight
age: Age of the patient
lbph: Log of benign prostatic hyperplasia amount
svi: Seminal vesicle invasion (binary)
lcp: Log of capsular penetration
gleason: Gleason score
pgg45: Percentage of Gleason scores 4 or 5
lpsa: Log of prostate-specific antigen

Source

Stamey, T.A., et al. (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.

References

Stamey, T.A., Kabalin, J.N., McNeal, J.E., Johnstone, I.M., Freiha, F., Redwine, E.A., and Yang, N. (1989). Journal of Urology, 141(5), 1076-1083.

Examples

data(Prostate)
summary(Prostate)
cor(Prostate$lpsa, Prostate$lcavol)
## Not run: 
  fit <- islasso(lpsa ~ ., data = Prostate, family = gaussian())
  summary(fit, pval = 0.05)
  lambda.aic <- aic.islasso(fit, method = "AIC")
  fit.aic <- update(fit, lambda = lambda.aic)
  summary(fit.aic, pval = 0.05)

## End(Not run)

Optimization for Lambda Selection

Description

Minimizes information criteria to select the optimal tuning parameter lambda for islasso models. Supports AIC, BIC, AICc, GCV, and GIC.

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

Fitted model of class "islasso".

method

Criterion to minimize. Options are "AIC", "BIC", "AICc", "GCV", "GIC".

interval

Numeric vector (length 2) giving lower and upper bounds for lambda optimization. Optional if object includes prior cross-validation.

g

Numeric in [0,1]. Governs BIC generalization: g = 0 is classic BIC, g = 0.5 is extended BIC.

y

Response vector. Required only if object is missing.

X

Design matrix. Required only if object is missing.

intercept

Logical. Whether to include intercept in X. Used if object is missing.

family

Error distribution. Accepted: gaussian, binomial, poisson. Uses canonical link.

alpha

Elastic-net mixing parameter, 0 <= alpha <= 1. Lasso: alpha = 1; Ridge: alpha = 0.

offset

Optional numeric vector. Adds known linear predictor component.

weights

Optional weights for observations. Defaults to 1.

unpenalized

Logical vector indicating variables to exclude from penalization.

control

List of control parameters. See is.control.

trace

Logical. If TRUE, prints progress of optimization. Default is TRUE.

Details

Instead of using cross-validation, this function selects the best lambda by minimizing criteria like AIC or BIC. Degrees of freedom are computed as the trace of the hat matrix (not necessarily an integer).

Value

Optimal lambda value as numeric.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

set.seed(1)
n <- 100; p <- 100
beta <- c(rep(2, 20), rep(0, p - 20))
sim1 <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
o <- islasso(y ~ ., data = sim1$data, family = gaussian())

## 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(0.1, 30), method = "BIC")

# Overwrites the evaluation interval for lambda using eBIC criterion
lambda_ebic <- aic.islasso(o, interval = c(0.1, 30), method = "BIC", g = 0.5)

## End(Not run)

General Linear Hypotheses for islasso Models

Description

Tests general linear hypotheses and computes confidence intervals for linear combinations of coefficients from a fitted islasso model.

Usage

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

Arguments

object

A fitted model object of class "islasso".

A

Hypothesis specification. Either:

A numeric matrix or vector with each row specifying a linear combination of coefficients,
Or a character vector with symbolic expressions (e.g. "X1 + X2 = 3").

b

Right-hand side vector for the null hypotheses A %*% beta = b. If omitted, defaults to zeros.

ci

Optional 2-column matrix of confidence intervals for coefficients.

...

Currently unused.

Details

The method tests the null hypothesis H_0: A \beta = b, where A and b define a linear constraint on model coefficients.

Symbolic expressions support natural syntax: coefficients may be added/subtracted, constants may be multiplied (e.g. "2 * X1 + 3 * X2 = 7"). Equations with omitted = assume zero on the right-hand side. See examples for syntax flexibility.

Value

An object of class "anova.islasso" containing:

Estimate

Linear combination estimates

SE

Standard errors

Wald

Wald statistics

p-value

Associated p-values

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

n <- 100; p <- 100
beta <- c(runif(10, -2, 2), rep(0, p - 10))
sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
fit <- islasso(y ~ . -1, data = sim$data, family = gaussian())

# Test if first 5 variables sum to -7.5
anova(fit, A = c("X1 + X2 + X3 + X4 + X5 = -7.5"))

# Test multiple hypotheses
anova(fit, A = c("X1 + X2 + X3 + X4 + X5", "X6 + X7 + X8 + X9 + X10"), b = c(-7.5, 8.75))

# Full diagonal comparison to true coefficients
anova(fit, A = diag(p), b = beta)

Breast Cancer microarray experiment

Description

This data set details a microarray experiment for 52 breast cancer patients. The binary variable status indicates whether or not the patient died of breast cancer (status = 0: did not die, status = 1: died). The other variables represent amplification or deletion of specific genes.

Format

A data frame with 52 rows and multiple variables, including a binary status and gene-level measurements.

Details

Unlike gene expression studies, this experiment focuses on measuring gene amplification or deletion-the number of DNA copies for a given genomic sequence. The goal is to identify key genomic markers distinguishing aggressive from non-aggressive breast cancer.

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

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.

Examples

data(breast)
str(breast)
table(breast$status)

## Not run: 
  fit <- islasso.path(status ~ ., data = breast, family = binomial(),
                      alpha = 0, control = is.control(trace = 2L))
  temp <- GoF.islasso.path(fit)
  lambda.aic <- temp$lambda.min["AIC"]
  fit.aic <- islasso(status ~ ., data = breast, family = binomial(),
                     alpha = 0, lambda = lambda.aic)
  summary(fit.aic, pval = 0.05)

## End(Not run)

confint method for `islasso` objects

Description

Computes confidence intervals for islasso objects using a Wald-type approach.

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

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

trace

Logical. If TRUE (default), a bar shows the iterations status.

...

Additional arguments for methods.

Details

confint method for islasso objects

Author(s)

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

Examples

n <- 100; p <- 100; p1 <- 10
beta.veri <- sort(round(c(seq(0.5, 3, length.out = p1 / 2),
                          seq(-1, -2, length.out = p1 / 2)), 2))
beta <- c(beta.veri, rep(0, p - p1))
sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
o <- islasso(y ~ ., data = sim$data, family = gaussian())

ci <- confint(o, type.ci = "wald", parm = 1:11)
ci
plot(ci)

Blood and other measurements in diabetics

Description

The diabetes data frame contains 442 observations used in the Efron et al. "Least Angle Regression" paper.

Format

A data frame with 442 rows and 3 columns:

x: Matrix with 10 numeric columns (standardized)
y: Numeric response vector
x2: Matrix with 64 columns including interactions

Details

The x matrix has been standardized to have unit L2 norm and zero mean in each column. The x2 matrix extends x by adding selected interaction terms.

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.

Examples

data(diabetes)
str(diabetes)
summary(diabetes$y)

## Not run: 
  fit <- islasso(y ~ ., data = data.frame(y = diabetes$y, diabetes$x2),
                 family = gaussian())
  summary(fit, pval = 0.05)
  lambda.aic <- aic.islasso(fit, interval = c(1, 100))
  fit.aic <- update(fit, lambda = lambda.aic)
  summary(fit.aic, pval = 0.05)

## End(Not run)

Control Settings for islasso Model Fitting

Description

Auxiliary function used to configure and customize the fitting process of islasso models.

Usage

is.control(
  sigma2 = -1,
  tol = 1e-05,
  itmax = 1000,
  stand = TRUE,
  trace = 0,
  nfolds = 5,
  seed = NULL,
  adaptive = FALSE,
  g = 0.5,
  b0 = NULL,
  V0 = NULL,
  c = 0.5
)

Arguments

sigma2

Numeric. Fixed value of the dispersion parameter. If -1 (default), it is estimated from data.

tol

Numeric. Tolerance level to declare convergence. Default is 1e-5.

itmax

Integer. Maximum number of iterations. Default is 1000.

stand

Logical. If TRUE (default), standardizes covariates before fitting. Returned coefficients remain on the original scale.

trace

Integer. Controls verbosity of the iterative procedure:

0 - no printing,
1 - compact printing,
2 - detailed printing,
3 - compact printing with Fisher scoring info (only for GLM).

nfolds

Integer. Number of folds for CV if lambda is missing in islasso. Defaults to 5.

seed

Optional. Integer seed for reproducibility in cross-validation.

adaptive

Logical. If TRUE, fits an adaptive LASSO. (Experimental)

g

Numeric in [0,1]. Governs BIC selection: g = 0 is standard BIC; g = 0.5 is extended BIC.

b0

Optional. Starting values for regression coefficients. If NULL, uses glmnet estimates.

V0

Optional. Initial covariance matrix. Defaults to identity matrix if NULL.

c

Numeric. Controls the weight in the induced smoothed LASSO. Default is 0.5; use -1 to recompute at every iteration.

Value

A list of control parameters for use in islasso.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Internal Functions

Description

Internal islasso functions.

Usage

qqNorm(x, probs = seq(0.005, 0.995, length.out = 200), centre = FALSE,
  scale = FALSE, leg = TRUE, mean = 0, sd = 1, dF = FALSE, ylab = NULL,
  color = "black", ...)
modelX(n, p, rho=.5, scale.data=TRUE)
lminfl(mod, tol = 1e-8)
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, ...)

.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)

interpolate(y1, y2, x1, x2, x.new)
create_coef_plot(coef1, loglambda, label, id.best, gof, dots, active,
  unactive, legend, nlambda)
create_se_plot(se1, coef1, loglambda, label, id.best, gof, dots,
  active, unactive, legend, nlambda)
create_weight_plot(weight1, coef1, loglambda, label, id.best, gof, dots,
  active, unactive, legend, nlambda)
calculate_gradient(object, lambda, nlambda, intercept)
create_gradient_plot(grad, coef1, lambda, label, id.best, gof, dots,
  active, unactive, legend, nlambda)
create_gof_plot(object, loglambda, id.best, gof, dots)

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)

Details

These functions are not intended for users.

Author(s)

Gianluca Sottile (gianluca.sottile@unipa.it)

Induced Smoothed Lasso Regularization Path

Description

Fits a sequence of penalized regression models using the Induced Smoothing Lasso approach over a grid of lambda values. Supports elastic-net penalties and generalized linear models: Gaussian, Binomial, Poisson, and Gamma.

Usage

islasso.path(
  formula,
  family = gaussian(),
  lambda = NULL,
  nlambda = 100,
  lambda.min.ratio = ifelse(nobs < nvars, 0.01, 1e-04),
  alpha = 1,
  data,
  weights,
  subset,
  offset,
  contrasts = NULL,
  unpenalized,
  control = is.control()
)

Arguments

formula

Model formula of type response ~ predictors.

family

Response distribution. Supported families: gaussian(), binomial(), poisson(), Gamma().

lambda

Optional numeric vector of lambda values. If not provided, a sequence is automatically generated.

nlambda

Integer. Number of lambda values to generate if lambda is missing. Default is 100.

lambda.min.ratio

Smallest lambda as a fraction of lambda.max. Default: 1e-2 if nobs < nvars, else 1e-3.

alpha

Elastic-net mixing parameter: alpha = 1 is lasso, alpha = 0 is ridge.

data

Data frame containing model variables.

weights

Optional observation weights.

subset

Optional logical or numeric vector to subset observations.

offset

Optional vector of prior known components for the linear predictor.

contrasts

Optional contrast settings for factor variables.

unpenalized

Optional vector of variable names or indices excluded from penalization.

control

A list of control parameters via is.control.

Details

This function fits a regularization path of models using the induced smoothing paradigm, replacing the non-smooth L1 penalty with a differentiable surrogate. Standard errors are returned for all lambda points, allowing for Wald-based hypothesis testing. The regularization path spans a range of lambda values, either user-defined or automatically computed.

Value

A list with components:

call

Matched function call.

Info

Matrix with diagnostics: lambda, deviance, degrees of freedom, dispersion, iterations, convergence status.

GoF

Model goodness-of-fit metrics: AIC, BIC, AICc, GCV, GIC, eBIC.

Coef

Matrix of coefficients across lambda values.

SE

Matrix of standard errors.

Weights

Matrix of mixing weights for the smoothed penalty.

Gradient

Matrix of gradients for the smoothed penalty.

Linear.predictors, Fitted.values, Residuals

Matrices of fitted quantities across the path.

Input

List of input arguments and design matrix.

control, formula, model, terms, data, xlevels, contrasts

Standard model components.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

References

Cilluffo G., Sottile G., La Grutta S., Muggeo V.M.R. (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 V.M.R. (2019). The R package islasso: estimation and hypothesis testing in lasso regression. Technical Report. DOI: 10.13140/RG.2.2.16360.11521

Examples

n <- 100; p <- 30; p1 <- 10  # number of nonzero coefficients

beta.veri <- sort(round(c(seq(.5, 3, length.out = p1/2),
                         seq(-1, -2, length.out = p1/2)), 2))
beta <- c(beta.veri, rep(0, p - p1))
sim1 <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
o <- islasso.path(y ~ ., data = sim1$data,
                  family = gaussian(), nlambda = 30L)
o

summary(o, lambda = 10, pval = 0.05)
coef(o, lambda = 10)
fitted(o, lambda = 10)
predict(o, type = "response", lambda = 10)
plot(o, yvar = "coef")
residuals(o, lambda = 10)
deviance(o, lambda = 10)
logLik(o, lambda = 10)
GoF.islasso.path(o)

## Not run: 
##### binomial ######
beta <- c(1, 1, 1, rep(0, p - 3))
sim2 <- simulXy(n = n, p = p, beta = beta, interc = 1, seed = 1,
                size = 100, family = binomial())
o2 <- islasso.path(cbind(y.success, y.failure) ~ ., data = sim2$data,
                   family = binomial(), lambda = seq(0.1, 100, l = 50L))
temp <- GoF.islasso.path(o2)
summary(o2, pval = 0.05, lambda = temp$lambda.min["BIC"])

##### poisson ######
beta <- c(1, 1, 1, rep(0, p - 3))
sim3 <- simulXy(n = n, p = p, beta = beta, interc = 1, seed = 1,
                family = poisson())
o3 <- islasso.path(y ~ ., data = sim3$data, family = poisson(), nlambda = 30L)
temp <- GoF.islasso.path(o3)
summary(o3, pval = 0.05, lambda = temp$lambda.min["BIC"])

##### Gamma ######
beta <- c(1, 1, 1, rep(0, p - 3))
sim4 <- simulXy(n = n, p = p, beta = beta, interc = -1, seed = 1,
                family = Gamma(link = "log"))
o4 <- islasso.path(y ~ ., data = sim4$data, family = Gamma(link = "log"),
                   nlambda = 30L)
temp <- GoF.islasso.path(o4)
summary(o4, pval = .05, lambda = temp$lambda.min["BIC"])

## End(Not run)

Diagnostic Plots for islasso Models

Description

Produces standard diagnostic plots for a fitted islasso model to assess residuals, model fit, and variance structure.

Usage

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

Arguments

x

An object of class "islasso", typically created via islasso.

...

Additional graphical parameters passed to the underlying plot() functions.

Details

Generates a 2x2 grid of diagnostic plots:

Top-left: Deviance residuals vs fitted values.
Top-right: Normal Q-Q plot of standardized deviance residuals (red line = reference).
Bottom-left: Squared standardized Pearson residuals vs fitted values.
Bottom-right: Working response vector vs linear predictor.

These plots help assess the assumptions of linearity, homoscedasticity, and residual normality in penalized regression.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

## Not run: 
  set.seed(1)
  n <- 100; p <- 100
  beta <- c(runif(20, -3, 3), rep(0, p - 20))
  sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
  fit <- islasso(y ~ ., data = sim$data, family = gaussian(), lambda = 2)
  plot(fit)

## End(Not run)

Coefficient Profile and Diagnostic Plots for islasso.path

Description

Generates plots of coefficient profiles, standard errors, gradients, weights, or goodness-of-fit criteria from a fitted islasso.path model.

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.path", typically created via islasso.path.

yvar

Character. Specifies what to display on the y-axis. Choices are:

"coefficients" - coefficient paths over log(lambda),
"se" - standard errors over log(lambda),
"gradient" - gradient values over log(lambda),
"weight" - mixture weights used in smoothing,
"gof" - goodness-of-fit values.

gof

Character. Criterion used for highlighting active variables. Choices: "none", "AIC", "BIC", "AICc", "eBIC", "GCV", "GIC".

label

Logical. Whether to annotate curves with variable names.

legend

Logical. Whether to display a plot legend.

...

Additional graphical parameters, e.g. main, xlab, ylab, xlim, ylim, lty, col, lwd, cex.axis, cex.lab, cex.main, gof_lty, gof_col, gof_lwd.

Details

This function visualizes the behavior of the solution path across a sequence of lambda values, helping diagnose coefficient shrinkage, influence of penalty, and variable selection stability.

Value

Produces plots. Does not return an object.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

## Not run: 
  n <- 100; p <- 30
  beta <- c(runif(10, -2, 2), rep(0, p - 10))
  sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
  fit <- islasso.path(y ~ ., data = sim$data, family = gaussian())

  plot(fit, yvar = "coefficients", gof = "AICc", label = TRUE)
  plot(fit, yvar = "se", gof = "AICc")
  plot(fit, yvar = "gradient", gof = "AICc")
  plot(fit, yvar = "gof", gof = "AICc")

## End(Not run)

Prediction Method for islasso Objects

Description

Computes predictions from a fitted islasso model object. Multiple output types supported, including response scale, linear predictor, and coefficient values.

Usage

## S3 method for class 'islasso'
predict(
  object,
  newdata = NULL,
  type = c("link", "response", "coefficients", "class", "terms"),
  se.fit = FALSE,
  ci = NULL,
  type.ci = c("wald", "score"),
  level = 0.95,
  terms = NULL,
  na.action = na.pass,
  ...
)

Arguments

object

A fitted model of class "islasso".

newdata

Optional data frame containing predictors for prediction. If omitted, the fitted model matrix is used.

type

Character. Specifies the prediction scale:

"link" (default): linear predictor scale;
"response": original response scale;
"coefficients": estimated coefficients;
"class": predicted class (only for binomial() family);
"terms": contribution of each term to the linear predictor.

se.fit

Logical. Whether to compute standard errors/confidence intervals.

ci

Optional. Precomputed matrix of confidence intervals (2 columns).

type.ci

Type of interval. Only "wald" is implemented.

level

Confidence level for intervals. Default is 0.95.

terms

If type = "terms", optionally specify which terms to extract.

na.action

Function to handle missing values in newdata. Default: na.pass.

...

Additional arguments passed to downstream methods.

Value

A numeric vector, matrix, or list depending on type.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

set.seed(1)
n <- 100; p <- 100
beta <- c(runif(20, -3, 3), rep(0, p - 20))
sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
fit <- islasso(y ~ ., data = sim$data, family = gaussian(), lambda = 2)
predict(fit, type = "response")

Prediction Method for islasso.path Objects

Description

Generates predictions from a fitted islasso.path model at one or more lambda values. Supports various output types including linear predictors, response scale, class labels, and coefficients.

Usage

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

Arguments

object

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

newdata

Optional data frame containing covariates for prediction. If omitted, returns fitted values from the original model.

type

Character. Type of prediction:

"link" (default) - linear predictor scale,
"response" - original response scale,
"coefficients" - estimated coefficients,
"class" - predicted class labels (only for binomial models).

lambda

Numeric value(s). Specific lambda value(s) at which predictions are required. If missing, predictions are computed for the full lambda sequence.

...

Additional arguments passed to lower-level methods.

Value

A vector, matrix, or list depending on the type requested.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

## Not run: 
  set.seed(1)
  n <- 100; p <- 30
  beta <- c(runif(10, -3, 3), rep(0, p - 10))
  sim <- simulXy(n = n, p = p, beta = beta, seed = 1, family = gaussian())
  fit <- islasso.path(y ~ ., data = sim$data, family = gaussian())
  optimal <- GoF.islasso.path(fit)
  pred <- predict(fit, type = "response", lambda = optimal$lambda.min)

## End(Not run)

Simulate Model Matrix and Response Vector

Description

Generates synthetic covariates and response vector from a specified distribution for simulation studies or method validation.

Usage

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

Arguments

n

Integer. Number of observations.

p

Integer. Total number of covariates in the model matrix.

interc

Numeric. Intercept to include in the linear predictor. Default is 0.

beta

Numeric vector of length p. Regression coefficients in the linear predictor.

family

Distribution and link function. Allowed: gaussian(), binomial(), poisson() and , Gamma(). Can be a string, function, or family object.

prop

Numeric in [0,1]. Used only if beta is missing; proportion of non-zero coefficients in p. Default is 0.1.

lim.b

Numeric vector of length 2. Range for coefficients if beta is missing. Default: c(-3, 3).

sigma

Standard deviation of Gaussian response. Default is 1.

size

Integer. Number of trials for binomial response. Default is 1.

rho

Numeric. Correlation coefficient for generating covariates. Used to create AR(1)-type covariance: rho^|i-j|. Default is 0.

scale.data

Logical. Whether to scale columns of the model matrix. Default is TRUE.

seed

Optional. Integer seed for reproducibility.

X

Optional. Custom model matrix. If supplied, it overrides the internally generated X.

dispersion

Dispersion parameter of Gamma response. Default is 0.1.

Value

A list with components:

X

Model matrix of dimension n x p

y

Simulated response vector

beta

True regression coefficients used

eta

Linear predictor

Examples

n <- 100; p <- 100
beta <- c(runif(10, -3, 3), rep(0, p - 10))
sim <- simulXy(n = n, p = p, beta = beta, seed = 1234)
o <- islasso(y ~ ., data = sim$data, family = gaussian())
summary(o, pval = 0.05)

Summarize islasso Fitted Model

Description

Provides a concise summary of a fitted islasso model, including p-values and optional filtering.

Usage

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

Arguments

object

A fitted model of class "islasso".

pval

Numeric threshold for displaying coefficients. Only those with p \le pval are printed. Unpenalized coefficients (like intercepts) are always shown.

which

Optional. Specifies a subset of coefficients to test. If missing, all parameters are evaluated.

use.t

Logical. If TRUE, p-values are computed using the t-distribution and residual degrees of freedom.

type.pval

Character. Type of p-value approximation. Only "wald" (default) is implemented.

...

Additional arguments (not currently used).

Value

An object of class "summary.islasso" containing:

coefficients

Coefficient estimates and related statistics

pval

Threshold used to filter coefficients

call

Original model call

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

## Not run: 
# Assuming object `o` from an islasso fit
summary(o, pval = 0.1)  # Show coefficients with p <= 0.1

## End(Not run)

Summarize islasso.path Model at Specific Lambda

Description

Extracts coefficient estimates, standard errors and p-values from an islasso.path fit at a given regularization level lambda.

Usage

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

Arguments

object

A fitted object of class "islasso.path".

pval

Numeric threshold for displaying coefficients. Only variables with p-value <= pval are printed. Unpenalized coefficients (like the intercept) are always shown.

use.t

Logical. If TRUE, p-values are computed using a t-distribution with residual degrees of freedom.

lambda

Numeric. Value of the regularization parameter at which the summary should be extracted.

...

Currently unused.

Value

An object of class "summary.islasso.path" containing filtered estimates and significance metrics.

Author(s)

Gianluca Sottile gianluca.sottile@unipa.it

Examples

## Not run: 
# Assuming object `o` is from islasso.path
summary(o, pval = 0.1, lambda = 5)

## End(Not run)

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

Description

Usage

Arguments

Details

Value

Author(s)

References

See Also

Examples

Select Optimal Lambda via Goodness-of-Fit Criteria

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Prostate Cancer Data

Description

Format

Source

References

Examples

Optimization for Lambda Selection

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

General Linear Hypotheses for islasso Models

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Breast Cancer microarray experiment

Description

Format

Details

Source

References

Examples

confint method for islasso objects

Description

Usage

Arguments

Details

Author(s)

See Also

Examples

Blood and other measurements in diabetics

Description

Format

Details

Source

References

Examples

Control Settings for islasso Model Fitting

Description

Usage

Arguments

Value

Author(s)

See Also

Internal Functions

Description

Usage

Details

Author(s)

Induced Smoothed Lasso Regularization Path

Description

Usage

confint method for `islasso` objects