Author: Gianluca Sottile
Date: 2025-11-15

# An Introduction to islasso

#### Gianluca Sottile

#### Giovanna Cilluffo

#### Vito M.R. Muggeo

#### November 15, 2025

# Abstract

In this short note we present and briefly discuss the R package islasso dealing with regression models having a large number of covariates. Estimation is carried out by penalizing the coefficients via a quasi-lasso penalty, wherein the nonsmooth lasso penalty is replaced by its smooth counterpart determined iteratively by data according to the induced smoothing idea. The package includes functions to estimate the model and to test for linear hypothesis on linear combinations of relevant coefficients. We illustrate R code throughout a worked example, by avoiding intentionally to report details and extended bibliography.

# Introduction

Let \(\mathbf{y} = \mathbf{X}\beta + \mathbf{\epsilon}\) be the linear model of interest with usual zero-means and homoscedastic errors. As usual, \(\mathbf{y} = (y_1,\ldots,y_n)^T\) is the response vector, \(\mathbf{X}\) is the \(n \times p\) design matrix (having \(p\) quite large) with regression coefficients \(\mathbf{\beta}\). When interest lies in selecting the non-noise covariates and estimating the relevant effect, one assumes the lasso penalized objective function (Tibshirani, 1996), \[\frac{1}{2}||\mathbf{y}-\mathbf{X}\mathbf{\beta}||_2^2+\lambda||\mathbf{\beta}||_1\]

# The R functions

The main function of the package are islasso() where the user supplies the model formula as in the usual lm or glm functions, i.e.

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

and islasso.path used to fit the regularization path via the induced smoothed lasso framework, i.e.

    islasso.path(formula, family = gaussian, lambda = NULL, nlambda = 100,
                 lambda.min.ratio = ifelse(nobs < nvars, 1E-3, 1E-05), alpha = 1,
                 data, weights, subset, offset, unpenalized, contrasts = NULL, control = is.control())

family accepts specification of family and link function as in Table 1, lambda is the tuning parameter, alpha is elastic-net mixing parameter, nlambda is the number of lambda values, lambda.min.ratio is the smallest value for lambda (as a fraction of lambda.max), and unpenalized allows to indicate covariates with unpenalized coefficients.

Table 1. Families and link functions allowed in islasso

family | link
--- | ---
gaussian | identity
binomial | logit, probit
poisson | log
gamma | identity, log, inverse

The fitter functions are and which reads as

    islasso.fit(X, y, family = gaussian(), lambda, alpha = 1, intercept = FALSE,
                weights = NULL, offset = NULL, unpenalized = NULL, control = is.control())

and

    islasso.path.fit(X, y, family = gaussian(), lambda, nlambda, lambda.min.ratio,
                     alpha = 1, intercept = FALSE, weights = NULL, offset = NULL,
                     unpenalized = NULL, control = is.control())

whose actually implements the estimating algorithm as described in the paper. The lambda argument in islasso.fit and islasso specifies the positive tuning parameter in the penalized objective. Any non-negative value can be provided, but if missing, it is computed via \(K\)-fold cross validation by the function cv.glmnet() from package glmnet. The number of folds being used can be specified via the argument nfolds of the auxiliary function is.control(). The lambda argument in islasso.path.fit and islasso.path specifies the sequence of positive tuning parameters, user supplied or automatically computed based on nlambda and lambda.min.ratio.

# A worked example: the Prostate data set

The `Prostate` dataset is a well-known benchmark in regression analysis and variable selection, originally published by Stamey et al. (1989). It contains clinical measurements from 97 male patients diagnosed with prostate cancer.

The primary goal is to model the level of prostate-specific antigen (PSA) based on several predictive variables, including:

- `lcavol`: log cancer volume
- `lweight`: log prostate weight
- `age`: patient’s age
- `lbph`: log benign prostatic hyperplasia amount
- `svi`: seminal vesicle invasion
- `lcp`: log capsular penetration
- `gleason`: Gleason score (histological grade of tumor)
- `pgg45`: percent of Gleason scores 4 or 5

The response variable is `lpsa`, the logarithm of PSA levels.

This dataset is particularly useful for testing due to its moderate size and the presence of multicollinearity among predictors.

To select the important terms in the regression equation we could simply apply the lasso using the R package glmnet

    library(islasso)
    #> Loading required package: glmnet
    #> Loading required package: Matrix
    #> Loaded glmnet 4.1-10
    #>
    #> +----------------------------------------------+
    #> |          Welcome to *islasso*                |
    #> |  The Induced Smoothed Lasso for R            |
    #> |  Hypothesis testing in high-dimensional data |
    #> +----------------------------------------------+

    data("Prostate", package = "islasso")
    x <- model.matrix(lpsa ~ ., data = Prostate)[, -1]
    y <- Prostate$lpsa
    a1 <- cv.glmnet(x, y, family = "gaussian")
    n <- nrow(Prostate)
    a1$lambda.min * n
    #> [1] 3.797398

    b <- drop(coef(a1, "lambda.min", exact = TRUE))
    length(b[b != 0])
    #> [1] 7

Ten-fold cross validation “selects” \(\lambda=\) 3.797. corresponding to 7 non null coefficients

    names(b[b != 0])
    #> [1] "(Intercept)" "lcavol"      "lweight"     "age"         "lbph"
    #> [6] "svi"         "pgg45"

The last three estimates are

    tail(b[b != 0], n = 3)
    #>        lbph         svi       pgg45
    #> 0.069318034 0.592326060 0.002184294

A reasonable question is if all the “selected” coefficients are significant in the model. Unfortunately lasso regression does not return standard errors due to nonsmoothness of objective, and some alternative approaches have been proposed., including the (Lockhart et al., 2013). Among the (few) strategies, including the ‘covariance test’, the ‘post-selection inference’ and the ‘(modified) residual bootstrap’, here we illustrate the R package islasso implementing the recent `quasi’ lasso approach based on the induced smoothing idea (Brown and Wang, 2005) as discussed in Cilluffo et al. (2019)

While the optimal lambda could be selected (without supplying any value to lambda), we use optimal value minimizing a specific criterion chosen between AIC, BIC, AICc, eBIC, GCV or GIC. From version 1.4.0 of the R package islasso optimal strategy is to built the regularization path

    out <- islasso.path(lpsa ~ ., data = Prostate, nlambda = 50L, family = gaussian())
    out
    #>
    #> Call:
    #> islasso.path(formula = lpsa ~ ., family = gaussian(), nlambda = 50L,
    #>     data = Prostate)
    #>
    #> Coefficients:
    #>       lambda     df    phi deviance    logLik
    #> 1     0.1333 8.9761 0.5018  44.1676 -100.4811
    #> 2     0.1608 8.9710 0.5018  44.1696 -100.4833
    #> 3     0.1941 8.9649 0.5018  44.1725 -100.4865
    #> 4     0.2342 8.9573 0.5018  44.1767 -100.4912
    #> 5     0.2825 8.9481 0.5018  44.1829 -100.4979
    #> 6     0.3409 8.9368 0.5018  44.1918 -100.5077
    #> 7     0.4113 8.9228 0.5019  44.2046 -100.5218
    #> 8     0.4962 8.9056 0.5020  44.2232 -100.5421
    #> 9     0.5987 8.8842 0.5022  44.2500 -100.5715
    #> 10    0.7224 8.8576 0.5025  44.2885 -100.6137
    #> 11    0.8716 8.8244 0.5029  44.3437 -100.6741
    #> 12    1.0516 8.7826 0.5036  44.4225 -100.7602
    #> 13    1.2688 8.7299 0.5045  44.5340 -100.8819
    #> 14    1.5309 8.0067 0.5021  44.6791 -101.0396
    #> 15    1.8471 7.9204 0.5038  44.8820 -101.2594
    #> 16    2.2286 7.8248 0.5065  45.1655 -101.5647
    #> 17    2.6889 7.7136 0.5102  45.5561 -101.9823
    #> 18    3.2443 6.9726 0.5114  46.0426 -102.4975
    #> 19    3.9145 6.7585 0.5158  46.5469 -103.0259
    #> 20    4.7230 6.6541 0.5230  47.2510 -103.7541
    #> 21    5.6985 6.5271 0.5330  48.2219 -104.7405
    #> 22    6.8755 6.6267 0.5486  49.5782 -106.0858
    #> 23    8.2956 6.1845 0.5656  51.3675 -107.8053
    #> 24   10.0091 5.7038 0.5838  53.2999 -109.5964
    #> 25   12.0764 5.2903 0.5997  54.9973 -111.1169
    #> 26   14.5708 4.9208 0.6146  56.5942 -112.5050
    #> 27   17.5803 4.6441 0.6288  58.0776 -113.7599
    #> 28   21.2115 3.7489 0.6423  59.8984 -115.2571
    #> 29   25.5927 3.4034 0.6649  62.2319 -117.1106
    #> 30   30.8788 3.3301 0.6981  65.3864 -119.5088
    #> 31   37.2567 3.2587 0.7418  69.5331 -122.4910
    #> 32   44.9520 3.6938 0.8058  75.1879 -126.2831
    #> 33   54.2367 3.5160 0.8870  82.9182 -131.0295
    #> 34   65.4392 3.1442 0.9892  92.8465 -136.5145
    #> 35   78.9555 2.6256 1.1001 103.8176 -141.9314
    #> 36   95.2636 2.4779 1.1127 105.1725 -142.5603
    #> 37  114.9401 2.3613 1.1124 105.2762 -142.6080
    #> 38  138.6807 2.1349 1.1106 105.3552 -142.6444
    #> 39  167.3248 2.0283 1.1105 105.4700 -142.6972
    #> 40  201.8854 2.0023 1.1120 105.6370 -142.7740
    #> 41  243.5843 1.9998 1.1145 105.8802 -142.8855
    #> 42  293.8961 1.9993 1.1182 106.2340 -143.0473
    #> 43  354.5996 1.9981 1.1236 106.7486 -143.2817
    #> 44  427.8413 2.0038 1.1316 107.4951 -143.6196
    #> 45  516.2109 1.9858 1.1427 108.5750 -144.1044
    #> 46  622.8330 1.9603 1.1586 110.1161 -144.7880
    #> 47  751.4777 1.9013 1.1806 112.2704 -145.7277
    #> 48  906.6936 1.7895 1.2099 115.1909 -146.9732
    #> 49 1093.9690 1.5932 1.2478 119.0497 -148.5713
    #> 50 1319.9257 1.2931 1.2953 123.9704 -150.5356

and then to choose the best tuning parameter through the one of the criteria listed above using the function GoF.islasso.path, e.g.,

    lmb.best <- GoF.islasso.path(out)

    lmb.best$lambda.min
    #>       AIC       BIC      AICc      eBIC       GCV       GIC
    #>  1.530907  3.244343 25.592682  3.914454  3.244343  3.244343

Using also the regularization path is very usefull to have more insights about coefficients, standard errors and gradient profile

    p1 <- plot(out, yvar = "coefficients")
    p2 <- plot(out, yvar = "se")
    p3 <- plot(out, yvar = "gradient")
    gridExtra::grid.arrange(p1, p2, p3, ncol = 1L)

Regularization path for coefficients, standard errors and gradients.

Once selected the best lambda value minimizing for example the AIC criterion, the last step of the strategy consists on fitting a new islasso model.

    lambda.aic <- lmb.best$lambda.min["AIC"]
    out2 <- islasso(lpsa ~ ., data = Prostate, lambda = lambda.aic, family = gaussian())
    out2
    #>
    #> Call:
    #> islasso(formula = lpsa ~ ., family = gaussian(), lambda = lambda.aic,
    #>     data = Prostate)
    #>
    #> Coefficients:
    #> (Intercept)       lcavol      lweight          age         lbph          svi
    #>    1.074451     0.573820     0.389617    -0.016542     0.100958     0.572844
    #>         lcp      gleason        pgg45
    #>   -0.046512     0.000001     0.004925
    #>
    #> Degrees of Freedom: 96 Total (i.e. Null); 88.99 Residual
    #> Null Deviance: 127.9
    #> Residual Deviance: 44.68
    #> AIC: 218.1
    #> Lambda: 1.531

The summary method quickly returns the main output of the fitted model, including point estimates, standard errors and \(p\)-values. Visualizing estimates for all covariates could be somewhat inconvenient, especially when the number of covariates is large, thus we decide to print estimates only if the pvalue is less than a threshold value. We use 0.10

    summary(out2, pval = 0.10)
    #>
    #> Call:
    #> islasso(formula = lpsa ~ ., family = gaussian(), lambda = lambda.aic,
    #>     data = Prostate)
    #>
    #> Residuals:
    #>      Min       1Q   Median       3Q      Max
    #> -1.65308 -0.37468 -0.01048  0.37197  1.64352
    #>
    #>             Estimate Std. Error    Df z value Pr(>|z|)
    #> (Intercept)  1.07445    0.83950 1.000   1.280   0.2006
    #> lcavol       0.57382    0.08430 1.000   6.807 9.96e-12 ***
    #> lweight      0.38962    0.16573 0.987   2.351   0.0187 *
    #> lbph         0.10096    0.05725 0.985   1.763   0.0778 .
    #> svi          0.57284    0.23416 0.985   2.446   0.0144 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for gaussian family taken to be 0.502056)
    #>
    #>     Null deviance: 127.92  on 96.00  degrees of freedom
    #> Residual deviance:  44.68  on 88.99  degrees of freedom
    #> AIC: 218.09
    #> Lambda: 1.5309
    #>
    #> Number of Newton-Raphson iterations: 10

In addition to the usual information printed by the summary method, the output also includes the column Df representing the degrees of freedom of each coefficient. Their sum is used to quantify the model complexity

    sum(out2$internal$hi)
    #> [1] 8.006674

and the corresponding residual degrees of freedom () as reported above. The Wald test (column z value) and \(p\)-values can be used to assess important or significant covariates. Results suggest that variables `lcavol`, `lweight` and `svi` are informative to predict the measure of the logarithm of PSA levels, while `lbph` is borderline informative. Just to be clear, another way to obtain a similar result without computing the regularization path, is to use the function aic.islasso which requires a preliminary islasso fit object and a specification of the criterion to be used. Hence

    lambda.aic2 <- aic.islasso(out2, method = "AIC", interval = c(.1, 50))
    #>
    #> Optimization through AIC
    #>
    #> lambda =   19.1601 AIC =   237.91885
    #> lambda =   30.9399 AIC =   245.77014
    #> lambda =   11.8798 AIC =   232.62143
    #> lambda =    7.3803 AIC =   226.34327
    #> lambda =    4.5995 AIC =   220.59405
    #> lambda =    2.8808 AIC =   219.66168
    #> lambda =    2.9398 AIC =   219.75098
    #> lambda =    1.8186 AIC =   218.32970
    #> lambda =    1.1622 AIC =   219.15069
    #> lambda =    1.9195 AIC =   218.43005
    #> lambda =    1.7013 AIC =   218.22646
    #> lambda =    1.4954 AIC =   218.06882
    #> lambda =    1.3681 AIC =   217.98620
    #> lambda =    1.2894 AIC =   219.23594
    #> lambda =    1.4167 AIC =   218.01779
    #> lambda =    1.3381 AIC =   217.96882
    #> lambda =    1.3195 AIC =   219.25523
    #> lambda =    1.3495 AIC =   217.96997
    #> lambda =    1.3418 AIC =   217.97021
    #> lambda =    1.3310 AIC =   219.26258
    #> lambda =    1.3354 AIC =   219.26402
    #> lambda =    1.3395 AIC =   217.96808
    #> lambda =    1.3404 AIC =   217.96696
    #> lambda =    1.3410 AIC =   217.97116
    #> lambda =    1.3401 AIC =   217.96616
    #> lambda =    1.3400 AIC =   217.96475
    #> lambda =    1.3398 AIC =   217.96822
    #> lambda =    1.3399 AIC =   217.97010
    #> lambda =    1.3400 AIC =   217.96475
    out3 <- update(out2, lambda = lambda.aic2)
    summary(out3, pval = .10)
    #>
    #> Call:
    #> islasso(formula = lpsa ~ ., family = gaussian(), lambda = lambda.aic2,
    #>     data = Prostate)
    #>
    #> Residuals:
    #>      Min       1Q   Median       3Q      Max
    #> -1.66294 -0.37527 -0.01031  0.38855  1.64233
    #>
    #>             Estimate Std. Error    Df z value Pr(>|z|)
    #> (Intercept)  1.05960    0.84619 1.000   1.252   0.2105
    #> lcavol       0.57584    0.08498 1.000   6.776 1.23e-11 ***
    #> lweight      0.39689    0.16643 0.990   2.385   0.0171 *
    #> lbph         0.10178    0.05749 0.987   1.771   0.0766 .
    #> svi          0.59518    0.23648 0.989   2.517   0.0118 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for gaussian family taken to be 0.5011429)
    #>
    #>     Null deviance: 127.918  on 96.00  degrees of freedom
    #> Residual deviance:  44.572  on 88.94  degrees of freedom
    #> AIC: 217.96
    #> Lambda: 1.34
    #>
    #> Number of Newton-Raphson iterations: 10

Comparisons between methods to select the tuning parameter and further discussions are out of the scope of this short note. We conclude this note by emphasizing that islasso also accepts the so-called elastic-net penalty, such that \[ \frac{1}{2}||\mathbf{y}- \mathbf{X\beta}||_2^{2}+\lambda \{ \alpha ||\mathbf{\beta} ||^{}_1 + \frac{1}{2}(1-\alpha) ||\mathbf{\beta} ||^{2}_2 \} \] where \(0\le \alpha\le 1\) is the mixing parameter to be specified in islasso() and islasso.path() via the argument alpha, e.g.

    # update the islasso path to fit an elastic-net model
    out4 <- update(out, alpha = .5)
    out4
    #>
    #> Call:
    #> islasso.path(formula = lpsa ~ ., family = gaussian(), nlambda = 50L,
    #>     alpha = 0.5, data = Prostate)
    #>
    #> Coefficients:
    #>       lambda     df    phi deviance    logLik
    #> 1     0.2666 8.9540 0.5017  44.1702 -100.4840
    #> 2     0.3217 8.9445 0.5017  44.1733 -100.4874
    #> 3     0.3881 8.9328 0.5016  44.1779 -100.4925
    #> 4     0.4683 8.9188 0.5016  44.1845 -100.4997
    #> 5     0.5650 8.9017 0.5016  44.1941 -100.5102
    #> 6     0.6817 8.8810 0.5017  44.2078 -100.5253
    #> 7     0.8226 8.8558 0.5018  44.2275 -100.5469
    #> 8     0.9925 8.8251 0.5019  44.2557 -100.5778
    #> 9     1.1974 8.7878 0.5022  44.2959 -100.6218
    #> 10    1.4448 8.7423 0.5025  44.3529 -100.6841
    #> 11    1.7432 8.6868 0.5031  44.4332 -100.7720
    #> 12    2.1032 8.6191 0.5040  44.5458 -100.8947
    #> 13    2.5377 8.5365 0.5053  44.7017 -101.0641
    #> 14    3.0618 7.8238 0.5036  44.9049 -101.2841
    #> 15    3.6942 7.7078 0.5060  45.1809 -101.5813
    #> 16    4.4573 7.5819 0.5094  45.5527 -101.9787
    #> 17    5.3779 6.8248 0.5098  45.9746 -102.4259
    #> 18    6.4887 6.6362 0.5138  46.4262 -102.8999
    #> 19    7.8289 6.6102 0.5205  47.0458 -103.5429
    #> 20    9.4459 6.8886 0.5319  47.9332 -104.4493
    #> 21   11.3970 6.6206 0.5438  49.1459 -105.6610
    #> 22   13.7510 6.3072 0.5588  50.6764 -107.1484
    #> 23   16.5912 5.9402 0.5774  52.5775 -108.9345
    #> 24   20.0181 5.5303 0.5989  54.7831 -110.9276
    #> 25   24.1528 5.1570 0.6207  57.0024 -112.8535
    #> 26   29.1415 4.8164 0.6435  59.3193 -114.7859
    #> 27   35.1606 3.8405 0.6614  61.6176 -116.6295
    #> 28   42.4230 3.5911 0.6886  64.3186 -118.7102
    #> 29   51.1854 3.4481 0.7246  67.7886 -121.2586
    #> 30   61.7576 3.6817 0.7731  72.1477 -124.2812
    #> 31   74.5135 3.4501 0.8260  77.2703 -127.6081
    #> 32   89.9040 3.3208 0.8884  83.2288 -131.2108
    #> 33  108.4735 3.1424 0.9596  90.0654 -135.0396
    #> 34  130.8784 2.8654 1.0348  97.4128 -138.8430
    #> 35  157.9111 2.5229 1.1029 104.2028 -142.1110
    #> 36  190.5272 2.3829 1.1127 105.2810 -142.6102
    #> 37  229.8802 2.2673 1.1132 105.4538 -142.6898
    #> 38  277.3614 2.0977 1.1128 105.6061 -142.7598
    #> 39  334.6497 1.9705 1.1136 105.8208 -142.8583
    #> 40  403.7708 1.9332 1.1163 106.1233 -142.9967
    #> 41  487.1687 1.9177 1.1206 106.5466 -143.1898
    #> 42  587.7922 1.9018 1.1266 107.1346 -143.4567
    #> 43  709.1992 1.8830 1.1349 107.9438 -143.8217
    #> 44  855.6826 1.8684 1.1462 109.0445 -144.3137
    #> 45 1032.4218 1.8282 1.1613 110.5205 -144.9658
    #> 46 1245.6661 1.7808 1.1810 112.4538 -145.8069
    #> 47 1502.9554 1.7051 1.2058 114.9033 -146.8519
    #> 48 1813.3873 1.5916 1.2355 117.8800 -148.0924
    #> 49 2187.9381 1.4229 1.2698 121.3653 -149.5056
    #> 50 2639.8515 1.1963 1.3073 125.2452 -151.0318

    # some diagnostic plot
    p4 <- plot(out4, yvar = "coefficients")
    p5 <- plot(out4, yvar = "se")
    gridExtra::grid.arrange(p4, p5, ncol = 1L)

    # select the best tuning parameter
    lmb.best2 <- GoF.islasso.path(out4)

    lmb.best2$lambda.min
    #>       AIC       BIC      AICc      eBIC       GCV       GIC
    #>  3.061814  5.377894 42.422996  6.488685  5.377894  5.377894

    # fit a new islasso model with elastic-net penalty
    lambda.aic3 <- lmb.best2$lambda.min["AIC"]
    out5 <- update(out2, alpha = .5, lambda = lambda.aic3)
    summary(out5, pval = .10)
    #>
    #> Call:
    #> islasso(formula = lpsa ~ ., family = gaussian(), lambda = lambda.aic3,
    #>     alpha = 0.5, data = Prostate)
    #>
    #> Residuals:
    #>      Min       1Q   Median       3Q      Max
    #> -1.63555 -0.36932  0.00011  0.39077  1.63160
    #>
    #>             Estimate Std. Error    Df z value Pr(>|z|)
    #> (Intercept)  1.09123    0.82617 1.000   1.321   0.1866
    #> lcavol       0.56168    0.08145 0.975   6.896 5.35e-12 ***
    #> lweight      0.38098    0.15899 0.948   2.396   0.0166 *
    #> lbph         0.09957    0.05610 0.971   1.775   0.0759 .
    #> svi          0.53984    0.21749 0.920   2.482   0.0131 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for gaussian family taken to be 0.503556)
    #>
    #>     Null deviance: 127.918  on 96.00  degrees of freedom
    #> Residual deviance:  44.905  on 89.18  degrees of freedom
    #> AIC: 218.22
    #> Lambda: 3.0618
    #>
    #> Number of Newton-Raphson iterations: 10

    # or select the best tuning parameter using AIC with an islasso object
    lambda.aic4 <- aic.islasso(out5, method = "AIC", interval = c(.1, 100))
    #>
    #> Optimization through AIC
    #>
    #> lambda =   38.2584 AIC =   242.28062
    #> lambda =   61.8416 AIC =   255.97650
    #> lambda =   23.6832 AIC =   235.66356
    #> lambda =   14.6752 AIC =   227.83485
    #> lambda =    9.1080 AIC =   222.34228
    #> lambda =    5.6672 AIC =   218.58817
    #> lambda =    3.5407 AIC =   218.48045
    #> lambda =    4.4685 AIC =   219.13421
    #> lambda =    2.2265 AIC =   219.07120
    #> lambda =    4.4297 AIC =   219.09585
    #> lambda =    3.0387 AIC =   218.20451
    #> lambda =    2.7285 AIC =   218.05736
    #> lambda =    2.5367 AIC =   219.19927
    #> lambda =    2.8470 AIC =   218.11420
    #> lambda =    2.6553 AIC =   219.24969
    #> lambda =    2.7738 AIC =   218.09241
    #> lambda =    2.7005 AIC =   218.04309
    #> lambda =    2.6832 AIC =   219.26164
    #> lambda =    2.7112 AIC =   218.04643
    #> lambda =    2.6922 AIC =   219.25501
    #> lambda =    2.7046 AIC =   218.04976
    #> lambda =    2.6973 AIC =   218.04238
    #> lambda =    2.6954 AIC =   218.04531
    #> lambda =    2.6986 AIC =   218.04668
    #> lambda =    2.6966 AIC =   218.04283
    #> lambda =    2.6978 AIC =   218.04661
    #> lambda =    2.6971 AIC =   218.04656
    #> lambda =    2.6975 AIC =   218.04465
    #> lambda =    2.6972 AIC =   218.04045
    #> lambda =    2.6972 AIC =   218.04733
    #> lambda =    2.6973 AIC =   218.04122
    #> lambda =    2.6972 AIC =   218.04045
    out6 <- update(out5, lambda = lambda.aic4)
    summary(out6, pval = .10)
    #>
    #> Call:
    #> islasso(formula = lpsa ~ ., family = gaussian(), lambda = lambda.aic4,
    #>     alpha = 0.5, data = Prostate)
    #>
    #> Residuals:
    #>      Min       1Q   Median       3Q      Max
    #> -1.64659 -0.37033 -0.00621  0.36601  1.63182
    #>
    #>             Estimate Std. Error    Df z value Pr(>|z|)
    #> (Intercept)  1.07554    0.83321 1.000   1.291   0.1968
    #> lcavol       0.56489    0.08237 0.977   6.858 6.99e-12 ***
    #> lweight      0.38879    0.16028 0.954   2.426   0.0153 *
    #> lbph         0.10048    0.05643 0.974   1.781   0.0750 .
    #> svi          0.56322    0.22116 0.930   2.547   0.0109 *
    #> ---
    #> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    #>
    #> (Dispersion parameter for gaussian family taken to be 0.5023252)
    #>
    #>     Null deviance: 127.918  on 96.00  degrees of freedom
    #> Residual deviance:  44.762  on 89.11  degrees of freedom
    #> AIC: 218.04
    #> Lambda: 2.6972
    #>
    #> Number of Newton-Raphson iterations: 10

# References

- Tibshirani R. Regression shrinkage and selection via the lasso. J R Stat Soc: Series B 1996; 58: 267–288
- Cilluffo, G, Sottile, G, 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, online doi: 10.1177/0962280219842890.
- Brown B and Wang Y. Standard errors and covariance matrices for smoothed rank estimators. Biometrika 2005; 92: 149–158.