# Using the `sparseR` package

#### Ryan A. Peterson

#### 2025-04-14

# 1 Introduction

The `sparseR` package aids in the implementation of ranked sparsity methods for model selection in the presence of interactions and polynomials.

The following methods are supported, with help from `ncvreg` on the backend:

1. The sparsity-ranked lasso (SRL)
2. Sparsity-ranked MCP/SCAD (SRM/SRS)
3. Sparsity-ranked elastic net (SREN)

Additionally, `sparseR` makes it easy to preprocess data and get it ready for looking at all candidate interactions of order `k`, or all possible polynomials of order `poly`.

This document goes through several use cases of the package on different datasets.

# 2 Use case: `iris` data

## 2.1 Data set description

To illustrate in a simple case, consider Fisher’s Iris data set, composed of 4 numeric variables and one categorical variable.

    data(iris)
    summary(iris)
    ##   Sepal.Length    Sepal.Width     Petal.Length    Petal.Width
    ##  Min.   :4.300   Min.   :2.000   Min.   :1.000   Min.   :0.100
    ##  1st Qu.:5.100   1st Qu.:2.800   1st Qu.:1.600   1st Qu.:0.300
    ##  Median :5.800   Median :3.000   Median :4.350   Median :1.300
    ##  Mean   :5.843   Mean   :3.057   Mean   :3.758   Mean   :1.199
    ##  3rd Qu.:6.400   3rd Qu.:3.300   3rd Qu.:5.100   3rd Qu.:1.800
    ##  Max.   :7.900   Max.   :4.400   Max.   :6.900   Max.   :2.500
    ##        Species
    ##  setosa    :50
    ##  versicolor:50
    ##  virginica :50
    ##
    ##
    ##

If we are interested in predicting `Sepal.Width` based on the other variables and their pairwise interactions, have a couple of options. We can either use the built-in preprocessing for building the model matrix and fitting the model, or we can manually build the model matrix and pass it to the `sparseR` function.

## 2.2 Using formula specification

Passing a `formula` to `sparseR` will tell the function which main effects should be in the model. These are also the main effects which will indicate which interactions/polynomials should be investigated. Note that the formula will only accept main effect terms, and will not allow specification of individual interaction effects (or other functions of the main effects).

The sparsity-ranked lasso model for all-pairwise interactions can be fit as follows (cross-validation is performed for `lambda` via the `ncvreg` package, so it is important to set the seed for reproducibililty).

    srl <- sparseR(Sepal.Width ~ ., data = iris, k = 1, seed = 1)

The print method will present useful results from these fits:

    srl
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=150, p=21
    ##   (At lambda=0.0023):
    ##     Nonzero coefficients: 7
    ##     Cross-validation error (deviance): 0.07
    ##     R-squared: 0.62
    ##     Signal-to-noise ratio: 1.64
    ##     Scale estimate (sigma): 0.267
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect     6        2      0.333    2.45
    ##  Order 1 interaction    12        3      0.250    3.46
    ##   Order 2 polynomial     3        2      0.667    3.00
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=150, p=21
    ##   (At lambda=0.0074):
    ##     Nonzero coefficients: 6
    ##     Cross-validation error (deviance): 0.08
    ##     R-squared: 0.57
    ##     Signal-to-noise ratio: 1.35
    ##     Scale estimate (sigma): 0.284
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect     6        2      0.333    2.45
    ##  Order 1 interaction    12        2      0.167    3.46
    ##   Order 2 polynomial     3        2      0.667    3.00

For more detailed model information, the `summary` method for `sparseR` objects displays more information about the fits (see `?ncvreg::summary.ncvreg` for more information).

    # At the lambda which minimizes CVE
    summary(srl, at = "cvmin")
    ## lasso-penalized linear regression with n=150, p=21
    ## At lambda=0.0023:
    ## -------------------------------------------------
    ##   Nonzero coefficients         :   7
    ##   Expected nonzero coefficients:   1.96
    ##   Average mfdr (7 features)    :   0.280
    ##
    ##                                 Estimate       z      mfdr Selected
    ## Species_setosa                  1.106390 25.6231   < 1e-04        *
    ## Sepal.Length:Species_setosa     0.376189 10.1686   < 1e-04        *
    ## Sepal.Length_poly_1             0.167953  8.4066   < 1e-04        *
    ## Petal.Width_poly_2              0.135230  5.6513   < 1e-04        *
    ## Petal.Width:Species_versicolor  0.467836  4.1960 0.0081893        *
    ## Sepal.Length_poly_2            -0.011020 -0.9665 0.9717917        *
    ## Sepal.Length:Petal.Length      -0.006347 -0.6651 0.9778049        *

    # At the lambda which is within 1 SE of the minimum CVE
    summary(srl, at = "cv1se")
    ## lasso-penalized linear regression with n=150, p=21
    ## At lambda=0.0074:
    ## -------------------------------------------------
    ##   Nonzero coefficients         :   6
    ##   Expected nonzero coefficients:   1.22
    ##   Average mfdr (6 features)    :   0.204
    ##
    ##                                Estimate      z      mfdr Selected
    ## Species_setosa                  0.80638 18.013   < 1e-04        *
    ## Sepal.Length_poly_1             0.19734  9.713   < 1e-04        *
    ## Petal.Width_poly_2              0.09871  4.614 0.0011579        *
    ## Petal.Width:Species_versicolor  0.27739  3.259 0.1328700        *
    ## Sepal.Length_poly_2            -0.03363 -2.804 0.3663287        *
    ## Sepal.Length:Species_setosa     0.04275  2.190 0.7229428        *

Finally, plots of either the cross-validation error across `lambda` values or the coefficient paths can be produced fairly easily:

    plot(srl, plot_type = "cv")

    plot(srl, plot_type = "path")

Importantly, since `sparseR` with a formula centers and scales the covariates prior to forming interactions, the coefficients should be interpreted this way (a 1-unit change in the covariate corresponds to a 1 SD change in the unit of the main effect, and parameters involving interactions must be interpreted at the mean value of involved covariates).

## 2.3 Using model matrix specification

Since the `sparseR` preprocessing has limits to its customizability, and often it will be of interest to create interactions that are centered at prespecified values (as opposed to their mean), it is also possible to feed in a matrix to `sparseR` that has already been preprocessed. The function also allows users to circumvent the recipes functionality and supply their own model matrix, composed of interactions and/or polynomials. This may be useful in cases where users wish to:

- specify their own missing data imputation methods,
- specify the intercept differently than a grand mean (avoid centering variables prior to making certain interactions of interest)
- scale variables differently

The key arguments for this purpose are to set `pre_process` to `FALSE`, and supply `model_matrix` and the outcome `y`. Users may also want to specify how the model matrix is specified in terms of polynomial order (the package requires a prefix, such as “poly_1”, or “poly_2” to denote polynomial terms of orders 1 and 2 respectively) and interaction seperator (by default, this is “\\:”, which is the default for `model.matrix`).

The code below creates a model matrix will all pairwise interactions, however as opposed to the formula, the numeric variables are not centered prior to forming the interactions, and the factor variable of species will be given a reference category (as opposed to a cell-means specification). Therefore, this method will lead to different results.

    X <- model.matrix(Sepal.Width ~ .*., data = iris)[,-1]

    set.seed(1)
    srl2 <- sparseR(pre_process = FALSE, model_matrix = X, y = iris$Sepal.Width)
    ## Warning in ncvreg(X = X, y = y, ...): Maximum number of iterations reached

Since we have a warning about iterations, note that we can pass through arguments to `ncvreg` and fix this issue:

    set.seed(1)
    srl2 <- sparseR(pre_process = FALSE, model_matrix = X, y = iris$Sepal.Width, max.iter = 1e6)

And we can use the same S3 methods to get more information about the SRL model:

    srl2
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=150, p=14
    ##   (At lambda=0.0005):
    ##     Nonzero coefficients: 8
    ##     Cross-validation error (deviance): 0.07
    ##     R-squared: 0.64
    ##     Signal-to-noise ratio: 1.77
    ##     Scale estimate (sigma): 0.261
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect     5        3      0.600    2.24
    ##  Order 1 interaction     9        5      0.556    3.00
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=150, p=14
    ##   (At lambda=0.0021):
    ##     Nonzero coefficients: 5
    ##     Cross-validation error (deviance): 0.08
    ##     R-squared: 0.59
    ##     Signal-to-noise ratio: 1.46
    ##     Scale estimate (sigma): 0.277
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect     5        4      0.800    2.24
    ##  Order 1 interaction     9        1      0.111    3.00
    summary(srl2, at = "cv1se")
    ## lasso-penalized linear regression with n=150, p=14
    ## At lambda=0.0021:
    ## -------------------------------------------------
    ##   Nonzero coefficients         :   5
    ##   Expected nonzero coefficients:   0.00
    ##   Average mfdr (5 features)    :   0.000
    ##
    ##                           Estimate      z    mfdr Selected
    ## Speciesvirginica           -1.1502 -25.15 < 1e-04        *
    ## Speciesversicolor          -1.0558 -23.10 < 1e-04        *
    ## Sepal.Length                0.4243  16.32 < 1e-04        *
    ## Petal.Width                 0.3390  12.06 < 1e-04        *
    ## Sepal.Length:Petal.Length  -0.0189 -11.68 < 1e-04        *

    plot(srl2)

Since we did not center/scale any of the variables prior to this fitting procedure, the interpretation of these coefficients should be on the original scale of the predictors, and the origin for the terms involved in interactions is 0, not at the mean of their constituent covariates.

Note also that there are fewer total covariates listed. This is because `model.matrix` uses reference cell dummy variables by default. Another way to do this without the reference category specification would be to pass pre-process options to `sparseR`, or to manually use `sparseR_prep` to create the model matrix.

    set.seed(1)
    srl3 <- sparseR(Sepal.Width ~ ., data = iris, k = 1,
                    pre_proc_opts = c("none"), max.iter = 1e6)
    srl3
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=150, p=21
    ##   (At lambda=0.0006):
    ##     Nonzero coefficients: 6
    ##     Cross-validation error (deviance): 0.07
    ##     R-squared: 0.64
    ##     Signal-to-noise ratio: 1.80
    ##     Scale estimate (sigma): 0.259
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect     6        2      0.333    2.45
    ##  Order 1 interaction    12        4      0.333    3.46
    ##   Order 2 polynomial     3        0      0.000    3.00
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=150, p=21
    ##   (At lambda=0.0041):
    ##     Nonzero coefficients: 5
    ##     Cross-validation error (deviance): 0.08
    ##     R-squared: 0.59
    ##     Signal-to-noise ratio: 1.47
    ##     Scale estimate (sigma): 0.277
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect     6        2      0.333    2.45
    ##  Order 1 interaction    12        2      0.167    3.46
    ##   Order 2 polynomial     3        1      0.333    3.00
    summary(srl3, at = "cv1se")
    ## lasso-penalized linear regression with n=150, p=21
    ## At lambda=0.0041:
    ## -------------------------------------------------
    ##   Nonzero coefficients         :   5
    ##   Expected nonzero coefficients:   1.81
    ##   Average mfdr (5 features)    :   0.363
    ##
    ##                             Estimate      z    mfdr Selected
    ## Sepal.Length:Species_setosa  0.18862 21.079 < 1e-04        *
    ## Sepal.Length_poly_1          0.23292  9.248 < 1e-04        *
    ## Petal.Width_poly_2           0.06659  6.116 < 1e-04        *
    ## Species_virginica           -0.08945 -2.387 0.84724        *
    ## Petal.Width:Species_setosa   0.14950  1.542 0.96687        *

## 2.4 Centering to different values

In many cases with interactions, it makes sense to center variables to certain values (or rather, it does not make sense to center variables to their mean or to zero). Therefore we built some flexibility into `sparseR` and `sparseR_prep` in order to allow users to change these as the context dictates.

Each covariate’s centering location (in this case their minima) can be passed into `sparseR_prep` as follows:

    cc <- iris %>%
        select(Sepal.Length, Petal.Length, Petal.Width) %>%
        apply(2, min, na.rm = TRUE)

    p1 <- sparseR_prep(Sepal.Width ~ ., iris, k = 0, extra_opts = list(centers = cc))
    (c2min <- bake(p1, iris))
    ## # A tibble: 150 × 6
    ##    Sepal.Length Petal.Length Petal.Width Species_setosa Species_versicolor
    ##           <dbl>        <dbl>       <dbl>          <dbl>              <dbl>
    ##  1        0.966        0.227       0.131              1                  0
    ##  2        0.725        0.227       0.131              1                  0
    ##  3        0.483        0.170       0.131              1                  0
    ##  4        0.362        0.283       0.131              1                  0
    ##  5        0.845        0.227       0.131              1                  0
    ##  6        1.33         0.397       0.394              1                  0
    ##  7        0.362        0.227       0.262              1                  0
    ##  8        0.845        0.283       0.131              1                  0
    ##  9        0.121        0.227       0.131              1                  0
    ## 10        0.725        0.283       0                  1                  0
    ## # ℹ 140 more rows
    ## # ℹ 1 more variable: Species_virginica <dbl>
    summary(c2min)
    ##   Sepal.Length     Petal.Length     Petal.Width     Species_setosa
    ##  Min.   :0.0000   Min.   :0.0000   Min.   :0.0000   Min.   :0.0000
    ##  1st Qu.:0.9661   1st Qu.:0.3399   1st Qu.:0.2624   1st Qu.:0.0000
    ##  Median :1.8115   Median :1.8977   Median :1.5743   Median :0.0000
    ##  Mean   :1.8638   Mean   :1.5623   Mean   :1.4422   Mean   :0.3333
    ##  3rd Qu.:2.5360   3rd Qu.:2.3226   3rd Qu.:2.2303   3rd Qu.:1.0000
    ##  Max.   :4.3475   Max.   :3.3422   Max.   :3.1486   Max.   :1.0000
    ##  Species_versicolor Species_virginica
    ##  Min.   :0.0000     Min.   :0.0000
    ##  1st Qu.:0.0000     1st Qu.:0.0000
    ##  Median :0.0000     Median :0.0000
    ##  Mean   :0.3333     Mean   :0.3333
    ##  3rd Qu.:1.0000     3rd Qu.:1.0000
    ##  Max.   :1.0000     Max.   :1.0000

Or directly into `sparseR`:

    srl_centered2min <- sparseR(Sepal.Width ~ ., iris, extra_opts = list(centers = cc), seed = 1)

Or, a function can be passed to evaluate the location for each covariate:

    p2 <- sparseR_prep(Sepal.Width ~ ., iris, k = 0, extra_opts = list(center_fn = function(x, ...) min(x, ...)))
    (c2min2 <- bake(p2, iris))
    ## # A tibble: 150 × 6
    ##    Sepal.Length Petal.Length Petal.Width Species_setosa Species_versicolor
    ##           <dbl>        <dbl>       <dbl>          <dbl>              <dbl>
    ##  1        0.966        0.227       0.131              1                  0
    ##  2        0.725        0.227       0.131              1                  0
    ##  3        0.483        0.170       0.131              1                  0
    ##  4        0.362        0.283       0.131              1                  0
    ##  5        0.845        0.227       0.131              1                  0
    ##  6        1.33         0.397       0.394              1                  0
    ##  7        0.362        0.227       0.262              1                  0
    ##  8        0.845        0.283       0.131              1                  0
    ##  9        0.121        0.227       0.131              1                  0
    ## 10        0.725        0.283       0                  1                  0
    ## # ℹ 140 more rows
    ## # ℹ 1 more variable: Species_virginica <dbl>
    identical(c2min2, c2min)
    ## [1] TRUE

## 2.5 Plotting the model

Interactions are notoriously difficult to understand and to communicate, and visualizations can often be helpful. The `sparseR` package has some limited functionality for plotting the results from these fits, using the `visreg` package as a rough guide.

    effect_plot(srl, "Petal.Width", by = "Species")

Notice how this changes if the covariates are centered to their minima instead:

    effect_plot(srl_centered2min, "Petal.Width", by = "Species")

It’s clear that the model itself can change quite a bit depending on the center location utilized prior to building the interactions. Unfortunately, optimizing the center location for each covariate is a difficult, p-dimensional problem (set in the context of another difficult optimization problem of regularized regression with multiple tuning parameters).

The `sparseR` package will allow users to compare certain choices in terms of their cross-validated error, which is a decent starting point. The plot below shows how the “centered to zero” model has slightly lower cross-validated error than the best mean-centered model and the best minimum-centered model.

    plot(srl3, plot_type = "cv", ylim = c(0,.2))
    abline(h = min(srl3$fit$cve), col = "red")
    plot(srl_centered2min, plot_type = "cv", ylim = c(0,.2))
    abline(h = min(srl3$fit$cve), col = "red")
    plot(srl, plot_type = "cv", ylim = c(0,.2))
    abline(h = min(srl3$fit$cve), col = "red")

    effect_plot(srl3, "Petal.Width", by = "Species",
                plot.args = list(ylim = c(1.5, 4.8)))
    effect_plot(srl_centered2min, "Petal.Width", by = "Species",
                plot.args = list(ylim = c(1.5, 4.8)))
    effect_plot(srl, "Petal.Width", by = "Species",
                plot.args = list(ylim = c(1.5, 4.8)))

Though these models look substantially different, it’s worth noting that the predictions obtained from the three models are very close to one another (since the models look similar in the neighborhood of existing data):

    p1 <- predict(srl3, at = "cvmin")
    p2 <- predict(srl_centered2min, at = "cvmin")
    p3 <- predict(srl, at = "cvmin")

    cor(cbind(p1, p2 ,p3))
    ##           p1        p2        p3
    ## p1 1.0000000 0.9903513 0.9873102
    ## p2 0.9903513 1.0000000 0.9898124
    ## p3 0.9873102 0.9898124 1.0000000
    pairs(cbind(p1, p2 ,p3))

At the CV1se lambda value, there is slightly less agreement in these predictions:

    # At CV1se
    p4 <- predict(srl3, at = "cv1se")
    p5 <- predict(srl_centered2min, at = "cv1se")
    p6 <- predict(srl, at = "cv1se")

    cor(cbind(p1, p2 ,p3, p4,p5,p6))
    ##           p1        p2        p3        p4        p5        p6
    ## p1 1.0000000 0.9903513 0.9873102 0.9781221 0.9586728 0.9752579
    ## p2 0.9903513 1.0000000 0.9898124 0.9714612 0.9784280 0.9627639
    ## p3 0.9873102 0.9898124 1.0000000 0.9706405 0.9671849 0.9821402
    ## p4 0.9781221 0.9714612 0.9706405 1.0000000 0.9709760 0.9845977
    ## p5 0.9586728 0.9784280 0.9671849 0.9709760 1.0000000 0.9504649
    ## p6 0.9752579 0.9627639 0.9821402 0.9845977 0.9504649 1.0000000
    pairs(cbind(p1, p2 ,p3, p4,p5,p6))

    effect_plot(srl3, "Petal.Width", by = "Species", at = "cv1se")
    effect_plot(srl_centered2min, "Petal.Width", by = "Species", at = "cv1se")
    effect_plot(srl, "Petal.Width", by = "Species", at = "cv1se")

## 2.6 Using stepwise RBIC

RBIC, or Ranked-sparsity Bayesian Information Criterion, is a ranked sparsity model selection criterion that will correctly penalize interactions (higher than main effects). The benefit of using a stepwise approach informed by RBIC is that there is no shrinkage, and it thus doesn’t matter whether or not (or where) the covariates are centered (usually). The exceptions here occur when the stepwise process is not sequential/hierarchical (i.e. main effects first, then interactions of the “active” main effects, etc.).

Inference on the parameters is still i) difficult because one must account for post-selection inference, and ii) affected by where and whether the covariates are centered. However, the lines themselves will not be affected by centering status.

We can see this in the examples below:

    ## Centered model
    (rbic1 <- sparseRBIC_step(Sepal.Width ~ ., iris, pre_proc_opts = c("center", "scale")))
    ## Note: sparseRBIC_step is currently experimental and may not behave as expected.
    ##
    ## Call:  glm(formula = y ~ Species_setosa + Sepal.Length + Petal.Width +
    ##     Species_versicolor + Petal.Length + `Sepal.Length:Species_setosa`,
    ##     family = family, data = X)
    ##
    ## Coefficients:
    ##                   (Intercept)                 Species_setosa
    ##                       2.40062                        2.14130
    ##                  Sepal.Length                    Petal.Width
    ##                       0.16506                        0.44726
    ##            Species_versicolor                   Petal.Length
    ##                       0.29340                       -0.05916
    ## `Sepal.Length:Species_setosa`
    ##                       0.45943
    ##
    ## Degrees of Freedom: 149 Total (i.e. Null);  143 Residual
    ## Null Deviance:       28.31
    ## Residual Deviance: 8.999     AIC: 19.65

    # Non-centered model
    (rbic2 <- sparseRBIC_step(Sepal.Width ~ ., iris, pre_proc_opts = c("scale")))
    ## Note: sparseRBIC_step is currently experimental and may not behave as expected.
    ##
    ## Call:  glm(formula = y ~ Species_setosa + Sepal.Length + Petal.Width +
    ##     Species_virginica + Petal.Length + `Sepal.Length:Species_setosa`,
    ##     family = family, data = X)
    ##
    ## Coefficients:
    ##                   (Intercept)                 Species_setosa
    ##                       0.95150                       -1.39414
    ##                  Sepal.Length                    Petal.Width
    ##                       0.16506                        0.44726
    ##             Species_virginica                   Petal.Length
    ##                      -0.29340                       -0.05916
    ## `Sepal.Length:Species_setosa`
    ##                       0.45943
    ##
    ## Degrees of Freedom: 149 Total (i.e. Null);  143 Residual
    ## Null Deviance:       28.31
    ## Residual Deviance: 8.999     AIC: 19.65

    effect_plot(rbic1, "Petal.Width", by = "Species", plot.args = list(ylim = c(1.5, 5)))
    effect_plot(rbic2, "Petal.Width", by = "Species", plot.args = list(ylim = c(1.5, 5)))
    effect_plot(rbic1, "Sepal.Length", by = "Species")
    effect_plot(rbic2, "Sepal.Length", by = "Species")

Again similar S3 methods are available (note how inference does change for certain parameters based on the centering status). A message is produced because these p-values do not account for post-selection inference.

    summary(rbic1)
    ## P-values have **not** been corrected for multiple comparisons
    ## Consider sparseRBIC_sampsplit() or sparseRBIC_bootstrap()
    ##
    ## Call:
    ## glm(formula = y ~ Species_setosa + Sepal.Length + Petal.Width +
    ##     Species_versicolor + Petal.Length + `Sepal.Length:Species_setosa`,
    ##     family = family, data = X)
    ##
    ## Coefficients:
    ##                               Estimate Std. Error t value Pr(>|t|)
    ## (Intercept)                    2.40062    0.12025  19.964  < 2e-16 ***
    ## Species_setosa                 2.14130    0.30584   7.001 9.10e-11 ***
    ## Sepal.Length                   0.16506    0.06017   2.743  0.00687 **
    ## Petal.Width                    0.44726    0.08833   5.064 1.25e-06 ***
    ## Species_versicolor             0.29340    0.09839   2.982  0.00337 **
    ## Petal.Length                  -0.05916    0.15021  -0.394  0.69428
    ## `Sepal.Length:Species_setosa`  0.45943    0.09997   4.596 9.40e-06 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## (Dispersion parameter for gaussian family taken to be 0.06292687)
    ##
    ##     Null deviance: 28.3069  on 149  degrees of freedom
    ## Residual deviance:  8.9985  on 143  degrees of freedom
    ## AIC: 19.646
    ##
    ## Number of Fisher Scoring iterations: 2
    summary(rbic2)
    ## P-values have **not** been corrected for multiple comparisons
    ## Consider sparseRBIC_sampsplit() or sparseRBIC_bootstrap()
    ##
    ## Call:
    ## glm(formula = y ~ Species_setosa + Sepal.Length + Petal.Width +
    ##     Species_virginica + Petal.Length + `Sepal.Length:Species_setosa`,
    ##     family = family, data = X)
    ##
    ## Coefficients:
    ##                               Estimate Std. Error t value Pr(>|t|)
    ## (Intercept)                    0.95150    0.26641   3.572 0.000484 ***
    ## Species_setosa                -1.39414    0.58460  -2.385 0.018401 *
    ## Sepal.Length                   0.16506    0.06017   2.743 0.006865 **
    ## Petal.Width                    0.44726    0.08833   5.064 1.25e-06 ***
    ## Species_virginica             -0.29340    0.09839  -2.982 0.003367 **
    ## Petal.Length                  -0.05916    0.15021  -0.394 0.694277
    ## `Sepal.Length:Species_setosa`  0.45943    0.09997   4.596 9.40e-06 ***
    ## ---
    ## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
    ##
    ## (Dispersion parameter for gaussian family taken to be 0.06292687)
    ##
    ##     Null deviance: 28.3069  on 149  degrees of freedom
    ## Residual deviance:  8.9985  on 143  degrees of freedom
    ## AIC: 19.646
    ##
    ## Number of Fisher Scoring iterations: 2

Alternatively, sample splitting can be utilized to achieve valid inference. This method divides the data into two samples of equal size, uses the first sample to select the model, and the second one to fit and achieve valid (though potentially low-power) inference. Coefficients which are not selected are bestowed a p-value of 1.

    s1 <- sparseRBIC_sampsplit(rbic1)

    ## Note: sparseRBIC_sampsplit is currently experimental and may not behave as expected.
    ## Warning: `progress_estimated()` was deprecated in dplyr 1.0.0.
    ## ℹ The deprecated feature was likely used in the sparseR package.
    ##   Please report the issue to the authors.
    ## This warning is displayed once every 8 hours.
    ## Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
    ## generated.

Coefficient | Median_p | Mean_p | Gmean_p | % selected
--- | --- | --- | --- | ---
(Intercept) | 0.00000 | 0.00000 | 0.00000 | 100
Species_setosa | 0.00000 | 0.00000 | 0.00000 | 100
Sepal.Length | 0.00003 | 0.11104 | 0.00007 | 90
Sepal.Length:Species_setosa | 0.00870 | 0.30440 | 0.01235 | 70
Petal.Width | 0.50777 | 0.50342 | 0.05012 | 50
Species_versicolor | 1.00000 | 0.70857 | 0.24735 | 30
Petal.Length | 1.00000 | 0.96673 | 0.96036 | 10
Species_virginica | 1.00000 | 0.90080 | 0.61697 | 10

    s2 <- sparseRBIC_sampsplit(rbic2)

    ## Note: sparseRBIC_sampsplit is currently experimental and may not behave as expected.

Coefficient | Median_p | Mean_p | Gmean_p | % selected
--- | --- | --- | --- | ---
(Intercept) | 0.04286 | 0.12971 | 0.01964 | 100
Species_setosa | 0.09819 | 0.14157 | 0.00595 | 100
Sepal.Length | 0.00063 | 0.00644 | 0.00008 | 100
Sepal.Length:Species_setosa | 0.01678 | 0.20926 | 0.00805 | 80
Petal.Width | 0.06679 | 0.41570 | 0.06695 | 60
Species_virginica | 1.00000 | 0.64516 | 0.41337 | 40
Petal.Width:Species_setosa | 1.00000 | 0.92315 | 0.90718 | 20
Species_versicolor | 1.00000 | 0.90350 | 0.71508 | 10
Petal.Length | 1.00000 | 0.98405 | 0.98277 | 10
Sepal.Length:Petal.Length | 1.00000 | 0.93700 | 0.90536 | 10
Sepal.Length:Petal.Width | 1.00000 | 0.90141 | 0.65295 | 10

Additionally, inferences based on bootstrapped samples of the data are available via the `sparseRBIC_bootstrap` function, which stores p-values for each coefficient after model selection has been performed. If a coefficient has not been selected, the p-value is set to one for that bootstrap iteration, so when the “mean” p-value is calculated across bootstraps, the underestimation of the p-value due to model selection is attenuated, as the estimate gets “pulled up” by the 1’s in the vector. Since this may be a conservative approach, we also display the “geometric” mean which is a measure of central tendency less sensitive to skew (which many p-values will be highly right skewed). More research must be done on these techniques to ensure adequate coverage properties, but given a high enough number of bootstrap samples they should both be more conservative than the naive, “pretend model selection never took place” approach. The downside here is that depending on the size of the data and the number of variables, this could take quite some time.

Note that the mean p-value across bootstraps is bounded to be greater than 1-P(selected). In some sense, this makes sense, as if a variable has a low probability of being selected, then that variable should have a higher adjusted p-value. However, for variables whose effects will be highly related (i.e. an interaction and a constituent main effect), this may be too conservative.

    set.seed(1)
    ## Centered model
    b1 <- sparseRBIC_bootstrap(rbic1)

Coefficient | Mean_p | Gmean_p | % selected
--- | --- | --- | ---
(Intercept) | 0.00000 | 0.00000 | 100
Species_setosa | 0.00011 | 0.00000 | 100
Sepal.Length | 0.00293 | 0.00000 | 100
Sepal.Length:Species_setosa | 0.20011 | 0.00002 | 80
Petal.Width | 0.50000 | 0.00008 | 50
Species_versicolor | 0.70010 | 0.07691 | 30
Petal.Length | 0.81087 | 0.54467 | 20
Sepal.Length:Petal.Width | 0.90000 | 0.32270 | 10
Sepal.Length:Species_versicolor | 0.90222 | 0.68331 | 10
Species_virginica | 0.90000 | 0.21139 | 10
Petal.Width:Species_setosa | 0.90182 | 0.66977 | 10
Sepal.Length:Petal.Length | 0.90000 | 0.22450 | 10

    set.seed(1)
    ## Uncentered model
    b2 <- sparseRBIC_bootstrap(rbic2)

    ## Note: sparseRBIC_bootstrap is currently experimental and may not behave as expected.

Coefficient | Mean_p | Gmean_p | % selected
--- | --- | --- | ---
(Intercept) | 0.16411 | 0.00016 | 100
Species_setosa | 0.08507 | 0.00002 | 100
Sepal.Length | 0.00293 | 0.00000 | 100
Sepal.Length:Species_setosa | 0.20011 | 0.00002 | 80
Petal.Width | 0.50000 | 0.00017 | 50
Species_versicolor | 0.60079 | 0.02124 | 40
Petal.Length | 0.80312 | 0.38407 | 20
Sepal.Length:Petal.Width | 0.90000 | 0.32270 | 10
Sepal.Length:Species_versicolor | 0.90222 | 0.68331 | 10
Petal.Width:Species_setosa | 0.90182 | 0.66977 | 10
Sepal.Length:Petal.Length | 0.90000 | 0.22450 | 10

This section is still under development.

# 3 Use case: lung data

## 3.1 Data set description

As a more realistic example, we take a data set that is often used to train models that can predict lung cancer status. It has 1027 observations and 14 covariates, including urban/rural, age, years of school, smoking years, number of children, etc.

First, let’s look at the data a little:

    data("irlcs_radon_syn")
    summary(irlcs_radon_syn)
    ##        ID         CITY       YRSHOME          AGE        CASE    SCHOOL
    ##  Min.   :   1.0   0:246   Min.   :20.0   Min.   :44.16   0:632   1: 92
    ##  1st Qu.: 260.5   1:781   1st Qu.:25.0   1st Qu.:61.38   1:395   2:506
    ##  Median : 492.0           Median :31.0   Median :67.84           3:293
    ##  Mean   : 505.2           Mean   :33.1   Mean   :67.45           4:106
    ##  3rd Qu.: 761.0           3rd Qu.:39.0   3rd Qu.:73.82           5: 30
    ##  Max.   :1026.0           Max.   :75.0   Max.   :84.80
    ##
    ##      SMKYRS         CHILDREN           PYR            PYRRATE
    ##  Min.   : 0.00   Min.   : 0.000   Min.   :  0.00   Min.   :0.00000
    ##  1st Qu.: 0.00   1st Qu.: 2.000   1st Qu.:  0.00   1st Qu.:0.00000
    ##  Median :13.00   Median : 3.000   Median :  3.60   Median :0.05368
    ##  Mean   :20.58   Mean   : 3.065   Mean   : 19.31   Mean   :0.31745
    ##  3rd Qu.:41.00   3rd Qu.: 4.000   3rd Qu.: 35.07   3rd Qu.:0.58024
    ##  Max.   :64.00   Max.   :13.000   Max.   :138.45   Max.   :2.55702
    ##
    ##     SMKQUIT            BMI            WLM20        PRELUNG SMKEVER SMKCUR
    ##  Min.   : 0.000   Min.   :16.64   Min.   : 2.176   0:692   0:476   0:705
    ##  1st Qu.: 0.000   1st Qu.:21.79   1st Qu.: 5.902   1:335   1:551   1:322
    ##  Median : 0.000   Median :23.78   Median : 8.683
    ##  Mean   : 4.733   Mean   :24.60   Mean   :10.914
    ##  3rd Qu.: 3.325   3rd Qu.:27.40   3rd Qu.:13.246
    ##  Max.   :57.355   Max.   :41.60   Max.   :77.457
    ##                   NA's   :51

Note:

- There are missing values for BMI - we will see how `sparseR` handles this.
- Everything is coded as numeric except the outcome, case (this is not necessary for `sparseR` to work), as we will see.
- `ID` is an arbitrary identifying variable which should not be used in the modeling phase

    irlcs_radon_syn <- select(irlcs_radon_syn, -ID)

At this point, we will split the data into a test and training set.

    set.seed(13)

    N <- nrow(irlcs_radon_syn)
    trainIDX <- sample(1:N, N * .75)
    train <- irlcs_radon_syn[trainIDX,]
    test <- irlcs_radon_syn[-trainIDX,]

## 3.2 Preprocessing

Although the `sparseR()` function will do it automatically, it’s generally a good idea to see how the data is being preprocessed and to examine how the model matrix is built before the regularization. We can see what `sparseR` does to preprocess the data by running the function `sparseR_prep`, which utilizes the `recipes` package to perform preprocessing. At no time is the outcome used to train any of the steps, except for the elimination of NA values if any exist.

    prep_obj <- sparseR_prep(CASE ~ ., data = train, k = 0, poly = 1)
    prep_obj
    ##
    ## ── Recipe ──────────────────────────────────────────────────────────────────────
    ##
    ## ── Inputs
    ## Number of variables by role
    ## outcome:    1
    ## predictor: 10
    ## dummy:      4
    ##
    ## ── Training information
    ## Training data contained 770 data points and 43 incomplete rows.
    ##
    ## ── Operations
    ## • Variables removed: <none> | Trained
    ## • Sparse, unbalanced variable filter removed: <none> | Trained
    ## • Scaling for: YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, ... | Trained
    ## • Collapsing factor levels for: SCHOOL | Trained
    ## • K-nearest neighbor imputation for: YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Removing rows with NA values in: CITY, YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Variables removed: CASE | Trained
    ## • Dummy variables from: CITY, SCHOOL, PRELUNG, SMKEVER, SMKCUR | Trained
    ## • Sparse, unbalanced variable filter removed: SCHOOL_other | Trained
    ## Centering to mean for YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, S... [trained]

We see that the preprocessor did many things, among which it imputed the data for the missing value using the other predictors in the model. Apparently, by default, this preprocessor will remove variables with “near” zero variance, and that as a result, SMKYRS is not considered (it’s eliminated in an early step).

If we look at this variable, we see that it may be informative:

    MASS::truehist(train$SMKYRS)

Therefore, we can tell the preprocessor not to filter this variable in two ways. First, we could tell it to use `zv` instead of `nzv`, in which case it will only remove zero variance predictors:

    sparseR_prep(CASE ~ ., data = train, k = 0, poly = 1, filter = "zv")
    ##
    ## ── Recipe ──────────────────────────────────────────────────────────────────────
    ##
    ## ── Inputs
    ## Number of variables by role
    ## outcome:    1
    ## predictor: 10
    ## dummy:      4
    ##
    ## ── Training information
    ## Training data contained 770 data points and 43 incomplete rows.
    ##
    ## ── Operations
    ## • Variables removed: <none> | Trained
    ## • Zero variance filter removed: <none> | Trained
    ## Centering to mean for YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, S... [trained]
    ## • Scaling for: YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, ... | Trained
    ## • Collapsing factor levels for: SCHOOL | Trained
    ## • K-nearest neighbor imputation for: YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Removing rows with NA values in: CITY, YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Variables removed: CASE | Trained
    ## • Dummy variables from: CITY, SCHOOL, PRELUNG, SMKEVER, SMKCUR | Trained
    ## • Zero variance filter removed: <none> | Trained

Or, we could adjust the options for the near-zero variance filter (see `?recipes::step_nzv` for details):

    sparseR_prep(CASE ~ ., data = train, k = 0, poly = 1, extra_opts = list(unique_cut = 5))
    ##
    ## ── Recipe ──────────────────────────────────────────────────────────────────────
    ##
    ## ── Inputs
    ## Number of variables by role
    ## outcome:    1
    ## predictor: 10
    ## dummy:      4
    ##
    ## ── Training information
    ## Training data contained 770 data points and 43 incomplete rows.
    ##
    ## ── Operations
    ## • Variables removed: <none> | Trained
    ## • Sparse, unbalanced variable filter removed: <none> | Trained
    ## Centering to mean for YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, S... [trained]
    ## • Scaling for: YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, ... | Trained
    ## • Collapsing factor levels for: SCHOOL | Trained
    ## • K-nearest neighbor imputation for: YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Removing rows with NA values in: CITY, YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Variables removed: CASE | Trained
    ## • Dummy variables from: CITY, SCHOOL, PRELUNG, SMKEVER, SMKCUR | Trained
    ## • Sparse, unbalanced variable filter removed: SCHOOL_other | Trained

Notice that we haven’t told the preprocessor to build any interactions or polynomials of higher degree than 1. we can do this simply by setting `k` and `poly` respectively to higher numbers:

    sparseR_prep(CASE ~ ., data = train, k = 1, poly = 1, extra_opts = list(unique_cut = 5))
    ##
    ## ── Recipe ──────────────────────────────────────────────────────────────────────
    ##
    ## ── Inputs
    ## Number of variables by role
    ## outcome:    1
    ## predictor: 10
    ## dummy:      4
    ##
    ## ── Training information
    ## Training data contained 770 data points and 43 incomplete rows.
    ##
    ## ── Operations
    ## • Variables removed: <none> | Trained
    ## • Sparse, unbalanced variable filter removed: <none> | Trained
    ## Centering to mean for YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, S... [trained]
    ## • Scaling for: YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, ... | Trained
    ## • Collapsing factor levels for: SCHOOL | Trained
    ## • K-nearest neighbor imputation for: YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Removing rows with NA values in: CITY, YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Variables removed: CASE | Trained
    ## • Dummy variables from: CITY, SCHOOL, PRELUNG, SMKEVER, SMKCUR | Trained
    ## • Interactions with: (YRSHOME + AGE + SMKYRS + CHILDREN + PYR + PYRRATE +
    ##   SMKQUIT + BMI + WLM20 + (CITY_X0 + CITY_X1 + SCHOOL_X1 + SCHOOL_X2 +
    ##   SCHOOL_X3 + SCHOOL_X4 + SCHOOL_other + PRELUNG_X0 + PRELUNG_X1 + SMKEVER_X0 +
    ##   SMKEVER_X1 + SMKCUR_X0 + SMKCUR_X1)) * (YRSHOME + AGE + SMKYRS + CHILDREN +
    ##   PYR + PYRRATE + SMKQUIT + BMI + WLM20 + (CITY_X0 + CITY_X1 + SCHOOL_X1 +
    ##   SCHOOL_X2 + SCHOOL_X3 + SCHOOL_X4 + SCHOOL_other + PRELUNG_X0 + PRELUNG_X1 +
    ##   SMKEVER_X0 + SMKEVER_X1 + SMKCUR_X0 + SMKCUR_X1)) | Trained
    ## • Sparse, unbalanced variable filter removed: SCHOOL_other, ... | Trained
    sparseR_prep(CASE ~ ., data = train, k = 1, poly = 2, extra_opts = list(unique_cut = 5))
    ##
    ## ── Recipe ──────────────────────────────────────────────────────────────────────
    ##
    ## ── Inputs
    ## Number of variables by role
    ## outcome:    1
    ## predictor: 10
    ## dummy:      4
    ##
    ## ── Training information
    ## Training data contained 770 data points and 43 incomplete rows.
    ##
    ## ── Operations
    ## • Variables removed: <none> | Trained
    ## • Sparse, unbalanced variable filter removed: <none> | Trained
    ## Centering to mean for YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, S... [trained]
    ## • Scaling for: YRSHOME, AGE, SMKYRS, CHILDREN, PYR, PYRRATE, ... | Trained
    ## • Collapsing factor levels for: SCHOOL | Trained
    ## • K-nearest neighbor imputation for: YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Removing rows with NA values in: CITY, YRSHOME, AGE, SCHOOL, ... | Trained
    ## • Variables removed: CASE | Trained
    ## • Dummy variables from: CITY, SCHOOL, PRELUNG, SMKEVER, SMKCUR | Trained
    ## • Interactions with: (YRSHOME + AGE + SMKYRS + CHILDREN + PYR + PYRRATE +
    ##   SMKQUIT + BMI + WLM20 + (CITY_X0 + CITY_X1 + SCHOOL_X1 + SCHOOL_X2 +
    ##   SCHOOL_X3 + SCHOOL_X4 + SCHOOL_other + PRELUNG_X0 + PRELUNG_X1 + SMKEVER_X0 +
    ##   SMKEVER_X1 + SMKCUR_X0 + SMKCUR_X1)) * (YRSHOME + AGE + SMKYRS + CHILDREN +
    ##   PYR + PYRRATE + SMKQUIT + BMI + WLM20 + (CITY_X0 + CITY_X1 + SCHOOL_X1 +
    ##   SCHOOL_X2 + SCHOOL_X3 + SCHOOL_X4 + SCHOOL_other + PRELUNG_X0 + PRELUNG_X1 +
    ##   SMKEVER_X0 + SMKEVER_X1 + SMKCUR_X0 + SMKCUR_X1)) | Trained
    ## • Orthogonal polynomials on: YRSHOME, AGE, SMKYRS, CHILDREN, ... | Trained
    ## • Sparse, unbalanced variable filter removed: SCHOOL_other, ... | Trained

Note: This preprocessor does not actually produce any data, it only prepares all of the steps. In the recipes package, this is known as `prep`ping the processor. the final step is called `bake`, where the data is fed through the processor and a new data set is returned.

## 3.3 Using the `sparseR` function

All of the preprocessing options can be accomplished through the main function for the package, `sparseR`. The following code runs a SRL model, an APL model, a main effects model, and an SRL with polynomials:

    lso <- list(
      SRL  = sparseR(CASE ~ ., train, seed = 1),            ## SRL model
      APL  = sparseR(CASE ~ ., train, seed = 1, gamma = 0), ## APL model
      ME   = sparseR(CASE ~ ., train, seed = 1, k = 0),     ## Main effects model
      SRLp = sparseR(CASE ~ ., train, seed = 1, poly = 2)   ## SRL + polynomials
    )

The simple print method allows one to investigate all of these models

    lso
    ## $SRL
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=204
    ##   (At lambda=0.0009):
    ##     Nonzero coefficients: 41
    ##     Cross-validation error (deviance): 0.13
    ##     R-squared: 0.43
    ##     Signal-to-noise ratio: 0.75
    ##     Scale estimate (sigma): 0.367
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    21       11      0.524    4.58
    ##  Order 1 interaction   174       24      0.138   13.19
    ##   Order 2 polynomial     9        6      0.667    5.48
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=204
    ##   (At lambda=0.0093):
    ##     Nonzero coefficients: 4
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.40
    ##     Signal-to-noise ratio: 0.66
    ##     Scale estimate (sigma): 0.377
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    21        3      0.143    4.58
    ##  Order 1 interaction   174        0      0.000   13.19
    ##   Order 2 polynomial     9        1      0.111    5.48
    ##
    ## $APL
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=204
    ##   (At lambda=0.0100):
    ##     Nonzero coefficients: 43
    ##     Cross-validation error (deviance): 0.13
    ##     R-squared: 0.43
    ##     Signal-to-noise ratio: 0.75
    ##     Scale estimate (sigma): 0.367
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    21        4      0.190       1
    ##  Order 1 interaction   174       37      0.213       1
    ##   Order 2 polynomial     9        2      0.222       1
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=204
    ##   (At lambda=0.0425):
    ##     Nonzero coefficients: 10
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.40
    ##     Signal-to-noise ratio: 0.66
    ##     Scale estimate (sigma): 0.377
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    21        3     0.1429       1
    ##  Order 1 interaction   174        6     0.0345       1
    ##   Order 2 polynomial     9        1     0.1111       1
    ##
    ## $ME
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=30
    ##   (At lambda=0.0034):
    ##     Nonzero coefficients: 11
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.42
    ##     Signal-to-noise ratio: 0.71
    ##     Scale estimate (sigma): 0.371
    ##
    ##   SR information:
    ##             Vartype Total Selected Saturation Penalty
    ##         Main effect    21       10      0.476    4.58
    ##  Order 2 polynomial     9        1      0.111    5.48
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=30
    ##   (At lambda=0.0142):
    ##     Nonzero coefficients: 3
    ##     Cross-validation error (deviance): 0.15
    ##     R-squared: 0.39
    ##     Signal-to-noise ratio: 0.63
    ##     Scale estimate (sigma): 0.381
    ##
    ##   SR information:
    ##             Vartype Total Selected Saturation Penalty
    ##         Main effect    21        3      0.143    4.58
    ##  Order 2 polynomial     9        0      0.000    5.48
    ##
    ## $SRLp
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=204
    ##   (At lambda=0.0009):
    ##     Nonzero coefficients: 41
    ##     Cross-validation error (deviance): 0.13
    ##     R-squared: 0.43
    ##     Signal-to-noise ratio: 0.75
    ##     Scale estimate (sigma): 0.367
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    21       11      0.524    4.58
    ##  Order 1 interaction   174       24      0.138   13.19
    ##   Order 2 polynomial     9        6      0.667    5.48
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=770, p=204
    ##   (At lambda=0.0093):
    ##     Nonzero coefficients: 4
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.40
    ##     Signal-to-noise ratio: 0.66
    ##     Scale estimate (sigma): 0.377
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    21        3      0.143    4.58
    ##  Order 1 interaction   174        0      0.000   13.19
    ##   Order 2 polynomial     9        1      0.111    5.48

Similarly, plots can be produced of all of these figures:

    n <- lapply(lso, plot, log.l = TRUE)

MCP:

    mcp <- list(
      SRM  = sparseR(CASE ~ ., train, seed = 1, penalty = "MCP"),            ## SRM model
      APM  = sparseR(CASE ~ ., train, seed = 1, gamma = 0, penalty = "MCP"), ## APM model
      MEM   = sparseR(CASE ~ ., train, seed = 1, k = 0, penalty = "MCP"),     ## Main effects MCP model
      SRMp = sparseR(CASE ~ ., train, seed = 1, poly = 2, penalty = "MCP")   ## SRM + polynomials
    )

    n <- lapply(mcp, plot, log.l = TRUE)

SCAD (not evaluated):

    scad <- list(
      SRS  = sparseR(CASE ~ ., train, seed = 1, penalty = "SCAD"),            ## SRS model
      APS  = sparseR(CASE ~ ., train, seed = 1, gamma = 0, penalty = "SCAD"), ## APS model
      MES   = sparseR(CASE ~ ., train, seed = 1, k = 0, penalty = "SCAD"),    ## Main effects SCAD model
      SRSp = sparseR(CASE ~ ., train, seed = 1, poly = 2, penalty = "SCAD")   ## SRS + polynomials
    )

    n <- lapply(scad, plot, log.l = TRUE)

Summaries (using `results_summary` and `results_1se_summary` values within each object; formatting omitted).

    lapply(lso, function(x) bind_rows(x$results_summary, x$results1se_summary))
    lapply(mcp, function(x) bind_rows(x$results_summary, x$results1se_summary))

| Min CV | CV1se
--- | --- | ---
Vartype | Penalty | Total | Selected | Saturation | Selected | Saturation
SRL
Main effect | 4.583 | 21 | 11 | 0.524 | 3 | 0.143
Order 1 interaction | 13.191 | 174 | 24 | 0.138 | 0 | 0.000
APL
Order 2 polynomial | 5.477 | 9 | 6 | 0.667 | 1 | 0.111
Main effect | 1.000 | 21 | 4 | 0.190 | 3 | 0.143
MEL
Order 1 interaction | 1.000 | 174 | 37 | 0.213 | 6 | 0.034
SRLp
Order 2 polynomial | 1.000 | 9 | 2 | 0.222 | 1 | 0.111
Main effect | 4.583 | 21 | 10 | 0.476 | 3 | 0.143
Order 2 polynomial | 5.477 | 9 | 1 | 0.111 | 0 | 0.000
Main effect | 4.583 | 21 | 11 | 0.524 | 3 | 0.143
Order 1 interaction | 13.191 | 174 | 24 | 0.138 | 0 | 0.000
Order 2 polynomial | 5.477 | 9 | 6 | 0.667 | 1 | 0.111

| Min CV | CV1se
--- | --- | ---
Vartype | Penalty | Total | Selected | Saturation | Selected | Saturation
SRM
Main effect | 4.583 | 21 | 5 | 0.238 | 1 | 0.048
Order 1 interaction | 13.191 | 174 | 8 | 0.046 | 0 | 0.000
APM
Order 2 polynomial | 5.477 | 9 | 5 | 0.556 | 0 | 0.000
Main effect | 1.000 | 21 | 1 | 0.048 | 1 | 0.048
MEM
Order 1 interaction | 1.000 | 174 | 13 | 0.075 | 0 | 0.000
SRMp
Order 2 polynomial | 1.000 | 9 | 1 | 0.111 | 0 | 0.000
Main effect | 4.583 | 21 | 4 | 0.190 | 1 | 0.048
Order 2 polynomial | 5.477 | 9 | 2 | 0.222 | 0 | 0.000
Main effect | 4.583 | 21 | 5 | 0.238 | 1 | 0.048
Order 1 interaction | 13.191 | 174 | 8 | 0.046 | 0 | 0.000
Order 2 polynomial | 5.477 | 9 | 5 | 0.556 | 0 | 0.000

# 4 Use case: heart data

    ## Load Data set, correctly code factors + outcome
    data("Detrano")
    cleveland$thal <- factor(cleveland$thal)
    cleveland$case <- 1*(cleveland$num > 0)

    # Convert variables into factor variables if necessary!
    summary(cleveland)
    ##       age             sex               cp           trestbps
    ##  Min.   :29.00   Min.   :0.0000   Min.   :1.000   Min.   : 94.0
    ##  1st Qu.:48.00   1st Qu.:0.0000   1st Qu.:3.000   1st Qu.:120.0
    ##  Median :56.00   Median :1.0000   Median :3.000   Median :130.0
    ##  Mean   :54.44   Mean   :0.6799   Mean   :3.158   Mean   :131.7
    ##  3rd Qu.:61.00   3rd Qu.:1.0000   3rd Qu.:4.000   3rd Qu.:140.0
    ##  Max.   :77.00   Max.   :1.0000   Max.   :4.000   Max.   :200.0
    ##
    ##       chol            fbs            restecg          thalach
    ##  Min.   :126.0   Min.   :0.0000   Min.   :0.0000   Min.   : 71.0
    ##  1st Qu.:211.0   1st Qu.:0.0000   1st Qu.:0.0000   1st Qu.:133.5
    ##  Median :241.0   Median :0.0000   Median :1.0000   Median :153.0
    ##  Mean   :246.7   Mean   :0.1485   Mean   :0.9901   Mean   :149.6
    ##  3rd Qu.:275.0   3rd Qu.:0.0000   3rd Qu.:2.0000   3rd Qu.:166.0
    ##  Max.   :564.0   Max.   :1.0000   Max.   :2.0000   Max.   :202.0
    ##
    ##      exang           oldpeak         slope             ca           thal
    ##  Min.   :0.0000   Min.   :0.00   Min.   :1.000   Min.   :0.0000   3   :166
    ##  1st Qu.:0.0000   1st Qu.:0.00   1st Qu.:1.000   1st Qu.:0.0000   6   : 18
    ##  Median :0.0000   Median :0.80   Median :2.000   Median :0.0000   7   :117
    ##  Mean   :0.3267   Mean   :1.04   Mean   :1.601   Mean   :0.6722   NA's:  2
    ##  3rd Qu.:1.0000   3rd Qu.:1.60   3rd Qu.:2.000   3rd Qu.:1.0000
    ##  Max.   :1.0000   Max.   :6.20   Max.   :3.000   Max.   :3.0000
    ##                                                  NA's   :4
    ##       num              case
    ##  Min.   :0.0000   Min.   :0.0000
    ##  1st Qu.:0.0000   1st Qu.:0.0000
    ##  Median :0.0000   Median :0.0000
    ##  Mean   :0.9373   Mean   :0.4587
    ##  3rd Qu.:2.0000   3rd Qu.:1.0000
    ##  Max.   :4.0000   Max.   :1.0000
    ##
    sapply(cleveland, function(x) length(unique(x)))
    ##      age      sex       cp trestbps     chol      fbs  restecg  thalach
    ##       41        2        4       50      152        2        3       91
    ##    exang  oldpeak    slope       ca     thal      num     case
    ##        2       40        3        5        4        5        2
    cleveland$sex <- factor(cleveland$sex)
    cleveland$fbs <- factor(cleveland$fbs)
    cleveland$exang <- factor(cleveland$exang)

    # Set seed for reproducibility
    set.seed(167)

    # Split data into test and train
    N <- nrow(cleveland)
    trainIDX <- sample(1:N, N*.5)
    trainDF <- cleveland[trainIDX,] %>%
      select(-num)
    testDF <- cleveland[-trainIDX,] %>%
      select(-num)

    # Simulate missing data
    trainDF$thal[2] <- trainDF$thalach[1] <- NA

    lso <- list(
      SRL  = sparseR(case ~ ., trainDF, seed = 1), ## SRL model
      APL  = sparseR(case ~ ., trainDF, seed = 1, gamma = 0), ## APL model
      ME   = sparseR(case ~ ., trainDF, seed = 1, k = 0), ## Main effects model
      SRLp = sparseR(case ~ ., trainDF, seed = 1, poly = 2) ## SRL + polynomials
    )

    lso
    ## $SRL
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=162
    ##   (At lambda=0.0033):
    ##     Nonzero coefficients: 24
    ##     Cross-validation error (deviance): 0.12
    ##     R-squared: 0.51
    ##     Signal-to-noise ratio: 1.04
    ##     Scale estimate (sigma): 0.348
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    18       14     0.7778    4.24
    ##  Order 1 interaction   135        7     0.0519   11.62
    ##   Order 2 polynomial     9        3     0.3333    5.20
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=162
    ##   (At lambda=0.0156):
    ##     Nonzero coefficients: 8
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.45
    ##     Signal-to-noise ratio: 0.82
    ##     Scale estimate (sigma): 0.368
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    18        8      0.444    4.24
    ##  Order 1 interaction   135        0      0.000   11.62
    ##   Order 2 polynomial     9        0      0.000    5.20
    ##
    ## $APL
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=162
    ##   (At lambda=0.0635):
    ##     Nonzero coefficients: 15
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.43
    ##     Signal-to-noise ratio: 0.74
    ##     Scale estimate (sigma): 0.376
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    18        6     0.3333       1
    ##  Order 1 interaction   135        9     0.0667       1
    ##   Order 2 polynomial     9        0     0.0000       1
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=162
    ##   (At lambda=0.1144):
    ##     Nonzero coefficients: 9
    ##     Cross-validation error (deviance): 0.15
    ##     R-squared: 0.37
    ##     Signal-to-noise ratio: 0.59
    ##     Scale estimate (sigma): 0.393
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    18        5     0.2778       1
    ##  Order 1 interaction   135        4     0.0296       1
    ##   Order 2 polynomial     9        0     0.0000       1
    ##
    ## $ME
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=27
    ##   (At lambda=0.0024):
    ##     Nonzero coefficients: 17
    ##     Cross-validation error (deviance): 0.12
    ##     R-squared: 0.52
    ##     Signal-to-noise ratio: 1.09
    ##     Scale estimate (sigma): 0.343
    ##
    ##   SR information:
    ##             Vartype Total Selected Saturation Penalty
    ##         Main effect    18       14      0.778    4.24
    ##  Order 2 polynomial     9        3      0.333    5.20
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=27
    ##   (At lambda=0.0133):
    ##     Nonzero coefficients: 11
    ##     Cross-validation error (deviance): 0.13
    ##     R-squared: 0.46
    ##     Signal-to-noise ratio: 0.86
    ##     Scale estimate (sigma): 0.364
    ##
    ##   SR information:
    ##             Vartype Total Selected Saturation Penalty
    ##         Main effect    18       11      0.611    4.24
    ##  Order 2 polynomial     9        0      0.000    5.20
    ##
    ## $SRLp
    ##
    ## Model summary @ min CV:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=162
    ##   (At lambda=0.0033):
    ##     Nonzero coefficients: 24
    ##     Cross-validation error (deviance): 0.12
    ##     R-squared: 0.51
    ##     Signal-to-noise ratio: 1.04
    ##     Scale estimate (sigma): 0.348
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    18       14     0.7778    4.24
    ##  Order 1 interaction   135        7     0.0519   11.62
    ##   Order 2 polynomial     9        3     0.3333    5.20
    ##
    ##
    ## Model summary @ CV1se:
    ## -----------------------------------------------------
    ##   lasso-penalized linear regression with n=151, p=162
    ##   (At lambda=0.0156):
    ##     Nonzero coefficients: 8
    ##     Cross-validation error (deviance): 0.14
    ##     R-squared: 0.45
    ##     Signal-to-noise ratio: 0.82
    ##     Scale estimate (sigma): 0.368
    ##
    ##   SR information:
    ##              Vartype Total Selected Saturation Penalty
    ##          Main effect    18        8      0.444    4.24
    ##  Order 1 interaction   135        0      0.000   11.62
    ##   Order 2 polynomial     9        0      0.000    5.20
    plot(lso$SRL)

    lapply(lso, plot, log.l = TRUE)