# Regularization Paths for SCAD and MCP Penalized Regression Models

`ncvreg` fits regularization paths for linear regression, GLM, and Cox regression models using lasso or nonconvex penalties, in particular the minimax concave penalty (MCP) and smoothly clipped absolute deviation (SCAD) penalty, with options for additional L2 penalties (the “elastic net” idea). Utilities for carrying out cross-validation as well as post-fitting visualization, summarization, inference, and prediction are also provided.

## Basic Usage

The basic usage of ncvreg is as follows:

``fit <- ncvreg(X, y)``

The default penalty here is the minimax concave penalty (MCP), but SCAD and lasso penalties are also available. This produces a path of coefficients, which we can plot with

``plot(fit)``

Notice that variables enter the model one at a time, and that at any given value of `lambda`, several coefficients are zero. The `summary` method can be used for post-selection summarization and inference:

```{r summary} summary(fit, lambda=0.05)

# Average mfdr (6 features) : 0.418

```

`summary(fit)` also returns the following table:

Estimate z mfdr
lcavol 0.5317899 8.880429 0.0000000
svi 0.6725610 3.945052 0.0018967
lweight 0.6038969 3.665874 0.0050683
lbph 0.0887456 1.928241 0.4998035
age -0.0153092 -1.788334 1.0000000
pgg45 0.0016804 1.159772 1.0000000

In this case, it would appear that `lcavol`, `svi`, and `lweight` are clearly associated with the response, even after adjusting for the other variables in the model, while `lbph`, `age`, and `pgg45` may be false positives included simply by chance.

Typically, one would carry out cross-validation for the purposes of assessing the predictive accuracy of the model at various values of `lambda`:

`{r cvplot, h=4, w=6, cache=TRUE} cvfit <- cv.ncvreg(X, y) plot(cvfit)`

At this point, `coef(cvfit)` will return the coefficients at the value of `lambda` minimizing the cross-validation error. Likewise,

``predict(cvfit, X=head(X))``

will return predictions for that model, while

``predict(cvfit, type="nvars")``

will return the number of nonzero coefficients. Note that the original fit (to the full data set) is returned as `cvfit\$fit`; it is not necessary to call both `ncvreg` and `cv.ncvreg` to analyze a data set. For example, `plot(cvfit\$fit)` will produce the same coefficient path plot as `plot(fit)` above.

## Documentation and Citation

For more on the usage and syntax of `ncvreg`, see the ncvreg homepage.

For more on the algorithms used by `ncvreg`, see the original article:

For more about the marginal false discovery rate idea used for post-selection inference, see

## Installation

• To install the latest release version from CRAN: `install.packages("ncvreg")`
• To install the latest development version from GitHub: `devtools::install_github("pbreheny/ncvreg")`