`ncvreg`

is an R package for fitting 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 L_{2} penalties (the “elastic net” idea). Utilities
for carrying out cross-validation as well as post-fitting visualization,
summarization, inference, and prediction are also provided.

This vignette offers a brief introduction to the basic use of
`ncvreg`

. For more details on the package, visit the
`ncvreg`

website at https://pbreheny.github.io/ncvreg/ and follow the links
under “Learn more”. 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

`ncvreg`

comes with a few example data sets; we’ll look at
`Prostate`

, which has 8 features and one continuous response,
the PSA levels (on the log scale) from men about to undergo radical
prostatectomy:

To fit a penalized regression model to this data:

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

Notice that variables enter the model one at a time, and that at any
given value of \(\lambda\), several
coefficients are zero. To see what the coefficients are, we could use
the `coef`

function:

```
coef(fit, lambda=0.05)
# (Intercept) lcavol lweight age lbph svi
# 0.35121089 0.53178994 0.60389694 -0.01530917 0.08874563 0.67256096
# lcp gleason pgg45
# 0.00000000 0.00000000 0.00168038
```

The `summary`

method can be used for post-selection
inference:

```
summary(fit, lambda=0.05)
# MCP-penalized linear regression with n=97, p=8
# At lambda=0.0500:
# -------------------------------------------------
# Nonzero coefficients : 6
# Expected nonzero coefficients: 2.54
# Average mfdr (6 features) : 0.424
#
# Estimate z mfdr Selected
# lcavol 0.53179 8.880 < 1e-04 *
# svi 0.67256 3.945 0.010189 *
# lweight 0.60390 3.666 0.027894 *
# lbph 0.08875 1.928 0.773014 *
# age -0.01531 -1.788 0.815269 *
# pgg45 0.00168 1.160 0.917570 *
```

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 \(\lambda\):

```
cvfit <- cv.ncvreg(X, y)
summary(cvfit)
# MCP-penalized linear regression with n=97, p=8
# At minimum cross-validation error (lambda=0.0169):
# -------------------------------------------------
# Nonzero coefficients: 7
# Cross-validation error (deviance): 0.55
# R-squared: 0.59
# Signal-to-noise ratio: 1.42
# Scale estimate (sigma): 0.739
# MCP-penalized linear regression with n=97, p=8
# At lambda=0.0169:
# -------------------------------------------------
# Nonzero coefficients : 7
# Expected nonzero coefficients: 2.71
# Average mfdr (7 features) : 0.386
#
# Estimate z mfdr Selected
# lcavol 0.569546 8.902 < 1e-04 *
# svi 0.752397 4.131 0.0039506 *
# lweight 0.614420 3.491 0.0352443 *
# age -0.020913 -2.065 0.6238332 *
# pgg45 0.005324 1.991 0.6577467 *
# lcp -0.104959 -1.946 0.6773510 *
# lbph 0.097353 1.873 0.7069026 *
plot(cvfit)
```

The value of \(\lambda\) that
minimizes the cross-validation error is given by
`cvfit$lambda.min`

, which in this case is 0.017. Applying
`coef`

to the output of `cv.ncvreg`

returns the
coefficients at that value of \(\lambda\):

```
coef(cvfit)
# (Intercept) lcavol lweight age lbph svi
# 0.494154801 0.569546027 0.614419811 -0.020913467 0.097352536 0.752397339
# lcp gleason pgg45
# -0.104959403 0.000000000 0.005324465
```

Predicted values can be obtained via `predict`

, which has
a number of options:

```
predict(cvfit, X=head(X)) # Prediction of response for new observations
# 1 2 3 4 5 6
# 0.8304040 0.7650906 0.4262072 0.6230117 1.7449492 0.8449595
predict(cvfit, type="nvars") # Number of nonzero coefficients
# 0.01695
# 7
predict(cvfit, type="vars") # Identity of the nonzero coefficients
# lcavol lweight age lbph svi lcp pgg45
# 1 2 3 4 5 6 8
```

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.