The **xrnet** R package is an extension of regularized regression (i.e. ridge regression) that enables the incorporation of external data that may be informative for the effects of predictors on an outcome of interest. Let (y) be an n-dimensional observed outcome vector, (X) be a set of *p* potential predictors observed on the *n* observations, and (Z) be a set of *q* external features available for the *p* predictors. Our model builds off the standard two-level hierarchical regression model,

but allows regularization of both the predictors and the external features, where beta is the vector of coefficients describing the association of each predictor with the outcome and alpha is the vector of coefficients describing the association of each external feature with the predictor coefficients, beta. As an example, assume that the outcome is continuous and that we want to apply a ridge penalty to the predictors and lasso penalty to the external features. We minimize the following objective function (ignoring intercept terms):

Note that our model allows for the predictor coefficients, beta, to shrink towards potentially informative values based on the matrix (Z). In the event the external data is not informative, we can shrink alpha towards zero, returning back to a standard regularized regression. To efficiently fit the model, we rewrite this convex optimization with the variable substitution (gamma = beta - Z * alpha). The problem is then solved as a standard regularized regression in which we allow the penalty value and type (ridge / lasso) to be variable-specific:

This package extends the coordinate descent algorithm of Friedman et al. 2010 (used in the R package **glmnet**) to allow for this variable-specific generalization and to fit the model described above. Currently, we allow for continuous and binary outcomes, but plan to extend to other outcomes (i.e. survival) in the next release.

- OS-specific prerequisites
- Install the R package devtools
- Install the
**xrnet**package with the*install_github()*function (optionally install potentially unstable development branch)

```
# Master branch
devtools::install_github("USCbiostats/xrnet")
# Development branch
devtools::install_github("USCbiostats/xrnet", ref = "development")
```

As an example of how you might use xrnet, we have provided a small set of simulated external data variables (ext), predictors (x), and a continuous outcome variable (y). First, load the package and the example data:

To fit a linear hierarchical regularized regression model, use the main `xrnet`

function. At a minimum, you should specify the predictor matrix `x`

, outcome variable `y`

, and `family`

(outcome distribution). The `external`

option allows you to incorporate external data in the regularized regression model. If you do not include external data, a standard regularized regression model will be fit. By default, a lasso penalty is applied to both the predictors and the external data.

To modify the regularization terms and penalty path associated with the predictors or external data, you can use the `define_penalty`

function. This function allows you to configure the following regularization attributes:

- Regularization type
- Ridge = 0
- Elastic Net = (0, 1)
- Lasso / Quantile = 1 (additional parameter
`quantile`

used to specify quantile, not currently implemented)

- Penalty path
- Number of penalty values in the full penalty path (default = 20)
- Ratio of min(penalty) / max(penalty)

- User-defined set of penalties

As an example, we may want to apply a ridge penalty to the x variables and a lasso penalty to the external data variables. In addition, we may want to have 30 penalty values computed for the regularization path associated with both x and external. We modify our model call to `xrnet`

follows.

`penalty_main`

is used to specify the regularization for the x variables`penalty_external`

is used to specify the regularization for the external variables

```
xrnet_model <- xrnet(
x = x_linear,
y = y_linear,
external = ext_linear,
family = "gaussian",
penalty_main = define_penalty(0, num_penalty = 30),
penalty_external = define_penalty(1, num_penalty = 30)
)
```

Helper functions are also available to define the available penalty types (`define_lasso`

, `define_ridge`

, and `define_enet`

). The example below exemplifies fitting a standard ridge regression model with 100 penalty values by using the `define_ridge`

helper function. As mentioned previously, a standard regularized regression is fit if no external data is provided.

```
xrnet_model <- xrnet(
x = x_linear,
y = y_linear,
family = "gaussian",
penalty_main = define_ridge(100)
)
```

In general, we need a method to determine the penalty values that produce the optimal out-of-sample prediction. We provide a simple two-dimensional grid search that uses k-fold cross-validation to determine the optimal values for the penalties. The cross-validation function `tune_xrnet`

is used as follows.

```
cv_xrnet <- tune_xrnet(
x = x_linear,
y = y_linear,
external = ext_linear,
family = "gaussian",
penalty_main = define_ridge(),
penalty_external = define_lasso()
)
```

To visualize the results of the cross-validation we provide a contour plot of the mean cross-validation error across the grid of penalties with the `plot`

function.

Cross-validation error curves can also be generated with `plot`

by fixing the value of either the penalty on `x`

or the external penalty on `external`

. By default, either penalty defaults the optimal penalty on `x`

or `external`

.

The `predict`

function can be used to predict responses and to obtain the coefficient estimates at the optimal penalty combination (the default) or any other penalty combination that is within the penalty path(s). `coef`

is a another help function that can be used to return the coefficients for a combination of penalty values as well.

As an example of using `bigmemory`

with `xrnet`

, we have a provided a ASCII file, `x_linear.txt`

, that contains the data for `x`

. The `bigmemory`

function `read.big.matrix()`

can be used to create a `big.matrix`

version of this file. The ASCII file is located under `inst/extdata`

in this repository and is also included when you install the R package. To access the file in the R package, use `system.file("extdata", "x_linear.txt", package = "xrnet")`

as shown in the example below.

`x_big <- bigmemory::read.big.matrix(system.file("extdata", "x_linear.txt", package = "xrnet"), type = "double")`

We can now fit a ridge regression model with the `big.matrix`

version of the data and verify that we get the same estimates:

```
xrnet_model_big <- xrnet(
x = x_big,
y = y_linear,
family = "gaussian",
penalty_main = define_ridge(100)
)
all.equal(xrnet_model$beta0, xrnet_model_big$beta0)
#> [1] TRUE
all.equal(xrnet_model$betas, xrnet_model_big$betas)
#> [1] TRUE
all.equal(xrnet_model$alphas, xrnet_model_big$alphas)
#> [1] TRUE
```

To report a bug, ask a question, or propose a feature, create a new issue here. This project is released with the following Contributor Code of Conduct. If you would like to contribute, please abide by its terms.

Supported by National Cancer Institute Grant #1P01CA196596.