# Introduction

The hierr 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-stage regression model,

$y = X\beta + \epsilon$

$\beta = Z\alpha + \gamma$

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):

$\min_{\beta, \alpha}\frac{1}{2}||y - X\beta||^2_2 + \frac{\lambda_1}{2}||\beta - Z\alpha||^2_2 + \lambda_2||\alpha||_1$

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 subsitution (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:

$\min_{\gamma, \alpha}\frac{1}{2}||y - X\gamma - XZ\alpha||^2_2 + \frac{\lambda_1}{2}||\gamma||^2_2 + \lambda_2||\alpha||_1$

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 outcomes, but plan to extend to other outcomes (i.e. binomial, count).

# Setup

1. For Windows users, install RTools (not an R package)
2. Install the R package devtools
3. Install hierr package with the install_github function (optionally you can install the most recent / potentially unstable development branch)
4. Load the package
``````library(devtools)

# Master branch
install_github("USCbiostats/hierr")

# Or the development branch
install_github("USCbiostats/hierr", ref = "development")
``````
``````library(hierr)
``````

# A First Example

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

``````data(GaussianExample)
``````

#### Fitting a Model

To fit a linear hierarchical regularized regression model, use the main `hierr` 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 ridge penalty is applied to the predictors and a lasso penalty is applied to the external data.

``````hierr_model <- hierr(x = x_linear,
y = y_linear,
external = ext_linear,
family = "gaussian")
``````

#### Modifying Regularization Terms

To modify the regularization terms and penalty path associated with the predictors or external data, you can use the `definePenalty` 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)
• 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 both the x variables and 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 fitting as follows.

``````myPenalty <- definePenalty(penalty_type = 0,
penalty_type_ext = 0,
num_penalty = 30,
num_penalty_ext = 30)

hierr_model <- hierr(x = x_linear,
y = y_linear,
external = ext_linear,
family = "gaussian",
penalty = myPenalty)
``````

#### Tuning Penalty Parameters by Cross-Validation

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 `cvhierr` is used as follows.

``````cv_hierr <- cvhierr(x = x_linear,
y = y_linear,
external = ext_linear,
family = "gaussian")
``````

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.

``````plot(cv_hierr)
``````

gmweaver/hierr documentation built on Dec. 2, 2018, 5:36 p.m.