R: Fit a group penalized regression path

grpreg {grpreg} | R Documentation

## Fit a group penalized regression path

### Description

Fit regularization paths for models with grouped penalties over a grid of values for the regularization parameter lambda. Fits linear and logistic regression models.

### Usage

    grpreg(X, y, group=1:ncol(X), penalty=c("grLasso", "grMCP", "grSCAD",
    "gel", "cMCP"), family=c("gaussian", "binomial", "poisson"),
    nlambda=100, lambda, lambda.min={if (nrow(X) > ncol(X)) 1e-4 else .05},
    log.lambda = TRUE, alpha=1, eps=1e-4, max.iter=10000, dfmax=p,
    gmax=length(unique(group)), gamma=ifelse(penalty == "grSCAD", 4, 3),
    tau = 1/3, group.multiplier, warn=TRUE, returnX = FALSE, ...)

### Arguments

`X` | The design matrix, without an intercept. `grpreg` standardizes the data and includes an intercept by default.
`y` | The response vector, or a matrix in the case of multitask learning (see details).
`group` | A vector describing the grouping of the coefficients. For greatest efficiency and least ambiguity (see details), it is best if `group` is a factor or vector of consecutive integers, although unordered groups and character vectors are also allowed. If there are coefficients to be included in the model without being penalized, assign them to group 0 (or `"0"`).
`penalty` | The penalty to be applied to the model. For group selection, one of `grLasso`, `grMCP`, or `grSCAD`. For bi-level selection, one of `gel` or `cMCP`. See below for details.
`family` | Either "gaussian" or "binomial", depending on the response.
`nlambda` | The number of `lambda` values. Default is 100.
`lambda` | A user supplied sequence of `lambda` values. Typically, this is left unspecified, and the function automatically computes a grid of lambda values that ranges uniformly on the log scale over the relevant range of lambda values.
`lambda.min` | The smallest value for `lambda`, as a fraction of `lambda.max`. Default is .0001 if the number of observations is larger than the number of covariates and .05 otherwise.
`log.lambda` | Whether compute the grid values of lambda on log scale (default) or linear scale.
`alpha` | `grpreg` allows for both a group penalty and an L2 (ridge) penalty; `alpha` controls the proportional weight of the regularization parameters of these two penalties. The group penalties' regularization parameter is `lambda*alpha`, while the regularization parameter of the ridge penalty is `lambda*(1-alpha)`. Default is 1: no ridge penalty.
`eps` | Convergence threshhold. The algorithm iterates until the RMSD for the change in linear predictors for each coefficient is less than `eps`. Default is `1e-4`. See details.
`max.iter` | Maximum number of iterations (total across entire path). Default is 10000. See details.
`dfmax` | Limit on the number of parameters allowed to be nonzero. If this limit is exceeded, the algorithm will exit early from the regularization path.
`gmax` | Limit on the number of groups allowed to have nonzero elements. If this limit is exceeded, the algorithm will exit early from the regularization path.
`gamma` | Tuning parameter of the group or composite MCP/SCAD penalty (see details). Default is 3 for MCP and 4 for SCAD.
`tau` | Tuning parameter for the group exponential lasso; defaults to 1/3.
`group.multiplier` | A vector of values representing multiplicative factors by which each group's penalty is to be multiplied. Often, this is a function (such as the square root) of the number of predictors in each group. The default is to use the square root of group size for the group selection methods, and a vector of 1's (i.e., no adjustment for group size) for bi-level selection.
`warn` | Should the function give a warning if it fails to converge? Default is TRUE. See details.
`returnX` | Return the standardized design matrix (and associated group structure information)? Default is FALSE.
`...` | Arguments passed to other functions (such as gBridge).

### Details

There are two general classes of methods involving grouped penalties: those that carry out bi-level selection and those that carry out group selection. Bi-level means carrying out variable selection at the group level as well as the level of individual covariates (i.e., selecting important groups as well as important members of those groups). Group selection selects important groups, and not members within the group – i.e., within a group, coefficients will either all be zero or all nonzero. The `grLasso`, `grMCP`, and `grSCAD` penalties carry out group selection, while the `gel` and `cMCP` penalties carry out bi-level selection. For bi-level selection, see also the `gBridge` function. For historical reasons and backwards compatibility, some of these penalties have aliases; e.g., `gLasso` will do the same thing as `grLasso`, but users are encouraged to use `grLasso`.

Please note the distinction between `grMCP` and `cMCP`. The former involves an MCP penalty being applied to an L2-norm of each group. The latter involves a hierarchical penalty which places an outer MCP penalty on a sum of inner MCP penalties for each group, as proposed in Breheny & Huang, 2009. Either penalty may be referred to as the "group MCP", depending on the publication. To resolve this confusion, Huang et al. (2012) proposed the name "composite MCP" for the `cMCP` penalty.

For more information about the penalties and their properties, please consult the references below, many of which contain discussion, case studies, and simulation studies comparing the methods. If you use `grpreg` for an analysis, please cite the appropriate reference.

In keeping with the notation from the original MCP paper, the tuning parameter of the MCP penalty is denoted 'gamma'. Note, however, that in Breheny and Huang (2009), `gamma` is denoted 'a'.

The objective function for `grpreg` optimization is defined to be

Q(β|X, y) = (1/n)*L(β|X, y) + P(β, λ),

where the loss function L is the deviance (-2 times the log likelihood) for the specified outcome distribution (gaussian/binomial/poisson). For more details, refer to the following:

- Models and loss functions

- Penalties

For the bi-level selection methods, a locally approximated coordinate descent algorithm is employed. For the group selection methods, group descent algorithms are employed.

The algorithms employed by `grpreg` are stable and generally converge quite rapidly to values close to the solution. However, especially when p is large compared with n, `grpreg` may fail to converge at low values of `lambda`, where models are nonidentifiable or nearly singular. Often, this is not the region of the coefficient path that is most interesting. The default behavior warning the user when convergence criteria are not met may be distracting in these cases, and can be modified with `warn` (convergence can always be checked later by inspecting the value of `iter`).

If models are not converging, increasing `max.iter` may not be the most efficient way to correct this problem. Consider increasing `n.lambda` or `lambda.min` in addition to increasing `max.iter`.

Although `grpreg` allows groups to be unordered and given arbitary names, it is recommended that you specify groups as consecutive integers. The first reason is efficiency: if groups are out of order, `X` must be reordered prior to fitting, then this process reversed to return coefficients according to the original order of `X`. This is inefficient if `X` is very large. The second reason is ambiguity with respect to other arguments such as `group.multiplier`. With consecutive integers, `group=3` unambiguously denotes the third element of `group.multiplier`.

Seemingly unrelated regressions/multitask learning can be carried out using `grpreg` by passing a matrix to `y`. In this case, `X` will be used in separate regressions for each column of `y`, with the coefficients grouped across the responses. In other words, each column of `X` will form a group with m members, where m is the number of columns of `y`. For multiple Gaussian responses, it is recommended to standardize the columns of `y` prior to fitting, in order to apply the penalization equally across columns.

`grpreg` requires groups to be non-overlapping.

### Value

An object with S3 class `"grpreg"` containing:

beta

  The fitted matrix of coefficients. The number of rows is equal to the number of coefficients, and the number of columns is equal to `nlambda`.

family

  Same as above.

group

  Same as above.

lambda

  The sequence of `lambda` values in the path.

alpha

  Same as above.

loss

  A vector containing either the residual sum of squares (`"gaussian"`) or negative log-likelihood (`"binomial"`) of the fitted model at each value of `lambda`.

n

  Number of observations.

penalty

  Same as above.

df

  A vector of length `nlambda` containing estimates of effective number of model parameters all the points along the regularization path. For details on how this is calculated, see Breheny and Huang (2009).

iter

  A vector of length `nlambda` containing the number of iterations until convergence at each value of `lambda`.

group.multiplier

  A named vector containing the multiplicative constant applied to each group's penalty.

### Author(s)

Patrick Breheny

### References

- Breheny P and Huang J. (2009) Penalized methods for bi-level variable selection. Statistics and its interface, 2: 369-380. doi: 10.4310/sii.2009.v2.n3.a10

- Huang J, Breheny P, and Ma S. (2012). A selective review of group selection in high dimensional models. Statistical Science, 27: 481-499. doi: 10.1214/12-sts392

- Breheny P and Huang J. (2015) Group descent algorithms for nonconvex penalized linear and logistic regression models with grouped predictors. Statistics and Computing, 25: 173-187. doi: 10.1007/s11222-013-9424-2

- Breheny P. (2015) The group exponential lasso for bi-level variable selection. Biometrics, 71: 731-740. doi: 10.1111/biom.12300

### See Also

`cv.grpreg`, as well as `plot` and `select` methods.

### Examples

    # Birthweight data
    data(Birthwt)
    X <- Birthwt$X
    group <- Birthwt$group

    # Linear regression
    y <- Birthwt$bwt
    fit <- grpreg(X, y, group, penalty="grLasso")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="grMCP")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="grSCAD")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="gel")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="cMCP")
    plot(fit)
    select(fit, "AIC")

    # Logistic regression
    y <- Birthwt$low
    fit <- grpreg(X, y, group, penalty="grLasso", family="binomial")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="grMCP", family="binomial")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="grSCAD", family="binomial")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="gel", family="binomial")
    plot(fit)
    fit <- grpreg(X, y, group, penalty="cMCP", family="binomial")
    plot(fit)
    select(fit, "BIC")

    # Multitask learning (simulated example)
    set.seed(1)
    n <- 50
    p <- 10
    k <- 5
    X <- matrix(runif(n*p), n, p)
    y <- matrix(rnorm(n*k, X[,1] + X[,2]), n, k)
    fit <- grpreg(X, y)
    # Note that group is set up automatically
    fit$group
    plot(fit)

[Package grpreg version 3.4.0 Index]