R: fit a GLM with lasso or elasticnet regularization

glmnet {glmnet} | R Documentation

## fit a GLM with lasso or elasticnet regularization

### Description

Fit a generalized linear model via penalized maximum likelihood. The regularization path is computed for the lasso or elasticnet penalty at a grid of values for the regularization parameter lambda. Can deal with all shapes of data, including very large sparse data matrices. Fits linear, logistic and multinomial, poisson, and Cox regression models.

### Usage

    glmnet(
      x,
      y,
      family = c("gaussian", "binomial", "poisson", "multinomial", "cox", "mgaussian"),
      weights = NULL,
      offset = NULL,
      alpha = 1,
      nlambda = 100,
      lambda.min.ratio = ifelse(nobs < nvars, 0.01, 1e-04),
      lambda = NULL,
      standardize = TRUE,
      intercept = TRUE,
      thresh = 1e-07,
      dfmax = nvars + 1,
      pmax = min(dfmax * 2 + 20, nvars),
      exclude = NULL,
      penalty.factor = rep(1, nvars),
      lower.limits = -Inf,
      upper.limits = Inf,
      maxit = 1e+05,
      type.gaussian = ifelse(nvars < 500, "covariance", "naive"),
      type.logistic = c("Newton", "modified.Newton"),
      standardize.response = FALSE,
      type.multinomial = c("ungrouped", "grouped"),
      relax = FALSE,
      trace.it = 0,
      ...
    )

    relax.glmnet(fit, x, ..., maxp = n - 3, path = FALSE, check.args = TRUE)

### Arguments

`x` | input matrix, of dimension nobs x nvars; each row is an observation vector. Can be in sparse matrix format (inherit from class `"sparseMatrix"` as in package `Matrix`)
`y` | response variable. Quantitative for `family="gaussian"`, or `family="poisson"` (non-negative counts). For `family="binomial"` should be either a factor with two levels, or a two-column matrix of counts or proportions (the second column is treated as the target class; for a factor, the last level in alphabetical order is the target class). For `family="multinomial"`, can be a `nc>=2` level factor, or a matrix with `nc` columns of counts or proportions. For either `"binomial"` or `"multinomial"`, if `y` is presented as a vector, it will be coerced into a factor. For `family="cox"`, preferably a `Surv` object from the survival package: see Details section for more information. For `family="mgaussian"`, `y` is a matrix of quantitative responses.
`family` | Either a character string representing one of the built-in families, or else a `glm()` family object. For more information, see Details section below or the documentation for response type (above).
`weights` | observation weights. Can be total counts if responses are proportion matrices. Default is 1 for each observation
`offset` | A vector of length `nobs` that is included in the linear predictor (a `nobs x nc` matrix for the `"multinomial"` family). Useful for the `"poisson"` family (e.g. log of exposure time), or for refining a model by starting at a current fit. Default is `NULL`. If supplied, then values must also be supplied to the `predict` function.
`alpha` | The elasticnet mixing parameter, with 0≤α≤ 1. The penalty is defined as (1-α)/2||β||_2^2+α||β||_1. `alpha=1` is the lasso penalty, and `alpha=0` the ridge penalty.
`nlambda` | The number of `lambda` values - default is 100.
`lambda.min.ratio` | Smallest value for `lambda`, as a fraction of `lambda.max`, the (data derived) entry value (i.e. the smallest value for which all coefficients are zero). The default depends on the sample size `nobs` relative to the number of variables `nvars`. If `nobs > nvars`, the default is `0.0001`, close to zero. If `nobs < nvars`, the default is `0.01`. A very small value of `lambda.min.ratio` will lead to a saturated fit in the `nobs < nvars` case. This is undefined for `"binomial"` and `"multinomial"` models, and `glmnet` will exit gracefully when the percentage deviance explained is almost 1.
`lambda` | A user supplied `lambda` sequence. Typical usage is to have the program compute its own `lambda` sequence based on `nlambda` and `lambda.min.ratio`. Supplying a value of `lambda` overrides this. WARNING: use with care. Avoid supplying a single value for `lambda` (for predictions after CV use `predict()` instead). Supply instead a decreasing sequence of `lambda` values. `glmnet` relies on its warms starts for speed, and its often faster to fit a whole path than compute a single fit.
`standardize` | Logical flag for x variable standardization, prior to fitting the model sequence. The coefficients are always returned on the original scale. Default is `standardize=TRUE`. If variables are in the same units already, you might not wish to standardize. See details below for y standardization with `family="gaussian"`.
`intercept` | Should intercept(s) be fitted (default=TRUE) or set to zero (FALSE)
`thresh` | Convergence threshold for coordinate descent. Each inner coordinate-descent loop continues until the maximum change in the objective after any coefficient update is less than `thresh` times the null deviance. Defaults value is `1E-7`.
`dfmax` | Limit the maximum number of variables in the model. Useful for very large `nvars`, if a partial path is desired.
`pmax` | Limit the maximum number of variables ever to be nonzero
`exclude` | Indices of variables to be excluded from the model. Default is none. Equivalent to an infinite penalty factor for the variables excluded (next item). Users can supply instead an `exclude` function that generates the list of indices. This function is most generally defined as `function(x, y, weights, ...)`, and is called inside `glmnet` to generate the indices for excluded variables. The `...` argument is required, the others are optional. This is useful for filtering wide data, and works correctly with `cv.glmnet`. See the vignette 'Introduction' for examples.
`penalty.factor` | Separate penalty factors can be applied to each coefficient. This is a number that multiplies `lambda` to allow differential shrinkage. Can be 0 for some variables, which implies no shrinkage, and that variable is always included in the model. Default is 1 for all variables (and implicitly infinity for variables listed in `exclude`). Note: the penalty factors are internally rescaled to sum to nvars, and the lambda sequence will reflect this change.
`lower.limits` | Vector of lower limits for each coefficient; default `-Inf`. Each of these must be non-positive. Can be presented as a single value (which will then be replicated), else a vector of length `nvars`
`upper.limits` | Vector of upper limits for each coefficient; default `Inf`. See `lower.limits`
`maxit` | Maximum number of passes over the data for all lambda values; default is 10^5.
`type.gaussian` | Two algorithm types are supported for (only) `family="gaussian"`. The default when `nvar<500` is `type.gaussian="covariance"`, and saves all inner-products ever computed. This can be much faster than `type.gaussian="naive"`, which loops through `nobs` every time an inner-product is computed. The latter can be far more efficient for `nvar >> nobs` situations, or when `nvar > 500`.
`type.logistic` | If `"Newton"` then the exact hessian is used (default), while `"modified.Newton"` uses an upper-bound on the hessian, and can be faster.
`standardize.response` | This is for the `family="mgaussian"` family, and allows the user to standardize the response variables
`type.multinomial` | If `"grouped"` then a grouped lasso penalty is used on the multinomial coefficients for a variable. This ensures they are all in our out together. The default is `"ungrouped"`
`relax` | If `TRUE` then for each active set in the path of solutions, the model is refit without any regularization. See `details` for more information. This argument is new, and users may experience convergence issues with small datasets, especially with non-gaussian families. Limiting the value of 'maxp' can alleviate these issues in some cases.
`trace.it` | If `trace.it=1`, then a progress bar is displayed; useful for big models that take a long time to fit.
`...` | Additional argument used in `relax.glmnet`. These include some of the original arguments to 'glmnet', and each must be named if used.
`fit` | For `relax.glmnet` a fitted 'glmnet' object
`maxp` | a limit on how many relaxed coefficients are allowed. Default is 'n-3', where 'n' is the sample size. This may not be sufficient for non-gaussian familes, in which case users should supply a smaller value. This argument can be supplied directly to 'glmnet'.
`path` | Since `glmnet` does not do stepsize optimization, the Newton algorithm can get stuck and not converge, especially with relaxed fits. With `path=TRUE`, each relaxed fit on a particular set of variables is computed pathwise using the original sequence of lambda values (with a zero attached to the end). Not needed for Gaussian models, and should not be used unless needed, since will lead to longer compute times. Default is `path=FALSE`. appropriate subset of variables
`check.args` | Should `relax.glmnet` make sure that all the data dependent arguments used in creating 'fit' have been resupplied. Default is 'TRUE'.

### Details

The sequence of models implied by `lambda` is fit by coordinate descent. For `family="gaussian"` this is the lasso sequence if `alpha=1`, else it is the elasticnet sequence.

The objective function for `"gaussian"` is

1/2 RSS/nobs + λ*penalty,

and for the other models it is

-loglik/nobs + λ*penalty.

Note also that for `"gaussian"`, `glmnet` standardizes y to have unit variance (using 1/n rather than 1/(n-1) formula) before computing its lambda sequence (and then unstandardizes the resulting coefficients); if you wish to reproduce/compare results with other software, best to supply a standardized y. The coefficients for any predictor variables with zero variance are set to zero for all values of lambda.

#### Details on `family` option

From version 4.0 onwards, glmnet supports both the original built-in families, as well as any family object as used by `stats:glm()`. This opens the door to a wide variety of additional models. For example `family=binomial(link=cloglog)` or `family=negative.binomial(theta=1.5)` (from the MASS library). Note that the code runs faster for the built-in families.

The built in families are specifed via a character string. For all families, the object produced is a lasso or elasticnet regularization path for fitting the generalized linear regression paths, by maximizing the appropriate penalized log-likelihood (partial likelihood for the "cox" model). Sometimes the sequence is truncated before `nlambda` values of `lambda` have been used, because of instabilities in the inverse link functions near a saturated fit. `glmnet(...,family="binomial")` fits a traditional logistic regression model for the log-odds. `glmnet(...,family="multinomial")` fits a symmetric multinomial model, where each class is represented by a linear model (on the log-scale). The penalties take care of redundancies. A two-class `"multinomial"` model will produce the same fit as the corresponding `"binomial"` model, except the pair of coefficient matrices will be equal in magnitude and opposite in sign, and half the `"binomial"` values. Two useful additional families are the `family="mgaussian"` family and the `type.multinomial="grouped"` option for multinomial fitting. The former allows a multi-response gaussian model to be fit, using a "group -lasso" penalty on the coefficients for each variable. Tying the responses together like this is called "multi-task" learning in some domains. The grouped multinomial allows the same penalty for the `family="multinomial"` model, which is also multi-responsed. For both of these the penalty on the coefficient vector for variable j is

(1-α)/2||β_j||_2^2+α||β_j||_2.

When `alpha=1` this is a group-lasso penalty, and otherwise it mixes with quadratic just like elasticnet. A small detail in the Cox model: if death times are tied with censored times, we assume the censored times occurred just before the death times in computing the Breslow approximation; if users prefer the usual convention of after, they can add a small number to all censoring times to achieve this effect.

#### Details on response for `family="cox"`

For Cox models, the response should preferably be a `Surv` object, created by the `Surv()` function in survival package. For right-censored data, this object should have type "right", and for (start, stop] data, it should have type "counting". To fit stratified Cox models, strata should be added to the response via the `stratifySurv()` function before passing the response to `glmnet()`. (For backward compatibility, right-censored data can also be passed as a two-column matrix with columns named 'time' and 'status'. The latter is a binary variable, with '1' indicating death, and '0' indicating right censored.)

#### Details on `relax` option

If `relax=TRUE` a duplicate sequence of models is produced, where each active set in the elastic-net path is refit without regularization. The result of this is a matching `"glmnet"` object which is stored on the original object in a component named `"relaxed"`, and is part of the glmnet output. Generally users will not call `relax.glmnet` directly, unless the original 'glmnet' object took a long time to fit. But if they do, they must supply the fit, and all the original arguments used to create that fit. They can limit the length of the relaxed path via 'maxp'.

### Value

An object with S3 class `"glmnet","*"` , where `"*"` is `"elnet"`, `"lognet"`, `"multnet"`, `"fishnet"` (poisson), `"coxnet"` or `"mrelnet"` for the various types of models. If the model was created with `relax=TRUE` then this class has a prefix class of `"relaxed"`.

`call` | the call that produced this object
`a0` | Intercept sequence of length `length(lambda)`
`beta` | For `"elnet"`, `"lognet"`, `"fishnet"` and `"coxnet"` models, a `nvars x length(lambda)` matrix of coefficients, stored in sparse column format (`"CsparseMatrix"`). For `"multnet"` and `"mgaussian"`, a list of `nc` such matrices, one for each class.
`lambda` | The actual sequence of `lambda` values used. When `alpha=0`, the largest lambda reported does not quite give the zero coefficients reported (`lambda=inf` would in principle). Instead, the largest `lambda` for `alpha=0.001` is used, and the sequence of `lambda` values is derived from this.
`dev.ratio` | The fraction of (null) deviance explained (for `"elnet"`, this is the R-square). The deviance calculations incorporate weights if present in the model. The deviance is defined to be 2*(loglike_sat - loglike), where loglike_sat is the log-likelihood for the saturated model (a model with a free parameter per observation). Hence dev.ratio=1-dev/nulldev.
`nulldev` | Null deviance (per observation). This is defined to be 2*(loglike_sat -loglike(Null)); The NULL model refers to the intercept model, except for the Cox, where it is the 0 model.
`df` | The number of nonzero coefficients for each value of `lambda`. For `"multnet"`, this is the number of variables with a nonzero coefficient for any class.
`dfmat` | For `"multnet"` and `"mrelnet"` only. A matrix consisting of the number of nonzero coefficients per class
`dim` | dimension of coefficient matrix (ices)
`nobs` | number of observations
`npasses` | total passes over the data summed over all lambda values
`offset` | a logical variable indicating whether an offset was included in the model
`jerr` | error flag, for warnings and errors (largely for internal debugging).
`relaxed` | If `relax=TRUE`, this additional item is another glmnet object with different values for `beta` and `dev.ratio`

### Author(s)

Jerome Friedman, Trevor Hastie, Balasubramanian Narasimhan, Noah Simon, Kenneth Tay and Rob Tibshirani
Maintainer: Trevor Hastie hastie@stanford.edu

### References

Friedman, J., Hastie, T. and Tibshirani, R. (2008) Regularization Paths for Generalized Linear Models via Coordinate Descent (2010), Journal of Statistical Software, Vol. 33(1), 1-22, https://web.stanford.edu/~hastie/Papers/glmnet.pdf.
Simon, N., Friedman, J., Hastie, T. and Tibshirani, R. (2011) Regularization Paths for Cox's Proportional Hazards Model via Coordinate Descent, Journal of Statistical Software, Vol. 39(5), 1-13, https://www.jstatsoft.org/v39/i05/.
Tibshirani, Robert, Bien, J., Friedman, J., Hastie, T.,Simon, N.,Taylor, J. and Tibshirani, Ryan. (2012) Strong Rules for Discarding Predictors in Lasso-type Problems, JRSSB, Vol. 74(2), 245-266, https://statweb.stanford.edu/~tibs/ftp/strong.pdf.
Hastie, T., Tibshirani, Robert and Tibshirani, Ryan. Extended Comparisons of Best Subset Selection, Forward Stepwise Selection, and the Lasso (2017), Stanford Statistics Technical Report, https://arxiv.org/abs/1707.08692.
Glmnet webpage with four vignettes, https://glmnet.stanford.edu.

### See Also

`print`, `predict`, `coef` and `plot` methods, and the `cv.glmnet` function.

### Examples

    # Gaussian
    x = matrix(rnorm(100 * 20), 100, 20)
    y = rnorm(100)
    fit1 = glmnet(x, y)
    print(fit1)
    coef(fit1, s = 0.01)  # extract coefficients at a single value of lambda
    predict(fit1, newx = x[1:10, ], s = c(0.01, 0.005))  # make predictions

    # Relaxed
    fit1r = glmnet(x, y, relax = TRUE)  # can be used with any model

    # multivariate gaussian
    y = matrix(rnorm(100 * 3), 100, 3)
    fit1m = glmnet(x, y, family = "mgaussian")
    plot(fit1m, type.coef = "2norm")

    # binomial
    g2 = sample(c(0,1), 100, replace = TRUE)
    fit2 = glmnet(x, g2, family = "binomial")
    fit2n = glmnet(x, g2, family = binomial(link=cloglog))
    fit2r = glmnet(x,g2, family = "binomial", relax=TRUE)
    fit2rp = glmnet(x,g2, family = "binomial", relax=TRUE, path=TRUE)

    # multinomial
    g4 = sample(1:4, 100, replace = TRUE)
    fit3 = glmnet(x, g4, family = "multinomial")
    fit3a = glmnet(x, g4, family = "multinomial", type.multinomial = "grouped")
    # poisson
    N = 500
    p = 20
    nzc = 5
    x = matrix(rnorm(N * p), N, p)
    beta = rnorm(nzc)
    f = x[, seq(nzc)] %*% beta
    mu = exp(f)
    y = rpois(N, mu)
    fit = glmnet(x, y, family = "poisson")
    plot(fit)
    pfit = predict(fit, x, s = 0.001, type = "response")
    plot(pfit, y)

    # Cox
    set.seed(10101)
    N = 1000
    p = 30
    nzc = p/3
    x = matrix(rnorm(N * p), N, p)
    beta = rnorm(nzc)
    fx = x[, seq(nzc)] %*% beta/3
    hx = exp(fx)
    ty = rexp(N, hx)
    tcens = rbinom(n = N, prob = 0.3, size = 1)  # censoring indicator
    y = cbind(time = ty, status = 1 - tcens)  # y=Surv(ty,1-tcens) with library(survival)
    fit = glmnet(x, y, family = "cox")
    plot(fit)

    # Cox example with (start, stop] data
    set.seed(2)
    nobs <- 100; nvars <- 15
    xvec <- rnorm(nobs * nvars)
    xvec[sample.int(nobs * nvars, size = 0.4 * nobs * nvars)] <- 0
    x <- matrix(xvec, nrow = nobs)
    start_time <- runif(100, min = 0, max = 5)
    stop_time <- start_time + runif(100, min = 0.1, max = 3)
    status <- rbinom(n = nobs, prob = 0.3, size = 1)
    jsurv_ss <- survival::Surv(start_time, stop_time, status)
    fit <- glmnet(x, jsurv_ss, family = "cox")

    # Cox example with strata
    jsurv_ss2 <- stratifySurv(jsurv_ss, rep(1:2, each = 50))
    fit <- glmnet(x, jsurv_ss2, family = "cox")

    # Sparse
    n = 10000
    p = 200
    nzc = trunc(p/10)
    x = matrix(rnorm(n * p), n, p)
    iz = sample(1:(n * p), size = n * p * 0.85, replace = FALSE)
    x[iz] = 0
    sx = Matrix(x, sparse = TRUE)
    inherits(sx, "sparseMatrix")  #confirm that it is sparse
    beta = rnorm(nzc)
    fx = x[, seq(nzc)] %*% beta
    eps = rnorm(n)
    y = fx + eps
    px = exp(fx)
    px = px/(1 + px)
    ly = rbinom(n = length(px), prob = px, size = 1)
    system.time(fit1 <- glmnet(sx, y))
    system.time(fit2n <- glmnet(x, y))

[Package glmnet version 4.1-2 Index]