`grpsurv.Rd`

Fit regularization paths for Cox models with grouped penalties over a grid of values for the regularization parameter lambda.

```
grpsurv(X, y, group=1:ncol(X), penalty=c("grLasso", "grMCP", "grSCAD",
"gel", "cMCP"), gamma=ifelse(penalty=="grSCAD", 4, 3), alpha=1,
nlambda=100, lambda, lambda.min={if (nrow(X) > ncol(X)) 0.001 else .05},
eps=.001, max.iter=10000, dfmax=p, gmax=length(unique(group)), tau=1/3,
group.multiplier, warn=TRUE, returnX=FALSE, ...)
```

- X
The design matrix.

- y
The time-to-event outcome, as a two-column matrix or

`Surv`

object. The first column should be time on study (follow up time); the second column should be a binary variable with 1 indicating that the event has occurred and 0 indicating (right) censoring.- 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.- gamma
Tuning parameter of the group or composite MCP/SCAD penalty (see details). Default is 3 for MCP and 4 for SCAD.

- alpha
`grpsurv`

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.- nlambda
The number of lambda values. Default is 100.

- lambda.min
The smallest value for lambda, as a fraction of lambda.max. Default is .001 if the number of observations is larger than the number of covariates and .05 otherwise.

- lambda
A user-specified sequence of lambda values. By default, a sequence of values of length

`nlambda`

is computed automatically, equally spaced on the log scale.- eps
Convergence threshhold. The algorithm iterates until the RMSD for the change in linear predictors for each coefficient is less than

`eps`

. Default is`0.001`

.- max.iter
Maximum number of iterations (total across entire path). Default is 10000.

- 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.

- 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
Return warning messages for failures to converge and model saturation? Default is TRUE.

- returnX
Return the standardized design matrix? Default is FALSE.

- ...
Not used.

The sequence of models indexed by the regularization parameter
`lambda`

is fit using a coordinate descent algorithm. In order
to accomplish this, the second derivative (Hessian) of the Cox partial
log-likelihood is diagonalized (see references for details). The
objective function is defined to be
$$Q(\beta|X, y) = \frac{1}{n} L(\beta|X, y) +
P_\lambda(\beta)$$
where the loss function L is the deviance (-2 times the partial
log-likelihood) from the Cox regression mode.
See
here for more details.

Presently, ties are not handled by `grpsurv`

in a particularly
sophisticated manner. This will be improved upon in a future release
of `grpreg`

.

An object with S3 class `"grpsurv"`

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`

.- group
Same as above.

- lambda
The sequence of

`lambda`

values in the path.- penalty
Same as above.

- gamma
Same as above.

- alpha
Same as above.

- loss
The negative partial log-likelihood of the fitted model at each value of

`lambda`

.- n
The number of observations.

- 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.

For Cox models, the following objects are also returned (and are necessary to
estimate baseline survival conditonal on the estimated regression coefficients),
all of which are ordered by time on study. I.e., the ith row of `W`

does
not correspond to the ith row of `X`

):

- W
Matrix of

`exp(beta)`

values for each subject over all`lambda`

values.- time
Times on study.

- fail
Failure event indicator.

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.a10Huang 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-sts392Breheny 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/s11222-013-9424-2Breheny P. (2015) The group exponential lasso for bi-level variable selection.

*Biometrics*,**71**: 731-740. doi:10.1111/biom.12300Simon N, Friedman JH, Hastie T, and Tibshirani R. (2011) Regularization Paths for Cox's Proportional Hazards Model via Coordinate Descent.

*Journal of Statistical Software*,**39**: 1-13. doi:10.18637/jss.v039.i05