The frailty function allows one to add a simple random effects term to a Cox model.

```
frailty(x, distribution="gamma", ...)
frailty.gamma(x, sparse = (nclass > 5), theta, df, eps = 1e-05,
method = c("em","aic", "df", "fixed"), ...)
frailty.gaussian(x, sparse = (nclass > 5), theta, df,
method =c("reml","aic", "df", "fixed"), ...)
frailty.t(x, sparse = (nclass > 5), theta, df, eps = 1e-05, tdf = 5,
method = c("aic", "df", "fixed"), ...)
```

this function is used in the model statement of either
`coxph`

or `survreg`

.
It's results are used internally.

- x
the variable to be entered as a random effect. It is always treated as a factor.

- distribution
either the

`gamma`

,`gaussian`

or`t`

distribution may be specified. The routines`frailty.gamma`

,`frailty.gaussian`

and`frailty.t`

do the actual work.- ...
Arguments for specific distribution, including (but not limited to)

- sparse
cutoff for using a sparse coding of the data matrix. If the total number of levels of

`x`

is larger than this value, then a sparse matrix approximation is used. The correct cutoff is still a matter of exploration: if the number of levels is very large (thousands) then the non-sparse calculation may not be feasible in terms of both memory and compute time. Likewise, the accuracy of the sparse approximation appears to be related to the maximum proportion of subjects in any one class, being best when no one class has a large membership.- theta
if specified, this fixes the variance of the random effect. If not, the variance is a parameter, and a best solution is sought. Specifying this implies

`method='fixed'`

.- df
if specified, this fixes the degrees of freedom for the random effect. Specifying this implies

`method='df'`

. Only one of`theta`

or`df`

should be specified.- method
the method used to select a solution for theta, the variance of the random effect. The

`fixed`

corresponds to a user-specified value, and no iteration is done. The`df`

selects the variance such that the degrees of freedom for the random effect matches a user specified value. The`aic`

method seeks to maximize Akaike's information criteria 2*(partial likelihood - df). The`em`

and`reml`

methods are specific to Cox models with gamma and gaussian random effects, respectively. Please see further discussion below.- tdf
the degrees of freedom for the t-distribution.

- eps
convergence criteria for the iteration on theta.

The `frailty`

plugs into the general penalized
modeling framework provided by the `coxph`

and `survreg`

routines.
This framework deals with likelihood, penalties, and degrees of freedom;
these aspects work well with either parent routine.

Therneau, Grambsch, and Pankratz show how maximum likelihood estimation for
the Cox model with a gamma frailty can be accomplished using a general
penalized routine, and Ripatti and Palmgren work through a similar argument
for the Cox model with a gaussian frailty. Both of these are specific to
the Cox model.
Use of gamma/ml or gaussian/reml with
`survreg`

does not lead to valid results.

The extensible structure of the penalized methods is such that the penalty
function, such as `frailty`

or
`pspine`

, is completely separate from the modeling
routine. The strength of this is that a user can plug in any penalization
routine they choose. A weakness is that it is very difficult for the
modeling routine to know whether a sensible penalty routine has been
supplied.

Note that use of a frailty term implies a mixed effects model and use of a cluster term implies a GEE approach; these cannot be mixed.

The `coxme`

package has superseded
this method. It is faster, more stable, and more flexible.

S Ripatti and J Palmgren, Estimation of multivariate frailty models using penalized partial likelihood, Biometrics, 56:1016-1022, 2000.

T Therneau, P Grambsch and VS Pankratz, Penalized survival models and frailty, J Computational and Graphical Statistics, 12:156-175, 2003.

coxph, survreg

```
# Random institutional effect
coxph(Surv(time, status) ~ age + frailty(inst, df=4), lung)
# Litter effects for the rats data
rfit2a <- coxph(Surv(time, status) ~ rx +
frailty.gaussian(litter, df=13, sparse=FALSE), rats,
subset= (sex=='f'))
rfit2b <- coxph(Surv(time, status) ~ rx +
frailty.gaussian(litter, df=13, sparse=TRUE), rats,
subset= (sex=='f'))
```

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