# frailty: Random effects terms In survival: Survival Analysis

## Description

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

## Usage

 ```1 2 3 4 5 6 7``` ```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"), ...) ```

## Arguments

 `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.

## Details

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.

## Value

this function is used in the model statement of either `coxph` or `survreg`. It's results are used internally.

## References

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

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10``` ```# Random institutional effect coxph(Surv(time, status) ~ age + frailty(inst, df=4), lung) # Litter effects for the rats data rfit2a <- survreg(Surv(time, status) ~ rx + frailty.gaussian(litter, df=13, sparse=FALSE), rats, subset= (sex=='f')) rfit2b <- survreg(Surv(time, status) ~ rx + frailty.gaussian(litter, df=13, sparse=TRUE), rats, subset= (sex=='f')) ```

### Example output

```Call:
coxph(formula = Surv(time, status) ~ age + frailty(inst, df = 4),
data = lung)

coef se(coef)     se2   Chisq   DF     p
age                   0.01937  0.00933 0.00925 4.31149 1.00 0.038
frailty(inst, df = 4)                          3.33459 3.99 0.501

Iterations: 3 outer, 10 Newton-Raphson
Variance of random effect= 0.038   I-likelihood = -743.6
Degrees of freedom for terms= 1 4
Likelihood ratio test=9.96  on 4.97 df, p=0.075
n=227 (1 observation deleted due to missingness)
```

survival documentation built on Jan. 11, 2020, 9:38 a.m.