Random effects terms

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

Usage

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.

Value

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

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.

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.

See also

coxph, survreg

Examples

# Random institutional effect
coxph(Surv(time, status) ~ age + frailty(inst, df=4), lung)
#> 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=0.08
#> n= 227, number of events= 164 
#>    (1 observation deleted due to missingness)

# 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'))