nltm {nltm}

Fit Non-Linear Transformation Model for analyzing survival data

Description

Fits a non-linear transformation model (nltm) for analyzing survival data, see Tsodikov (2003). The class of nltm includes the following currently supported models: Cox proportional hazard, proportional hazard cure, proportional odds, proportional hazard - proportional hazard cure, proportional hazard - proportional odds cure, Gamma frailty, and proportional hazard - proportional odds.

Usage

nltm(formula1=formula(data), formula2=formula(data),
     data=parent.frame(), subset, na.action, init=NULL, control,
     nlt.model=c("PH","PHC","PO","PHPHC","PHPOC","GFM","PHPO"),
     model=FALSE, x=FALSE, y=FALSE, verbose=FALSE, ...)

Arguments

formula1

A formula object with the response on the left of a ~ operator, and the terms on the right. The response must be a survival object as returned by the Surv function. In models with two predictors, this corresponds to the long term effect.

formula2

A formula corresponding to the short term effect. Will be ignored in models with only one predictor. If not present in models with two predictors, then formula1 will be used both for the long and short term effect.

data

A data.frame in which to interpret the variables named in formula1 and formula2, or in the subset argument.

subset

Expression saying that only a subset of the rows of the data should be used in the fit.

na.action

A missing-data filter function, applied to the model.frame, after any subset argument has been used. Default is options()$na.action.

init

Vector of initial values for the calculation of the maximum likelihood estimator of the regression parameters. Default is zero.

control

Object specifying iteration limit and other control options. Default is nltm.control(...).

nlt.model

A character string specifying a non-linear transformation model. Default is Proportional Hazards Model.

The conditional survival function S(t|z) given the covariates z of each of the models currently supported are given below. Let S_0(t) be the non-parametric baseline survival function, and \theta(z) and \eta(z) predictors. We take \theta(z)=\exp(\beta_\theta z) and \eta(z)=\exp(\beta_\eta z), where \beta_\theta and \beta_\eta are the regresssion coefficients. In cure models, there is an additional regression parameter \beta_c and \theta(z)=\exp(\beta_\theta z+\beta_c).

• Proportional hazard model (PH):

  S(t|z)=S_0(t)^{\theta(z)}.

• Proportional hazard cure model (PHC):

  S(t|z)=\exp\bigl(-\theta(z)(1-S_0(t))\bigr).

• Proportional odds model (PO):

  S(t|z)=\frac{\theta(z)}{\theta(z)-\ln(S_0(t))}.

• Proportional hazard - proportional hazard cure model (PHPHC):

  S(t|z)=\exp\bigl(-\theta(z)(1-S_0^{\eta(z)}(t))\bigr).

• Proportional hazard - proportional odds cure model (PHPOC):

  S(t|z)=\exp\biggl(-\frac{\theta(z)(1-S_0(t))} {1-(1-\eta(z))S_0(t)}\biggr).

• Gamma frailty model (GFM):

  S(t|z)=\biggl(\frac{\theta(z)^{\eta(z)}} {\theta(z)-\ln(S_0(t))}\biggr)^{\eta(z)}.

• Proportional hazard - proportional odds model (PHPO):

  S(t|z)=\frac {\theta(z) S_0^{\eta(z)}(t)} {1-(1-\theta(z))S_0^{\eta(z)}(t)}.

model

If TRUE the model frame is stored. Default is FALSE.

x

If TRUE the model matrix is stored. Default is FALSE.

y

If TRUE the response matrix is stored. Default is FALSE.

verbose

If a file name is given, it stores information from maximization of likelihood and calculation of information matrix in a file. Otherwise, verbose=FALSE. Default is FALSE.

...

Other arguments.

Value

An object of class "nltm". See nltm.object for details.

Author(s)

Gilda Garibotti ([e-mail]) and Alexander Tsodikov.

References

Tsodikov A. (2003) "Semiparametric models: a generalized self-consistency approach". Journal of the Royal Statistical Society B, 65, Part 3, 759-774.

Tsodikov A. (2002) "Semi-parametric models of long- and short-term survival: an application to the analysis of breast cancer survival in Utah by age and stage". Statistics in Medicine, 21, 895-920.

Tsodikov A., Garibotti G. (2006) "Profile information matrix for nonlinear transformation models". to appear in Journal of Lifetime Data Analysis.

Tsodikov A., Ibrahim J., Yakovlev A. (2003) "Estimating cure rates from survival data: an alternative to two-component mixture models". Journal of the American Statistical Association, Vol. 98, No. 464, 1063-1078.

Wendland M., Tsodikov A., Glenn M., Gaffney D. (2004) "Time interval to the development of breast carcinoma after treatment for Hodgkin disease". Cancer Vol. 101, No. 6, 1275-1282.

See Also

nltm.object, summary.nltm, nltm.control

Examples

# fit a Proportional Odds Model 
data(melanoma, package="nltm")
fit <- nltm(Surv(time,status) ~ size + age, data=melanoma, nlt.model="PO")
summary(fit)