coxph_mpl {survivalMPL}

Fit Cox Proportional Hazards Regression Model Via MPL

Description

Simultaneously estimate the regression coefficients and the baseline hazard function of proportional hazard Cox models using maximum penalised likelihood (MPL).

Usage

coxph_mpl(formula, data, subset, na.action, control, ...)

## S3 method for class 'coxph_mpl'
print(x,...)

Arguments

formula

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 the Surv, use type = "interval2" with interval censored data).

data

a data.frame in which to interpret the variables named in the formula, or in the subset argument. If no dataset is indicated, variables will be taken from the global environment.

subset

expression indicating which subset of the rows of data should be used in the fit. All observations are included by default.

na.action

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

x

an object inheriting from class coxph_mpl, representing a fitted Cox proportional hazard model.

control

Object of class coxph_mpl.control specifying control options like basis choice, smoothing parameter value and maximum number of itereations, for example. Refer to coxph_mpl.control to see the defaults.

...

Other arguments. In coxph_mpl, these elements, will be passed to coxph_mpl.control. In print.coxph_mpl, these elements will be passed to the print function.

Details

coxph_mpl allows to simultaneously estimate the regression coefficients and baseline hazard function of Cox proportional hazard models, with right censored data and independent censoring, by maximising a penalised likelihood, in which a penalty function is used to smooth the baseline hazard estimate.

Optimisation is achieved using a new iterative algorithm, which combines Newton's method and the multiplicative iterative algorithm proposed by Ma (2010), and respects the non-negativity constraints on the baseline hazard estimate (refer to Ma, Couturier, Heritier and Marschner (2021)).

The centered X matrix is used in the optimisation process to get a better shaped (penalised) log-likelihood. Baseline hazard parameter estimates and covariance matrix are then respectively corrected using a correction factor and the delta method.

When the chosen basis is not uniform, estimates of zero are possible for baseline hazard parameters and will correspond to active constraints as defined by Moore and Sadler (2008). Inference, as described by Ma, Heritier and Lo (2014), is then corrected accordingly (refer to Moore and Sadler (2008)) by adequately 'cutting' the corresponding covariance matrix.

There are currently 3 ways to perform inference on model parameters:

Let H denote the Hessian matrix of the unpenalised likelihood, Q denote the product of the first order derivative of the penalised likelihood by its transpose, and M_{2} denote the second order derivative of the penalised likelihood. Then,

• 'H' refers to H^{-1}, the inverse of the Hessian matrix,

• 'M2QM2', refers to the sandwich formula M_{2}^{-1} Q M_{2}^{-1},

• 'M2HM2', refers to the sandwich formula M_{2}^{-1} H M_{2}^{-1}.

Simulations analysing the coverage levels of confidence intervals for the regression parameters seem to indicate that M_{2}^{-1} H M_{2}^{-1} performs better when using the uniform basis, and that M_{2}^{-1} Q M_{2}^{-1} performs when using other bases.

Value

an object of class coxph_mpl representing the fit. See coxph_mpl.object for details.

Author(s)

Dominique-Laurent Couturier, Jun Ma, Stephane Heritier, Maurizio Manuguerra. Design inspired by the function coxph of the survival package.

References

Ma, J. and Couturier, D.-L., and Heritier, S. and Marschner, I.C. (2021), Penalized likelihood estimation of the proportional hazards model for survival data with interval censoring. International Journal of Biostatistics,doi:10.1515/ijb-2020-0104.

Ma, J. and Heritier, S. and Lo, S. (2014), On the Maximum Penalised Likelihood Approach for Proportional Hazard Models with Right Censored Survival Data. Computational Statistics and Data Analysis 74, 142-156.

Ma, J. (2010), Positively constrained multiplicative iterative algorithm for maximum penalised likelihood tomographic reconstruction. IEEE Transactions On Signal Processing 57, 181-192.

Moore, T. J. and Sadler, B. M. and Kozick R. J. (2008), Maximum-Likelihood Estimation, the Cramer-Rao Bound, and the Method of Scoring With Parameter Constraints, IEEE Transactions On Signal Processing 56, 3, 895-907.

Moore, D. K. (2016), Applied Survival Analysis Using R, Springer .

See Also

coxph_mpl.object, coxph_mpl.control, summary.coxph_mpl and plot.coxph_mpl.

Examples

## Not run: 
## right censoring example based on the dataset 'lung' 
## of the survival package (refer to ?lung)
## with hazard approximated by means of a step function (default).
data(lung)

fit_mpl <- coxph_mpl(Surv(time, status == 2) ~ age + sex + ph.karno + wt.loss, data = lung)

summary(fit_mpl)

## interval censoring example 
## (refer to ?bcos2 and to Moore (2016, example 12.2))
## with hazard approximated by means of m-splines

data(bcos2)

fit_mpl <- coxph_mpl(Surv(left, right, type="interval2") ~ treatment, 
                     data = bcos2, basis="m")
summary(fit_mpl)


## End(Not run)