Profile Likelihood Variance and Confidence Intervals for Semi-Parametric Models

Usage

# S3 method for class 'ic_sp2'
vcov(object, type = "oim_curvature", fixed = 5, typical = 1, large = 2, ...)

# S3 method for class 'ic_sp2'
confint(object, parm, level = 0.95, ...)

Arguments

object: Fitted model object from ic_sp
type: One of "oim_curvature" (default), "oim_fixed", or "opg_fixed". See details for explanation.
fixed: A fixed factor to multiply by n^(-1/2) to determin the perturbation size for fixed types.
typical: A typical value for the regression parameters, used to determine the scale of h_n. Default is 1. This is required for the "oim_curvature" type.
large: A large value for the regression parameters, used to determine the scale of h_n. Default is 2. This is required for the "oim_curvature" type.
...: Unused in vcov(). In confint() it is used to specify the vcov parameters such as type, fixed, typical, and large.
parm: a specification of which parameters are to be given confidence intervals, either a vector of numbers or a vector of names. If missing, all parameters are considered.
level: the confidence level required.

Value

Variance-covariance matrix of the regression parameters or the confidence interval matrix.

Details

The covariance matrix is calculated using the profile likelihood approach. (Murphey and Vand Der Vaart 2000). This method involves perturbing the regression parameters, updating the baseline hazard estimates using the profile_fit function with the perturbed parameters, and calculating the change in log-likelihood for each perturbation, which is then used to compute the covariance matrix. We borrowing the naming convention from the Stata stintcox manual https://www.stata.com/manuals/ststintcox.pdf.

Type "oim_curvature" (Boruvka and Cook 2015) uses the curvature of the log-likelihood function to determine the perturbation size for the profile likelihood calculations, as described in Boruvka and Cook (2014). The typical and large parameters are used to determine the scale of the perturbations.

Type "oim_fixed" (Zeng et al 2016) uses a fixed perturbation size based on the fixed parameter and the sample size n.

Type "opg_fixed" (Zeng et al 2017) uses a fixed perturbation size based on the fixed parameter and the sample size n, but uses the outer product of gradients instead of the observed information matrix for the covariance calculation.

For larged values of fixed the model fitting may fail to converge.

The confint() method gives a Wald type CI and is based on stats::confint but allows the pass through of the vcov.ic_sp2 parameters.

References

Murphy, S. A., & Van Der Vaart, A. W. (2000). On Profile Likelihood. Journal of the American Statistical Association, 95(450), 449–465. https://doi.org/10.1080/01621459.2000.10474219

Boruvka, A., and Cook, R. J. (2015), A Cox-Aalen Model for Interval-censored Data. Scand J Statist, 42, 414–426. doi: 10.1111/sjos.12113.

Donglin Zeng, Lu Mao, D. Y. Lin, Maximum likelihood estimation for semiparametric transformation models with interval-censored data, Biometrika, Volume 103, Issue 2, June 2016, Pages 253–271, https://doi.org/10.1093/biomet/asw013

Donglin Zeng, Fei Gao, D. Y. Lin, Maximum likelihood estimation for semiparametric regression models with multivariate interval-censored data, Biometrika, Volume 104, Issue 3, September 2017, Pages 505–525, https://doi.org/10.1093/biomet/asx029