bayesCureRateModel (version 1.5)

complete_log_likelihood_general: Logarithm of the complete log-likelihood for the general cure rate model.

Description

Compute the logarithm of the complete likelihood, given a realization of the latent binary vector of cure indicators I_sim and current values of the model parameters g, lambda, b and promotion time parameters (\(\boldsymbol\alpha\)) which yield log-density values (one per observation) stored to the vector log_f and log-cdf values stored to the vector log_F.

Usage

complete_log_likelihood_general(y, X, Censoring_status, 
	g, lambda, log_f, log_F, b, I_sim, alpha)

Value

A list with the following entries

cll

the complete log-likelihood for the current parameter values.

logS

Vector of logS values (one for each observation).

logP0

Vector of logP0 values (one for each observation).

Arguments

y

observed data (time-to-event or censored time)

X

design matrix. Should contain a column of 1's if the model has a constant term.

Censoring_status

binary variables corresponding to time-to-event and censoring.

g

The \(\gamma\) parameter of the model (real).

lambda

The \(\lambda\) parameter of the model (positive).

log_f

A vector containing the natural logarithm of the density function of the promotion time distribution per observation, for the current set of parameters. Its length should be equal to the sample size.

log_F

A vector containing the natural logarithm of the cumulative density function of the promotion time distribution per observation, for the current set of parameters. Its length should be equal to the sample size.

b

Vector of regression coefficients. Its length should be equal to the number of columns of the design matrix.

I_sim

Binary vector of the current value of the latent cured status per observation. Its length should be equal to the sample size. A time-to-event cannot be cured.

alpha

A parameter between 0 and 1, corresponding to the temperature of the complete posterior distribution.

Author

Panagiotis Papastamoulis

Details

The complete likelihood of the model is $$L_c(\boldsymbol{\theta};\boldsymbol{y}, \boldsymbol{I}) = \prod_{i\in\Delta_1}(1-p_0(\boldsymbol{x}_i,\boldsymbol\theta))f_U(y_i;\boldsymbol\theta,\boldsymbol{x}_i)\\ \prod_{i\in\Delta_0}p_0(\boldsymbol{x}_i,\boldsymbol\theta)^{1-I_i}\{(1-p_0(\boldsymbol{x}_i,\boldsymbol\theta))S_U(y_i;\boldsymbol\theta,\boldsymbol{x}_i)\}^{I_i}.$$ \(f_U\) and \(S_U\) denote the probability density and survival function of the susceptibles, respectively, that is $$ S_U(y_i;\boldsymbol\theta,{\boldsymbol x}_i)=\frac{S_P(y_i;\boldsymbol{\theta},{\boldsymbol x}_i)-p_0({\boldsymbol x}_i;\boldsymbol\theta)}{1-p_0({\boldsymbol x}_i;\boldsymbol\theta)}, f_U(y_i;\boldsymbol\theta,{\boldsymbol x}_i)=\frac{f_P(y_i;\boldsymbol\theta,{\boldsymbol x}_i)}{1-p_0({\boldsymbol x}_i;\boldsymbol\theta)}.$$

References

Papastamoulis and Milienos (2023). Bayesian inference and cure rate modeling for event history data. arXiv:2310.06926.

Examples

# simulate toy data 
	set.seed(1)
	n = 4
	stat = rbinom(n, size = 1, prob = 0.5)
	x <- cbind(1, matrix(rnorm(n), n, 1))
	y <- rexp(n)
	lw <- log_weibull(y, a1 = 1, a2 = 1, c_under = 1e-9)
# compute complete log-likelihood
complete_log_likelihood_general(y = y, X = x, 
	Censoring_status = stat, 
	g = 1, lambda = 1, 
	log_f = lw$log_f, log_F = lw$log_F, 
	b = c(-0.5,0.5), 
	I_sim = stat, alpha = 1)