Let \(\boldsymbol{y} = (y_1,\ldots,y_n)\) denote the observed data, which correspond to time-to-event data or censoring times. Let also \(\boldsymbol{x}_i = (x_{i1},\ldots,x_{x_{ip}})'\) denote the covariates for subject \(i\), \(i=1,\ldots,n\).
Assuming that the \(n\) observations are independent, the observed likelihood is defined as
$$
L=L({\boldsymbol \theta}; {\boldsymbol y}, {\boldsymbol x})=\prod_{i=1}^{n}f_P(y_i;{\boldsymbol\theta},{\boldsymbol x}_i)^{\delta_i}S_P(y_i;{\boldsymbol \theta},{\boldsymbol x}_i)^{1-\delta_i},
$$
where \(\delta_i=1\) if the \(i\)-th observation corresponds to time-to-event while \(\delta_i=0\) indicates censoring time. The parameter vector \(\boldsymbol\theta\) is decomposed as
$$
\boldsymbol\theta = (\boldsymbol\alpha', \boldsymbol\beta', \gamma,\lambda)
$$
where
\(\boldsymbol\alpha = (\alpha_1,\ldots,\alpha_d)'\in\mathcal A\) are the parameters of the promotion time distribution whose cumulative distribution and density functions are denoted as \(F(\cdot,\boldsymbol\alpha)\) and \(f(\cdot,\boldsymbol\alpha)\), respectively.
\(\boldsymbol\beta\in\mathbf R^{k}\) are the regression coefficients with \(k\) denoting the number of columns in the design matrix (it may include a constant term or not).
\(\gamma\in\mathbf R\)
\(\lambda > 0\).
The population survival and density functions are defined as
$$S_P(y;\boldsymbol\theta) = \left(1 + \gamma\exp\{\boldsymbol{x}_i\boldsymbol{\beta}'\}c^{\gamma\exp\{\boldsymbol{x}_i\boldsymbol{\beta}'\}}F(y;\boldsymbol\alpha)^\lambda\right)^{-1/\gamma}$$
whereas,
$$f_P(y;\boldsymbol\theta)=-\frac{\partial S_P(y;\boldsymbol\theta)}{\partial y}.$$
Finally, the cure rate is affected through covariates and parameters as follows
$$p_0(\boldsymbol{x}_i;\boldsymbol{\theta}) = \left(1 + \gamma\exp\{\boldsymbol{x}_i\boldsymbol{\beta}'\}c^{\gamma\exp\{\boldsymbol{x}_i\boldsymbol{\beta}'\}}\right)^{-1/\gamma}$$
where \(c = e^{e^{-1}}\).
The promotion time distribution can be a member of standard families (currently available are the following: Exponential, Weibull, Gamma, Lomax, Gompertz, log-Logistic) and in this case \(\alpha = (\alpha_1,\alpha_2)\in (0,\infty)^2\). Also considered is the Dagum distribution, which has three parameters \((\alpha_1,\alpha_2,\alpha_3)\in(0,\infty)^3\). In case that the previous parametric assumptions are not justified, the promotion time can belong to the more flexible family of finite mixtures of Gamma distributions. For example, assume a mixture of two Gamma distributions of the form
$$
f(y;\boldsymbol \alpha) = \alpha_5 f_{\mathcal G}(y;\alpha_1,\alpha_3) + (1-\alpha_5) f_{\mathcal G}(y;\alpha_2,\alpha_4),
$$
where $$f_\mathcal{G}(y;\alpha,\beta)=\frac{\beta^{\alpha}}{\Gamma(\alpha)}y^{\alpha-1}\exp\{-\beta y\}, y>0$$ denotes the density of the Gamma distribution with parameters \(\alpha > 0\) (shape) and \(\beta > 0\) (rate).
For the previous model, the parameter vector is $$\boldsymbol\alpha = (\alpha_1,\alpha_2,\alpha_3,\alpha_4,\alpha_5)'\in\mathcal A$$
where \(\mathcal A = (0,\infty)^4\times (0,1)\).
User defined promotion time distributions and finite mixtures of them can be also fitted using the options 'user' and 'user_mixture', respectively. The appropriate model can be selected according to information criteria such as the BIC.
The binary vector \(\boldsymbol{I} = (I_1,\ldots,I_n)\) contains the (latent) cure indicators, that is, \(I_i = 1\) if the \(i\)-th subject is susceptible and \(I_i = 0\) if the \(i\)-th subject is cured. \(\Delta_0\) denotes the subset of \(\{1,\ldots,n\}\) containing the censored subjects, whereas \(\Delta_1 = \Delta_0^c\) is the (complementary) subset of uncensored subjects. 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
$$
S_U(y_i;\boldsymbol\theta,{\boldsymbol x}_i)=\frac{S_P(y_i;\boldsymbol{\theta},{\boldsymbol x}_i)-p_0({\boldsymbol x}_i;\boldsymbol\theta)}{1-p_0({\boldsymbol x}_i;\boldsymbol\theta)}, f_U(y_i;\boldsymbol\theta,{\boldsymbol x}_i)=\frac{f_P(y_i;\boldsymbol\theta,{\boldsymbol x}_i)}{1-p_0({\boldsymbol x}_i;\boldsymbol\theta)}.$$
tools:::Rd_package_indices("bayesCureRateModel")