pscEst: Function for performing estimation procedures in 'pscfit'

Description

Function for performing estimation procedures in 'pscfit'

Usage

pscEst(CFM, DC_clean, nsim, start, start.se, trt)

Value

A matrix containing the draws form the posterior distribution

Arguments

CFM: a model object supplied to pscfit
DC_clean: a cleaned dataset ontained using dataComb().
nsim: the number of MCMC simulations to run
start: the stating value for
start.se: the stating value for
trt: an optional vector denoting treatment allocations where mulitple treatment comparisons are bieng made

Details

Define the set of model parameters $B$ to contain $\Gamma$ which summarize the parameters of the CFM. Prior distributions are defined for B using a multivariate normal distribution $\pi (B) \sim MVN(\mu ,\Sigma)$ where $\mu|$ is the vector of coefficient estimates from the validated model and $\Sigma$ is the variance-covariance matrix. This information is taken directly from the outputs of the parametric model and no further elicitation is required. The prior distirbution for the efficacy parameter ($\pi{(\beta)}$) is set as an uniformative $N(0,1000)$.

Ultimately the aim is to estimate the posterior distribution for $\beta$ conditional on the distribution of B and the observed data. A full form for the posterior distribution is then given as

$$P(\beta \vert B,D) \propto L(D \vert B,\beta) \pi(B) \pi(\beta)$$

Please see 'pscfit' for more details on liklihood formation.

For each iteration of the MCMC procedure, the following algorithm is performed

Set and indicator s=1, and define an initial state based on prior hyperparameters for $\pi(B)$ and $\pi(\beta)$ such that $b_s = \mu and \tau_s=0$
Update $s = s+1$ and draw model parameters $b_s$ from $\pi(B)$ and an draw a proposal estimate of $\beta$ from some target distribution
Estimate $\Gamma_(i,S)=\nu^T x_i$ where $\nu$ is the subset of parameters from $b_s$ which relate to the model covariates and define 2 new likelihood functions $\Theta_(s,1)=L(D \vert B=b_s,\beta=\tau_(s-1) )$ & $\Theta_(s,2)= L(D \vert B=b_s,\beta=\tau_s)$
Draw a single value $\psi$ from a Uniform (0,1) distribution and estimate the condition $\omega= \Theta_(s,1)/\Theta_(s,2)$. If $\omega > \psi$ then accept $\tau_s$ as belonging to the posterior distribution $P(\beta \vert B,D)$ otherwise retain $\tau_(s-1)$
Repeat steps 2 – 4 for the required number of iterations

The result of the algorithm is a posterior distribution for the log hazard ratio, $\beta$, captures the variability in B through the defined priors $\pi{(\beta)}$.