// fit standard cure rate model: S(t) = pcure + (1 - pcure) * survival function for PWE model
functions {
  //' PWE log likelihood
  //' @param y               failure / censoring time
  //' @param eta             linear predictor
  //' @param lambda          vector of baseline hazards
  //' @param log_lambda      vector of log(lambda)
  //' @param breaks          (J+1)-dim vector giving intervals
  //' @param j               index of the interval for which the individual failed / was censored: j \in {1, ..., J}
  //' @param death_ind       integer giving whether individual died (death_ind == 1) or was censored (death_ind == 0)
  //' @param lambda_d_breaks (J-1)-dim vector giving lambda[j] * (s[j] - s[j-1]), j = 1, ..., J-1
  real pwe_lpdf(real y, real eta, vector lambda, vector log_lambda, vector breaks, int j, int death_ind, vector cumblhaz) {
    real loglik;

    // Initialize cumhaz to lambda[j] ( y - s[j] )
    real cumhaz = lambda[j] * ( y - breaks[j] );

    // add on (sum_{g=1}^{j-1} lambda[j] * ( s[j] - s[j-1] )
    if ( j > 1 )
      cumhaz += cumblhaz[j-1];

    // Initialize loglik = - cumhaz * exp(x'beta) = log(Survival)
    loglik = -cumhaz * exp(eta);

    // If individual experienced event, add log hazard: log(lambda[j]) + x'beta
    if ( death_ind == 1 )
      loglik += log_lambda[j] + eta;

    return(loglik);
  }
}
data {
  int<lower=0>                    n1;                 // current data sample size
  int<lower=0>                    J;                  // number of time intervals
  int<lower=0>                    p;                  // number of regression coefficients
  vector[n1]                      y1;                 // event times in current data
  matrix[n1,p]                    X1;                 // current data design matrix (no intercept for current data likelihood)
  array[n1] int<lower=1,upper=J>  intindx;            // index giving interval into which obs i failed / was censored for current data
  array[n1] int<lower=0,upper=1>  death_ind;          // event indicator (1 = event; 0 = censored) for current data
  vector[J+1]                     breaks;             // J+1-dim interval of breaks
  vector[p]                       beta_mean;          // mean for normal initial prior on coefficients
  vector<lower=0>[p]              beta_sd;            // sd for normal initial prior on coefficients
  vector[J]                       hazard_mean;        // location parameter for half-normal prior on baseline hazards
  vector<lower=0>[J]              hazard_sd;          // scale parameter for half-normal prior on baseline hazards
  real                            logit_p_cured_mean; // mean for normal prior on logit_p_cured
  real<lower=0>                   logit_p_cured_sd;   // sd for normal prior on logit_p_cured
  int<lower=0,upper=1>            get_loglik;         // whether to generate log-likelihood matrix
}
transformed data{
  vector[n1] logcens;
  real       lognc_hazard = normal_lccdf(0 | hazard_mean, hazard_sd);

  // compute censoring indicator
  for ( i in 1:n1 )
    logcens[i] = log(1 - death_ind[i]);
}
parameters{
  real                  logit_p_cured; // logit of proportion of cured individuals
  vector<lower=0>[J]    lambda;        // the J hazards for each interval
  vector[p]             beta;          // regression coefficients
}
transformed parameters{
  real<lower=0,upper=1> p_cured = inv_logit(logit_p_cured); // proportion of cured individuals
}
model{
  vector[n1]  eta        = X1 * beta;
  vector[J]   log_lambda = log(lambda);
  vector[J-1] cumblhaz;

  // compute lambda[j] * (s[j] * s[j-1])
  cumblhaz = cumulative_sum( lambda[1:(J-1)] .* ( breaks[2:J] - breaks[1:(J-1)] ) );

  // half-normal prior on baseline hazards
  target += normal_lpdf(lambda | hazard_mean, hazard_sd) - lognc_hazard;

  // prior on beta
  target += normal_lpdf(beta | beta_mean, beta_sd);

  // prior on logit_p_cured
  target += normal_lpdf(logit_p_cured | logit_p_cured_mean, logit_p_cured_sd);

  // current data likelihood
  for ( i in 1:n1 ){
    target += log_mix(
      p_cured,
      logcens[i],
      pwe_lpdf(y1[i] | eta[i], lambda, log_lambda, breaks, intindx[i], death_ind[i], cumblhaz)
    );
  }
}
generated quantities{
  vector[(get_loglik == 1) ? n1 : 0] log_lik; //to generate log likelihood matrices
  vector[n1]  eta        = X1 * beta;
  vector[J]   log_lambda = log(lambda);
  vector[J-1] cumblhaz   = cumulative_sum( lambda[1:(J-1)] .* ( breaks[2:J] - breaks[1:(J-1)] ) );

  if ( get_loglik == 1 ) {
    for( i in 1:n1 ){
      log_lik[i] = log_mix(
        p_cured,
        logcens[i],
        pwe_lpdf(y1[i] | eta[i], lambda, log_lambda, breaks, intindx[i], death_ind[i], cumblhaz)
      );
    }
  }
}

