I have a number of studies describing families tested for a genetic condition. For each study the following data are described:

• np, number of probands (the proband is the first person in a family to be diagnosed with the genetic condition, so this is generally equal to the number of families)
• nr, number of relatives identified (this is the total number of people identified across families minus the number of probands)
• nc, number of relatives given genetic counseling
• nt, number of relatives given genetic testing
  

Relatives must be identified before they can be given genetic counseling, and they must be given genetic counseling before they can be given genetic testing.

If I assume that nr～r×np, nc～pc×nr and nt～pt×nc how can I infer r, pc and pt given that some values of np, nr, nc and nt are missing (not reported)?

model {
  # Where all four variables are observed
  for (i in 1:N_PRCT) {
    # N_{r,i} | N_{p,i} = n_{p,i} ~ Poisson(r * n_{p,i})
    lam[Q_PRCT] <- r * N_p[Q_PRCT]
    N_r[Q_PRCT] ~ dpois(lam[Q_PRCT])
    # N_{c,i} | N_{r,i} = n_{r,i} ~ Bin(p_c, n_{r,i})
    N_c[Q_PRCT] ~ dbin(p_c, N_r[Q_PRCT])
    # N_{t,i} | N_{c,i} = n_{c,i} ~ Bin(p_t, n_{c,i})
    N_t[Q_PRCT] ~ dbin(p_t, N_c[Q_PRCT])
  }


  # Where the number being counseled is not observed
  for (i in 1:N_PRT) {
    # N_{r,i} | N_{p,i} = n_{p,i} ~ Poisson(r * n_{p,i})
    lam[Q_PRT] <- r * N_p[Q_PRT]
    N_r[Q_PRT] ~ dpois(lam[Q_PRT])
    # N_{t,i} | N_{r,i} = n_{r,i} ~ Bin(p_c * p_t, n_{r,i})
    N_t[Q_PRT] ~ dbin(p_cp_t, N_r[Q_PRT])
  }


  # Where only the number of probands and the number of
  # relatives tested are observed
  for (i in 1:N_PT) {
    # N_{t,i} | N_{p,i} = n_{p,i} ~ Poisson(r * p_c * p_t * n_{p,i})
    lam[Q_PT] <- rp_cp_t * N_p[Q_PT]
    N_t[Q_PT] ~ dpois(lam[Q_PT])
  }