set_datamod_miscount: Specify Miscount Data Model

Description

Specify a data model for the outcome in a Poisson model, where the outcome is subject to undercount and overcount.

Usage

set_datamod_miscount(mod, prob, rate)

Value

A revised version of mod.

Arguments

mod: An object of class "bage_mod_pois", created with mod_pois().
prob: The prior for the probability that a person or event in the target population will correctly enumerated. A data frame with a variable called "mean", a variable called "disp", and, optionally, one or more 'by' variables.
rate: The prior for the overcoverage rate. A data frame with a variable called "mean", a variable called "disp", and, optionally, one or more 'by' variables.

The <code>prob</code> argument

The prob argument specifies a prior distribution for the probability that a person or event in the target population is included in the reported outcome. prob is a data frame with a variable called "mean", a variable called "disp", and, optionally, one or more 'by' variables. For instance, a prob of

data.frame(sex = c("Female", "Male"),
           mean = c(0.95, 0.92),
           disp = c(0.02, 0.015))

implies that the expected value for the inclusion probability is 0.95 for females and 0.92 for males, with slightly more uncertainty for females than for males.

The <code>rate</code> argument

The rate argument specifies a prior distribution for the overcoverage rate. rate is a data frame with a variable called "mean", a variable called "disp", and, optionally, one or more 'by' variables. For instance, a rate of

data.frame(mean = 0.03, disp = 0.1)

implies that the expected value for the overcoverage rate is 0.03, with a dispersion of 0.1. Since no 'by' variables are included, the same mean and dispersion values are applied to all cells.

Mathematical details

The model for the observed outcome is

$$y_i^{\text{obs}} = u_i + v_i$$ $$u_i \sim \text{Binomial}(y_i^{\text{true}}, \pi_{g[i]})$$ $$v_i \sim \text{Poisson}(\kappa_{h[i]} \gamma_i w_i)$$ $$\pi_g \sim \text{Beta}(m_g^{(\pi)} / d_g^{(\pi)}, (1-m_g^{(\pi)}) / d_g^{(\pi)})$$ $$\kappa_h \sim \text{Gamma}(1/d_h^{(\kappa)}, 1/(d_h^{(\kappa)} m_h^{(\kappa)}))$$

where

$y_i^{\text{obs}}$ is the observed outcome for cell $i$;
$y_i^{\text{true}}$ is the true outcome for cell $i$;
$\gamma_i$ is the rate for cell $i$;
$w_i$ is exposure for cell $i$;
$\pi_{g[i]}$ is the probability that a member of the target population in cell $i$ is correctly enumerated in that cell;
$\kappa_{h[i]}$ is the overcoverage rate for cell $i$;
$m_g^{(\pi)}$ is the expected value for $\pi_g$ (specified via prob);
$d_g^{(\pi)}$ is disperson for $\pi_g$ (specified via prob);
$m_h^{(\kappa)}$ is the expected value for $\kappa_h$ (specified via rate); and
$d_h^{(\kappa)}$ is disperson for $\kappa_h$ (specified via rate).

Details

The miscount data model is essentially a combination of the undercount and overcount data models. It assumes that reported outcome is the sum of two quantities:

Units from target population, undercounted People or events belonging to the target population, in which each unit's inclusion probability is less than 1.
Overcount People or events that do not belong to target population, or that are counted more than once.

If, for instance, a census enumerates 91 people from a true population of 100, but also mistakenly enumerates a further 6 people, then

the true value for the outcome variable is 100
the value for the undercounted target population is 91,
the value for the overcount is 6, and
the observed value for the outcome variable is 91 + 6 = 97.

Examples

Run this code

## specify 'prob' and 'rate'
prob <- data.frame(sex = c("Female", "Male"),
                   mean = c(0.95, 0.97),
                   disp = c(0.05, 0.05))
rate <- data.frame(mean = 0.03, disp = 0.15)

## specify model
mod <- mod_pois(divorces ~ age * sex + time,
                data = nzl_divorces,
                exposure = population) |>
  set_datamod_miscount(prob = prob, rate = rate)
mod

## fit model
mod <- mod |>
  fit()
mod

## original data, plus imputed values for outcome
mod |>
  augment()

## parameter estimates
library(dplyr)
mod |>
  components() |>
  filter(term == "datamod")

## the data have in fact been confidentialized,
## so we account for that, in addition
## to accounting for undercoverage and
## overcoverage
mod <- mod |>
 set_confidential_rr3() |>
 fit()
mod

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