Specify a data model for the outcome in a Poisson or binomial model, where the outcome is subject to undercount.
Arguments
- mod
An object of class
"bage_mod", created withmod_pois()ormod_binom().- 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.
Details
The undercount data model assumes that reported values for the outcome variable understate the true values, because the reported values miss some people or events in the target population. In other words, the probability that any given unit in the target population will be included in the reported outcome is less than 1.
The prob 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
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.
Mathematical details
The model for the observed outcome is
$$y_i^{\text{obs}} \sim \text{Binomial}(y_i^{\text{true}}, \pi_{g[i]})$$ $$\pi_g \sim \text{Beta}(m_g^{(\pi)} / d_g^{(\pi)}, (1-m_g^{(\pi)}) / d_g^{(\pi)})$$
where
\(y_i^{\text{obs}}\) is the observed outcome for cell \(i\);
\(y_i^{\text{true}}\) is the true outcome 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;
\(m_g\) is the expected value for \(\pi_g\) (specified via
prob); and\(d_g\) is disperson for \(\pi_g\) (specified via
prob).
See also
mod_pois()Specify a Poisson modelmod_binom()Specify a binomial modelaugment()Original data plus estimated values, including estimates of true value for the outcome variablecomponents()Estimated values for model parameters, including inclusion probabilities and overcount ratesset_datamod_overcount()An overcount-only data modelset_datamod_miscount()An undercount-and-overcount data modeldatamods All data models implemented in
bageconfidential Confidentialization procedures modeled in
bageMathematical Details vignette
Examples
## specify 'prob'
prob <- data.frame(sex = c("Female", "Male"),
mean = c(0.95, 0.97),
disp = c(0.05, 0.05))
## specify model
mod <- mod_pois(divorces ~ age * sex + time,
data = nzl_divorces,
exposure = population) |>
set_datamod_undercount(prob)
mod
#>
#> ------ Unfitted Poisson model ------
#>
#> divorces ~ age * sex + time
#>
#> exposure: population
#> data model: undercount
#>
#> term prior along n_par n_par_free
#> (Intercept) NFix() - 1 1
#> age RW() age 11 11
#> sex NFix() - 2 2
#> time RW() time 11 11
#> age:sex RW() age 22 22
#>
#> disp: mean = 1
#>
#> n_draw var_time var_age var_sexgender
#> 1000 time age sex
#>
## fit model
mod <- mod |>
fit()
#> Building log-posterior function...
#> Finding maximum...
#> Drawing values for hyper-parameters...
mod
#>
#> ------ Fitted Poisson model ------
#>
#> divorces ~ age * sex + time
#>
#> exposure: population
#> data model: undercount
#>
#> term prior along n_par n_par_free std_dev
#> (Intercept) NFix() - 1 1 -
#> age RW() age 11 11 2.00
#> sex NFix() - 2 2 0.36
#> time RW() time 11 11 0.13
#> age:sex RW() age 22 22 0.33
#>
#> disp: mean = 1
#>
#> n_draw var_time var_age var_sexgender optimizer
#> 1000 time age sex nlminb
#>
#> time_total time_max time_draw iter converged message
#> 0.31 0.16 0.12 17 TRUE relative convergence (4)
#>
## original data, plus imputed values for outcome
mod |>
augment()
#> ℹ Adding variable `.divorces` with true values for `divorces`.
#> # A tibble: 242 × 9
#> age sex time divorces .divorces population .observed
#> <fct> <chr> <int> <dbl> <rdbl<1000>> <dbl> <dbl>
#> 1 15-19 Female 2011 0 0 (0, 1) 154460 0
#> 2 15-19 Female 2012 6 6 (6, 7) 153060 0.0000392
#> 3 15-19 Female 2013 3 3 (3, 4) 152250 0.0000197
#> 4 15-19 Female 2014 3 3 (3, 4) 152020 0.0000197
#> 5 15-19 Female 2015 3 3 (3, 4) 152970 0.0000196
#> 6 15-19 Female 2016 3 3 (3, 4) 154170 0.0000195
#> 7 15-19 Female 2017 6 6 (6, 7) 154450 0.0000388
#> 8 15-19 Female 2018 0 0 (0, 1) 154170 0
#> 9 15-19 Female 2019 3 3 (3, 4) 154760 0.0000194
#> 10 15-19 Female 2020 0 0 (0, 1) 154480 0
#> # ℹ 232 more rows
#> # ℹ 2 more variables: .fitted <rdbl<1000>>, .expected <rdbl<1000>>
## parameter estimates
library(dplyr)
mod |>
components() |>
filter(term == "datamod")
#> # A tibble: 2 × 4
#> term component level .fitted
#> <chr> <chr> <chr> <rdbl<1000>>
#> 1 datamod prob Female 0.96 (0.81, 1)
#> 2 datamod prob Male 0.98 (0.86, 1)
## the data have in fact been confidentialized,
## so we account for that, in addition
## to accounting for undercoverage
mod <- mod |>
set_confidential_rr3() |>
fit()
#> Building log-posterior function...
#> Finding maximum...
#> Drawing values for hyper-parameters...
mod
#>
#> ------ Fitted Poisson model ------
#>
#> divorces ~ age * sex + time
#>
#> exposure: population
#> data model: undercount
#> confidentialization: rr3
#>
#> term prior along n_par n_par_free std_dev
#> (Intercept) NFix() - 1 1 -
#> age RW() age 11 11 1.94
#> sex NFix() - 2 2 0.32
#> time RW() time 11 11 0.13
#> age:sex RW() age 22 22 0.30
#>
#> disp: mean = 1
#>
#> n_draw var_time var_age var_sexgender optimizer
#> 1000 time age sex nlminb
#>
#> time_total time_max time_draw iter converged message
#> 0.71 0.37 0.30 19 TRUE relative convergence (4)
#>