Specify Noise Data Model

Specify a data model in which

observed outcome = true outcome + error,

where the error has a symmetric distribution with mean 0.

If the true outcome has a normal distribution, then the error has a normal distribution. If the true outcome has a Poisson distribution, then the error has a symmetric Skellam distribution.

Usage

set_datamod_noise(mod, sd)

Arguments

mod: An object of class "bage_mod", created with mod_norm() or mod_pois().
sd: Standard deviation of measurement errors. A single number, or a data frame with 'by' variables.

Value

A revised version of mod.

Details

The model assumes that the outcome variable is unbiased. If there is in fact evidence of biases, then this evidence should be used to create a de-biased version of the outcome variable in data, and this de-biased version should be used by mod_norm() or mod_pois().

If set_datamod_noise() is used with a Poisson model, then the dispersion term for the Poisson rates must be set to zero. This can be done using set_disp(), though set_datamod_noise() will also do so.

The Skellam distribution

The Skellam distribution is confined to integers, but can take positive and negative values.

$$X_1 \sim \text{Poisson}(\mu_1)$$ $$X_2 \sim \text{Poisson}(\mu_2)$$

then

$$Y = X_1 - X_2$$

has a $\text{Skellam}(\mu_1, \mu_2)$ distribution. If $\mu_1 = \mu_2$, then the distribution is symmetric.

The `sd` argument

sd can be a single number, in which case the same standard deviation is used for all cells. sd can also be a data frame with a with a variable called "sd" and one or more columns with 'by' variables. For instance, a sd of

data.frame(sex = c("Female", "Male"),
           sd = c(330, 240))

implies that measurement errors have standard deviation 330 for females and 240 for males.

Mathematical details

The model for the observed outcome is

$$y_i^{\text{obs}} = y_i^{\text{true}} + \epsilon_i$$

with

$$\epsilon_i \sim \text{N}(0, s_{g[i]}^2)$$

if $y_i^{\text{true}}$ has a normal distribution, and

$$\epsilon_i \sim \text{Skellam}(0.5 s_{g[i]}^2, 0.5 s_{g[i]}^2)$$

if $y_i^{\text{true}}$ has a Poisson distribution, where

$y_i^{\text{obs}}$ is the observed outcome for cell $i$;
$y_i^{\text{true}}$ is the true outcome for cell $i$;
$\epsilon_i$ is the measurement error for cell $i$; and
$s_{g\lbrack i\rbrack }$ is the standard deviation of the measurement error for cell $i$.

Examples

## Normal model ------------------------------

## prepare outcome variable
library(dplyr, warn.conflicts = FALSE)
spend <- nld_expenditure |>
  mutate(log_spend = log(value + 1))

## specify model
mod <- mod_norm(log_spend ~ age * diag + year,
                data = spend,
                weights = 1) |>
  set_datamod_noise(sd = 0.1)

## fit model
mod <- mod |>
  fit()
#> Building log-posterior function...
#> Finding maximum...
#> Drawing values for hyper-parameters...
mod
#> 
#>     ------ Fitted normal model ------
#> 
#>    log_spend ~ age * diag + year
#> 
#>                data model: noise
#> 
#>         term  prior along n_par n_par_free std_dev
#>  (Intercept) NFix()     -     1          1       -
#>          age   RW()   age    18         18    0.28
#>         diag    N()     -    18         18    0.05
#>         year   RW()  year     4          4    0.06
#>     age:diag   RW()   age   324        324    0.93
#> 
#>  disp: mean = 1
#> 
#>  n_draw var_time var_age optimizer
#>    1000     year     age    nlminb
#> 
#>  time_total time_max time_draw iter converged                    message
#>        0.38     0.14      0.19   20      TRUE   relative convergence (4)
#> 

## create new aggregated diagnositic
## group variable
library(dplyr, warn.conflicts = FALSE)
spend <- spend |>
  mutate(diag_ag = case_when(
    diag == "Neoplasms" ~ diag,
    diag == "Not allocated" ~ diag,
    TRUE ~ "Other"
  ))

## assume size of measurement errors
## varies across these aggregated groups
sd_diag <- data.frame(diag_ag = c("Neoplasms",
                                  "Not allocated",
                                  "Other"),
                      sd = c(0.05, 0.2, 0.1))

## fit model that uses diagnostic-specific
## standard deviations
mod <- mod_norm(log_spend ~ age * diag + year,
                data = spend,
                weights = 1) |>
  set_datamod_noise(sd = sd_diag)


## Poisson model -----------------------------

mod <- mod_pois(deaths ~ month,
                data = usa_deaths,
                exposure = 1) |>
  set_datamod_noise(sd = 200)
#> → Setting dispersion to zero. (Required when using noise data model with Poisson rates model.)