Marginalization over Incomplete Auxiliaries

Implements methods to estimate conditional outcome means in settings with missingness-not-at-random and incomplete auxiliary variables. Specifically, this package implements the marginalization over incomplete auxiliaries (MIA) method proposed by Mathur et al. (2026) . The package supports the iterative conditional expectation estimator and the noniterative conditional expectation estimator.

miapack

The miapack package implements methods to estimate conditional outcome means in settings with missingness-not-at-random and incomplete auxiliary variables. Specifically, this package implements the marginalization over incomplete auxiliaries (MIA) method proposed by Mathur et al. (2026).

Installation

You can install the development version of miapack from GitHub with:

# install.packages("devtools")
devtools::install_github("stmcg/miapack")

Example

We first load the package.

library(miapack)

Data Set

We will use the example dataset dat.sim included in the package. The dataset contains 9,297 observations with a continuous outcome Y, a binary auxiliary variable W, and binary predictors X1 and X2. The first 10 rows of dat.sim are:

dat.sim[1:10,]
#>           Y X1 X2  W
#> 1        NA  0 NA  0
#> 2        NA  1  1 NA
#> 3  6.066826  1  1  1
#> 4  6.113787  1  1  1
#> 5        NA  1  1 NA
#> 6        NA NA NA  0
#> 7        NA NA NA  0
#> 8        NA  1 NA  0
#> 9  6.439700  1  1 NA
#> 10 6.859992  1 NA  1

MIA Method

The MIA method estimates the conditional outcome mean $\mu_{\text{MIA}}(x)$, which is identified by

\int_{w} E [ Y | X=x, W=w, M=1 ] p( w | X=x, R_W = R_X = 1 ) dw

where $R_W$ and $R_X$ are indicators of non-missing values of $W$ and $X$, respectively, and $M$ is an indicator of a complete case pattern (i.e., $Y$, $X$, and $W$ are non-missing).

The package provides two estimators of this identifying functional, referred to as the noniterative conditional expectation (NICE) and iterative conditional expectation (ICE) approaches. The mia_nice function applies the NICE approach and the mia_ice function applies the ICE approach. Both functions obtain point estimates of the identifying functionals of $\mu_{\text{MIA}}(x_1)$ and $\mu_{\text{MIA}}(x_2)$ as well as contrasts between them (differences, ratios).

MIA Method via Noniterative Conditional Expectation

The mia_nice function implements the NICE approach to estimate $\mu_{\text{MIA}}(x)$, which is based on fitting models for the conditional mean of $Y$ and the conditional density of $W$ and performing Monte Carlo integration to compute the integral. The mia_nice function requires specifying the following regression models:

• Y_model: Formula for the outcome model
• W_model: Formula for the auxiliary model when the auxiliary variable is univariate, or a list of formulas for each component of the auxiliary variable

It also requires specifying the names of the variable(s) $X$ by X_names and their values $x_1$ and $x_2$ by X_values_1 and X_values_2, respectively.

An application of mia_nice to estimate $\mu_{\text{MIA}}(x_{1,1} = 0, x_{1,2} = 1)$ and $\mu_{\text{MIA}}(x_{2,1} = 0, x_{2,2} = 0)$ as well as their difference is given below. Note that we set a random number seed because the function involves performing Monte Carlo integration.

set.seed(1234)
res <- mia_nice(data = dat.sim,
                X_names = c("X1", "X2"), 
                X_values_1 = c(0, 1), X_values_2 = c(0, 0),
                Y_model = Y ~ W + X1 + X2, W_model = W ~ X1 + X2)
res
#> MIA METHOD FOR CONDITIONAL MEAN ESTIMATION
#> ==========================================
#> 
#> Setting:
#>   Method:                      Noniterative conditional expectation (NICE)
#>   Outcome variable type:       continuous
#>   Auxiliary variable(s) type:  binary (W)
#> 
#> Results:
#>   Predictor values:            X1=0, X2=1
#>   Mean estimate:               2.1335
#> 
#>   Predictor values:            X1=0, X2=0
#>   Mean estimate:               -0.1636
#> 
#>   Mean difference estimate:    2.2971

We can obtain 95% confidence intervals around our estimates by applying the get_CI function to the output of the mia_nice function. The get_CI function performs nonparametric bootstrap. Here, we use the percentile method with 100 bootstrap replicates for ease of computation.

get_CI(res, n_boot = 100, type = 'perc')
#> BOOTSTRAP CONFIDENCE INTERVALS FOR MIA METHOD
#> =============================================
#> 
#> Setting:
#>   Confidence level:        0.95
#>   Interval type:           perc
#>   Number of replicates:    100
#> 
#> Results:
#>   Predictor values:        X1=0, X2=1
#>   CI for mean:             (2.0350, 2.2495)
#> 
#>   Predictor values:        X1=0, X2=0
#>   CI for mean:             (-0.2638, -0.0588)
#> 
#>   CI for difference:       (2.1565, 2.4539)

MIA Method via Iterative Conditional Expectation

The mia_ice function implements the ICE approach to estimate $\mu_{\text{MIA}}(x)$. The ICE approach is based on re-expressing the identifying functional as

\mu_{\text{MIA}}(x) = E \left[ \, E [ Y | X, W, M=1 ] \mid X=x, R_W = R_X = 1 \, \right]

and fitting regression models for each of the conditional expectations. Specifically, the ICE approach involves the following steps: (1) fit a model for the conditional outcome mean $g(x, w) = E [ Y | X=x, W=w, M=1 ]$ using the complete cases; (2) for every unit with $X$ and $W$ observed, compute the fitted outcome mean $\hat{g}(X, W)$ at the unit’s observed $X$ and $W$; and (3) regress these fitted outcome means on $X$ and take the prediction at $X = x$ as the estimate.

The mia_ice function requires specifying the following regression models:

• Y_model: Formula for the outcome model
• outer_model: Formula for the outer regression of the fitted outcome mean on $X$. The left-hand side must be g_hat, which denotes the fitted outcome mean $\hat{E}[ Y | X, W, M = 1 ]$, and the right-hand side may depend only on the variable(s) in X_names.

As with the mia_nice function, the mia_ice function requires specifying the names of the variable(s) $X$ by X_names and their values $x_1$ and $x_2$ by X_values_1 and X_values_2.

An application of mia_ice to the same estimands is given below. Here we use an outer_model that is saturated with respect to the binary predictors X1 and X2.

res_ice <- mia_ice(data = dat.sim,
                   X_names = c("X1", "X2"), 
                   X_values_1 = c(0, 1), X_values_2 = c(0, 0),
                   Y_model = Y ~ W + X1 + X2, outer_model = g_hat ~ X1 * X2)
res_ice
#> MIA METHOD FOR CONDITIONAL MEAN ESTIMATION
#> ==========================================
#> 
#> Setting:
#>   Method:                      Iterative conditional expectation (ICE)
#>   Outcome variable type:       continuous
#> 
#> Results:
#>   Predictor values:            X1=0, X2=1
#>   Mean estimate:               2.1337
#> 
#>   Predictor values:            X1=0, X2=0
#>   Mean estimate:               -0.1637
#> 
#>   Mean difference estimate:    2.2974

As with mia_nice, confidence intervals can be obtained by applying the get_CI function to the output of mia_ice.

set.seed(1234)
get_CI(res_ice, n_boot = 100, type = 'perc')
#> BOOTSTRAP CONFIDENCE INTERVALS FOR MIA METHOD
#> =============================================
#> 
#> Setting:
#>   Confidence level:        0.95
#>   Interval type:           perc
#>   Number of replicates:    100
#> 
#> Results:
#>   Predictor values:        X1=0, X2=1
#>   CI for mean:             (2.0392, 2.2467)
#> 
#>   Predictor values:        X1=0, X2=0
#>   CI for mean:             (-0.2624, -0.0551)
#> 
#>   CI for difference:       (2.1676, 2.4621)