Mass imputation using predictive mean matching method

Model for the outcome for the mass imputation estimator. The implementation is currently based on RANN::nn2 function and thus it uses Euclidean distance for matching units from $S_A$ (non-probability) to $S_B$ (probability) based on predicted values from model $\boldsymbol{x}_i$ based either on method_glm or method_npar. Estimation of the mean is done using $S_B$ sample.

This implementation extends Yang et al. (2021) approach as described in Chlebicki et al. (2025), namely:

pmm_weights: if k>1 weighted aggregation of the mean for a given unit is used. We use distance matrix returned by RANN::nn2 function (pmm_weights from the control_out() function)
nn_exact_se: if the non-probability sample is small we recommend using a mini-bootstrap approach to estimate variance from the non-probability sample (nn_exact_se from the control_inf() function)
pmm_k_choice: the main nonprob function allows for dynamic selection of k neighbours based on the variance minimization procedure (pmm_k_choice from the control_out() function)

Usage

method_pmm(
  y_nons,
  X_nons,
  X_rand,
  svydesign,
  weights = NULL,
  family_outcome = "gaussian",
  start_outcome = NULL,
  vars_selection = FALSE,
  pop_totals = NULL,
  pop_size = NULL,
  control_outcome = control_out(),
  control_inference = control_inf(),
  verbose = FALSE,
  se = TRUE
)

Arguments

y_nons: target variable from non-probability sample
X_nons: a model.matrix with auxiliary variables from non-probability sample
X_rand: a model.matrix with auxiliary variables from non-probability sample
svydesign: a svydesign object
weights: case / frequency weights from non-probability sample
family_outcome: family for the glm model
start_outcome: start parameters
vars_selection: whether variable selection should be conducted
pop_totals: a place holder (not used in method_pmm)
pop_size: population size from the nonprob function
control_outcome: controls passed by the control_out function
control_inference: controls passed by the control_inf function
verbose: parameter passed from the main nonprob function
se: whether standard errors should be calculated

Value

an nonprob_method class which is a list with the following entries

model_fitted: fitted model either an glm.fit or cv.ncvreg object
y_nons_pred: predicted values for the non-probablity sample
y_rand_pred: predicted values for the probability sample or population totals
coefficients: coefficients for the model (if available)
svydesign: an updated surveydesign2 object (new column y_hat_MI is added)
y_mi_hat: estimated population mean for the target variable
vars_selection: whether variable selection was performed
var_prob: variance for the probability sample component (if available)
var_nonprob: variance for the non-probability sampl component
model: model type (character "pmm")
family: depends on the method selected for estimating E(Y|X)

Details

Matching

In the package we support two types of matching:

$\hat{y} - \hat{y}$ matching (default; control_out(pmm_match_type = 1)).
$\hat{y} - y$ matching (control_out(pmm_match_type = 2)).

Analytical variance

The variance of the mean is estimated based on the following approach (a) non-probability part ( $S_A$ with size $n_A$ ; denoted as var_nonprob in the result) is currently estimated using the non-parametric mini-bootstrap estimator proposed by Chlebicki et al. (2025, Algorithm 2). It is not proved to be consistent but with good finite population properties. This bootstrap can be applied using control_inference(nn_exact_se=TRUE) and can be summarized as follows:

Sample $n_A$ units from $S_A$ with replacement to create $S_A'$ (if pseudo-weights are present inclusion probabilities should be proportional to their inverses).
Estimate regression model $\mathbb{E}[Y|\boldsymbol{X}]=m(\boldsymbol{X}, \cdot)$ based on $S_{A}'$ from step 1.
Compute $\hat{\nu}'(i,t)$ for $t=1,\dots,k, i\in S_{B}$ using estimated $m(\boldsymbol{x}', \cdot)$ and $\left\lbrace(y_{j},\boldsymbol{x}_{j})| j\in S_{A}'\right\rbrace$ .
Compute $\displaystyle\frac{1}{k}\sum_{t=1}^{k}y_{\hat{\nu}'(i)}$ using $Y$ values from $S_{A}'$ .
Repeat steps 1-4 $M$ times (we set (hard-coded) $M=50$ in our code).
Estimate $\hat{V}_1=\text{var}({\hat{\boldsymbol{\mu}}})$ obtained from simulations and save it as var_nonprob.

(b) probability part ( $S_B$ with size $n_B$ ; denoted as var_prob in the result)

This part uses functionalities of the {survey} package and the variance is estimated using the following equation:

$\hat{V}_2=\frac{1}{N^2} \sum_{i=1}^{n_B} \sum_{j=1}^{n_B} \frac{\pi_{i j}-\pi_i \pi_j}{\pi_{i j}} \frac{m(\boldsymbol{x}_i; \hat{\boldsymbol{\beta}})}{\pi_i} \frac{m(\boldsymbol{x}_i; \hat{\boldsymbol{\beta}})}{\pi_j}.$

Note that $\hat{V}_2$ in principle can be estimated in various ways depending on the type of the design and whether population size is known or not.

Examples


data(admin)
data(jvs)
jvs_svy <- svydesign(ids = ~ 1,  weights = ~ weight, strata = ~ size + nace + region, data = jvs)

res_pmm <- method_pmm(y_nons = admin$single_shift,
                      X_nons = model.matrix(~ region + private + nace + size, admin),
                      X_rand = model.matrix(~ region + private + nace + size, jvs),
                      svydesign = jvs_svy)

res_pmm
#> Mass imputation model (PMM approach). Estimated mean: 0.6969 (se: 0.0146)