Mass imputation using the generalized linear model method

Model for the outcome for the mass imputation estimator using generalized linear models via the stats::glm function. Estimation of the mean is done using $S_B$ probability sample or known population totals.

Usage

method_glm(
  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 (default NULL)
vars_selection: whether variable selection should be conducted
pop_totals: population totals from the nonprob function
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 (currently not used, for further development)
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
var_total: total variance, if possible it should be var_prob+var_nonprob if not, just a scalar
model: model type (character "glm")
family: family type (character "glm")

Details

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)

$$ \hat{V}_1 = \frac{1}{n_A^2}\sum_{i=1}^{n_A} \hat{e}_i \left\lbrace \boldsymbol{h}(\boldsymbol{x}_i; \hat{\boldsymbol{\beta}})^\prime\hat{\boldsymbol{c}}\right\rbrace, $$

where $\hat{e}_i = y_i - m(\boldsymbol{x}_i; \hat{\boldsymbol{\beta}})$ and $$\widehat{\boldsymbol{c}}=\left\lbrace n_B^{-1} \sum_{i \in B} \dot{\boldsymbol{m}}\left(\boldsymbol{x}_i ; \boldsymbol{\beta}^*\right) \boldsymbol{h}\left(\boldsymbol{x}_i ; \boldsymbol{\beta}^*\right)^{\prime}\right\rbrace^{-1} N^{-1} \sum_{i \in A} w_i \dot{\boldsymbol{m}}\left(\boldsymbol{x}_i ; \boldsymbol{\beta}^*\right)$$.

Under the linear regression model $\boldsymbol{h}\left(\boldsymbol{x}_i ; \widehat{\boldsymbol{\beta}}\right)=\boldsymbol{x}_i$ and $\widehat{\boldsymbol{c}}=\left(n_A^{-1} \sum_{i \in A} \boldsymbol{x}_i \boldsymbol{x}_i^{\prime}\right)^{-1} N^{-1} \sum_{i \in B} w_i \boldsymbol{x}_i .$

(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.

Furthermore, if only population totals/means are known and assumed to be fixed we set $\hat{V}_2=0$.

References

Kim, J. K., Park, S., Chen, Y., & Wu, C. (2021). Combining non-probability and probability survey samples through mass imputation. Journal of the Royal Statistical Society Series A: Statistics in Society, 184(3), 941-963.

Examples


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

res_glm <- method_glm(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_glm
#> Mass imputation model (GLM approach). Estimated mean: 0.7039 (se: 0.0115)