nonprobsvy (version 0.2.0)

method_npar: Mass imputation using non-parametric model method

Description

Model for the outcome for the mass imputation estimator using loess via stats::loess. Estimation of the mean is done using the \(S_B\) probability sample.

Usage

method_npar(
  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
)

Value

an nonprob_method class which is a list with the following entries

model_fitted

fitted model object returned by stats::loess

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 "npar")

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 (default NULL)

family_outcome

family for the glm model)

start_outcome

a place holder (not used in method_npar)

vars_selection

whether variable selection should be conducted

pop_totals

a place holder (not used in method_npar)

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

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)

$$ \hat{V}_1 = \frac{1}{N^2} \sum_{i=1}^{n_A} \left\lbrace\hat{g}_B(\boldsymbol{x}_i)\right\rbrace^{2} \hat{e}_i^2, $$

where \(\hat{e}_i=y_i - \hat{m}(x_i)\) is the residual and \(\hat{g}_B(\boldsymbol{x}_i) = \left\lbrace \pi_B(\boldsymbol{x}_i) \right\rbrace^{-1}\) can be estimated various ways. In the package we estimate \(\hat{g}_B(\boldsymbol{x}_i)\) using \(\pi_B(\boldsymbol{x}_i)=E(R | \boldsymbol{x})\) as suggested by Chen et al. (2022, p. 6). In particular, we currently support this using stats::loesswith"gaussian"` family.

(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:

$$ \hat{V}_2=\frac{1}{N^2} \sum_{i=1}^{n_B} \sum_{j=1}^{n_B} \frac{\pi_{i j}-\pi_i \pi_j}{\pi_{i j}} \frac{\hat{m}(x_i)}{\pi_i} \frac{\hat{m}(x_j)}{\pi_j}. $$

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.

References

Chen, S., Yang, S., & Kim, J. K. (2022). Nonparametric mass imputation for data integration. Journal of Survey Statistics and Methodology, 10(1), 1-24.

Examples

set.seed(123123123)
N <- 10000
n_a <- 500
n_b <- 1000
n_b1 <- 0.7*n_b
n_b2 <- 0.3*n_b
x1 <- rnorm(N, 2, 1)
x2 <- rnorm(N, 2, 1)
y1 <- rnorm(N, 0.3 + 2*x1+ 2*x2, 1)
y2 <- rnorm(N, 0.3 + 0.5*x1^2+ 0.5*x2^2, 1)
strata <- x1 <= 2
pop <- data.frame(x1, x2, y1, y2, strata)
sample_a <- pop[sample(1:N, n_a),]
sample_a$w_a <- N/n_a
sample_a_svy <- svydesign(ids=~1, weights=~w_a, data=sample_a)
pop1 <- subset(pop, strata == TRUE)
pop2 <- subset(pop, strata == FALSE)
sample_b <- rbind(pop1[sample(1:nrow(pop1), n_b1), ],
                  pop2[sample(1:nrow(pop2), n_b2), ])
res_y_npar <- nonprob(outcome = y1 + y2 ~ x1 + x2,
                      data = sample_b,
                      svydesign = sample_a_svy,
                      method_outcome = "npar")
res_y_npar