Inference Based on Non-Probability Samples

Statistical inference with non-probability samples when auxiliary information from external sources such as probability samples or population totals or means is available. The package implements various methods such as inverse probability (propensity score) weighting, mass imputation and doubly robust approach. Details can be found in: Chen et al. (2020) , Yang et al. (2020) , Kim et al. (2021) , Yang et al. (2021) < https://www150.statcan.gc.ca/n1/pub/12-001-x/2021001/article/00004-eng.htm> and Wu (2022) < https://www150.statcan.gc.ca/n1/pub/12-001-x/2022002/article/00002-eng.htm>. For details on the package and its functionalities see .

nonprobsvy: an R package for modern statistical inference methods based on non-probability samples

Basic information

The goal of this package is to provide R users access to modern methods for non-probability samples when auxiliary information from the population or probability sample is available:

• inverse probability weighting estimators with possible calibration constraints (Chen et al. 2020),
• mass imputation estimators based on nearest neighbours (Yang et al. 2021), predictive mean matching (Chlebicki et al. 2025), non-parametric (Chen et al. 2022) and regression imputation (Kim et al. 2021),
• doubly robust estimators (Chen et al. 2020) with bias minimization (Yang et al. 2020).

The package allows for:

• variable section in high-dimensional space using SCAD (Yang et al. 2020), Lasso and MCP penalty (via the ncvreg, Rcpp, RcppArmadillo packages),
• estimation of variance using analytical and bootstrap approach (see Wu (2023)),
• integration with the survey and srvyr packages when probability sample is available (Lumley 2004, 2023; Freedman Ellis and Schneider 2024),
• different links for selection (logit, probit and cloglog) and outcome (gaussian, binomial and poisson) variables.

Details on the use of the package can be found:

• please cite the following paper: Chrostowski, Ł., Chlebicki, P., & Beręsewicz, M(2026). nonprobsvy: An R package for modern methods for non-probability surveys. Journal of Statistical Software, 117(2), 1-37. https://doi.org/10.18637/jss.v117.i02
• in example codes that reproduce papers available on github in the repository software tutorials.

Installation

You can install the recent version of nonprobsvy package from main branch Github with:

pak::pkg_install("ncn-foreigners/nonprobsvy")

or install the stable version from CRAN

install.packages("nonprobsvy")

or development version from the dev branch

pak::pkg_install("ncn-foreigners/nonprobsvy@dev")

Basic idea

Consider the following setting where two samples are available: non-probability (denoted as $S_{\text{NP}}$) and probability (denoted as $S_{\text{P}}$) where a set of auxiliary variables (denoted as $\boldsymbol{X}$) is available for both sources, the target variable $Y$ is observed in the non-probability sample, and design or calibrated weights ($\boldsymbol{d}$ or $\boldsymbol{w}$) are observed in the probability sample.

The current implementation does not use target-variable values from the probability sample. Data structures where $Y$ is observed in both samples, or where overlapping units must be linked across samples, are not currently implemented. The dependence and key arguments in control_sel() are reserved for future overlap handling and currently raise a “not yet implemented” error if set.

Basic functionalities

Suppose $Y$ is the target variable, $\boldsymbol{X}$ is a matrix of auxiliary variables, $R$ is the inclusion indicator. Then, if we are interested in estimating the mean $\bar{\tau}_Y$ or the sum $\tau_Y$ of the of the target variable given the observed data set $(y_k, \boldsymbol{x}k, I{\text{NP}, k})$, we can approach this problem with the possible scenarios:

• unit-level data is available for the non-probability sample $S_{\text{NP}}$, i.e. $(y_k,\boldsymbol{x}k)$ is available for all units $k \in S{\text{NP}}$, and population-level data is available for $\boldsymbol{x}1,\ldots,\boldsymbol{x}p$, denoted as $\tau{x{1}},\tau_{x_{2}},\ldots,\tau_{x_{p}}$ and population size $N$ is known. We can also consider situations where population data are estimated (e.g. on the basis of a survey to which we do not have access),
• unit-level data is available for the non-probability sample $S_{\text{NP}}$ and the probability sample $S_{\text{P}}$, i.e. $(\boldsymbol{x}k,I{\text{NP}, k})$ is determined by the data: $I_{\text{NP}, k}=1$ if $k \in S_{\text{NP}}$ otherwise $I_{\text{NP}, k}=0$, $y_k$ is observed only for sample $S_{\text{NP}}$ and $\boldsymbol{x}k$ is observed in both $S{\text{NP}}$ and $S_{\text{P}}$.

Supported target-variable types depend on the estimator family:

The compact examples below use the built-in admin non-probability sample and jvs probability sample.

library(survey)
library(nonprobsvy)
data(admin)
data(jvs)

prob <- svydesign(
  ids = ~1,
  weights = ~weight,
  strata = ~size + nace + region,
  data = jvs
)
pop_totals <- colSums(model.matrix(~region + private + nace + size, jvs) * jvs$weight)

When unit-level data is available for non-probability survey only

nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  pop_totals = pop_totals,
  method_outcome = "glm",
  family_outcome = "binomial",
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  pop_totals = pop_totals,
  method_selection = "logit",
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  pop_totals = pop_totals,
  method_selection = "logit",
  control_selection = control_sel(est_method = "gee", gee_h_fun = 1),
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  pop_totals = pop_totals,
  method_outcome = "glm", 
  family_outcome = "binomial",
  se = FALSE
)

When unit-level data are available for both surveys

nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "binomial",
  se = FALSE
)

nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "nn", 
  control_outcome = control_out(k = 2),
  se = FALSE
)

nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "pmm", 
  se = FALSE
)

nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "glm",
  family_outcome = "binomial",
  control_outcome = control_out(penalty = "lasso"), 
  control_inference = control_inf(vars_selection = TRUE),
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  svydesign = prob,
  method_selection = "logit",
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  svydesign = prob,
  method_selection = "logit", 
  control_selection = control_sel(est_method = "gee", gee_h_fun = 1),
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  svydesign = prob,
  method_selection = "logit",
  control_selection = control_sel(penalty = "SCAD"),
  control_inference = control_inf(vars_selection = TRUE),
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "binomial",
  se = FALSE
)

nonprob(
  selection = ~region + private + nace + size,
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "binomial",
  control_inference = control_inf(
    vars_selection = TRUE, 
    vars_combine = TRUE,
    bias_correction = TRUE
  ),
  se = FALSE
)

Examples

Simulate example data from the following paper: Kim, Jae Kwang, and Zhonglei Wang. “Sampling techniques for big data analysis.” International Statistical Review 87 (2019): S177-S191 [section 5.2]

library(survey)
library(nonprobsvy)

set.seed(1234567890)
N <- 1e6 ## 1000000
n <- 1000
x1 <- rnorm(n = N, mean = 1, sd = 1)
x2 <- rexp(n = N, rate = 1)
epsilon <- rnorm(n = N) # rnorm(N)
y1 <- 1 + x1 + x2 + epsilon
y2 <- 0.5*(x1 - 0.5)^2 + x2 + epsilon
p1 <- exp(x2)/(1+exp(x2))
p2 <- exp(-0.5+0.5*(x2-2)^2)/(1+exp(-0.5+0.5*(x2-2)^2))
flag_bd1 <- rbinom(n = N, size = 1, prob = p1)
flag_srs <- as.numeric(1:N %in% sample(1:N, size = n))
base_w_srs <- N/n
population <- data.frame(x1,x2,y1,y2,p1,p2,base_w_srs, flag_bd1, flag_srs, pop_size = N)
base_w_bd <- N/sum(population$flag_bd1)

Declare svydesign object with survey package

sample_prob <- svydesign(ids= ~1, weights = ~ base_w_srs, 
                         data = subset(population, flag_srs == 1),
                         fpc = ~ pop_size)

sample_prob
#> Independent Sampling design
#> svydesign(ids = ~1, weights = ~base_w_srs, data = subset(population, 
#>     flag_srs == 1), fpc = ~pop_size)

or with the srvyr package

sample_prob <- srvyr::as_survey_design(.data = subset(population, flag_srs == 1),
                                       weights = base_w_srs)

sample_prob

Independent Sampling design (with replacement)
Called via srvyr
Sampling variables:
Data variables: 
  - x1 (dbl), x2 (dbl), y1 (dbl), y2 (dbl), p1 (dbl), p2 (dbl), base_w_srs (dbl), flag_bd1 (int), flag_srs (dbl)

Estimate population mean of y1 based on doubly robust estimator using IPW with calibration constraints and we specify that auxiliary variables should not be combined for the inference.

result_dr <- nonprob(
  selection = ~ x2,
  outcome = y1 + y2 ~ x1 + x2,
  data = subset(population, flag_bd1 == 1),
  svydesign = sample_prob
)

Results

result_dr
#> A nonprob object
#>  - estimator type: doubly robust
#>  - method: glm (gaussian)
#>  - IPW point estimator: Hajek (denominator: 1025063)
#>  - auxiliary variables source: survey
#>  - vars selection: false
#>  - variance estimator: analytic
#>  - population size fixed: false
#>  - naive (uncorrected) estimators:
#>    - variable y1: 3.1817
#>    - variable y2: 1.8087
#>  - selected estimators:
#>    - variable y1: 2.9500 (se=0.0414, ci=(2.8689, 3.0312))
#>    - variable y2: 1.5762 (se=0.0313, ci=(1.5150, 1.6375))

Mass imputation estimator

result_mi <- nonprob(
  outcome = y1 + y2 ~ x1 + x2,
  data = subset(population, flag_bd1 == 1),
  svydesign = sample_prob
)

Results

result_mi
#> A nonprob object
#>  - estimator type: mass imputation
#>  - method: glm (gaussian)
#>  - auxiliary variables source: survey
#>  - vars selection: false
#>  - variance estimator: analytic
#>  - population size fixed: false
#>  - naive (uncorrected) estimators:
#>    - variable y1: 3.1817
#>    - variable y2: 1.8087
#>  - selected estimators:
#>    - variable y1: 2.9498 (se=0.0420, ci=(2.8675, 3.0321))
#>    - variable y2: 1.5760 (se=0.0326, ci=(1.5122, 1.6398))

Inverse probability weighting estimator

For IPW, MLE without a fixed pop_size, pop_totals, or pop_means uses the Hajek-type estimator. Supplying a fixed population size uses the Horvitz-Thompson-type estimator, while IPW-GEE with a reference survey uses sum(weights(svydesign)) as the denominator.

result_ipw <- nonprob(
  selection = ~ x2,
  target = ~y1+y2,
  data = subset(population, flag_bd1 == 1),
  svydesign = sample_prob)

Results

result_ipw
#> A nonprob object
#>  - estimator type: IPW (Hajek, denominator: 1025063)
#>  - method: logit (mle)
#>  - auxiliary variables source: survey
#>  - vars selection: false
#>  - variance estimator: analytic
#>  - population size fixed: false
#>  - naive (uncorrected) estimators:
#>    - variable y1: 3.1817
#>    - variable y2: 1.8087
#>  - selected estimators:
#>    - variable y1: 2.9248 (se=0.0500, ci=(2.8269, 3.0227))
#>    - variable y2: 1.5517 (se=0.0499, ci=(1.4539, 1.6496))

Funding

Work on this package is supported by the National Science Centre, OPUS 20 grant no. 2020/39/B/HS4/00941.

References (selected)