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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:

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:

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.

Sample Auxiliary variables \boldsymbol{X} Target variable Y Design (\boldsymbol{d}) or calibrated (\boldsymbol{w}) weights
S_{\text{NP}} (non-probability) 1 \checkmark \checkmark ?
… \checkmark \checkmark ?
n_{\text{NP}} \checkmark \checkmark ?
S_{\text{P}} (probability) n_{\text{NP}}+1 \checkmark ? \checkmark
… \checkmark ? \checkmark
n_{\text{NP}}+n_{\text{P}} \checkmark ? \checkmark

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:

Estimator family Supported target variable Y
IPW Numeric targets whose population mean is meaningful, including continuous, count, and 0/1 binary variables. No outcome model is fitted.
Mass imputation with method_outcome = "glm" Continuous, count, or binary variables through family_outcome = "gaussian", "poisson", or "binomial".
Mass imputation with method_outcome = "nn", "pmm", or "npar" Numeric targets; categorical, ordinal, survival, and other structured outcomes are not supported.
Doubly robust GLM outcome models only; use family_outcome = "gaussian", "poisson", or "binomial".

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

Estimator Example code
Mass imputation based on regression imputation 
nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  pop_totals = pop_totals,
  method_outcome = "glm",
  family_outcome = "binomial",
  se = FALSE
)
Inverse probability weighting 
nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  pop_totals = pop_totals,
  method_selection = "logit",
  se = FALSE
)
Inverse probability weighting with calibration constraint 
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
)
Doubly robust estimator 
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

Estimator Example code
Mass imputation based on regression imputation 
nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "binomial",
  se = FALSE
)
Mass imputation based on nearest neighbour imputation 
nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "nn", 
  control_outcome = control_out(k = 2),
  se = FALSE
)
Mass imputation based on predictive mean matching 
nonprob(
  outcome = single_shift ~ region + private + nace + size,
  data = admin,
  svydesign = prob,
  method_outcome = "pmm", 
  se = FALSE
)
Mass imputation based on regression imputation with variable selection (LASSO) 
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
)
Inverse probability weighting 
nonprob(
  selection = ~region + private + nace + size,
  target = ~single_shift,
  data = admin,
  svydesign = prob,
  method_selection = "logit",
  se = FALSE
)
Inverse probability weighting with calibration constraint 
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
)
Inverse probability weighting with calibration constraint with variable selection (SCAD) 
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
)
Doubly robust estimator 
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
)
Doubly robust estimator with variable selection (SCAD) and bias minimization 
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)
#>  - 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: inverse probability weighting
#>  - method: logit (mle)
#>  - IPW point estimator: Hajek (denominator: estimated IPW weights = 1025062.6981)
#>  - 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)

Chen, Sixia, Shu Yang, and Jae Kwang Kim. 2022. “Nonparametric Mass Imputation for Data Integration.” Journal of Survey Statistics and Methodology 10 (1): 1–24.
Chen, Yilin, Pengfei Li, and Changbao Wu. 2020. “Doubly Robust Inference with Nonprobability Survey Samples.” Journal of the American Statistical Association 115 (532): 2011–21. https://doi.org/10.1080/01621459.2019.1677241.
Chlebicki, Piotr, Łukasz Chrostowski, and Maciej Beręsewicz. 2025. Data Integration of Non-Probability and Probability Samples with Predictive Mean Matching. https://arxiv.org/abs/2403.13750.
Freedman Ellis, Greg, and Ben Schneider. 2024. Srvyr: ’Dplyr’-Like Syntax for Summary Statistics of Survey Data. https://CRAN.R-project.org/package=srvyr.
Kim, Jae Kwang, Seho Park, Yilin Chen, and Changbao Wu. 2021. “Combining Non-Probability and Probability Survey Samples Through Mass Imputation.” Journal of the Royal Statistical Society Series A: Statistics in Society 184 (3): 941–63. https://doi.org/10.1111/rssa.12696.
Lumley, Thomas. 2004. “Analysis of Complex Survey Samples.” Journal of Statistical Software 9 (1): 1–19.
Lumley, Thomas. 2023. Survey: Analysis of Complex Survey Samples.
Wu, Changbao. 2023. “Statistical Inference with Non-Probability Survey Samples.” Survey Methodology 48 (2): 283–311. https://www150.statcan.gc.ca/n1/pub/12-001-x/2022002/article/00002-eng.htm.
Yang, Shu, Jae Kwang Kim, and Youngdeok Hwang. 2021. “Integration of Data from Probability Surveys and Big Found Data for Finite Population Inference Using Mass Imputation.” Survey Methodology 47 (1): 29–58. https://www150.statcan.gc.ca/n1/pub/12-001-x/2021001/article/00004-eng.htm.
Yang, Shu, Jae Kwang Kim, and Rui Song. 2020. “Doubly Robust Inference When Combining Probability and Non-Probability Samples with High Dimensional Data.” Journal of the Royal Statistical Society Series B: Statistical Methodology 82 (2): 445–65. https://doi.org/10.1111/rssb.12354.