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 (Y. Chen, Li, and Wu 2020),
• mass imputation estimators based on nearest neighbours (Yang, Kim, and Hwang 2021), predictive mean matching (Chlebicki, Chrostowski, and Beręsewicz 2025), non-parametric (S. Chen, Yang, and Kim 2022) and regression imputation (Kim et al. 2021),
• doubly robust estimators (Y. Chen, Li, and Wu 2020) with bias minimization (Yang, Kim, and Song 2020).

The package allows for:

• variable section in high-dimensional space using SCAD (Yang, Kim, and Song 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:

• in the draft (and not proofread) version of the book Modern inference methods for non-probability samples with R,
• 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:

remotes::install_github("ncn-foreigners/nonprobsvy")

or install the stable version from CRAN

install.packages("nonprobsvy")

or development version from the dev branch

remotes::install_github("ncn-foreigners/nonprobsvy@dev")

Basic idea

Consider the following setting where two samples are available: non-probability (denoted as \(S_A\) ) and probability (denoted as \(S_B\)) where set of auxiliary variables (denoted as \(\boldsymbol{X}\)) is available for both sources while \(Y\) and \(\boldsymbol{d}\) (or \(\boldsymbol{w}\)) is present only in probability sample.

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 \((y_k, \boldsymbol{x}_k, R_k)\), we can approach this problem with the possible scenarios:

• unit-level data is available for the non-probability sample \(S_{A}\), i.e. \((y_{k}, \boldsymbol{x}_{k})\) is available for all units \(k \in S_{A}\), and population-level data is available for \(\boldsymbol{x}_{1}, ..., \boldsymbol{x}_{p}\), denoted as \(\tau_{x_{1}}, \tau_{x_{2}}, ..., \tau_{x_{p}}\) 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 \(S_A\) and the probability sample \(S_B\), i.e. \((y_k, \boldsymbol{x}_k, R_k)\) is determined by the data. is determined by the data: \(R_k=1\) if \(k \in S_A\) otherwise \(R_k=0\), \(y_k\) is observed only for sample \(S_A\) and \(\boldsymbol{x}_k\) is observed in both in both \(S_A\) and \(S_B\),

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

nonprob(
  outcome = y ~ x1 + x2 + ... + xk,
  data = nonprob,
  pop_totals = c(`(Intercept)`= N,
                 x1 = tau_x1,
                 x2 = tau_x2,
                 ...,
                 xk = tau_xk),
  method_outcome = "glm",
  family_outcome = "gaussian"
)

nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  pop_totals = c(`(Intercept)` = N, 
                 x1 = tau_x1, 
                 x2 = tau_x2, 
                 ..., 
                 xk = tau_xk), 
  method_selection = "logit"
)

nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  pop_totals = c(`(Intercept)`= N, 
                 x1 = tau_x1, 
                 x2 = tau_x2, 
                 ..., 
                 xk = tau_xk), 
  method_selection = "logit", 
  control_selection = control_sel(est_method = "gee", gee_h_fun = 1)
)

nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + …, + xk, 
  pop_totals = c(`(Intercept)` = N, 
                 x1 = tau_x1, 
                 x2 = tau_x2, 
                 ..., 
                 xk = tau_xk), 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)

When unit-level data are available for both surveys

nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)

nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "nn", 
  family_outcome = "gaussian", 
  control_outcome = control_outcome(k = 2)
)

nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian"
)

nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian", 
  control_outcome = control_out(penalty = "lasso"), 
  control_inference = control_inf(vars_selection = TRUE)
)

nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_selection = "logit"
)

nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_selection = "logit", 
  control_selection = control_sel(est_method = "gee", gee_h_fun = 1)
)

nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian", 
  control_inference = control_inf(vars_selection = TRUE)
)

nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)

nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "gaussian", 
  control_inference = control_inf(
    vars_selection = TRUE, 
    bias_correction = TRUE
  )
)

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=0.0498, ci=(1.4786, 1.6739))

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

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)
#>  - 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: 2.9981 (se=0.0137, ci=(2.9713, 3.0249))
#>    - variable y2: 1.5906 (se=0.0137, ci=(1.5639, 1.6174))

Funding

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

References (selected)

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