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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:

  • inverse probability weighting estimators with possible calibration constraints (Chen, Li, and Wu 2020),
  • mass imputation estimators based in nearest neighbours (Yang, Kim, and Hwang 2021), predictive mean matching and regression imputation (Kim et al. 2021),
  • doubly robust estimators 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 package when probability sample is available Lumley (2023),
  • different links for selection (logit, probit and cloglog) and outcome (gaussian, binomial and poisson) variables.

Details on use of the package be found:

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 SA ) and probability (denoted as SB) where set of auxiliary variables (denoted as X) is available for both sources while Y and d (or w) is present only in probability sample.

Sample Auxiliary variables X Target variable Y Design (d) or calibrated (w) weights
SA (non-probability) 1 ✓ ✓ ?
… ✓ ✓ ?
nA ✓ ✓ ?
SB (probability) nA + 1 ✓ ? ✓
… ✓ ? ✓
nA + nB ✓ ? ✓

Basic functionalities

Suppose Y is the target variable, X is a matrix of auxiliary variables, R is the inclusion indicator. Then, if we are interested in estimating the mean τ̄Y or the sum τY of the of the target variable given the observed data set (yk,xk,Rk), we can approach this problem with the possible scenarios:

  • unit-level data is available for the non-probability sample SA, i.e. (yk,xk) is available for all units k ∈ SA, and population-level data is available for x1, ..., xp, denoted as τx1, τx2, ..., τxp 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 SA and the probability sample SB, i.e. (yk,xk,Rk) is determined by the data. is determined by the data: Rk = 1 if k ∈ SA otherwise Rk = 0, yk is observed only for sample SA and xk is observed in both in both SA and SB,

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

Estimator Example code
Mass imputation based on regression imputation 
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"
)
Inverse probability weighting 
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"
)
Inverse probability weighting with calibration constraint 
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 = controlSel(est_method_sel = "gee", h = 1)
)
Doubly robust estimator 
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

Estimator Example code
Mass imputation based on regression imputation 
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)
Mass imputation based on nearest neighbour imputation 
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "nn", 
  family_outcome = "gaussian", 
  control_outcome = controlOutcome(k = 2)
)
Mass imputation based on predictive mean matching 
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian"
)
Mass imputation based on regression imputation with variable selection (LASSO) 
nonprob(
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian", 
  control_outcome = controlOut(penalty = "lasso"), 
  control_inference = controlInf(vars_selection = TRUE)
)
Inverse probability weighting 
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_selection = "logit"
)
Inverse probability weighting with calibration constraint 
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_selection = "logit", 
  control_selection = controlSel(est_method_sel = "gee", h = 1)
)
Inverse probability weighting with calibration constraint with variable selection (SCAD) 
nonprob(
  selection =  ~ x1 + x2 + ... + xk, 
  target = ~ y, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "pmm", 
  family_outcome = "gaussian", 
  control_inference = controlInf(vars_selection = TRUE)
)
Doubly robust estimator 
nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob, 
  method_outcome = "glm", 
  family_outcome = "gaussian"
)
Doubly robust estimator with variable selection (SCAD) and bias minimization 
nonprob(
  selection = ~ x1 + x2 + ... + xk, 
  outcome = y ~ x1 + x2 + ... + xk, 
  data = nonprob, 
  svydesign = prob,
  method_outcome = "glm", 
  family_outcome = "gaussian", 
  control_inference = controlInf(
    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)
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))

Estimate population mean of y1 based on doubly robust estimator using IPW with calibration constraints.

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

Results

summary(result_dr)
#> 
#> Call:
#> nonprob(data = subset(population, flag_bd1 == 1), selection = ~x2, 
#>     outcome = y1 ~ x1 + x2, svydesign = sample_prob)
#> 
#> -------------------------
#> Estimated population mean: 2.95 with overall std.err of: 0.04195
#> And std.err for nonprobability and probability samples being respectively:
#> 0.000783 and 0.04195
#> 
#> 95% Confidence inverval for popualtion mean:
#>    lower_bound upper_bound
#> y1    2.867789     3.03224
#> 
#> 
#> Based on: Doubly-Robust method
#> For a population of estimate size: 1025063
#> Obtained on a nonprobability sample of size: 693011
#> With an auxiliary probability sample of size: 1000
#> -------------------------
#> 
#> Regression coefficients:
#> -----------------------
#> For glm regression on outcome variable:
#>             Estimate Std. Error z value P(>|z|)    
#> (Intercept) 0.996282   0.002139   465.8  <2e-16 ***
#> x1          1.001931   0.001200   835.3  <2e-16 ***
#> x2          0.999125   0.001098   910.2  <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> -----------------------
#> For glm regression on selection variable:
#>              Estimate Std. Error z value P(>|z|)    
#> (Intercept) -0.498997   0.003702  -134.8  <2e-16 ***
#> x2           1.885629   0.005303   355.6  <2e-16 ***
#> -------------------------
#> 
#> Weights:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   1.000   1.071   1.313   1.479   1.798   2.647 
#> -------------------------
#> 
#> Residuals:
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> -0.99999  0.06603  0.23778  0.26046  0.44358  0.62222 
#> 
#> AIC: 1010622
#> BIC: 1010645
#> Log-Likelihood: -505309 on 694009 Degrees of freedom

Mass imputation estimator

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

Results

summary(result_mi)
#> 
#> Call:
#> nonprob(data = subset(population, flag_bd1 == 1), outcome = y1 ~ 
#>     x1 + x2, svydesign = sample_prob)
#> 
#> -------------------------
#> Estimated population mean: 2.95 with overall std.err of: 0.04203
#> And std.err for nonprobability and probability samples being respectively:
#> 0.001227 and 0.04201
#> 
#> 95% Confidence inverval for popualtion mean:
#>    lower_bound upper_bound
#> y1    2.867433    3.032186
#> 
#> 
#> Based on: Mass Imputation method
#> For a population of estimate size: 1e+06
#> Obtained on a nonprobability sample of size: 693011
#> With an auxiliary probability sample of size: 1000
#> -------------------------
#> 
#> Regression coefficients:
#> -----------------------
#> For glm regression on outcome variable:
#>             Estimate Std. Error z value P(>|z|)    
#> (Intercept) 0.996282   0.002139   465.8  <2e-16 ***
#> x1          1.001931   0.001200   835.3  <2e-16 ***
#> x2          0.999125   0.001098   910.2  <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> -------------------------

Inverse probability weighting estimator

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

Results

summary(result_ipw)
#> 
#> Call:
#> nonprob(data = subset(population, flag_bd1 == 1), selection = ~x2, 
#>     target = ~y1, svydesign = sample_prob)
#> 
#> -------------------------
#> Estimated population mean: 2.925 with overall std.err of: 0.05
#> And std.err for nonprobability and probability samples being respectively:
#> 0.001586 and 0.04997
#> 
#> 95% Confidence inverval for popualtion mean:
#>    lower_bound upper_bound
#> y1     2.82679    3.022776
#> 
#> 
#> Based on: Inverse probability weighted method
#> For a population of estimate size: 1025063
#> Obtained on a nonprobability sample of size: 693011
#> With an auxiliary probability sample of size: 1000
#> -------------------------
#> 
#> Regression coefficients:
#> -----------------------
#> For glm regression on selection variable:
#>              Estimate Std. Error z value P(>|z|)    
#> (Intercept) -0.498997   0.003702  -134.8  <2e-16 ***
#> x2           1.885629   0.005303   355.6  <2e-16 ***
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> -------------------------
#> 
#> Weights:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   1.000   1.071   1.313   1.479   1.798   2.647 
#> -------------------------
#> 
#> Residuals:
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> -0.99999  0.06603  0.23778  0.26046  0.44358  0.62222 
#> 
#> AIC: 1010622
#> BIC: 1010645
#> Log-Likelihood: -505309 on 694009 Degrees of freedom

Funding

Work on this package is supported by the National Science Centre, OPUS 22 grant no. 2020/39/B/HS4/00941.

References (selected)

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.
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.
———. 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.