Inference with the non-probability survey samples

nonprob fits model for inference based on non-probability surveys (including big data) using various methods. The function allows you to estimate the population mean with access to a reference probability sample, as well as sums and means of covariates.

The package implements state-of-the-art approaches recently proposed in the literature: Chen et al. (2020), Yang et al. (2020), Wu (2022) and use the Lumley 2004 survey package for inference.

It provides propensity score weighting (e.g. with calibration constraints), mass imputation (e.g. nearest neighbor) and doubly robust estimators that take into account minimisation of the asymptotic bias of the population mean estimators, variable selection or overlap between probability and non-probability samples. The package uses survey package functionality when a probability sample is available.

Usage

nonprob(
  data,
  selection = NULL,
  outcome = NULL,
  target = NULL,
  svydesign = NULL,
  pop_totals = NULL,
  pop_means = NULL,
  pop_size = NULL,
  method_selection = c("logit", "cloglog", "probit"),
  method_outcome = c("glm", "nn", "pmm"),
  family_outcome = c("gaussian", "binomial", "poisson"),
  subset = NULL,
  strata = NULL,
  weights = NULL,
  na_action = NULL,
  control_selection = controlSel(),
  control_outcome = controlOut(),
  control_inference = controlInf(),
  start_selection = NULL,
  start_outcome = NULL,
  verbose = FALSE,
  x = TRUE,
  y = TRUE,
  se = TRUE,
  ...
)

Arguments

data: data.frame with data from the non-probability sample.
selection: formula, the selection (propensity) equation.
outcome: formula, the outcome equation.
target: formula with target variables.
svydesign: an optional svydesign object (from the survey package) containing probability sample and design weights.
pop_totals: an optional named vector with population totals of the covariates.
pop_means: an optional named vector with population means of the covariates.
pop_size: an optional double with population size.
method_selection: a character with method for propensity scores estimation
method_outcome: a character with method for response variable estimation
family_outcome: a character string describing the error distribution and link function to be used in the model. Default is "gaussian". Currently supports: gaussian with identity link, poisson and binomial.
subset: an optional vector specifying a subset of observations to be used in the fitting process.
strata: an optional vector specifying strata.
weights: an optional vector of prior weights to be used in the fitting process. Should be NULL or a numeric vector. It is assumed that this vector contains frequency or analytic weights
na_action: a function which indicates what should happen when the data contain NAs.
control_selection: a list indicating parameters to use in fitting selection model for propensity scores
control_outcome: a list indicating parameters to use in fitting model for outcome variable
control_inference: a list indicating parameters to use in inference based on probability and non-probability samples, contains parameters such as estimation method or variance method
start_selection: an optional vector with starting values for the parameters of the selection equation
start_outcome: an optional vector with starting values for the parameters of the outcome equation
verbose: verbose, numeric
x: Logical value indicating whether to return model matrix of covariates as a part of output.
y: Logical value indicating whether to return vector of outcome variable as a part of output.
se: Logical value indicating whether to calculate and return standard error of estimated mean.
...: Additional, optional arguments.

Value

Returns an object of class c("nonprobsvy", "nonprobsvy_dr") in case of doubly robust estimator, c("nonprobsvy", "nonprobsvy_mi") in case of mass imputation estimator and c("nonprobsvy", "nonprobsvy_ipw") in case of inverse probability weighting estimator with type list containing:

X -- model matrix containing data from probability and non-probability samples if specified at a function call.
y -- list of vector of outcome variables if specified at a function call.
prob -- vector of estimated propensity scores for non-probability sample.
weights -- vector of estimated weights for non-probability sample.
control -- list of control functions.
output -- output of the model with information on the estimated population mean and standard errors.
SE -- standard error of the estimator of the population mean, divided into errors from probability and non-probability samples.
confidence_interval -- confidence interval of population mean estimator
nonprob_size -- size of non-probability sample
prob_size -- size of probability sample
pop_size -- estimated population size derived from estimated weights (non-probability sample) or known design weights (probability sample)
outcome -- list containing information about the fitting of the mass imputation model, in the case of regression model the object containing the list returned by stats::glm(), in the case of the nearest neighbour imputation the object containing list returned by RANN::nn2(). If bias_correction in controlInf() is set to TRUE, the estimation is based on the joint estimating equations for the selection and outcome model and therefore, the list is different from the one returned by the stats::glm() function and contains elements such as
- coefficients -- estimated coefficients of the regression model
- std_err -- standard errors of the estimated coefficients
- residuals -- The response residuals
- variance_covariance -- The variance-covariance matrix of the coefficient estimates
- df_residual -- The degrees of freedom for residuals
- family -- specifies the error distribution and link function to be used in the model
- fitted.values -- The predicted values of the response variable based on the fitted model
- linear.predictors -- The linear fit on link scale
- X -- The design matrix
- method -- set on glm, since the regression method
In addition, if the variable selection model for the outcome variable is fitting, the list includes the
- cve -- the error for each value of lambda, averaged across the cross-validation folds.
selection -- list containing information about fitting of propensity score model, such as
- coefficients -- a named vector of coefficients
- std_err -- standard errors of the estimated model coefficients
- residuals -- the response residuals
- variance -- the root mean square error
- fitted_values -- the fitted mean values, obtained by transforming the linear predictors by the inverse of the link function.
- link -- the link object used.
- linear_predictors -- the linear fit on link scale.
- aic -- A version of Akaike's An Information Criterion, minus twice the maximized log-likelihood plus twice the number of parameters.
- weights -- vector of estimated weights for non-probability sample.
- prior.weights -- the weights initially supplied, a vector of 1s if none were.
- formula -- the formula supplied.
- df_residual -- the residual degrees of freedom.
- log_likelihood -- value of log-likelihood function if mle method, in the other case NA.
- cve -- the error for each value of the lambda, averaged across the cross-validation folds for the variable selection model when the propensity score model is fitting. Returned only if selection of variables for the model is used.
stat -- matrix of the estimated population means in each bootstrap iteration. Returned only if a bootstrap method is used to estimate the variance and keep_boot in controlInf() is set on TRUE.

Details

Let \(y\) be the response variable for which we want to estimate the population mean, given by \[\mu_{y} = \frac{1}{N} \sum_{i=1}^N y_{i}.\] For this purpose we consider data integration with the following structure. Let \(S_A\) be the non-probability sample with the design matrix of covariates as \[ \boldsymbol{X}_A = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \cr x_{21} & x_{22} & \cdots & x_{2p} \cr \vdots & \vdots & \ddots & \vdots \cr x_{n_{A}1} & x_{n_{A}2} & \cdots & x_{n_{A}p} \cr \end{bmatrix} \] and vector of outcome variable \[ \boldsymbol{y} = \begin{bmatrix} y_{1} \cr y_{2} \cr \vdots \cr y_{n_{A}}. \end{bmatrix} \] On the other hand, let \(S_B\) be the probability sample with design matrix of covariates be \[ \boldsymbol{X}_B = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \cr x_{21} & x_{22} & \cdots & x_{2p} \cr \vdots & \vdots & \ddots & \vdots \cr x_{n_{B}1} & x_{n_{B}2} & \cdots & x_{n_{B}p}. \cr \end{bmatrix} \] Instead of a sample of units we can consider a vector of population sums in the form of \(\tau_x = (\sum_{i \in \mathcal{U}}\boldsymbol{x}_{i1}, \sum_{i \in \mathcal{U}}\boldsymbol{x}_{i2}, ..., \sum_{i \in \mathcal{U}}\boldsymbol{x}_{ip})\) or means \(\frac{\tau_x}{N}\), where \(\mathcal{U}\) refers to a finite population. Note that we do not assume access to the response variable for \(S_B\). In general we make the following assumptions:

The selection indicator of belonging to non-probability sample \(R_{i}\) and the response variable \(y_i\) are independent given the set of covariates \(\boldsymbol{x}_i\).
All units have a non-zero propensity score, i.e., \(\pi_{i}^{A} > 0\) for all i.
The indicator variables \(R_{i}^{A}\) and \(R_{j}^{A}\) are independent for given \(\boldsymbol{x}_i\) and \(\boldsymbol{x}_j\) for \(i \neq j\).

There are three possible approaches to the problem of estimating population mean using non-probability samples:

Inverse probability weighting - The main drawback of non-probability sampling is the unknown selection mechanism for a unit to be included in the sample. This is why we talk about the so-called "biased sample" problem. The inverse probability approach is based on the assumption that a reference probability sample is available and therefore we can estimate the propensity score of the selection mechanism. The estimator has the following form: \[\hat{\mu}_{IPW} = \frac{1}{N^{A}}\sum_{i \in S_{A}} \frac{y_{i}}{\hat{\pi}_{i}^{A}}.\] For this purpose several estimation methods can be considered. The first approach is maximum likelihood estimation with a corrected log-likelihood function, which is given by the following formula \[ \ell^{*}(\boldsymbol{\theta}) = \sum_{i \in S_{A}}\log \left\lbrace \frac{\pi(\boldsymbol{x}_{i}, \boldsymbol{\theta})}{1 - \pi(\boldsymbol{x}_{i},\boldsymbol{\theta})}\right\rbrace + \sum_{i \in S_{B}}d_{i}^{B}\log \left\lbrace 1 - \pi({\boldsymbol{x}_{i},\boldsymbol{\theta})}\right\rbrace.\] In the literature, the main approach to modelling propensity scores is based on the logit link function. However, we extend the propensity score model with the additional link functions such as cloglog and probit. The pseudo-score equations derived from ML methods can be replaced by the idea of generalised estimating equations with calibration constraints defined by equations. \[ \mathbf{U}(\boldsymbol{\theta})=\sum_{i \in S_A} \mathbf{h}\left(\mathbf{x}_i, \boldsymbol{\theta}\right)-\sum_{i \in S_B} d_i^B \pi\left(\mathbf{x}_i, \boldsymbol{\theta}\right) \mathbf{h}\left(\mathbf{x}_i, \boldsymbol{\theta}\right).\] Notice that for \( \mathbf{h}\left(\mathbf{x}_i, \boldsymbol{\theta}\right) = \frac{\pi(\boldsymbol{x}, \boldsymbol{\theta})}{\boldsymbol{x}}\) We do not need a probability sample and can use a vector of population totals/means.
Mass imputation -- This method is based on a framework where imputed values of outcome variables are created for the entire probability sample. In this case, we treat the large sample as a training data set that is used to build an imputation model. Using the imputed values for the probability sample and the (known) design weights, we can build a population mean estimator of the form: \[\hat{\mu}_{MI} = \frac{1}{N^B}\sum_{i \in S_{B}} d_{i}^{B} \hat{y}_i.\] It opens the the door to a very flexible method for imputation models. The package uses generalized linear models from stats::glm(), the nearest neighbour algorithm using RANN::nn2() and predictive mean matching.
Doubly robust estimation -- The IPW and MI estimators are sensitive to misspecified models for the propensity score and outcome variable, respectively. To this end, so-called doubly robust methods are presented that take these problems into account. It is a simple idea to combine propensity score and imputation models during inference, leading to the following estimator \[\hat{\mu}_{DR} = \frac{1}{N^A}\sum_{i \in S_A} \hat{d}_i^A (y_i - \hat{y}_i) + \frac{1}{N^B}\sum_{i \in S_B} d_i^B \hat{y}_i.\] In addition, an approach based directly on bias minimisation has been implemented. The following formula \[ \begin{aligned} bias(\hat{\mu}_{DR}) = & \mathbb{E} (\hat{\mu}_{DR} - \mu) \cr = & \mathbb{E} \left\lbrace \frac{1}{N} \sum_{i=1}^N (\frac{R_i^A}{\pi_i^A (\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\theta})} - 1 ) (y_i - \operatorname{m}(\boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta})) \right\rbrace \cr + & \mathbb{E} \left\lbrace \frac{1}{N} \sum_{i=1}^N (R_i^B d_i^B - 1) \operatorname{m}( \boldsymbol{x}_i^{\mathrm{T}} \boldsymbol{\beta}) \right\rbrace, \end{aligned} \] lead us to system of equations \[ \begin{aligned} J(\theta, \beta) = \left\lbrace \begin{array}{c} J_1(\theta, \beta) \cr J_2(\theta, \beta) \end{array}\right\rbrace = \left\lbrace \begin{array}{c} \sum_{i=1}^N R_i^A\ \left\lbrace \frac{1}{\pi(\boldsymbol{x}_i, \boldsymbol{\theta})}-1 \right\rbrace \left\lbrace y_i-m(\boldsymbol{x}_i, \boldsymbol{\beta}) \right\rbrace \boldsymbol{x}_i \cr \sum_{i=1}^N \frac{R_i^A}{\pi(\boldsymbol{x}_i, \boldsymbol{\theta})} \frac{\partial m(\boldsymbol{x}_i, \boldsymbol{\beta})}{\partial \boldsymbol{\beta}} - \sum_{i \in \mathcal{S}_{\mathrm{B}}} d_i^{\mathrm{B}} \frac{\partial m(\boldsymbol{x}_i, \boldsymbol{\beta})}{\partial \boldsymbol{\beta}} \end{array} \right\rbrace, \end{aligned} \] where \(m\left(\boldsymbol{x}_{i}, \boldsymbol{\beta}\right)\) is a mass imputation (regression) model for the outcome variable and propensity scores \(\pi_i^A\) are estimated using a logit function for the model. As with the MLE and GEE approaches we have extended this method to cloglog and probit links.

As it is not straightforward to calculate the variances of these estimators, asymptotic equivalents of the variances derived using the Taylor approximation have been proposed in the literature. Details can be found here. In addition, a bootstrap approach can be used for variance estimation.

The function also allows variables selection using known methods that have been implemented to handle the integration of probability and non-probability sampling. In the presence of high-dimensional data, variable selection is important, because it can reduce the variability in the estimate that results from using irrelevant variables to build the model. Let \(\operatorname{U}\left( \boldsymbol{\theta}, \boldsymbol{\beta} \right)\) be the joint estimating function for \(\left( \boldsymbol{\theta}, \boldsymbol{\beta} \right)\). We define the penalized estimating functions as \[ \operatorname{U}^p \left(\boldsymbol{\theta}, \boldsymbol{\beta}\right) = \operatorname{U}\left(\boldsymbol{\theta}, \boldsymbol{\beta}\right) - \left\lbrace \begin{array}{c} q_{\lambda_\theta}(|\boldsymbol{\theta}|) \operatorname{sgn}(\boldsymbol{\theta}) \ q_{\lambda_\beta}(|\boldsymbol{\beta}|) \operatorname{sgn}(\boldsymbol{\beta}) \end{array} \right\rbrace, \] where \(\lambda_{\theta}\) and \(q_{\lambda_{\beta}}\) are some smooth functions. We let \(q_{\lambda} \left(x\right) = \frac{\partial p_{\lambda}}{\partial x}\), where \(p_{\lambda}\) is some penalization function. Details of penalization functions and techniques for solving this type of equation can be found here. To use the variable selection model, set the vars_selection parameter in the controlInf() function to TRUE. In addition, in the other control functions such as controlSel() and controlOut() you can set parameters for the selection of the relevant variables, such as the number of folds during cross-validation algorithm or the lambda value for penalizations. Details can be found in the documentation of the control functions for nonprob.

References

Kim JK, Park S, Chen Y, Wu C. Combining non-probability and probability survey samples through mass imputation. J R Stat Soc Series A. 2021;184:941– 963.

Shu Yang, Jae Kwang Kim, Rui Song. Doubly robust inference when combining probability and non-probability samples with high dimensional data. J. R. Statist. Soc. B (2020)

Yilin Chen , Pengfei Li & Changbao Wu (2020) Doubly Robust Inference With Nonprobability Survey Samples, Journal of the American Statistical Association, 115:532, 2011-2021

Shu Yang, Jae Kwang Kim and Youngdeok Hwang Integration of data from probability surveys and big found data for finite population inference using mass imputation. Survey Methodology, June 2021 29 Vol. 47, No. 1, pp. 29-58

Author

Łukasz Chrostowski, Maciej Beręsewicz

Examples

# \donttest{
# generate data based on Doubly Robust Inference With Non-probability Survey Samples (2021)
# Yilin Chen , Pengfei Li & Changbao Wu
library(sampling)
#> 
#> Attaching package: ‘sampling’
#> The following objects are masked from ‘package:survival’:
#> 
#>     cluster, strata
set.seed(123)
# sizes of population and probability sample
N <- 20000 # population
n_b <- 1000 # probability
# data
z1 <- rbinom(N, 1, 0.7)
z2 <- runif(N, 0, 2)
z3 <- rexp(N, 1)
z4 <- rchisq(N, 4)

# covariates
x1 <- z1
x2 <- z2 + 0.3 * z2
x3 <- z3 + 0.2 * (z1 + z2)
x4 <- z4 + 0.1 * (z1 + z2 + z3)
epsilon <- rnorm(N)
sigma_30 <- 10.4
sigma_50 <- 5.2
sigma_80 <- 2.4

# response variables
y30 <- 2 + x1 + x2 + x3 + x4 + sigma_30 * epsilon
y50 <- 2 + x1 + x2 + x3 + x4 + sigma_50 * epsilon
y80 <- 2 + x1 + x2 + x3 + x4 + sigma_80 * epsilon

# population
sim_data <- data.frame(y30, y50, y80, x1, x2, x3, x4)
## propensity score model for non-probability sample (sum to 1000)
eta <- -4.461 + 0.1 * x1 + 0.2 * x2 + 0.1 * x3 + 0.2 * x4
rho <- plogis(eta)

# inclusion probabilities for probability sample
z_prob <- x3 + 0.2051
sim_data$p_prob <- inclusionprobabilities(z_prob, n = n_b)

# data
sim_data$flag_nonprob <- UPpoisson(rho) ## sampling nonprob
sim_data$flag_prob <- UPpoisson(sim_data$p_prob) ## sampling prob
nonprob_df <- subset(sim_data, flag_nonprob == 1) ## non-probability sample
svyprob <- svydesign(
  ids = ~1, probs = ~p_prob,
  data = subset(sim_data, flag_prob == 1),
  pps = "brewer"
) ## probability sample

## mass imputation estimator
MI_res <- nonprob(
  outcome = y80 ~ x1 + x2 + x3 + x4,
  data = nonprob_df,
  svydesign = svyprob
)
summary(MI_res)
#> 
#> Call:
#> nonprob(data = nonprob_df, outcome = y80 ~ x1 + x2 + x3 + x4, 
#>     svydesign = svyprob)
#> 
#> -------------------------
#> Estimated population mean: 9.518 with overall std.err of: 0.151
#> And std.err for nonprobability and probability samples being respectively:
#> 0.08679 and 0.1236
#> 
#> 95% Confidence inverval for popualtion mean:
#>     lower_bound upper_bound
#> y80    9.222349    9.814346
#> 
#> 
#> Based on: Mass Imputation method
#> For a population of estimate size: 21631.63
#> Obtained on a nonprobability sample of size: 1032
#> With an auxiliary probability sample of size: 1044
#> -------------------------
#> 
#> Regression coefficients:
#> -----------------------
#> For glm regression on outcome variable:
#>             Estimate Std. Error z value  P(>|z|)    
#> (Intercept)  1.93113    0.24859   7.768 7.95e-15 ***
#> x1           1.06616    0.16954   6.289 3.20e-10 ***
#> x2           1.04125    0.09731  10.700  < 2e-16 ***
#> x3           0.98891    0.06927  14.277  < 2e-16 ***
#> x4           0.98930    0.01904  51.946  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> -------------------------
#> 
## inverse probability weighted estimator
IPW_res <- nonprob(
  selection = ~ x1 + x2 + x3 + x4,
  target = ~y80,
  data = nonprob_df,
  svydesign = svyprob
)
summary(IPW_res)
#> 
#> Call:
#> nonprob(data = nonprob_df, selection = ~x1 + x2 + x3 + x4, target = ~y80, 
#>     svydesign = svyprob)
#> 
#> -------------------------
#> Estimated population mean: 9.718 with overall std.err of: 0.2375
#> And std.err for nonprobability and probability samples being respectively:
#> 0.1887 and 0.1442
#> 
#> 95% Confidence inverval for popualtion mean:
#>     lower_bound upper_bound
#> y80    9.252143    10.18302
#> 
#> 
#> Based on: Inverse probability weighted method
#> For a population of estimate size: 21127.35
#> Obtained on a nonprobability sample of size: 1032
#> With an auxiliary probability sample of size: 1044
#> -------------------------
#> 
#> Regression coefficients:
#> -----------------------
#> For glm regression on selection variable:
#>              Estimate Std. Error z value  P(>|z|)    
#> (Intercept) -4.582627   0.105508 -43.434  < 2e-16 ***
#> x1           0.102629   0.074416   1.379    0.168    
#> x2           0.234843   0.042871   5.478 4.30e-08 ***
#> x3           0.181632   0.029253   6.209 5.33e-10 ***
#> x4           0.184285   0.008568  21.508  < 2e-16 ***
#> ---
#> Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
#> -------------------------
#> 
#> Weights:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   1.172  10.583  18.137  20.472  27.939  79.562 
#> -------------------------
#> 
#> Residuals:
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> -0.56121 -0.04204 -0.01457  0.43052  0.94475  0.98743 
#> 
#> AIC: 7797.97
#> BIC: 7826.161
#> Log-Likelihood: -3893.985 on 2071 Degrees of freedom
## doubly robust estimator
DR_res <- nonprob(
  outcome = y80 ~ x1 + x2 + x3 + x4,
  selection = ~ x1 + x2 + x3 + x4,
  data = nonprob_df,
  svydesign = svyprob
)
summary(DR_res)
#> 
#> Call:
#> nonprob(data = nonprob_df, selection = ~x1 + x2 + x3 + x4, outcome = y80 ~ 
#>     x1 + x2 + x3 + x4, svydesign = svyprob)
#> 
#> -------------------------
#> Estimated population mean: 9.483 with overall std.err of: 0.1525
#> And std.err for nonprobability and probability samples being respectively:
#> 0.08508 and 0.1265
#> 
#> 95% Confidence inverval for popualtion mean:
#>     lower_bound upper_bound
#> y80    9.183858    9.781461
#> 
#> 
#> Based on: Doubly-Robust method
#> For a population of estimate size: 21127.35
#> Obtained on a nonprobability sample of size: 1032
#> With an auxiliary probability sample of size: 1044
#> -------------------------
#> 
#> Regression coefficients:
#> -----------------------
#> For glm regression on outcome variable:
#>             Estimate Std. Error z value  P(>|z|)    
#> (Intercept)  1.93113    0.24859   7.768 7.95e-15 ***
#> x1           1.06616    0.16954   6.289 3.20e-10 ***
#> x2           1.04125    0.09731  10.700  < 2e-16 ***
#> x3           0.98891    0.06927  14.277  < 2e-16 ***
#> x4           0.98930    0.01904  51.946  < 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) -4.582627   0.105508 -43.434  < 2e-16 ***
#> x1           0.102629   0.074416   1.379    0.168    
#> x2           0.234843   0.042871   5.478 4.30e-08 ***
#> x3           0.181632   0.029253   6.209 5.33e-10 ***
#> x4           0.184285   0.008568  21.508  < 2e-16 ***
#> -------------------------
#> 
#> Weights:
#>    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
#>   1.172  10.583  18.137  20.472  27.939  79.562 
#> -------------------------
#> 
#> Residuals:
#>     Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
#> -0.56121 -0.04204 -0.01457  0.43052  0.94475  0.98743 
#> 
#> AIC: 7797.97
#> BIC: 7826.161
#> Log-Likelihood: -3893.985 on 2071 Degrees of freedom
# }