Inference with non-probability survey samples

nonprob function provides an access to the various methods for inference based on non-probability surveys (including big data). The function allows to estimate the population mean based on the access to a reference probability sample (via the survey package), as well as totals or 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 uses the Lumley 2004 survey package for inference (if a reference probability sample is provided).

It provides various inverse probability weighting (e.g. with calibration constraints), mass imputation (e.g. nearest neighbour, predictive mean matching) and doubly robust estimators (e.g. that take into account minimisation of the asymptotic bias of the population mean estimators).

The package uses the survey package functionality when a probability sample is available.

All optional parameters are set to NULL. The obligatory ones include data as well as one of the following three: selection, outcome, or target – depending on which method has been selected. In the case of outcome and target multiple $y$ variables can be specified.

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", "npar"),
  family_outcome = c("gaussian", "binomial", "poisson"),
  subset = NULL,
  strata = NULL,
  case_weights = NULL,
  na_action = na.omit,
  control_selection = control_sel(),
  control_outcome = control_out(),
  control_inference = control_inf(),
  start_selection = NULL,
  start_outcome = NULL,
  verbose = FALSE,
  se = TRUE,
  ...
)

Arguments

data: a data.frame with dataset containing the non-probability sample
selection: a formula (default NULL) for the selection (propensity) score model
outcome: a formula (default NULL) for the outcome (target) model
target: a formula (default NULL) with target variable(s). We allow multiple target variables (e.g. ~y1 + y2 + y3)
svydesign: an optional svydesign2 class object containing a 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 value with population size
method_selection: a character (default logit) indicating the method for the propensity score link function.
method_outcome: a character (default glm) indicating the method for the outcome model.
family_outcome: a character (default gaussian) describing the error distribution and the link function to be used in the model. Currently supports: gaussian with the 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 (not yet supported, for further development)
case_weights: an optional vector of prior weights to be used in the fitting process. It is assumed that this vector contains frequency or analytic weights (i.e. rows of the data argument are repeated according to the values of the case_weights argument), not probability/design weights.
na_action: a function which indicates what should happen when the data contain NAs (default na.omit and it is the only method currently supported)
control_selection: a list (default control_sel() result) indicating parameters to be used when fitting the selection model for propensity scores. To change the parameters one should use the control_sel() function
control_outcome: a list (default control_out() result) indicating parameters to be used when fitting the model for the outcome variable. To change the parameters one should use the control_out() function
control_inference: a list (default control_inf() result) indicating parameters to be used for inference based on probability and non-probability samples. To change the parameters one should use the control_inf() function
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: a numerical value (default TRUE) whether detailed information on the fitting should be presented
se: Logical value (default TRUE) indicating whether to calculate and return standard error of estimated mean.
...: Additional, optional arguments

Value

Returns an object of the nonprob class (it is actually a list) which contains the following elements:

call – the call of the nonprob function
data – a data.frame passed from the nonprob function data argument
X – a model.matrix containing data from probability (first $n_{S_B}$ rows) and non-probability samples (next $n_{S_B}$ rows) if specified at a function call
y – a list of vector of outcome variables if specified at a function call
R – a numeric vector indicating whether a unit belongs to the probability (0) or non-probability (1) units in the matrix X
ps_scores – a numeric vector of estimated propensity scores for probability and non-probability sample
case_weights – a vector of case weights for non-probability sample based on the call
ipw_weights – a vector of inverse probability weights for non-probability sample (if applicable)
control – a list of control functions based on the call
output – a data.frame with the estimated means and standard errors for the variables specified in the target or outcome arguments
SE – a data.frame with standard error of the estimator of the population mean, divided into errors from probability and non-probability samples (if applicable)
confidence_interval – a data.frame with confidence interval of population mean estimator
nonprob_size – a scalar numeric vector denoting the size of non-probability sample
prob_size – a scalar numeric vector denoting the size of probability sample
pop_size – a scalar numeric vector estimated population size derived from estimated weights (non-probability sample) or known design weights (probability sample)
pop_size_fixed – a logical value whether the population size was fixed (known) or estimated (unknown)
pop_totals – a numeric vector with the total values of the auxiliary variables derived from a probability sample or based on the call
pop_means – a numeric vector with the mean values of the auxiliary variables derived from a probability sample or based on the call
outcome – a list containing information about the fitting of the mass imputation model. Structure of the object is based on the method_outcome and family_outcome arguments which point to specific methods as defined by functions method_* (if specified in the call)
selection – a list containing information about the fitting of the propensity score model. Structure of the object is based on the method_selection argument which point to specific methods as defined by functions method_ps (if specified in the call)
boot_sample – a matrix with bootstrap estimates of the target variable(s) (if specified)
svydesign – a svydesign2 object (if specified)
ys_rand_pred – a list of predicted values for the target variable(s) for the probability sample (for the MI and DR estimator)
ys_nons_pred – a list of predicted values for the target variable(s) for the non-probability sample (for the MI and DR estimator)
ys_resid – a list of residuals for the target variable(s) for the non-probability sample (for the MI and DR estimator)
estimator – a character vector with information what type of estimator was selected (one of c("ipw", "mi", "dr")).
selection_formula – a formula based on the selection argument (if specified)
estimator_method – a character vector with information on the detailed method applied (for the print method)

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 $$ \boldsymbol{X}_A = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \cr x_{21} & x_{22} & \cdots & x_{2p} \cr \vdots & \vdots & \ddots & \vdots \cr x_{n_{A1}} & x_{n_{A2}} & \cdots & x_{n_{Ap}} \cr \end{bmatrix}, $$ 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 $$ \boldsymbol{X}_B = \begin{bmatrix} x_{11} & x_{12} & \cdots & x_{1p} \cr x_{21} & x_{22} & \cdots & x_{2p} \cr \vdots & \vdots & \ddots & \vdots \cr x_{n_{B1}} & x_{n_{B2}} & \cdots & x_{n_{Bp}}\cr \end{bmatrix}. $$ 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 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, the 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 control_inf() function to TRUE. In addition, in the other control functions such as control_sel() and control_out() 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, Piotr Chlebicki

Examples

# \donttest{
# generate data based on Doubly Robust Inference With Non-probability Survey Samples (2021)
# Yilin Chen , Pengfei Li & Changbao Wu
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 <- n_b* z_prob/sum(z_prob)

# data
sim_data$flag_nonprob <- as.numeric(runif(N) < rho) ## sampling nonprob
sim_data$flag_prob <- as.numeric(runif(n_b) < 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 = y30 + y50 + y80 ~ x1 + x2 + x3 + x4,
  data = nonprob_df,
  svydesign = svyprob
)
mi_res
#> 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 y30: 11.8211
#>    - variable y50: 11.9242
#>    - variable y80: 11.9797
#>  - selected estimators:
#>    - variable y30: 9.4791 (se=0.3927, ci=(8.7094, 10.2488))
#>    - variable y50: 9.5458 (se=0.2241, ci=(9.1067, 9.9850))
#>    - variable y80: 9.5818 (se=0.1517, ci=(9.2845, 9.8791))
## inverse probability weighted estimator
ipw_res <- nonprob(
  selection = ~ x1 + x2 + x3 + x4,
  target = ~y30 + y50 + y80,
  data = nonprob_df,
  svydesign = svyprob
)
ipw_res
#> 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 y30: 11.8211
#>    - variable y50: 11.9242
#>    - variable y80: 11.9797
#>  - selected estimators:
#>    - variable y30: 9.3798 (se=0.5655, ci=(8.2714, 10.4881))
#>    - variable y50: 9.5155 (se=0.5655, ci=(8.4071, 10.6238))
#>    - variable y80: 9.5886 (se=0.5655, ci=(8.4802, 10.6969))
## doubly robust estimator
dr_res <- nonprob(
  outcome = y30 + y50 + y80 ~ x1 + x2 + x3 + x4,
  selection = ~ x1 + x2 + x3 + x4,
  data = nonprob_df,
  svydesign = svyprob
)
dr_res
#> 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 y30: 11.8211
#>    - variable y50: 11.9242
#>    - variable y80: 11.9797
#>  - selected estimators:
#>    - variable y30: 9.3388 (se=0.4548, ci=(8.4473, 10.2302))
#>    - variable y50: 9.4757 (se=0.2578, ci=(8.9704, 9.9810))
#>    - variable y80: 9.5494 (se=0.1745, ci=(9.2075, 9.8913))
# }