README.md

In shuyang1987/IntegrativeFPM: Integrative Analysis of Finite Population Mean Combining Probability and Nonprobability Samples

IntegrativeFPM

Disclaimer

This R package/function is intended to provide a template for implementing the new methods. Modification of default options may be needed for a specific application. Please read the associated article for method details.

To proficient R programmers who are interested in advancing this R package/function, please contact me and I am happy to work together.

Goal and descripton

The goal of IntegrativeFPM is to implement integrative analyses for the finite population mean (FPM) parameters combining a non-probability sample with a probability sample which provides high-dimensional representative covariate information of the target population.

Two datasets

• The non-probability sample contains observations on (X,Y); however, the sampling mechanism is unknown.

• The probability sample with sampling weights represents the finite population; however, it contains observations only on X.

The function IntegrativeFPM implements a two-step approach for integrating the probability sample and the nonprobability sample for finite population inference.

• Step 1 selects important variables in the sampling score model and the outcome model by penalization.

• Step 2 conducts a doubly robust inference for the finite population mean of interest. In this step, the nuisance model parameters are refitted based on the selected variables by minimizing the asymptotic squared bias of the doubly robust estimator. This estimating strategy mitigates the possible first-step selection error and renders the doubly robust estimator root-n consistent if either the sampling probability or the outcome model is correctly specified.

Main Paper: Yang, Kim, and Song (2019)

Yang, S., Kim, J.K., and Song, R. (2019). Doubly Robust Inference when Combining Probability and Non-probability Samples with High-dimensional Data.

Usage

IntegrativeFPM(y, x, deltaB, sw, family, lambda_a, lambda_b)

|Note | Defaults | | --------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------- | |1| the default probability sampling design is Bernoulli PPS (probability proportional to size) sampling

Arguments

| Argument | | | --------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------- | | y | is a vector of outcome; NA for the probability sample (n x 1) | | x | is a matrix of covariates without intercept (n x p) | | deltaB | is a vector of the binary indicator of belonging to the nonprobability sample; i.e., 1 if the unit belongs to the nonprobability sample, and 0 otherwise (n x 1) | | sw | is a vector of weights: 1 for the unit in the nonprobability sample and the design weight for the unit in the probability sample (n x 1) | | family | specifies the outcome model | | – | “gaussian”: a linear regression model for the continuous outcome | | – | “binomial”: a logistic regression model for the binary outcome | | lambda_a | is a scalar tuning parameter in the penalized estimating equation for alpha (the sampling score parameter) | | – | The sampling score is a logistic regression model for the probability of selection into the nonprobability sample given X | | | lambda_b | is a scalar tuning parameter in the penalized estimation for beta (the outcome model parameter) | | – | The outcome model is a linear regression model for continuous outcome or a logistic regression model for binary outcome |

Value

| | | | -------------- | -------------------------------------------------------- | | est | estimate of the finite population mean of Y | | se | standard error estimate for est | | alpha.selected | the column(s) of X selected for the sampling score model | | beta.selected | the column(s) of X selected for the outcome model |

Example

This is an example for illustration.

set.seed(1234)

## population size

N <- 10000

## x is a p-dimensional covariate

p <- 50
x <- matrix( rnorm(N*p,0,1),N,p)

## y is a continuous outcome 

beta0 <- c(1,1,1,1,1,rep(0,p-4))
y <- cbind(1,x)%*%beta0 + rnorm(N,0,1)
true <- mean(y)

## y2 is a binary outcome

ly2 <- (cbind(1,x)%*%beta0)
ply <- exp(ly2)/(1+exp(ly2))
y2 <- rbinom(N,1,ply)
true2 <- mean(y2)

## A.set is a prob sample: SRS
## sampling probability into A is known when estimation

nAexp <- 1000
probA <- rep(nAexp/N,N)
A.index <- rbinom(N,size = 1,prob = probA)
A.loc <- which(A.index == 1)
nA <- sum(A.index == 1)
sw.A <- 1/probA[A.loc]
x.A <- x[A.loc,]
y.A <- rep(NA,nA) # y is not observed in Sample A
y2.A <- rep(NA,nA)

## B.set is a nonprob sample
## sampling probability into B is unknown when estimation

nBexp <- 2000
alpha0 <- c(-2,1,1,1,1,rep(0,p-4))
probB <- (1+exp(-cbind(1,x)%*%alpha0))^(-1) 
B.index <- rbinom(N,size = 1,prob = probB)
B.loc <- which(B.index == 1)
nB <- sum(B.index)
x.B <- x[B.loc,]
y.B <- y[B.loc]
y2.B <- y2[B.loc]

## combined dataset

y.AB <- c(y.A,y.B)
y2.AB <- c(y2.A,y2.B)
x.AB <- rbind(x.A,x.B)
deltaB <- c(rep(0,nA),rep(1,nB))
sw <- c(sw.A,rep(1,nB))

## specify tuning parameters

lambda_a <- 1.25
lambda_b <- 0.25
lambda2_b <- 0.02

true
#> [1] 1.000452
IntegrativeFPM::IntegrativeFPM(y=y.AB, x=x.AB, deltaB, sw, family="gaussian",lambda_a, lambda_b)
#> $est
#> [1] 1.057844
#> 
#> $se
#> [1] 0.09016443
#> 
#> $alpha.selected
#> [1] 1 2 3 4
#> 
#> $beta.selected
#> [1] 1 2 3 4

true2
#> [1] 0.6551
IntegrativeFPM::IntegrativeFPM(y=y2.AB, x=x.AB, deltaB, sw, family="binomial",lambda_a, lambda2_b)
#> $est
#> [1] 0.6618466
#> 
#> $se
#> [1] 0.02876472
#> 
#> $alpha.selected
#> [1] 1 2 3 4
#> 
#> $beta.selected
#> [1] 1 2 3 4

shuyang1987/IntegrativeFPM documentation built on April 28, 2021, 8:24 p.m.