This R package implements the methods proposed by Atsusaka and Stevenson (2023) "A Bias-Corrected Estimator for the Crosswise Model with Inattentive Respondents" (https://doi.org/10.1017/pan.2021.43). Our workhorse function is bc.est which generates a bias-corrected crosswise estimate of the proportion of individuals with sensitive attributes. cmBound applies our sensitivity analysis to crosswise data without the anchor question. cmreg and cmreg.p implement crosswise regressions in which the latent sensitive trait can be used as an outcome or as a predictor, respectively. cmpredict generates predicted proportions of having sensitive traits given specific covariate values with uncertainty estimates via parametric bootstrap, whereas cmpredict.p yields predicted values for a given outcome variable and specific covariate values in the presence and absence of sensitive traits. Simulated crosswise data are saved as cmdata, cmdata2, and cmdata3.
To install the latest development version of cWise directly from
GitHub use:
library(devtools)
devtools::install_github("YukiAtsusaka/cWise")First, load the package.
library(cWise)The following examples use a toy data set (cmdata) that comes with
the package. This data contains artificially generated information in a survey using the crosswise model.
data(cmdata)
head(cmdata)
#> Y A p p.prime
#> 1 1 1 0.15 0.15
#> 2 0 0 0.15 0.15
#> 3 0 0 0.15 0.15
#> 4 1 0 0.15 0.15
#> 5 0 1 0.15 0.15
#> 6 1 1 0.15 0.15Here, Y is a binary response in the crosswise question (Y=1 if TRUE-TRUE or FALSE-FALSE), and A is a binary response in the anchor question.
p and p.prime are auxiliary probabilities in the crosswise and anchor questions, respectively.
While researchers can directly input the values of p and p.prime in the function below (without including them in data), we include them for an illustrative purpose.
Generate a bias-corrected crosswise estimate using a crosswise data as follows:
prev <- bc.est(Y=Y, A=A, p=0.15, p.prime=0.15, data=cmdata)
prev
#> $Results
#> Estimate Std. Error 95%CI(Low) 95%CI(Up)
#> Naive Crosswise 0.1950 0.0144 0.1667 0.2233
#> Bias-Corrected 0.1054 0.0208 0.0649 0.1394
#>
#> $Stats
#> Attentive Rate Sample Size
#> 0.7729 2000The output is a list containing two elements. $Results is a matrix containing the point estimate, (estimated) standard error, the lower and upper bounds of 95% confidence intervals for the naive crosswise estimate and bias-corrected estimate, respectively.
$Stats is a vector containing the estimated attentive rate and sample size used for estimation. In this example, it is estimated that about 77% of respondents are attentive and followed the instructions under the design.
When using unrepresentative samples, it is crucial to consider weighting to estimate the prevalence of sensitive attributes at the (real) population of interest. Great news: sample weights can be easily incorporated in our bias-corrected estimator by specifying the optional argument weight as follows:
bc.est(Y=Y, A=A, weight=weight, p=0.15, p.prime=0.15, data=cmdata)
#> $Results
#> Estimate Std. Error 95%CI(Low) 95%CI(Up)
#> Naive Crosswise 0.1921 0.0144 0.1639 0.2204
#> Bias-Corrected 0.1097 0.0250 0.0596 0.1503
#>
#> $Stats
#> Attentive Rate Sample Size
#> 0.7888 2000Apply a sensitivity analysis and generate sensitivity bounds for naive crosswise estimates.
p <- cmBound(lambda.hat=0.6385, p=0.25, N=310, dq=0.073, N.dq=310)
p
Since the output is a ggplot object, one can easily add additional information by using "+". For example, to add a title with a specific font:
p <- p + ggtitle("Sensitivity Analysis") +
theme(plot.title = element_text(size=20, face="bold"))
p 
For an illustration, let's load and see cmdata2 that contains the main and anchor response variables along with two covariates.
data(cmdata2)
head(cmdata2)
#> Y A female age p p.prime
#> 1 1 1 0 23 0.1 0.15
#> 2 1 1 1 31 0.1 0.15
#> 3 0 1 1 32 0.1 0.15
#> 4 1 0 1 19 0.1 0.15
#> 5 0 1 1 19 0.1 0.15
#> 6 0 1 1 25 0.1 0.15To run a crosswise regression, one can specify the model by writing a formula: Crosswise Response ~ var1 + ... + varN + Anchor Response as follows:
m <- cmreg(Y~female+age+A, p=0.1, p.prime=0.15, data=cmdata2)
m
#> $Call
#> Y ~ female + age + A
#>
#> $Coefficients
#> Estimate Std. Error z score Pr(>|z|)
#> (intercept) -1.6509 0.4268 -3.868 0.000
#> female 0.2816 0.1427 1.974 0.048
#> age 0.0326 0.0133 2.450 0.014
#>
#> $AuxiliaryCoef
#> Estimate Std. Error z score Pr(>|z|)
#> (intercept) 0.1387 1.1347 0.122 0.903
#> female -0.2044 0.4119 -0.496 0.620
#> age 0.0595 0.0394 1.511 0.131$Coefficients shows main restuls. They suggest that female and older respondents are more likely to possess the sensitive trait of interest. $AuxiliaryCoef lists esimated coefficients for being attentive in the crosswise model.
For a demonstration, let's load cmdata3 that contains an outcome variable of interest (V), two covariates (female and age), and crosswise and anchor responses (Y and A).
data(cmdata3)
head(cmdata3)
#> V Y female age A p p.prime
#> 1 -0.38350925 1 0 23 1 0.1 0.15
#> 2 -0.05965305 1 1 31 1 0.1 0.15
#> 3 0.72655660 0 1 32 1 0.1 0.15
#> 4 0.79845870 1 1 19 0 0.1 0.15
#> 5 -0.19410532 0 1 19 1 0.1 0.15
#> 6 -0.34926673 0 1 25 1 0.1 0.15To run a regression with the sensitive trait as a predictor, one can specify the formula: Outcome ~ Cov1 + ... + CovN + CrosswiseResponse + AnchorResponse as follows:
m2 <- cmreg.p(V~age+female+Y+A, p=0.1, p.prime=0.15, data=cmdata3)
m2
#> $Call
#> V ~ age + female + Y + A
#>
#> $Coefficients
#> Estimate Std. Error z score Pr(>|z|)
#> (intercept) 0.0235 0.1478 0.159 0.874
#> age 0.0096 0.0048 2.006 0.045
#> female 0.2473 0.0520 4.753 0.000
#> Y 0.9858 0.0756 13.035 0.000
#>
#> $AuxiliaryCoef
#> Estimate Std. Error z score Pr(>|z|)
#> (intercept) -1.7338 0.4009 -4.324 0.000
#> age 0.0352 0.0126 2.794 0.005
#> female 0.2878 0.1356 2.123 0.034
#>
#> $AuxiliaryCoef2
#> Estimate Std. Error z score Pr(>|z|)
#> (intercept) 0.2481 1.0680 0.232 0.816
#> age 0.0548 0.0370 1.481 0.139
#> female -0.1075 0.3779 -0.284 0.776$Coefficients returns a list of coefficients that associte each covariate (including the sensitive trait of interest) and the outcome variable. Our primary quantities of interest are:
#> Y 0.9858 0.0756 13.035 0.000cmpredict offers an easy way to compute predicted probabilities (proportions) of having sensitive attributes. It does so with three arguments: out = output of cmreg, typical = a vector of typical values for control variables, and zval = a specific value that the main explanatory variable (first listed variable in cmreg) takes.
pred.nonfem = cmpredict(out=m, typical=30, zval=0)
pred.female = cmpredict(out=m, typical=30, zval=1)
par(mmfrow=c(1,2))
hist(pred.nonfem, main="Among non-Female", xlab="Proportion w/ Sensitive Traits")
hist(pred.female, main="Among Female", xlab="Proportion w/ Sensitive Traits")
cmpredict.p provides an easy way to compute the predicted values of the outcome variable after applying cmreg.p. One can only specify a vector of typical values.
pred <- cmpredict.p(out=m2, typical=c(1,30))
par(mfrow=c(1,2))
hist(pred[1,], main="No Sensitive Trait", xlab="Outcome Value", breaks=40)
hist(pred[2,], main="With Sensitive Trait", xlab="Outcome Value", breaks=40)