Surveys of income are complicated by the sensitive nature of the topic. The problem researchers face is how to encourage participants to respond and to provide truthful responses in surveys. To correct biases induced by nonresponse or underreporting, we propose a two-stage multilevel randomized response (MRR) technique to investigate the true level of income and to protect personal privacy. For a wide range of applications, we present a proportional odds model for two-stage MRR data and apply inverse probability weighting and multiple imputation methods to deal with covariates on some subjects that are missing at random. A simulation study is conducted to investigate the effects of missing covariates and to evaluate the performance of the proposed methods. The practicality of the proposed methods is illustrated with the regular monthly income data collected in the Taiwan Social Change Survey. Furthermore, we provide an estimate of personal regular monthly mean income.
AßmannC.WürbachA.GoßmannS.GeisslerF.BelaA.. 2017. “Nonparametic Multiple Imputation for Questionnaires with Individual Skip Patterns and Constraints: The Case of Income Imputation in the National Educational Panel Study.” Sociological Methods & Research46:864–97.
2.
AnthonyB. A. 2002. “Performing Logistic Regression on Survey Data with the New SURVEYLOGISTIC Procedure.” Paper presented at the Twenty-seventh Annual SAS User Group International Conference, Cary, NC.
3.
BöckenholtU.Van der HeijdenP. G. M.. 2007. “Item Randomized-response Models for Measuring Noncompliance: Risk-return Perceptions, Social Influences, and Self-protective Responses.” Psychometrika72:245–62.
4.
BoruchR. F.1971. “Assuring Confidentiality of Responses in Social Research: A Note on Strategies.” The American Sociologist6:308–11.
CorstangeD.2009. “Sensitive Questions, Truthful Answers? Modeling the List Experiment with LISTIT.” Political Analysis17:45–63.
8.
CruyffM. J. L. F.BöckenholtU.Van den HoutA.Van der HeijdenP. G. M.. 2008. “Accounting for Self-protective Responses in Randomized Response Data from a Social Security Survey Using the Zero-inflated Poisson Model.” The Annals of Applied Statistics2:316–31.
9.
CruyffM. J. L. F.Van den HoutA.Van der HeijdenP. G. M.. 2008. “The Analysis of Randomized-response Sum Score Variables.” Journal of the Royal Statistical Society: Series B70:21–30.
10.
EltingeJ. L.YansanchI. S.. 1997. “Diagnostics for Formation of Nonresponse Adjustment Cells, with an Application to Income Nonresponse in the U.S. Consumer Expenditure Survey.” Survey Methodology23:33–40.
11.
EspondaF.HuertaK.GuerreroV. M.. 2016. “A Statistical Approach to Provide Individualized Privacy for Surveys.” PLoS One1–14. doi: 10.1371/journal.pone.0147314.
12.
FlandersW. D.GreenlandS.. 1991. “Analytic Methods for Two-stage Case-control Studies and Other Stratified Designs.” Statistics in Medicine10:739–47.
13.
FranklinL. A.1989. “A Comparison of Estimators for Randomized Response Sampling with Continuous Distributions from a Dichotomous Population.” Communication in Statistics Theory and Methods18:489–505.
14.
FrickJ. R.GrabkaM. M.. 2005. “Item Non-response on Income Questions in Panel Surveys: Incidence, Imputation and the Impact on Inequality and Mobility.” Allgemeines Statistiches Archiv89:49–61.
15.
GlonekG. F. V.McCullaghP.. 1995. “Multivariate Logistic Models.” Journal of the Royal Statistical Society: Series B57:533–46.
16.
GodambeV. P.1960. “An Optimum Property of Regular Maximum Likelihood Estimation.” Annals of Mathematical Statistics31:1208–12.
17.
GreenbergB. G.Abul-ElaA.SimmonsW. R.HorvitzD. G.. 1969. “The Unrelated Question Randomized Response Model: Theoretical Framework.” Journal of the American Statistical Association64:520–39.
18.
HazizaD.BeaumontJ. F.. 2007. “On the Construction of Imputation Classes in Surveys.” International Statistical Review72:147–48.
19.
HortonN. J.LairdN. M.. 1999. “Maximum Likelihood Analysis of Generalized Linear Models with Missing Covariates.” Statistical Methods in Medical Research8:37–50.
20.
HoustonJ.TranA.. 2001. “A Survey of Tax Evasion using the Randomized Response Technique.” Advances in Taxation13:69–94.
21.
HsiehS. H.LeeS. M.ShenP. S.. 2009. “Semiparametric Analysis of Randomized Response Data with Missing Covariates in Logistic Regression.” Computational Statistics and Data Analysis53:2673–92.
22.
HsiehS. H.LeeS. M.ShenP. S.. 2010. “Logistic Regression Analysis of Randomized Response Data with Missing Covariates.” Journal of Statistical Planning and Inference140:927–40.
23.
HsiehS. H.LeeS. M.TuS. H.. 2018. “Randomized Response Techniques for a Multi-level Attribute using a Single Sensitive Question.” Statistical Papers59:291–306.
24.
IbrahimJ. G.ChenM. H.LipsitzS. R.HerringA. H.. 2005. “Missing Data Methods for Generalized Linear Methods: A Comparative Review.” Journal of the American Statistical Association100:332–46.
25.
LeeS. M.GeeM. J.HsiehS. H.. 2011. “Semiparametric Methods in the Proportional Odds Model for Ordinal Response Data with Missing Covariates.” Biometrics67:788–98.
26.
LeeS. M.HwangW. H.de Dieu TapsobaJ.. 2016. “Estimation in Closed Capture-recapture Models When Covariates Are Missing at Random.” Biometrics72:1294–1304.
27.
Lensvelt-MuldersG. J. L. M.HoxJ. J.Van der HeijdenP. G. M.MaasC. J. M.. 2005. “Meta-analysis of Randomized Response Research: Thirty-five Years of Validation.” Sociological Methods & Research33:319–48.
28.
Lensvelt-MuldersG. J. L. M.Van der HeijdenP. G. M.LaudyO.Van GilsG.. 2006. “A Validation of a Computer-assisted Randomized Response Survey to Estimate the Prevalence of Fraud in Social Security.” Journal of the Royal Statistical Society: Series A169:305–18.
29.
LillardL.SmithJ. P.WelchF.. 1986. “What Do We Really Know about Wages? The Importance of Nonreporting and Census Imputation.” Journal of Political Economy94:489–506.
30.
LittleR. J. A.1986. “Survey Nonresponse Adjustments for Estimates of Means.” International Statistical Review54:139–57.
31.
LittleR. J. A.1992. “Regression with Missing X’s: A Review.” Journal of the American Statistical Association87:1227–37.
32.
LittleR. J. A.RubinD. B.. 2002. Statistical Analysis with Missing Data. 2nd ed. Hoboken, NJ: Wiley.
33.
LuiX.KoiralaH.. 2013. “Fitting Proportional Odds Models to Educational Data with Complex Sampling Design in Ordinal Logistic Regression.” Journal of Modern Applied Statistical Methods12:26.
34.
MangatN. S.1994. “An Improved Randomized Response Strategy.” Journal of the Royal Statistical Society: Series B56:93–95.
35.
MoorsJ. J.1971. “Optimization of the Unrelated Question Randomized Response Model.” Journal of American Statistical Association66:627–29.
36.
ParkerR. N.FenwickR.. 1983. “The Pareto Curve and Its Utility for Open-ended Income Distributions in Survey Research.” Social Forces61:872–85.
37.
PerriP. F.CoboB.RuedaM.. 2017. “A Mixed-mode Sensitive Research on Cannabis Use and Sexual Addiction: Improving Self-reporting by Means of Indirect Questioning Techniques.” Quality & Quantity52:1593–1611.
38.
RighiP.FalorsiS.FasuloA.. 2014. “Methods for Variance Estimation under Random Hot Deck Imputation in Business Surveys.”Rivista di statistica ufficialeItalian: National Institute of Statistics(Rome, Italy)16:45–64.
39.
RiphahnR. T.SerflingO.. 2002. “Item Non-response on Income and Wealth Questions.” IZA Discussion Paper No. 573, Bonn, Germany.
40.
RonningG.2005. “Randomized Response and the Binary Probit Model.” Economics Letters86:221–28.
41.
RubinD. B.1976. “Inference and Missing Data.” Biometrika63:581–92.
42.
RubinD. B.1978. “Multiple Imputation in Sample Surveys: A Phenomenological Bayesian Approach to Nonresponse.” Proceedings Survey Research Method Section of the American Statistical Association, 20–34.
43.
RubinD. B.1987. Multiple Imputation for Nonresponse in Surveys. New York: Wiley.
44.
RubinD. B.SchenkerN.. 1986. “Multiple Imputation for Interval Estimation from Simple Random Samples with Ignorable Nonresponse.” Journal of the American Statistical Association81:366–74.
45.
ScheersN. J.DaytonC. M.. 1988. “Covariate Randomized Response Models.” Journal of the American Statistical Association83:969–74.
46.
ThomsenI.1973. “A Note on the Efficiency of Weighting Subclass Means to Reduce the Effects of Nonresponse when Analyzing Survey Data.” Statistik Tidskrift11:278–83.
47.
TianG. L.YuenK. C.TangM. L.TanM. T.. 2009. “Bayesian Non-randomized Response Models for Surveys with Sensitive Questions.” Statistics and Its Interface2:13–25.
48.
Van den HoutA.Van der HeijdenP. G. M.GilchristR.. 2007. “The Logistic Regression Model with Response Variables Subject to Randomized Response.” Computation Statistics and Data Analysis51:6060–69.
49.
WangC. Y.ChenJ. C.LeeS. M.OuS. T.. 2002. “Joint Conditional Likelihood Estimator in Logistic Regression with Missing Covariate Data.” Statistica Sinica12:555–74.
50.
WangD.ChenS. X.. 2009. “Empirical Likelihood for Estimating Equations with Missing Values.” The Annals of Statistics37:490–517.
51.
WangC. Y.WangS.. 1997. “Semiparametric Methods in Logistic Regression with Measurement Error.” Statistica Sinica7:1103–20.
52.
WangC. Y.WangS.ZhaoL.OuS. T.. 1997. “Weighted Semiparametric Estimation in Regression Analysis with Missing Covariate Data.” Journal of the American Statistical Association92:512–25.
53.
WarnerS. L.1965. “Randomized Response: A Survey Technique for Eliminating Evasive Answer Bias.” Journal of the American Statistical Association60:63–69.
54.
YuR. R.LiL. A.. 2011. “Imputation of Non-ignorable Nonresponse for Income Analysis of a Panel Study on Taiwan.” Quality and Quantity45:875–84.
55.
ZhaoL.LipsitzS.. 1992. “Designs and Analysis of Two-stage Studies.” Statistics in Medicine11:769–82.
56.
ZweimüllerJ.1992. “Survey Non-response and Biases in Wage Regressions.” Economics Letters39:105–9.
Supplementary Material
Please find the following supplemental material available below.
For Open Access articles published under a Creative Commons License, all supplemental material carries the same license as the article it is associated with.
For non-Open Access articles published, all supplemental material carries a non-exclusive license, and permission requests for re-use of supplemental material or any part of supplemental material shall be sent directly to the copyright owner as specified in the copyright notice associated with the article.
