A Mixed Effects Randomized Item Response Model

The randomized response technique ensures that individual item responses, denoted as true item responses, are randomized before observing them and so-called randomized item responses are observed. A relationship is specified between randomized item response data and true item response data. True item response data are modeled with a (non)linear mixed effects and/or item response theory model. Although the individual true item responses are masked through randomizing the responses, the model extension enables the computation of individual true item response probabilities and estimates of individuals’ sensitive behavior/attitude and their relationships with background variables taking into account any clustering of respondents. Results are presented from a College Alcohol Problem Scale (CAPS) where students were interviewed via direct questioning or via a randomized response technique. A Markov Chain Monte Carlo algorithm is given for estimating simultaneously all model parameters given hierarchical structured binary or polytomous randomized item response data and background variables.

[1]  S. Chib,et al.  Bayesian analysis of binary and polychotomous response data , 1993 .

[2]  W. R. Simmons,et al.  The Unrelated Question Randomized Response Model: Theoretical Framework , 1969 .

[3]  J. Fox.  Beta-binomial ANOVA for multivariate randomized response data. , 2008, The British journal of mathematical and statistical psychology.

[4]  Brian W. Junker,et al.  Applications and Extensions of MCMC in IRT: Multiple Item Types, Missing Data, and Rated Responses , 1999 .

[5]  Richard J. Patz,et al.  A Straightforward Approach to Markov Chain Monte Carlo Methods for Item Response Models , 1999 .

[6]  Siddhartha Chib,et al.  A Gibbs sampling approach , 1993 .

[7]  L. Wasserman,et al.  Computing Bayes Factors by Combining Simulation and Asymptotic Approximations , 1997 .

[8]  H. Goldstein.  Multilevel Statistical Models , 2006 .

[9]  C. Mitchell Dayton,et al.  Covariate Randomized Response Models , 1988 .

[10]  F. Samejima.  Estimation of latent ability using a response pattern of graded scores , 1968 .

[11]  Paul E. Tracy,et al.  The Validity of Randomized Response for Sensitive Measurements , 1981 .

[12]  P. Boeck,et al.  Explanatory item response models : a generalized linear and nonlinear approach , 2004 .

[13]  J-P Fox,et al.  Multilevel IRT using dichotomous and polytomous response data. , 2005, The British journal of mathematical and statistical psychology.

[14]  Scott L. Zeger,et al.  Generalized linear models with random e ects: a Gibbs sampling approach , 1991 .

[15]  Peter G. M. van der Heijden,et al.  A validation of a computer‐assisted randomized response survey to estimate the prevalence of fraud in social security , 2006 .

[16]  Adrian F. M. Smith,et al.  Sampling-Based Approaches to Calculating Marginal Densities , 1990 .

[17]  Andrew Gelman,et al.  General methods for monitoring convergence of iterative simulations , 1998 .

[18]  S L Warner,et al.  Randomized response: a survey technique for eliminating evasive answer bias. , 1965, Journal of the American Statistical Association.

[19]  Peter G. M. van der Heijden,et al.  Meta-Analysis of Randomized Response Research , 2005 .

[20]  J. Ware,et al.  Random-effects models for longitudinal data. , 1982, Biometrics.

[21]  Gregory J. Cizek,et al.  Cheating on Tests : How To Do It, Detect It, and Prevent It , 1999 .

[22]  Cora J. M. Maas,et al.  Meta-Analysis of Randomized Response Research: 35 Years of Validation. , 2010 .

[23]  D. Hedeker,et al.  A random-effects ordinal regression model for multilevel analysis. , 1994, Biometrics.

[24]  P. Schmidt,et al.  Limited-Dependent and Qualitative Variables in Econometrics. , 1984 .

[25]  Donald Geman,et al.  Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[26]  M. Karim.  Generalized Linear Models With Random Effects , 1991 .

[27]  G. Maddala.  Limited-dependent and qualitative variables in econometrics: Introduction , 1983 .

[28]  G. C. Tiao,et al.  Bayesian inference in statistical analysis , 1973 .

[29]  S. Edgell,et al.  Validity of Forced Responses in a Randomized Response Model , 1982 .

[30]  J. Fox,et al.  Randomized Item Response Theory Models , 2005 .

[31]  T. O’Hare,et al.  Measuring problem drinking in first time offenders. Development and validation of the College Alcohol Problem Scale (CAPS). , 1997, Journal of substance abuse treatment.

[32]  Stan Lipovetsky,et al.  Generalized Latent Variable Modeling: Multilevel,Longitudinal, and Structural Equation Models , 2005, Technometrics.

[33]  Ulf Böckenholt,et al.  Item Randomized-Response Models for Measuring Noncompliance: Risk-Return Perceptions, Social Influences, and Self-Protective Responses , 2007 .

[34]  Risto Lethonen.  Multilevel Statistical Models (3rd ed.) , 2005 .

[35]  N. Longford.  A fast scoring algorithm for maximum likelihood estimation in unbalanced mixed models with nested random effects , 1987 .