The randomized response log linear model as a composite link model

Abstract

In randomized response (RR) designs, misclassification is used to protect the privacy of respondents when sensitive questions are asked. A generalized linear model with a composite link function is presented to formulate log linear models that take the RR design into account. The approach is extended to model the situation where some respondents do not follow the instructions of the RR design. For example, if there are three binary RR variables with regard to practicing fraud, the 2 × 2 × 2 cross-classification of the true answers is latent due to the misclassification. Using composite link functions, log linear models can be specified for the latent table to investigate possible association between the variables. Fast iteratively re-weighted least squares algorithms are presented.

Keywords

iteratively re-weighted least squares misclassification sensitive questions

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References

Böckenholt U and van der Heijden PGM (2007) Item randomized-response models for measuring noncompliance: risk-return perceptions, social influences, and self-protective responses . Psychometrika, 72, 245–62 .

Boruch RF (1971) Assuring confidentiality of responses in social research: a note on strategies . The American Sociologist, 6, 308–11 .

Chen TT (1989) A review of methods for misclassified categorical data in epidemiology . Statistics in Medicine, 8, 1095–106 .

Cruyff MJLF , van den Hout A , van der Heijden PGM and Böckenholt U (2007) Log-linear randomized response models taking cheating into account . Sociological Methods and Research, 36, 266–82 .

Eilers PHC (2007) Ill-posed problems with counts, the composite link model, and penalized likelihood . Statistical Modelling, 7, 239–54 .

Gjestvang C and Singh S (2007) Forced quantitative randomized response model: a new device . Metrika, 66, 243–57 .

Greenberg BG , Abul-Ela ALA , Simmons WR and Horvitz DG (1969) The unrelated question RR model: theoretical framework . Journal of the American Statistical Association, 64, 520–39 .

Huang KC (2007) Constructing optimal randomized response designs with consideration for the level of privacy protection . Statistica Neerlandica, 61, 284–91 .

Kuk AYC (1990) Asking sensitive questions indirectly . Biometrika, 77, 436–38 .

10.

Moriarty M and Wiseman F (1976) On the choice of a randomization technique with the randomized response model. ‘Proceedings of the social statistics section of the American Statistical Association’. Washington DC, American Statistical Association , pp. 624–26.

11.

Pregibon D (1980) Goodness of link test for generalized linear models . Applied Statistics, 29, 15–24 .

12.

Rabe-Hesketh S and Skondral A (2007) Multilevel and latent variable modeling with composite links and exploded likelihoods . Psychometrika, 72, 123–40 .

13.

Scallan A , Gilchrist R and Green M (1984) Fitting parametric link functions in generalised linear models . Computational Statistics and Data Analysis, 2, 37–49 .

14.

Thompson R and Baker RJ (1981) Composite link functions in generalized linear models . Applied Statistics, 30, 125–31 .

15.

Van den Hout A and van der Heijden PGM (2004) The analysis of multivariate misclassified data with special attention to randomized response data . Sociological Methods and Research, 32, 310–36 .

16.

Van Gils G , van der Heijden PGM and Landsheer H (1996) Rapportage van uitkeringsfraude in surveys. Den Haag: Ministerie van Sociale Zaken en Werkgelegenheid , Report 43. (In Dutch)

17.

Warner SL (1965) Randomized response: a survey technique for eliminating answer bias . Journal of the American Statistical Association, 60, 63–69 .