×

zbMATH — the first resource for mathematics

Cultural consensus theory for the ordinal data case. (English) Zbl 1314.62267
Summary: A Cultural Consensus Theory approach for ordinal data is developed, leading to a new model for ordered polytomous data. The model introduces a novel way of measuring response biases and also measures consensus item values, a consensus response scale, item difficulty, and informant knowledge. The model is extended as a finite mixture model to fit both simulated and real multicultural data, in which subgroups of informants have different sets of consensus item values. The extension is thus a form of model-based clustering for ordinal data. The hierarchical Bayesian framework is utilized for inference, and two posterior predictive checks are developed to verify the central assumptions of the model.

MSC:
62P15 Applications of statistics to psychology
62H30 Classification and discrimination; cluster analysis (statistical aspects)
62F15 Bayesian inference
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anders, R. (2013). CCTpack: cultural consensus theory applications to data. R package version 0.9.
[2] Anders, R.; Batchelder, W.H., Cultural consensus theory for multiple consensus truths, Journal for Mathematical Psychology, 56, 452-469, (2012)
[3] Batchelder, W.H.; Anders, R., Cultural consensus theory: comparing different concepts of cultural truth, Journal of Mathematical Psychology, 56, 316-332, (2012) · Zbl 1282.91293
[4] Batchelder, W.H.; Romney, A.K.; Grofman, B. (ed.); Owen, G. (ed.), The statistical analysis of a general Condorcet model for dichotomous choice situations, 103-112, (1986), Greenwich
[5] Batchelder, W.H.; Romney, A.K., Test theory without an answer key, Psychometrika, 53, 71-92, (1988) · Zbl 0718.62260
[6] Batchelder, W.H.; Romney, A.K.; Roskam (ed.), New results in test theory without an answer key, 229-248, (1989), Heidelberg
[7] Buhrmester, M.; Kwang, T.; Gosling, S.D., Amazon’s mechanical turk a new source of inexpensive, yet high-quality, data?, Perspectives on Psychological Science, 6, 3-5, (2011)
[8] Comrey, A.L., The minimum residual method of factor analysis, Psychological Reports, 11, 15-18, (1962)
[9] Boeck, P., Random item IRT models, Psychometrika, 73, 533-559, (2008) · Zbl 1284.62699
[10] DeCarlo, L.T., A model of rater behavior in essay grading based on signal detection theory, Journal of Educational Measurement, 42, 53-76, (2005)
[11] Fischer, G.H., & Molenaar, I.W. (1995). Rasch models: recent developments and applications. New York: Springer. · Zbl 0815.00010
[12] Fox, C.R.; Tversky, A., Ambiguity aversion and comparative ignorance, The Quarterly Journal of Economics, 110, 585-603, (1995) · Zbl 0836.90004
[13] Fox, J. (2013). Polycor: polychoric and polyserial correlations. R package version 0.7-8.
[14] Gelman, A., Carlin, J.B., Stern, H.S., & Rubin, D.B. (2004). Bayesian data analysis (2nd ed.). Boca Raton: Chapman and Hall/CRC.
[15] Gonzalez, R.; Wu, G., On the shape of the probability weighting function, Cognitive Psychology, 38, 129-166, (1999)
[16] Green, D.M., & Swets, J.A. (1966). Signal detection theory and psychophysics. New York: Wiley.
[17] Hruschka, D.J.; Kalim, N.; Edmonds, J.; Sibley, L., When there is more than one answer key: cultural theories of postpartum hemorrhage in Matlab, bangladesh, Field Methods, 20, 315-337, (2008)
[18] Johnson, V.E., & Albert, J.H. (1999). Ordinal data modeling. Statistics for social science and public policy. Berlin: Springer.
[19] Karabatsos, G.; Batchelder, W.H., Markov chain estimation methods for test theory without an answer key, Psychometrika, 68, 373-389, (2003) · Zbl 1306.62447
[20] Kruschke, J.K. (2011). Doing Bayesian data analysis: a tutorial with R and BUGS. Amsterdam: Elsevier/Academic Press. · Zbl 1301.62001
[21] Lancaster, H.; Hamdan, M., Estimation of the correlation coefficient in contingency tables with possibly nonmetrical characters, Psychometrika, 29, 383-391, (1964) · Zbl 0127.35904
[22] Lee, M.D., How cognitive modeling can benefit from hierarchical Bayesian models, Journal of Mathematical Psychology, 55, 1-7, (2011) · Zbl 1208.91123
[23] Lord, F.M., Novick, M.R., & Birnbaum, A. (1968). Statistical theories of mental test scores (Vol. 47). Reading: Addison-Wesley. · Zbl 0186.53701
[24] Macmillan, N.A., & Creelman, C.D. (2005). Detection theory: a users guide (2nd ed.). Mahwah: Erlbaum.
[25] Nering, M.L., & Ostini, R. (2011). Handbook of polytomous item response theory models. New York: Taylor and Francis.
[26] Patz, R.J.; Junker, B.W.; Johnson, M.S.; Mariano, L.T., The hierarchical rater model for rated test items and its application to large-scale educational assessment data, Journal of Educational and Behavioral Statistics, 27, 341-384, (2002)
[27] Plummer, M. (2003). JAGS: a program for analysis of Bayesian graphical models using Gibbs sampling.
[28] Plummer, M. (2012). Rjags: Bayesian graphic models using MCMC. R package version 3.2.0. http://CRAN.R-project.org/package=rjags. · Zbl 0718.62260
[29] Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Denmarks Paedagogiske Institute.
[30] Revelle, W. (2012). psych: procedures for psychological, psychometric, and personality research. Northwestern University Evanston, Illinois. R package version 1.2.1.
[31] Rigdon, E.E.; Salkind, N.J. (ed.), Polychoric correlation coefficient, 1046-1049, (2010), Thonsand Oaks
[32] Romney, A.K.; Batchelder, W.H.; Wilson, R. (ed.); Keil, F. (ed.), Cultural consensus theory, 208-209, (1999), Cambridge
[33] Romney, A.K.; Batchelder, W.H.; Weller, S.C., Recent applications of cultural consensus theory, American Behavioral Scientist, 31, 163-177, (1987)
[34] Romney, A.K.; Weller, S.C.; Batchelder, W.H., Culture as consensus: a theory of culture and informant accuracy, American Anthropologist, 88, 313-338, (1986)
[35] Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph Supplement. · Zbl 0718.62260
[36] Spearman, C.E., ‘general intelligence’ objectively determined and measured, The American Journal of Psychology, 15, 72-101, (1904)
[37] Spiegelhalter, D.J.; Best, N.G.; Carlin, B.P.; Linde, A., Bayesian measures of model complexity and fit (with discussion), Journal of the Royal Statistical Society, Series B, 6, 583-640, (2002) · Zbl 1067.62010
[38] Sprouse, J.; Wagers, M.; Phillips, C., A test of the relation between working-memory capacity and syntactic island effects, Language, 88, 82-123, (2012)
[39] Stephens, M., Dealing with label switching in mixture models, Journal of the Royal Statistical Society. Series B. Statistical Methodology, 62, 795-809, (2000) · Zbl 0957.62020
[40] Takane, Y.; Leeuw, J., On the relationship between item response theory and factor analysis of discretized variables, Psychometrika, 52, 393-408, (1987) · Zbl 0628.62104
[41] van der Linden, W.J., & Hambleton, R.K. (1997). Handbook of modern item response theory. Berlin: Springer. · Zbl 0872.62099
[42] Weller, S.W., Cultural consensus theory: applications and frequently asked questions, Field Methods, 19, 339-368, (2007)
[43] Zhang, H., & Maloney, L.T. (2012). Ubiquitous log odds: a common representation of probability and frequency distortion in perception, action and cognition. Frontiers in Neuroscience, \(6\).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.