Abstract
In this chapter we will investigate and improve two methods of estimating reliability that were originally proposed by Mokken (1971). Based on these methods, a third way of estimating reliability is proposed. The three methods are estimates of the classical reliability concept, which is the ratio of true score variance to observed score variance (see, for example, Lord and Novick, 1968), in the context of nonparametric item response theory (Mokken, 1971; Henning, 1976; Stokman and van Schuur, 1980; Mokken and Lewis, 1982). Apart from new theoretical developments, results from a Monte Carlo study will also be reported and discussed. In this study the three methods already mentioned are compared with four ‘classical’ methods of estimating the reliability of psychological measures. These four methods are the well-known lower bounds alpha (Cronbach, 1951) and lambda-2 (Guttman, 1945), and the lower bounds mu–2 and mu–3 (ten Berge and Zegers, 1978). Finally, some remarks will be made concerning the use of the estimation methods.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
BERGE, J. M. F. ten and ZEGERS, F. E., ‘A Series of Lower Bounds to the Reliability of a Test’, Psychometrika, 43 (1978) 575–9.
BIRNBAUM, A., Part v in LORD, F. M. and NOVICK, M. R., Statistical Theories of Mental Test Scores (Reading, Mass.: Addison-Wesley, 1968).
CRONBACH, L. J., ‘Coefficient Alpha and the Internal Structure of Tests’, Psychometrika, 16 (1951) 297–334.
FELDT, L. S., ‘The Approximate Sampling Distribution of Kuder-Richard-son Reliability Coefficient Twenty’, Psychometrika, 30 (1965) 357–70.
FISCHER, G. H., Einführung in die Theorie psychologischer Tests (Bern: Huber, 1974).
GUTTMAN, L., ‘A Basis for Analyzing Test-Retest Reliability’, Psychometrika, 10 (1945) 255–82.
—— ‘The Basis for Scalogram Analysis’, in STOUFFER, S. A., GUTTMAN, L., SUCHMAN, E. A., LAZARSFELD, P. F., STAR, S. A. and CLAUSEN, J. A. (eds), Measurement and Prediction (Princeton, N.J.: Princeton University Press, 1950).
HAMBLETON, R. K. and COOK, L. L., ‘Latent Trait Models and Their Use in the Analysis of Educational Test Data’, Journal of Educational Measurement, 14 (1977) 75–96.
HENNING, H. J., ‘Die Technik der Mokken-Skalenanalyse’, Psychologische Beiträge, 18 (1976) 410–30.
JACKSON, P. H. and AGUNWAMBA, C. C., ‘Lower Bounds for the Reliability of the Total Score on a Test Composed of Non-homogeneous Items: I. Algebraic Lower Bounds’, Psychometrika, 42 (1977) 567–78.
JANSEN, P. G. W., ‘Homogenitätsmessung mit Hilfe des Koeffizienten H von Loevinger: Eine Kritische Diskussion’, Psychologische Beiträge, 24 (1982a) 96–105.
—— ‘De Onbruikbaarheid van Mokkenschaal-analyse’, Tijdschrift voor Onderwijsresearch, 7 (1982b) 11–24.
—— Rasch Analysis of Attitudinal Data (Den Haag: Rijks Psychologische Dienst, 1983) (dissertation).
—— ROSKAM, E. E. Ch. I. and WOLLENBERG, A. L. van den, ‘De Mokkenschaal gewogen’, Tijdschrift voor Onderwijsresearch, 7 (1982) 31–42.
KRISTOF, W., ‘The Statistical Theory of Stepped-up Reliability Coefficients When a Test has been Divided into Several Equivalent Parts’, Psychometrika, 28 (1963) 221–38.
LOEVINGER, J., ‘The Technique of Homogeneous Tests Compared with Some Aspects of “Scale Analysis” and Factor Analysis’, Psychological Bulletin, 45 (1948) 507–30.
LORD, F. M., ‘Practical Applications of Item Characteristic Curve Theory’, Journal of Educational Measurement, 14 (1977) 117–38.
—— Applications of Item Response Theory to Practical Testing Problems (Hillsdale, N.J.: Lawrence Erlbaum, 1980).
—— and NOVICK, M. R., Statistical Theories of Mental Test Scores (Reading, Mass.: Addison-Wesley, 1968).
LUMSDEN, J., ‘Test Theory’, Annual Review of Psychology, 27 (1976) 251–80.
MOKKEN, R. J., A Theory and Procedure of Scale Analysis (The Hague: Mouton, 1971).
—— and LEWIS, C., ‘A Nonparametric Approach to the Analysis of Dichotomous Item Responses’, Applied Psychological Measurement, 6 (1982) 417–30.
MOLENAAR, I. W., ‘Mokken Scaling Revisited’, Kwantitatieve Methoden, vol. 3, 8 (1982a) 145–64.
—— ‘Een Tweede Weging van de Mokken-Schaal’, Tijdschrift voor Onder-wijsresearch, 7 (1982b) 172–81.
—— ‘De Beperkte Bruikbaarheid van Jansen’s Kritiek’, Tijdschrift voor Onderwijsresearch, 7 (1982c) 25–30.
—— and SIJTSMA, K., ‘Internal Consistency and Reliability in Mokken’s Nonparametric Item Response Model’, Tijdschrift voor Onderwijsresearch, 9 (1984) 257–68.
SEDERE, M. U. and FELDT, L. S., ‘The Sampling Distributions of the Kristof Reliability Coefficient, the Feldt Coefficient, and Guttman’s lambda-2’, Journal of Educational Measurement, 14 (1977) 53–62.
SIJTSMA, K., ‘Useful Nonparametric Scaling: A Reply to Jansen’, Psychologische Beiträge, 26 (1984) 423–37.
—— and MOLENAAR, I. W., ‘Reliability of Test Scores in Nonparametric Item Response Theory’, Psychometrika, (in press).
STOKMAN, F. N. and SCHUUR, W. H. van, ‘Basic Scaling’, Quality and Quantity, 14 (1980) 5–30.
WEISS, D. J. and DAVISON, M. L., ‘Test Theory and Methods’, Annual Review of Psychology, 32 (1981) 629–58.
WOOD, R., ‘Fitting the Rasch Model — A Heady Tale’, British Journal of Mathematical and Statistical Psychology, 31 (1978) 27–32.
Editor information
Editors and Affiliations
Copyright information
© 1988 Willem E. Saris and Irmtraud N. Gallhofer
About this chapter
Cite this chapter
Sijtsma, K. (1988). Reliability Estimation in Mokken’s Nonparametric Item Response Model. In: Saris, W.E., Gallhofer, I.N. (eds) Sociometric Research. Palgrave Macmillan, London. https://doi.org/10.1007/978-1-349-19051-5_10
Download citation
DOI: https://doi.org/10.1007/978-1-349-19051-5_10
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-1-349-19053-9
Online ISBN: 978-1-349-19051-5
eBook Packages: Palgrave Social & Cultural Studies CollectionSocial Sciences (R0)