Point-Biserial Fit Indices

Richardson and Stalnaker (1933) derived rpbis, the point-biserial correlation between an ordinal scale with only two values and a continuous, interval scale, based on the biserial correlation of Karl Pearson (1909). Nowadays, rpbis is often the "corrected" correlation between respondents' right/wrong responses to a target item and their ordinal raw scores on the test (without the target item) (Henrysson). rpbis indicates the extent to which an item cooperates with the rest of the test. It is useful in Rasch analysis because negative values detect miskeyed MCQ responses and negatively worded survey questions. Other diagnostic use, such as detecting redundancy by means of overly high positive values, is problematic because of deficiencies in rpbis.

An obvious improvement in rpbis would be to replace ordinal raw scores by interval ability measures. This produces the point- bimeasure correlation, rpbim:

where
M1 is the mean measure of the n1 respondents answering the item correctly
M0 is the mean measure of the n0 respondents answering the item incorrectly
SDM is the standard deviation of all n1+n0=n respondent measures.

Even rpbim, however, is sensitive to targeting. With a uniform distribution of person abilities, the maximum rpbim could be .71 for an item with p-value .95, but .87 for an item with p-value .50. The corresponding minimum values could be -0.71 and -0.87. But ranges need not be symmetrical about 0. Fortunately rpbim's sensitivity to targeting can be reduced by standardizing its range to -1.0 to 1.0.

The maximum rpbim attainable for any item p-value is the one produced by its Guttman pattern (all 1's by high ability respondents, all 0's by low). The minimum rpbim is the anti- Guttman pattern (all 0's by high ability respondents, all 1's by low). This provides a standardized rpbig ("g" for Guttman):

Negative values continue to indicate that this item is "working backwards", contradicting the construct. But values close to 1.0 are now diagnostic. They indicate a local lack of stochasticity in the data.

Jack Stenner

Note: the biserial correlation originated in Karl Pearson, '"On a New Method of Determining Correlation ....", Biometrika, Vol. VII, pp. 96-105, 1909, and the point-biserial correlation originated in Richardson, M.W. & Stalnaker, J.M. (1933). "A note on the use of bi-serial r in test research". Journal of General Psychology, 8, 463-465.

Henrysson, S. (1963). Correction for item-total correlations in item analysis. Psychometrika, 28, 211-218.

Point-biserial fit indices. Stenner AJ. … Rasch Measurement Transactions, 1995, 9:1 p.416

Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn. George Engelhard, Jr. & Jue Wang	Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene	Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver	Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone	Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes	Statistical Analyses for Language Testers (Facets), Rita Green	Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind	Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M	Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind	Rasch Measurement: Applications, Khine	Winsteps Tutorials - free Facets Tutorials - free	Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre	Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
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Rasch Measurement Transactions & Rasch Measurement research papers - free	An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse	Rasch Measurement Theory Analysis in R, Wind, Hua	Applying the Rasch Model in Social Sciences Using R, Lamprianou	El modelo métrico de Rasch: Fundamentación, implementación e interpretación de la medida en ciencias sociales (Spanish Edition), Manuel González-Montesinos M.
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar	Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch	Rasch Models for Measurement, David Andrich	Constructing Measures, Mark Wilson	Best Test Design - free, Wright & Stone Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias	Diseño de Mejores Pruebas - free, Spanish Best Test Design	A Course in Rasch Measurement Theory, Andrich, Marais	Rasch Models in Health, Christensen, Kreiner, Mesba	Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen

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