Interpreting the observed value of a point-biserial correlation is made easier if we can compare the observed value with its expected value. Is the observed value much higher than the expected value (indicating dependency in the data) or much lower than expected (indicating unmodeled noise)? With knowledge of how the observed value compares with its expected value, there is no need for arbitrary rules such as "Delete items with point-biserials less than 0.2."
The general formula for a Pearson correlation coefficient is:
(1) |
Point-Biserial Correlation (including all observations in the correlated raw score)
Suppose that Xn is Xni the observation of person n on item i. Yn is Rn, the raw score of person n, then the point-biserial correlation is:
(2) |
where X. is the mean of the {Xni} for item i, and R. is the mean of the Rn.
According to the Rasch model, the expected value of Xni is Eni and the model variance of Xni around its expectation is Wni. The model variances of X.i, Rn, R. are ignored here. S(Eni) = S(Xni), so that E.i = X.i.
Thus an estimate of the expected value of the point-measure correlation is given by the Rasch model proposition that: Xni = Eni±√Wni
(3) |
Since ±√Wni is a random residual, its cross-product with any other variable is modeled to be zero. Thus
(4) |
which provides a convenient formula for computing the expected value of the point-biserial correlation. Also see Note below
Point-Biserial Correlation (excluding current observation from the correlated raw score)
(5) |
where R.'is the mean of the R_{n}-X_{ni}.
(6) |
(7) |
is the expected value of the point-biserial correlation excluding the current observation.
Point-Measure Correlation
Similarly, suppose that Yn is Bn, the ability measure of person n, then the point-measure correlation is:
(8) |
where B. is the mean of the Bn.
Thus an estimate of the expected value of the point-measure correlation is:
(9) |
which provides a convenient formula for computing the expected value of a point-measure correlation.
John Michael Linacre
Here is a worked example for a point-measure correlation:
Later note: Experience suggests that W_{ni}*(N-2)/N is a better term in the divisors than W_{ni}, so that the expected correlation for 2 observations becomes its observed value of ±1.0. The "expected value" derivation is asymptotic for large N. When we have only small N, in particular only two points, the derivation degrades. The correction of (N-2)/N makes the expected value more reasonable especially in the boundary condition of only two points. When there are only two points, (N-2) makes the W_{ni} term disappear, so that the expected value of the correlation becomes +1.0 or -1.0 (or undefined), which will generally match its empirical value.
The Expected Value of a Point-Biserial (or Similar) Correlation. Linacre J.M. … Rasch Measurement Transactions, 2008, 22:1 p. 1154
Rasch Publications | ||||
---|---|---|---|---|
Rasch Measurement Transactions (free, online) | Rasch Measurement research papers (free, online) | Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch | Applying the Rasch Model 3rd. Ed., Bond & Fox | Best Test Design, Wright & Stone |
Rating Scale Analysis, Wright & Masters | Introduction to Rasch Measurement, E. Smith & R. Smith | Introduction to Many-Facet Rasch Measurement, Thomas Eckes | Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. | Statistical Analyses for Language Testers, Rita Green |
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar | Journal of Applied Measurement | Rasch models for measurement, David Andrich | Constructing Measures, Mark Wilson | Rasch Analysis in the Human Sciences, Boone, Stave, Yale |
in Spanish: | Análisis de Rasch para todos, Agustín Tristán | Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez |
Forum | Rasch Measurement Forum to discuss any Rasch-related topic |
Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement
Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.
Coming Rasch-related Events | |
---|---|
June 26 - July 24, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
June 29 - July 1, 2020, Mon.-Wed. | Measurement at the Crossroads 2020, Milan, Italy , https://convegni.unicatt.it/mac-home |
July - November, 2020 | On-line course: An Introduction to Rasch Measurement Theory and RUMM2030Plus (Andrich & Marais), http://www.education.uwa.edu.au/ppl/courses |
July 1 - July 3, 2020, Wed.-Fri. | International Measurement Confederation (IMEKO) Joint Symposium, Warsaw, Poland, http://www.imeko-warsaw-2020.org/ |
Aug. 7 - Sept. 4, 2020, Fri.-Fri. | On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com |
Oct. 9 - Nov. 6, 2020, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com |
June 25 - July 23, 2021, Fri.-Fri. | On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com |
The URL of this page is www.rasch.org/rmt/rmt221e.htm
Website: www.rasch.org/rmt/contents.htm