Interpreting the observed value of a pointbiserial 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 pointbiserials less than 0.2."
The general formula for a Pearson correlation coefficient is:
(1) 
PointBiserial 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 pointbiserial 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 pointmeasure correlation is given by the Rasch model proposition that: Xni = Eni±√Wni
(3) 
Since ±√Wni is a random residual, its crossproduct with any other variable is modeled to be zero. Thus
(4) 
which provides a convenient formula for computing the expected value of the pointbiserial correlation. Also see Note below
PointBiserial 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 pointbiserial correlation excluding the current observation.
PointMeasure Correlation
Similarly, suppose that Yn is Bn, the ability measure of person n, then the pointmeasure correlation is:
(8) 
where B. is the mean of the Bn.
Thus an estimate of the expected value of the pointmeasure correlation is:
Similarly, suppose that Yn is Bn, the ability measure of person n, then the pointmeasure correlation is:
(9) 
which provides a convenient formula for computing the expected value of a pointmeasure correlation.
John Michael Linacre
Here is a worked example for a pointmeasure correlation:
Later note: Experience suggests that W_{ni}*(N2)/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 (N2)/N makes the expected value more reasonable especially in the boundary condition of only two points. When there are only two points, (N2) 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 PointBiserial (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 2nd. Ed., Bond & Fox  Best Test Design, Wright & Stone 
Rating Scale Analysis, Wright & Masters  Introduction to Rasch Measurement, E. Smith & R. Smith  Introduction to ManyFacet 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 


Forum  Rasch Measurement Forum to discuss any Raschrelated 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 Raschrelated Events  

Feb 23  June 19, 2015, Mon.Fri.  Online graduatelevel course: Advanced Rasch Measurement Theory (D. Andrich, I. Marais, RUMM), www.education.uwa.edu.au/ppl/courses 
March 1113, 2015, Wed.Fri.  Inperson workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
March 1113, 2015, Wed.Fri.  Inperson workshop: Einführung in die Raschanalyse mit RUMM2030 (Böcker, Salzberger, RUMM), Aachen, Germany, www.psychometrikon.de 
March 20, 2015, Fri.  UK Rasch User Group Meeting, London, United Kingdom, www.rasch.org.uk 
March 2627, 2015, Thur.Fri.  Inperson workshop: Introduction to Rasch Measurement with Winsteps (W. Boone), Cincinnati, raschmeasurementanalysis.com 
April 1620, 2015, Thurs.Mon.  AERA Annual Meeting, Chicago IL www.aera.net 
April 2122, 2015, Tues.Wed.  IOMC 2015: International Outcomes Measurement Conference, Chicago IL www.jampress.org 
May 1315, 2015, Wed.Fri.  Inperson workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
May 1820, 2015, Mon.Wed.  Inperson workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
May 29  June 26, 2015, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
July 331, 2015, Fri.Fri.  Online workshop: Practical Rasch Measurement  Further Topics (E. Smith, Winsteps), www.statistics.com 
July 27  Nov. 20, 2015, Mon.Fri.  Online course: Introduction to Rasch Measurement Theory (D. Andrich, I. Marais, RUMM), www.education.uwa.edu.au/ppl/courses 
Aug. 14  Sept. 11, 2015, Fri.Fri.  Online workshop: ManyFacet Rasch Measurement (E. Smith, Facets), www.statistics.com 
Sept. 911, 2015, Wed.Fri.  Inperson workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
Sept. 1416, 2015, Mon.Wed.  Inperson workshop: Intermediate Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
Sept. 1718, 2015, Thur.Fri.  Inperson workshop: Advanced Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
Oct. 16  Nov. 13, 2015, Fri.Fri.  Online workshop: Practical Rasch Measurement  Core Topics (E. Smith, Winsteps), www.statistics.com 
Sept. 4  Oct. 16, 2015, Fri.Fri.  Online workshop: Rasch Applications, Part 1: How to Construct a Rasch Scale (W. Fisher), www.statistics.com 
Oct. 23  Nov. 20, 2015, Fri.Fri.  Online workshop: Rasch Applications, Part 2: Clinical Assessment, Survey Research, and Educational Measurement (W. Fisher), www.statistics.com 
Dec. 24, 2015, Wed.Fri.  Inperson workshop: Introductory Rasch (A. Tennant, RUMM), Leeds, UK, www.leeds.ac.uk/medicine/rehabmed/psychometric 
Aug. 12  Sept. 9, 2016, Fri.Fri.  Online workshop: ManyFacet Rasch Measurement (E. Smith, Facets), www.statistics.com 
The HTML to add "Coming Raschrelated Events" to your webpage is: <script type="text/javascript" src="http://www.rasch.org/events.txt"></script> 
The URL of this page is www.rasch.org/rmt/rmt221e.htm
Website: www.rasch.org/rmt/contents.htm