This reports the correlation between the raw-score or measure for each element.
Pt-Biserial = Yes or P or B or Omit
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Point-biserial correlation of observation for the current element with its average observation omitting the observation.
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Pt-Biserial = Include or All
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Point-biserial correlation of observation for the current element with its average observation including the observation.
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Pt-Biserial = Measure
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Point-measure correlation of observation for the current element with the measure sum for the observation, and also outputs the expected value of the correlation
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Pt-Biserial = No or blank
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No correlation is reported, but the point-measure correlation is computed for Scorefile=
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The point-biserial correlation is a many-facet version of the Pearson point-biserial correlation, rpbis. It takes an extra iteration to calculate, but is useful on new data to check that all element scores work in the same direction along the variable. Negative point-biserial correlations usually signify miskeyed or miscoded data, or negatively worded items. More of a variable (however defined) is always intended to correspond to a higher score.
Let's start with a 2-facet (rectangle, Winsteps) situation. The point-biserial for an item is the correlation of the person scores with their responses to the item. The point-biserial for a person is the correlation of the item scores (p-values) with the responses by the person to those items.
Then to 3 facets (persons, items, raters), we have to extrapolate from the 2-facet procedure.
The point biserial for element x in facet 1 is the correlation of the total score for each element in facet 2 with the response by element x to that element in facet 2, combined with the correlation of the total score for each element in facet 3 with the response by element x to that element in facet. Notice that the element's responses are scanned twice. For the point-measure correlation, they are scanned once: the element's response correlated with sum of the measures for the response.
When a Model= statement specifies measure reversal with "-", i.e., Model=?,-?,R4, then the category value is reversed for the reversed facet, by subtracting the observation from the specified maximum. Thus a value of "3" is treated as "3" for the first facet, "?", but as "4-3"="1" for the second facet, "-?", when computing the point-biserial.
Pt-biserial = Yes: For three facets, i,j,k, the formula for this product-moment correlation coefficient is
Ai = (Ti - Wijk*Xijk) / (Ci - Wijk)
where Ti is the weighted total score for element i = sum (Wijk*Xijk). Ci is the weighted count of observations for element i = Sum(Wijk). Ai is the average observation for element i omitting element Wijk*Xijk. Ai is correlated with Wijk*Xijk
Wijk = 1 unless a weight is specified in Models=, Labels= or Data=. For instance,
Facets = 3
Data=
R5.24 , 1, 2, 3, 7
then
X123=7 ; the observation is 7
W123=5.24 ; its weight is 5.24
Pt-biserial = Include: For three facets, i,j,k, the formula for this product-moment correlation coefficient is
Ai = Ti / Ci
Then the point-biserial correlation for element k is:
PBSk = Correlation ( {Ai, Aj}, Xijk ) for i = 1,Ni and j = 1,Nj
Since the point-biserial is poorly defined for missing data, rating scales (or partial credit items) and multiple facets, please regard this correlation as an indication, not definitive.
See also: Linacre, J.M. (2003). Computing the “Single Rater-Rest of Raters” (SR/ROR) Correlations. Appendix A in C. Myford & E. Wolfe: Detecting and Measuring Rater Effects . Journal of Applied Measurement, 4, 421-2.
Example: A complete 3-facet dataset. We want the point-biserial and point-measure correlations for element j1.
The Residual/Response file contains the needed values. For each observation of item 1, we have (a) the value of the observation (Stp), (b) the sum of the measures for that observation (Meas), (c) the weight of the observation (wt). The element measures are all oriented or reversed so that higher score = higher measure for every measure. The correlation is then the standard correlation formula with weighting. www.reddit.com/r/excel/comments/h7ngzu/calculate_correlation_with_frequencyweighting/ suggests how to do this in Excel.
Data:
|
j1
|
j2
|
j3
|
i1
|
i2
|
i3
|
i1
|
i2
|
i3
|
i1
|
i2
|
i3
|
p1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
1
|
0
|
p2
|
1
|
1
|
0
|
0
|
0
|
0
|
1
|
1
|
0
|
p3
|
1
|
0
|
0
|
1
|
1
|
0
|
1
|
1
|
1
|
p4
|
1
|
0
|
0
|
1
|
0
|
0
|
1
|
0
|
0
|
Element totals and counts:
Element
|
Ti Total
|
Ci Count
|
Measures (+ve)
|
p1
|
8
|
9
|
3.61
|
p2
|
4
|
9
|
-0.42
|
p3
|
6
|
9
|
1.43
|
p4
|
3
|
9
|
-1.39
|
i1
|
11
|
12
|
2.78
|
i2
|
7
|
12
|
-0.09
|
i3
|
3
|
12
|
-2.69
|
j1
|
7
|
12
|
-0.02
|
j2
|
6
|
12
|
-0.77
|
j3
|
8
|
12
|
0.79
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Computation of Ai and the point-biserial correlation for element j1:
For j1
|
Observation
|
Total
|
Count
|
Pt-biserial=
Yes, Exclude
|
Pt-biserial=
All, Include
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for persons
|
Xnij
|
Ti
|
Ci
|
Ai
|
Ai
|
p1
|
i1
|
1
|
8
|
9
|
0.88
|
0.89
|
p2
|
i1
|
1
|
4
|
9
|
0.38
|
0.44
|
p3
|
i1
|
1
|
6
|
9
|
0.63
|
0.67
|
p4
|
i1
|
1
|
3
|
9
|
0.25
|
0.33
|
p1
|
i2
|
1
|
8
|
9
|
0.88
|
0.89
|
p2
|
i2
|
1
|
4
|
9
|
0.38
|
0.44
|
p3
|
i2
|
0
|
6
|
9
|
0.75
|
0.67
|
p4
|
i2
|
0
|
3
|
9
|
0.38
|
0.33
|
p1
|
i3
|
1
|
8
|
9
|
0.88
|
0.89
|
p2
|
i3
|
0
|
4
|
9
|
0.5
|
0.44
|
p3
|
i3
|
0
|
6
|
9
|
0.75
|
0.67
|
p4
|
i3
|
0
|
3
|
9
|
0.38
|
0.33
|
for items
|
|
|
|
|
|
p1
|
i1
|
1
|
11
|
12
|
0.91
|
0.92
|
p2
|
i1
|
1
|
11
|
12
|
0.91
|
0.92
|
p3
|
i1
|
1
|
11
|
12
|
0.91
|
0.92
|
p4
|
i1
|
1
|
11
|
12
|
0.91
|
0.92
|
p1
|
i2
|
1
|
7
|
12
|
0.55
|
0.58
|
p2
|
i2
|
1
|
7
|
12
|
0.55
|
0.58
|
p3
|
i2
|
0
|
7
|
12
|
0.64
|
0.58
|
p4
|
i2
|
0
|
7
|
12
|
0.64
|
0.58
|
p1
|
i3
|
1
|
3
|
12
|
0.18
|
0.25
|
p2
|
i3
|
0
|
3
|
12
|
0.27
|
0.25
|
p3
|
i3
|
0
|
3
|
12
|
0.27
|
0.25
|
p4
|
i3
|
0
|
3
|
12
|
0.27
|
0.25
|
|
|
^
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Facets Table 7:
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PtBis = Yes 0.34
|
Ptbis=Inc
0.51
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Computation of point-measure correlation for element j1:
For j1
|
Observation
Xnij
|
Sum of
Measures
|
Expected Observation
Enij
|
Model Variance of
Xnij around Enij
|
p1
|
i1
|
1
|
6.37
|
1
|
0
|
p1
|
i2
|
1
|
3.5
|
0.97
|
0.03
|
p1
|
i3
|
1
|
0.91
|
0.71
|
0.2
|
p2
|
i1
|
1
|
2.34
|
0.91
|
0.08
|
p2
|
i2
|
1
|
-0.53
|
0.37
|
0.23
|
p2
|
i3
|
0
|
-3.12
|
0.04
|
0.04
|
p3
|
i1
|
1
|
4.19
|
0.99
|
0.01
|
p3
|
i2
|
0
|
1.32
|
0.79
|
0.17
|
p3
|
i3
|
0
|
-1.27
|
0.22
|
0.17
|
p4
|
i1
|
1
|
1.38
|
0.8
|
0.16
|
p4
|
i2
|
0
|
-1.49
|
0.18
|
0.15
|
p4
|
i3
|
0
|
-4.09
|
0.02
|
0.02
|
PtBis=Measure: PtMea = 0.73
|
Variance of Enij = 0.14
|
Average
Model Variance =
0.11
|
Correlation of Enij with Measures:
|
0.94
|
Attenuation of correlation due to error =
Sqrt( Variance of Enij / (Variance of Enij + Average Model Variance) )=
|
0.75
|
Expected Point-measure Correlation = 0.94 * 0.75 = PtExp =
|
0.71
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