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Pt-biserial correlation = Measure |
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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. |
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. |
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 the expected value of the correlation |
Pt-Biserial = No or blank |
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No correlation is reported, but the point-measure correlation is computed for Scorefile= |
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.
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. Ci is the count of observations for element i. Ai is the average observation for element i omitting element Xijk of weight Wijk..
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= j = 1,Nj
Since the point-biserial is poorly defined for missing data, rating scales (or partial credit items) and multiple facets, regard this correlation as an indication, not definitive.
Example: A complete 3-facet dataset. We want the point-biserial correlation for element j1.
Data:
|
j1 |
j2 |
j3 |
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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:
|
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 |
Computation of Ai and the point-biserial correlation for element j1:
For j1 |
Observation |
Total |
Count |
Pt-biserial= Yes |
Pt-biserial= Include |
|
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 |
|
|
^ |
Facets Table 7: |
PtBis = Yes 0.34 |
Ptbis=Inc |
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Computation of point-measure correlation for element j1:
For j1 |
Observation |
Sum of |
Expected Observation |
Model Variance of |
|
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 |
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Correlation of Enij with Measures: |
0.94 |
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Attenuation of correlation due to error = |
0.75 |
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Expected Point-measure Correlation = 0.94 * 0.75 = PtExp = |
0.71 |
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Help for Facets Rasch Measurement Software: www.winsteps.com.