Pt-biserial correlation = Measure

This reports the correlation between the raw-score or measure for each element.

 

Pt-Biserial = Yes or P or B or Omit

Point-biserial correlation of observation for the current element with its average observation omitting the observation.

Pt-Biserial = Include or All

Point-biserial correlation of observation for the current element with its average observation including the observation.

Pt-Biserial = Measure

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

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.

 

See also: Linacre, J.M. (2003). Computing the “Single Rater-Rest of Rasters” (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.

 

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

 

Computation of Ai and the point-biserial correlation for element j1:

 

For j1

Observation

Total

Count

Pt-biserial=
Yes, Exclude

Pt-biserial=
All, 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
0.51

 

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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