﻿ Pt-biserial correlation = Measure

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.

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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