The point-biserial correlation (or the point-polyserial correlation) is the Pearson correlation between on the observations on an item (or by a person) and the person raw scores or measures (or item marginal scores or measures). These are crucial for evaluating whether the coding scheme and person responses accord with the requirement that "higher observations correspond to more of the latent variable" (and vice-versa). These are reported in Tables 14 for items and Table 18 for items persons. They correlate an item's (or person's) responses with the measures of the encountered persons (or items). In Rasch analysis, the point-biserial correlation, rpbis, is a useful diagnostic indicator of data mis-coding or item mis keying: negative or zero values indicate items or persons with response strings that contradict the variable. Positive values are less informative than INFIT and OUTFIT statistics.
Point-Biserial and Point-Measure Correlations
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Control Instruction
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Column Label
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Explanation
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Expected value (EXP.)
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PTBISERIAL= Yes
PTBISERIAL= Exclude
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PBSX
PTBISERL-EX
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Point-biserial (or point-polyserial) correlation excluding the current observation from the raw score. Computes the point-biserial or point-polyserial correlation coefficients, rpbis, for persons and items. This is the Pearson product-moment correlation between the scored responses (dichotomies and polytomies) and the "rest scores", the corresponding total (marginal) scores excluding the scored responses to be correlated. This is a point-biserial correlation for dichotomies, or a point-polyserial correlation for polytomies. Extreme (perfect, maximum possible and zero, minimum possible) scores are included in the computation, but missing observations are omitted pairwise. The Biserial correlation can be computed from the Point-biserial. This correlation loses its meaning when there are missing data or with CUTLO= or CUTHI=. Specify PTBISERIAL=X instead.
In Table 14.1, etc., and IFILE=
For each item, this correlates the current observation with the (raw score - current observation)
The actual number of observations summed into the raw score is ignored.
In Table 14.2, etc., and DISFILE=
For each item, each possible response option is evaluated in turn. If the option matches the current observation, then the option is score 1, if not the option is scored 0. Then the scored option is correlated with
if the current observation is not missing: the (raw score - current observation) / (number of non-missing observations - 1)
if the current observation is missing: the (raw score) / (number of non-missing observations)
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PBSX-E
PBX-E
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PTBISERIAL= All
PTBISERIAL= Include
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PBSA
PTBISERL-AL
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Point-biserial correlation for all observations including the current observation in the raw score. Computes the Pearson correlation between the total (marginal) scores including all responses and the responses to the targeted item and person. This is a point-biserial correlation for dichotomies, or a point-polyserial correlation for polytomies. This correlation loses its meaning when there are missing data or with CUTLO= or CUTHI=. Specify PTBISERIAL=N instead.
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PBSA-E
PBA-E
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PTBISERIAL= No
PTBISERIAL= RPM
PTBISERIAL= Measure
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PTMA
PTMEASUR-AL
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Point-measure correlation for all observations. Computes the Pearson point-measure correlation coefficients, rpm between the observations and the measures, estimated from the raw scores including the current observation or the anchored values. Measures corresponding to extreme scores are included in the computation.
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PTMA-E
PMA-E
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PTBISERIAL= X
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PTMX
PTMEASUR-EX
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Point-measure correlation excluding the current observation from the estimation of the measure. Computes the Pearson point-measure correlation coefficients, rpm between the observations and the measures or anchor values adjusted to exclude the current observation. Measures corresponding to extreme scores are included in the computation.
This is an approximation to the exact value. See "Numerical Example"
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PTMX-E
PMX-E
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PTBISERIAL= Skip
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----
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Point- correlations are not computed. This speeds up estimation process.
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----
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Point-correlations are always reported for items. Point-correlations are reported for persons when (i) all the items are dichotomies, (ii) all the items have three categories, or (iii) all the items are in the same item group (Andrich Rating Scale Model). Otherwise, the observed and expected correlations are reported as zero in PFILE=.
Here's how these correlations work:
Think of an item (or a person).
That item has a string of responses.
Each response ("point") is made by a person who has a raw score and a Rasch measure.
1. Correlate the raw scores with the responses. This is the point-biserial correlation (including the current response), PTBSA. (PTBIS=All)
2. Correlate the raw scores (less the current response) with the responses. This is the point-biserial correlation corrected for auto-correlation, PTBSE. (PTBIS=Yes)
3. Correlate the Rasch measures (estimated including the current response) with the responses. This is the point-measure correlation, PTMEA. (PTBIS=No)
4. Correlate the Rasch measures (estimated without the current response) with the responses. This is the point-measure correlation: corrected for autocorrelation, PTMEX. (PTBIS=X)
The "expected correlations" (the values of the correlations we would expect when the data fit the Rasch model perfectly) were introduced to avoid incorrect inferences based on the point-biserial correlations. For instance, eliminating items with low point-biserial correlations in situations where it is impossible for the point-biserial correlations to be high. In fact, without an "expected" reference point, it can be impossible to identify whether a reported correlation is too high, about right or too low.
For tests of different lengths including the same items we would first need to compare the items' "expected" correlation values, no matter which correlation we chose to report. This would provide the baseline for discussion about the idiosyncrasies of each item in each test.
Example 1: For rank-order or paired-comparison data, point-biserials are all -1. So specify Point-measure correlations.
PTBISERIAL=NO
Example 2: Winsteps results are to be compared to previous results for which the point-biserial correlation was based on the total marginal scores, including the responses to the targeted item.
PTBISERIAL=ALL
Example 3: Numerical example:
title=check
ptbiserial=N ; A, E
ni=4
item1=1
name1=6
codes=01
tfile=*
14.1 ; item table
*
; reported PTMX - first analysis
; ptbiserial=X
; ifile=if.txt
; sfile=sf.txt
; exact PTMX - second analysis
; ptbiserial=N
; iafile=if.txt
; safile=sf.txt
; iweight=*
; 4 0 ; target item for PTMX
; *
&END
Item 1
Item 2
Item 3
Current Item
END LABELS
0111 Jose
1010 Mary
0100 Robert
110 Kim
Column In Table 14.1
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Person
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Item 4
Response to current item
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Person Measure
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Person
Raw Score
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Person
Raw score less current response
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Person Measure S.E.
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Expected Response:
Item Measure = 0.87
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PTMX Measure = Person
Measure - (Response - Expected)*S.E.2
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Exact Measure (estimated without current response)
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Jose
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1
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1.21
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3
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2
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1.20
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0.586
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0.61
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0.46
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Mary
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0
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0.01
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2
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2
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1.06
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0.297
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0.34
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0.46
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Robert
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0
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-1.21
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1
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1
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1.22
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0.110
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-1.05
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-1.03
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Kim
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missing
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0.46
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2
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2
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1.26
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0.399
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0.46
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0.46
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PTMEASUR-AL
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Point-measure correlation: PTBISERIAL = N
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0.86
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PTBISERL-AL
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Point-biserial correlation (All responses): PTBISERIAL = A
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0.87
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PTBISERL-EX
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Point-biserial correlation (Excluding current response): PTBISERIAL= E
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0.50
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PTMEASUR-EX
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Point-measure correlation (eXcluding current response from measure) : PTBISERIAL = X
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items
1 -0.43
2 -0.39
3 -0.06
4 0.63
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items
1 -0.53
2 -0.47
3 -0.14
4 0.50
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