﻿ PTBISERIAL= compute point-biserial correlation coefficient

# PTBISERIAL= point-biserial-type correlation coefficient = No

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 Control Instruction Observed value Explanation Expected value (EXP.) PTBISERIAL= Yes PTBISERIAL= Exclude PBSX PTBISERL-EX 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. PBSX-E PBX-E PTBISERIAL= All PTBISERIAL= Include PBSA PTBISERL-AL 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. PBSA-E PBA-E PTBISERIAL= No PTBISERIAL= RPM PTBISERIAL= Measure PTMA PTMEASUR-AL 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. PTMA-E PMA-E PTBISERIAL= X PTMX PTMEASUR-EX 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. PTMX-E PMX-E

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)

Numerical example:

 Person Response to item Measure Raw Score Raw score less current response Measure (estimated without current response) Jose 1 2.00 21 20 1.9 Mary 0 1.00 13 13 1.1 Robert 0 0.00 7 7 0.1 Point-measure correlation: PTBIS = N 0.87 Point-biserial correlation (All responses): PTBIS = A 0.90 Point-biserial correlation (Excluding current response): PTBIS= E 0.89 Point-measure correlation (excluding current response from measure) : PTBIS = X 0.83

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

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Rasch Measurement Transactions (free, online) Rasch Measurement research papers (free, online) Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Applying the Rasch Model 3rd. Ed., Bond & Fox Best Test Design, Wright & Stone
Rating Scale Analysis, Wright & Masters Introduction to Rasch Measurement, E. Smith & R. Smith Introduction to Many-Facet Rasch Measurement, Thomas Eckes Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments, George Engelhard, Jr. & Stefanie Wind Statistical Analyses for Language Testers, Rita Green
Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Journal of Applied Measurement Rasch models for measurement, David Andrich Constructing Measures, Mark Wilson Rasch Analysis in the Human Sciences, Boone, Stave, Yale
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