According to the Rasch model, extreme or "perfect" scores (i.e., 0 or minimum possible or maximum possible or perfect scores) imply an infinite estimate for the parameter of the corresponding element. Accordingly they are flagged as "Minimum" or "Maximum" on the output. A reasonable finite estimate is also supplied for the measure corresponding to each extreme score by adjusting extreme scores by a fraction of a score point. The fraction is specified by the first value following Xtreme=. The standard value of 0.3 tends to give a reasonable logit measure corresponding to extreme scores. A similar adjustment procedure can be applied to extreme scores in the bias estimation procedure using the second value following Xtreme=. Here, the standard adjustment is 0.5, because scores within 0.5 score points of their expected values are as unbiased as can be observed. 0.9 is about the highest reasonable value for the extreme score adjustment, giving the most central extreme measures.
If the observations are weighted in Models=, Labels= or Data=, then the extreme-score adjustment is multiplied by the smallest weight applied to an observation.
Example 1: Produce estimates for extreme scores based on increasing all 0 scores by 0.4 score points and decreasing all maximum scores by that same amount of 0.4 score points:
Xtreme = .4
Example 2: You want to produce more extreme measures for extreme scores by assigning a 0.2 score point correction to extreme scores in the main analysis, but a correction of .4 score points in the bias analysis:
Xtreme = .2, .4
Example 3: You want measures for extreme scores to produce extreme Fair Scores:
Xtreme = .01
Table 7.1.1 Students Measurement Report (arranged by mN).
+----------------------------------------------------------
| Total Total Obsvd Fair(M)| Model | Infit
| Score Count Average Average|Measure S.E. | MnSq ZStd
|--------------------------------+--------------+----------
| 60 10 6.00 6.00 |( 12.22 9.80)|Maximum
| 58 10 5.80 5.71 | 6.70 .80 | .74 -.4
| 56 10 5.60 5.47 | 5.70 .65 | .82 -.6
Extreme scores for elements
"Xtreme= fraction" controls the estimation of measures for extreme scores. Facets provides several options for processing extreme scores. These scores are minimum possible (usually 0) or maximum possible scores attained by some element. Extreme scores are problematic in measurement construction, because they imply out-of-bounds, infinite measures.
Facets first drops all responses for which two or more elements have extreme scores. Responses for which only one element has an extreme score are counted only for that element. Responses for which no element has an extreme score are counted for all elements. Facets reports elements with extreme scores as follows:
1) Ignore the fact that the measures are extreme.
This is done when analyzed responses are reported. In Table 7.1 etc, elements with extreme scores are included in the count, score, average score, and point-biserial bar-charts. In Table 8, elements with extreme scores are included in the count, score, average score, and point-biserial summary statistics.
2) Do not report (drop) elements with extreme scores.
In Table 7.1 etc, elements with extreme scores are omitted from the "Logit", "Logit S.E." and Fit bar-charts. In Table 8, elements with extreme scores are omitted from the "Logit", "Logit S.E." and Fit summary statistics.
3) Report elements with extreme scores at the extremes of the measurement continuum.
This is done for Table 7, the vertical rulers. Elements with extreme scores are shown at the extreme top, or bottom, of the column for their facets.
4) Introduce a fractional point score adjustment to the extreme score (of size specified by "Xtreme=fraction"). Then estimate a finite measure for the amended score.
For example, you may feel that even the worst performance would be worth 0.3 score points. This is done with "Xtreme = 0.3" (the standard). Extreme scores will be given measures, but these measures will not alter the measures of other elements, affect the summary statistics, or be used in centering or bias estimation. These extreme measures and their putative standard errors are listed in Table 8.
Extreme scores for interaction terms.
"Xtreme= fraction" controls bias measures. Measures for interaction bias on extreme scores are problematical. How much is Item 1 biased in favor of the boys, if all boys succeeded on it, but only half the girls?
Facets surmounts this hurdle by introducing a fractional point score adjustment to the off-target extreme score (of size specified by "Xtreme= ,fraction"). This enables the estimation of a finite measure for the amended score. The standard fraction is 0.5 score-points because observed scores are always integers, so that amounts less than this can not be observed.
Measures and standard errors for all bias terms (extreme or not) are included in the summary statistics for Table 13.
