Applying The Rasch Rating-Scale Model
To Set Multiple Cut-Offs

Setting multiple cut-offs in educational contexts where score points are required to indicate transitions from one ability level to the next is a challenging issue. One such context is the Common European Framework of Reference for Languages (CEF). The CEF is a six-point proficiency scale with descriptors for each band in the form of 'can-do statements'. The levels on the CEF are A1, A2, B1, B2, C1 and C2, A1 being the lowest level and C2 the highest. Linking tests to the CEF and validating the claims of links to the CEF is an important issue that European language testers are wrestling with. In this paper a methodology for linking tests to the CEF or any other similar proficiency scale is suggested.

Procedures
The material was a reading comprehension test comprising 75 items. A group of 20 raters who were familiar with the CEF proficiency scale and its descriptors were asked to rate the 75 items, indicating what minimum ability level on the six-point CEF proficiency scale a student should exhibit to get each item right. The items were rated from 1 to 6, 1 indicating the lowest CEF level and 6 the highest. The items were then calibrated on the basis of these ratings using Andrich's (1978) rating scale model. The next step was to calibrate the items on the basis of actual student performances. The following tables show the descriptive statistics for the item measures based on the two analyses.
Table 1. Item measure summary.
Item measure summary statisticsRater-based analysisStudent-based analysis
N7575
Mean.26-.00
Median.16-.05
Std. Deviation2.721.80
Range10.408.42
Minimum-5.97-3.28
Maximum4.435.14
Reference difficulty0.00item mean

Figure 1. Cross-plot of item measures from rater-based and student-based analyzes

The cross plot of the item calibrations from the two analyses is shown in Figure 1. The two sets of item calibrations, i.e., those based on raters and those based on the students' performances, correlated at 0.80. It can be seen that there are a few conspicuous outliers, and there may be two trendlines, one for the upper half of the plot, and the other for the lower, but the overall pattern is clear. The slope of an empirical joint "best fit" line (through the two means, and two means+1 S.D.) is 0.66. The mean difference between the average item measures is 0.26 logits. Thus the person measures were converted into the rater frame-of-reference by means of the equating formula:

M2 = (M1 -mean(1))*SD(2)/SD(1) + mean(2)

i.e., Adjusted measure = (measure-.00)/0.66 + 0.26

Rater analysis equated with student analysis
When the person measures are equated for both the intercept and the slope of the trendline, they are mapped into the framework of the rater-based analysis. Table 2 shows the descriptive statistics for the 160 person measures in three different modes: (1) unequated, (2) equated with the rater-based analysis by correction for intercept only, and (3) equated with the rater-based analysis by correction for both intercept and slope.

Table 2: Descriptive statistics for 160 persons in three different modes.
 Person Measures
Unequated
Person Measures
Equated for Intercept
Person Measures
Equated for Intercept & Slope
N160160160
Mean.07.33.37
Median.26.52.65
Mode-1.51-1.25-2.02
Std. Deviation1.341.342.03
Range5.785.788.73
Minimum-3.28-3.02-4.70
Maximum2.502.764.04

Setting cut-points
Half-score-point thresholds on the reference item at zero logits in the rater analysis set the cut-off scores. Since the person measures have been brought to the framework of the rater-based analysis these half-score-point thresholds are directly applicable to the person measures after equating. The expected score ICC for the reference item is shown in Figure 2. The half-score point intervals are indicated on the latent variable. The locations of the 6 proficiency levels are indicated by their codes, A1, etc.

Figure 2: Expected score ICC: means.

Cross validation
In order to check the accuracy of the link, a small sample of students at different locations along the ability scale can be selected. It is better to select students whose ability measures on the test (after being equated with the raterbased analysis) fall well in the middle of the bands and students who fall very close to the transition points. Then the group of expert raters who rated the items can interview these students and try to rate them on the proficiency scale, they rated the items on. Agreements between rater judgments of where the students fall on the proficiency scale and students' measures, which empirically put them at certain levels on the scale, confirm the equating. Disagreements can be examined in case they indicate the need for slight adjustments to the criterion levels thresholds.

Purya Baghaei


Applying The Rasch Rating-Scale Model To Set Multiple Cut-Offs, Baghaea, P. … Rasch Measurement Transactions, 2007, 20:4 p. 1075-6



Rasch Publications
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: Using Rasch Models in the Social, Behavioral, and Health Sciences, George Engelhard, Jr. 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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