## Construct Deficiency?

Is your construct in trouble? While developing a new version of a widely used achievement test, we noticed several flags that could signal a poorly implemented construct.

```                    Achievement Test
Examinees          Logits  Level      Items
-------------------------------------------------------------
High Achievement   5.0            Difficult Items
X
XX
XXX                     4.0
XXXX
XXXXX
XXXX                    3.0         5-A
XXXXXX                         5    5-B
XXXXXXX                             5-C 5-D
XXXXXXXXX               2.0  -----  5-E 4-A
XXXXXXX                             4-B 4-C
XXXXXXXXX                           4-D          gap in
XXXXXXXXXXX             1.0    4              <= construct
XXXXXXXXXXX                         5-F 3-A      realization
XXXXXXX                             4-E
XXXXX                   0.0  -----  3-B
XXX                                 4-F 3-C
X                              3    3-C 3-D 2-A
XX                     -1.0  -----  3-E 3-F 2-B 1-A
2-C 2-D
X                              2    2-E 2-F
-2.0  -----  1-B 1-C
X                                   1-D 1-E
1    1-F
Low Achievement   -3.0                   Easy Items
-------------------------------------------------------------
```

Look at the simplified person and item map. We analyzed the responses of a sample of convenience, whose performance happened to cover 7 logits (+5.0 to -2.0). The test is intended to place each examinee into one of 5 achievement levels (1-5), with 5 indicating the highest performance. These levels have been defined by contents experts. The test contains 30 items with 6 items intended to be at each level. The items are shown by level number (1-5) and empirical difficulty ranking within level (A-F). The most difficult item, 5-A, is a level 5 item as expected. Item 1-A is about a logit harder than the other level 1 items. Perhaps this was because it is the first item on the test, where examinees could experience start-up problems.

Scanning down the item map, items written to be at higher levels are usually more difficult than items at lower levels. Around levels 3 and 4, however, there are contradictions. The construct is in trouble!

Notice the item disordering between +1.0 and -1.0 logits. The use of levels to define the construct is uncertain in this area. Examination of the experts' definitions of each level provides clues. Each definition is about 100 words long. Level 1 definitions include words like "simple, uncomplicated"; Level 2, "important details"; Level 3, "more details, greater complexity"; Level 4, "advanced, generalized"; Level 5, "more difficult, general principles". Even though levels 3 and 4 have the longest definitions, the item writers have failed to operationalize these levels as strata of difficulty. Now that the test developers have the benefit of empirical item difficulties, they can write more precise, more discriminating and shorter level definitions. This would enable item reclassification and rewriting that would produce a clearer level hierarchy.

This examinee sample happens to contain higher achievers with a mode of about 1.0 logit, i.e., at level 4. Yet there is a gap in the item map exactly at this point. Gaps not only reduce the effectiveness of a test, they also show points at which the construct is poorly defined. The test could be improved by rewriting item 4-F to have difficulty 1.0 logits. This could be accomplished by asking the item writers to write pilot items with predicted difficulties between 4-D and 4-E, a task which will be easier with improved level definitions.

Construct deficiency? Schulz EM. … Rasch Measurement Transactions, 1995, 9:3 p.447

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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: 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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