Subsets: Nested facet designs: remedying ambiguities

Here is a typical nested design. There are 7 prompts, but candidates only respond to one. From the measurement perspective, we cannot tell whether Mary scored higher than George was because Mary is the more able or because she answered the easier prompt.

 

There are several reasonable measurements solutions to this problem.

 

(A) Random Equivalence of Nested Subsets.

 

If the candidates are assigned to each prompt randomly, then the candidate-prompt subsets can be thought of as randomly equivalent. This implies that we can assert that the subsets have the same mean. This is implemented with group-anchoring.

 


 

A little word-processing of Table 7 can turn the "disjoint subset number" into the "group number":

 

Table 7.3.1  rater Measurement Report  (arranged by MN).

 

--------------------------------------------------------------------------------

| Obsvd   Obsvd  Obsvd  Fair-M|        Model | Infit      Outfit   |           |

| Score   Count Average Avrage|Measure  S.E. |MnSq ZStd  MnSq ZStd | Num rater |

--------------------------------------------------------------------------------

|    124     18     6.9   6.64|  -2.49   .38 |  .4  -1     .4  -1  |  74 Mary  | in subset: 12

|    123     18     6.8   6.58|  -2.35   .31 |  .4  -1     .4  -1  |  83 Fred  | in subset: 14

|    124     18     6.9   6.40|  -2.05   .39 |  .4  -1     .4  -1  | 430 Jose  | in subset: 29

|    121     18     6.7   6.31|  -1.93   .27 |  .4  -2     .4  -2  | 180 Chin  | in subset: 27

 

becomes

 

Labels=

.....

3,Raters,G

 74, Mary, 0, 12

 83, Fred, 0, 14

430, Jose, 0, 29

180, Chin, 0, 27

 

This can be done with the Output files menu, Subset file.

 


Example of a specification file for group-anchoring:

 

Facets=3  ; Prompt, Candidate, Rater

Positive=2  ; Candidates are oriented: larger score=>greater message

Noncenter=1  ; Prompts float

Labels=

1,Prompts

1=Prompt 1

2=Prompt 2

...

7=Prompt 7

*

2,Candidates,G  ; group-anchoring is requested: label groups by their prompt number.

5433=George,0,3 ; George, candidate 5433, is anchored at 0, as part of group 3

7629=Mary,0,2  ; Mary, candidate 7629, is anchored at 0, as part of group 2

3245=Anne,0,4  ; Anne responded to prompt 4

6751=Jose,0,2  ; Jose responded to prompt 2

.....

4765=Zoe,0,5  ; Zoe responded to prompt 7

*

3,Rater   ; The raters rated all prompts. More than one rater per candidate response.

1,Dr. Jones

2,Ms. Smith

3,Sra. Lopez

....

*

Data=

2,3,2,4   Prompt 2, Mary: Ms. Smith gives rating of 4.

......

 

Facet 2, Candidates, is specified with ",G". This implements group-anchoring. The members of each group are allowed to float relative to each other, but their mean measure is constrained to be the mean of their anchor values. In this case, the group means of candidates are set to zero. The differences between the groups are forced into the prompt difficulties. High scoring subsets answered easy prompts. Low scoring groups answered more challenging prompts.

 

(B) Equally difficult or known measures for the prompts.

group-anchoring is not used, but the prompts are anchored.

 

Facets=3  ; Prompt, Candidate, Rater

Positive=2  ; Candidates are oriented: larger score=>greater message

Noncenter=2  ; Candidates float

Labels=

1,Prompts,A

1=Prompt 1,0  ; Prompts all anchored to be the same difficulty

2=Prompt 2,0

...

7=Prompt 7,0

*

2,Candidates

5433=George

7629=Mary

3245=Anne

6751=Jose

.....

4765=Zoe

*

3,Rater   ; The raters rated all prompts. More than one rater per candidate response.

1,Dr. Jones

2,Ms. Smith

3,Sra. Lopez

....

*

Data=

2,3,2,4   Prompt 2, Mary: Ms. Smith gives rating of 4.

......

 

The prompts will all have the same measure of zero. This is equivalent to ignoring the prompt facet for measurement, but each prompt is reported with fit statistics commenting on the consistency with which it is rated.

 

(3) Candidates choose prompt.

In this situation, less able candidates may choose one prompt, "My day at the zoo", but more able ones may choose another, "Einstein's theory of relativity." This requires further information beyond the data.

 

"Virtual equating" is a useful post-hoc technique if the candidate essays and content experts are available. Each candidate-prompt subset is analyzed separately. For each prompt, a set of consistently-rated pieces of work is chosen, located about half-logit apart along the variable. These form substantive "prompt rulers". The experts then slide these rulers relative to each other to align pieces of work representing the same latent ability.

This provides the anchor values for the prompts.

 

(4) Equivalent elements.

If two elements are functionally the same, then the principle of "Virtual equating" applies. A convenient way of assigning two different elements the same element number is the "Target element number" in Labels=.


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