Weighting the data

There are 3 methods of weighting:

1) Models= model weight: Model = ?,?,..., R, model weight

2) Labels= element weight: element number = element label, anchor value, group number, element weight

3) Data= observation weight: R..,

 

These multiply to give a combined weight to each observation.

 

The true reliability of the measures is from the unweighted analysis. Weighting introduces an arbitrariness into the analysis. One solution is to adjust the weights to maintain the unweighted reliability = Ru. The reliability of the weighted analysis, using an initial set of weights, = Rw. We can then scale  the weights using the Spearman-Brown Prophecy Formula:  S = Ru * (1-Rw) / ((1-Ru)*Rw)). Multiply the initial set of weights by S. Then the weighted and unweighted reliabilities should be the same.

 

Weighting sub-tests: Example: Two Cases: A and B. Four aspects: Taste, Touch, Sound, Sight.

Case A Taste weight twice as important as the rest.

Case B Sound weight twice as important as the rest.

 

Labels =

1, Examinees

1-1000

*

2, Case

1=A

2=B

*

3, Aspect

1=Taste

2=Touch

3=Sound

4=Sight

*

Models=

?, 1, 1, MyScale, 2 ; Case A Taste weighted 2

?, 2, 3, MyScale, 2 ; Case B Sound weighted 2

?, ?, ?, MyScale, 1 ; everything else weighted 1

*

Rating scale = MyScale, R9, General ; this rating scale is the same for all models

 

If you want to keep the "reliabilities" and standard errors meaningful then adjust the weights:

 

Original total weights = 2 cases x 4 aspects = 8

New total weights = 2 + 2 + 6 = 10

Weight adjustment to maintain total weight is 8/10.

 

So adjusted weighting is:

Models=

?, 1, 1, MyScale, 1.6 ; Case A Taste

?, 2, 3, MyScale, 1.6 ; Case B Sound

?, ?, ?, MyScale, 0.8 ; everything else

*

 

Replication of a data point: can be specified by R (or another replication character) and the number of replications, for instance:

R3,2,23,6,4 means that the value of 4 was observed in this context 3 times.

Fractional replication permits flexible observation-weighting:

R3.5,2,23,6,4 means that the value of 4 was observed in this context 3.5 times.

 

Weighting observations: We want to give some incorrect answers a smaller penalty than other incorrect answers. There are two ways to do this:

 

1) in the data:

 

3 facets + correct

2,3,4, 1

3 facets + incorrect

2,3,4, 0

3 facets + half-weight incorrect

R0.5, 2,3,4, 0

 

2) with a Models= specification

 

Models =

; 3 facets + dummy indicator facet + correct/incorrect

?,?,?,1,D,1   ; full weight

?,?,?,2,D,0.5   ; half weight

*

Labels=

....

*

4, Weighting, A

1 = Full weight, 0

2 = Half weight, 0

*

 

Data =

 

3 facets + indicator +correct

2,3,4, 1, 1

3 facets + indicator + incorrect

2,3,4, 1, 0

3 facets + indicator +half-weight incorrect

2,3,4,  2, 0


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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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