Weighting items and persons

There are circumstances in which certain items are to be given more influence in constructing the measures than others. For instance, certain items may be considered critical to the demonstration of competence. Though Winsteps supports several methods, IWEIGHT= is simplest for items, and PWEIGHT= for persons. Another approach is to replicate the data for particular items. This can be done with FORMAT= without changing the data file. Items can also be rescored from say, 0-1 to 0-2, but this makes variable maps difficult to interpret.

 

Unweighted and Weighted analysis: unweighted data is preferable for calibrating the Rasch items. This is because each observation is modeled to contribute one unit of independent statistical information. The effect of weighting is to distort the distribution of independent statistical information in the data. A practical approach is:

 

Step 1. Analyze the data without weighting. Investigate misfit, construct validity,  etc.

 

Step 2. Weight the items. Compare the item calibrations with weighted and unweighted data to identify where there are discrepancies.

 

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.

 

Standard errors and fit statistics: The weights applied to items or persons are used in computing the measure estimates, standard errors and fit statistics. When using significance tests with weighting, normalize the weights so that the total amount of independent statistical information in the data is not over- or under-inflated, i.e., when using PWEIGHT= with an observed sample size of N, multiply all PWEIGHT= values by N / (sum of all weights).

 

The standard is weights = 1.

 

When an item or person is weighted as 2, then the expected score and observed score for the item or person is doubled..

 

When an item or person is weighted as 0, then that person does not influence the JMLE estimates, standard errors or fit statistics of other persons and items, but does have measure, standard error and fit statistics computed on all observations for itself. This is useful for evaluating pilot or off-dimensional items, or measuring idiosyncratic persons.

 

Estimation with weighting

 

Estimation Method

Raw score with weighting

Estimated when

JMLE

 

Person raw score = observations * IWEIGHT

Item raw score = observations * PWEIGHT

Item and Person: Expected score = raw score

CMLE

 

For item estimation: Person raw score = observations with PWEIGHT= occurrences (IWEIGHT= ignored)­ †

For item estimation: Item raw score = observations*PWEIGHT

For person estimation: Person raw score = observations * IWEIGHT

Item: Expected score = raw score

Person (AMLE):: Expected score = raw score

 

† all items are weighted 1 for CMLE item estimation, so that the conditional person scores are convenient integers. 0-weighted items (pilot items) are also weighted 1. But for CMLE-based person estimates (AMLE), item weighting applies, so 0-weighted items are ignored for person estimation.

 

Observation = Xni

Expected value (computed using the Rasch model) = Eni

Accumulated raw score = Accumulated raw score = Xni * IWEIGHT * PWEIGHT

Accumulated expected score = Accumulated expected score = Eni * IWEIGHT * PWEIGHT

Accumulated marginal count for item = Accumulated marginal count for item + IWEIGHT * PWEIGHT

Accumulated marginal count for person = Accumulated marginal count for person + IWEIGHT * PWEIGHT

 

Special rules apply when IWEIGHT=0 or PWEIGHT=0.

IWEIGHT=0 the item totals are incremented by PWEIGHT. The person totals are not incremented.

PWEIGHT=0 the person totals are incremented by IWEIGHT. The item totals are not incremented.

 

JMLE Estimation Accumulated expected score (for each person and each item) = Accumulated raw score (for each person and each item).

 


 

Weight Selection for Tables 23 and 24: On the output tables menu, these are the options for persons and/or items. When IWEIGHT= or PWEIGHT= are used in estimation, reports can be adjusted to reflect those weights or not. Weights of zero are useful for pilot items, variant items or persons with unusual characteristics. These can be reported exclusively or excluded from reports.

 

(1) all items or persons are reported, with their weights (the standard). Tables 23 and 24 are computed as though all weights are 1.

 

(2) items or persons with a weight of 0 are excluded from the reporting. Tables 23 and 24 are computed as though all weights are 1, but zero weights are omitted.

 

(3) only items or persons with a weight of 0 are reported. Tables 23 and 24 are computed only from items or persons with a weight of 0.

 

(4) all items or persons are reported as though they have a weight of 1.

 


 

Example 1: MCQ items are scored 0-1. CR items are scored 0-0.5-1. How can we combine them in one Winsteps analysis.

 

Approach A (recommended). Double the scores of the CR items to 0-1-2, and then IWEIGHT= them by 0.5.

 

Approach B (not recommended). Double the scores of the MCQ items to 0-(1)-2, and double the scores of the CR items to 0-1-2. IWEIGHT= them all by 0.5. This gives the MCQ items a rating scale in which the middle category is not observed, making their ICCs steeper.

 

Example 2: What is the cut-score in a weighted analysis corresponding to a cut-score in an unweighted analysis?

 

Here is an approach:

1.In the unweighted analysis, identify the logit value of the cut-score. Save the person measures to Excel.

2.In the weighted analysis, save the person measures to Excel.

3.Cross-plot the weighted person measures (y-axis) against the unweighted person measures (x-axis)

4.Use the Excel "trend line" function to obtain a reasonable curve through the person-measure points.

5.Identify the value on the y-axis (weighted cut-score measure) corresponding to the value on the x-axis of the unweighted cut-score measure.


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Rasch Books and Publications
Invariant Measurement: Using Rasch Models in the Social, Behavioral, and Health Sciences, 2nd Edn, 2024 George Engelhard, Jr. & Jue Wang Applying the Rasch Model (Winsteps, Facets) 4th Ed., Bond, Yan, Heene Advances in Rasch Analyses in the Human Sciences (Winsteps, Facets) 1st Ed., Boone, Staver Advances in Applications of Rasch Measurement in Science Education, X. Liu & W. J. Boone Rasch Analysis in the Human Sciences (Winsteps) Boone, Staver, Yale
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Rasch Models: Foundations, Recent Developments, and Applications, Fischer & Molenaar Probabilistic Models for Some Intelligence and Attainment Tests, Georg Rasch Rasch Models for Measurement, David Andrich Constructing Measures, Mark Wilson Best Test Design - free, Wright & Stone
Rating Scale Analysis - free, Wright & Masters
Virtual Standard Setting: Setting Cut Scores, Charalambos Kollias Diseño de Mejores Pruebas - free, Spanish Best Test Design A Course in Rasch Measurement Theory, Andrich, Marais Rasch Models in Health, Christensen, Kreiner, Mesba Multivariate and Mixture Distribution Rasch Models, von Davier, Carstensen
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