Table 23.99 Largest residual correlations for items

These Tables show items (Table 23.99) that may be locally dependent. Specify PRCOMP=R (for score residuals) or PRCOMP=S or Y (for standardized residuals) or PRCOMP=L (for logit residuals) to obtain this Table. Residuals are those parts of the data not explained by the Rasch model. High correlation of residuals for two items (or persons) indicates that they may not be locally independent, either because they duplicate some feature of each other or because they both incorporate some other shared dimension.

 

Table 23.0 Variance components scree plot for items

Table 23.1, 23.11 Principal components plots of item loadings

Table 23.2, 23.12 Item Principal components analysis/contrast of residuals

Table 23.3, 23.13 Item contrast by persons

Table 23.4, 23.14 Item contrast loadings sorted by measure

Table 23.5, 23.15 Item contrast loadings sorted by entry number

Table 23.6, 23.16 Person measures for item clusters in contrast. Cluster Measure Plot for Table 23.6.

Table 23.99 Largest residual correlations for items

 

Missing data are deleted pairwise if both of a pair are missing or PRCOMP=O (for observations), otherwise missing data are replaced by their Rasch expected residuals of 0.

 

     LARGEST STANDARDIZED RESIDUAL CORRELATIONS

     USED TO IDENTIFY DEPENDENT TAP

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

|CORREL-| ENTRY             | ENTRY             |

|  ATION|NUMBER TAP         |NUMBER TAP         |

|-------+-------------------+-------------------|

|  1.00 |    16 1-4-2-3-1-4 |    17 1-4-3-1-2-4 |

|   .52 |     6 3-4-1       |     7 1-4-3-2     |

|   .45 |     7 1-4-3-2     |     8 1-4-2-3     |

|   .28 |     8 1-4-2-3     |     9 1-3-2-4     |

|-------+-------------------+-------------------|

|  -.67 |     8 1-4-2-3     |    14 1-4-2-3-4-1 |

|  -.58 |     7 1-4-3-2     |    14 1-4-2-3-4-1 |

|  -.49 |     6 3-4-1       |    12 1-3-2-4-3   |

|  -.48 |     7 1-4-3-2     |    11 1-3-1-2-4   |

|  -.43 |    15 1-3-2-4-1-3 |    16 1-4-2-3-1-4 |

|  -.43 |    15 1-3-2-4-1-3 |    17 1-4-3-1-2-4 |

|  -.38 |     7 1-4-3-2     |    12 1-3-2-4-3   |

|  -.37 |    12 1-3-2-4-3   |    16 1-4-2-3-1-4 |

|  -.37 |    12 1-3-2-4-3   |    17 1-4-3-1-2-4 |

|  -.32 |     5 2-1-4       |    10 2-4-3-1     |

|  -.31 |     6 3-4-1       |    10 2-4-3-1     |

|  -.29 |    12 1-3-2-4-3   |    13 1-4-3-2-4   |

|  -.29 |    11 1-3-1-2-4   |    13 1-4-3-2-4   |

|  -.28 |     6 3-4-1       |    11 1-3-1-2-4   |

|  -.28 |     4 1-3-4       |     8 1-4-2-3     |

|  -.26 |     4 1-3-4       |     5 2-1-4       |

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

 

Note: Redundant correlations of 1.0 are not printed. If A has a correlation of 1.0 with B, and also with C, assume that B and C also have a correlation of 1.0. After eliminating redundant correlations, the largest correlations are shown in the Table. To see all correlations, output ICORFILE= or PCORFILE=

 

In this Table, high positive residual correlations may indicate local item dependency (LID) between pairs of items or persons. When raw score residual correlations are computed, PRCOMP=R, it corresponds to Wendy Yen's Q3 statistic. is used to detect dependency between pairs of items or persons. Wendy Yen suggests a small positive adjustment to the correlation of size 1/(L-1) where L is the test length.

Yen, W. M. (1984). Effects of local item dependence on the fit and equating performance of the three-parameter logistic model. Applied Psychological Measurement, 8, 125-145.

Yen, W. M. (1993). Scaling performance assessments: Strategies for managing local item dependence. Journal of Educational Measurement, 30, 187-213.

 

Local dependence would be a large positive correlation. Highly locally dependent items (Corr. > +.7), such as items "Q." and "R." share more than half their "random" variance, suggesting that only one of the two items is needed for measurement. But, in classical test theory terms, these items may have the highest point-biserial correlations and so be the "best" items.

 

A large negative correlation indicates the opposite of local dependence, as usually conceptualized. If you look at the item fit tables, item "J." or "R." is likely to have large misfit.

 

Remember that "common variance = correlation^2", so items 10 and 11 only share .40*.40 = 16% of the variance in their residuals in common. 84% of each of their residual variances differ. In this Table we are usually only interested in correlations that approach 1.0 or -1.0, because that may indicate that the pairs of items are duplicative or are dominated by a shared factor.

 

Suggestion: simulate Rasch-fitting data like yours using the Winsteps SIFILE= option. Analyze these data with Winsteps. Compare your correlation range with that of the simulated data.

 


 

Locally-Dependent Items

 

In practical terms, a correlation of r=0.40 is low dependency. The two items only have 0.4*0.4=0.16 of their variance in common. Correlations need to be around 0.7 before we are really concerned about dependency.

 

If you want to create one super-item out of two dependent items, then use Excel (or similar) to add the scored responses on the two item together. Include the super-item in the data file instead of the dependent items. Change CODES= and use ISGROUPS= to model the additional super-item. You will notice a very small reduction in the variances of the measures.

 

Procedure:

1. Winsteps analyze the original data

2. Output the scored responses: "Output file", "RFILE=", responses.xls

3. In Excel, open responses.xls

4. Sum the scored responses of dependent items into a new item

5. Delete the original dependent items

6. Save responses.xls

7. In Winsteps, "Excel/RSSST" menu, "Excel", Import responses.xls

8. Create Winsteps file of the new set of items (you may need edit ISGROUPS=)

9. Analyze the new Winsteps control file

 


 

Item Calibration without  Local Item Dependency (LID)

 

If you need item difficulties from your dataset, but without the effect of LID, here is a procedure:

 

1. Analyze all the dataset with Winsteps with PRCOMP=R to produce the raw residuals.

Output IFILE=iforig.txt as a reference for the original data.

2. Output ICORFILE= for the raw residuals to Excel  in list format

3. Sort the list by correlation

4. Look at the top and bottom of the list. How many inter-item residual correlations are >.02 (or your correlation cut-off value)? (In my empirical dataset about .1%)

Save these item pairings to avoid administering these pairs of item together in CAT or test forms.

5. Make an IDFILE=LID.txt list of one item from each item pair with inter-item residual correlation >.02  (or your correlation cut-off value)

 

6. Reanalyze all the data with Winsteps and IDFILE=LID.txt

7. Output PFILE=pf.txt  - these are the person measures matching only the LID-free items

 

8. Reanalyze all the data with Winsteps without IDFILE=.  Include PAFILE=pf.txt

This forces all the item difficulties and rating-scale structures to conform with the LID-free person measures.

Anchoring the person measures prevents LID from impacting the item difficulties.

9. Output IFILE=if.txt, SFILE=sf.txt - these are the item difficulty and rating-scale threshold values for the item bank or whatever.

 

10. Scatterplot if.txt against iforig.txt. How much is the impact of LID on iforig.txt? Was all this work worth it?


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