## Local Dependency, Correlations and Principal Components

```   COG.       PRINCIPAL COMPONENTS (LOGIT RESIDUAL) FACTOR PLOT
F     +-------------------------------------------------------------------+
A  .7 +                   Expression                                      |
C  .6 +            Com  Mem ProbSolve                                     |
T  .5 +           Social     |                                            |
O  .4 +                      |                                            |
R  .3 +                      |                                            |
.2 +                      |                                            |
.0 +-------------------------------------------------------------------+
L -.1 +      Eating          |                                            |
O -.2 +            Groom     |                                            |
A -.3 +                      |   Walk                                     |
D -.4 +                      |  Toileting    Tub         Stairs           |
I -.5 +                  UppBedBexLower                                   |
N -.6 +                      | ToiletTransfer                             |
G     +----------------------+--------------------------------------------+
MOTOR-1                     0                     1                     2
EASY                  ITEM CALIBRATION                         HARD
```

Successful implementation of Rasch measurement requires items that approximate local independence. Statistical independence in data occurs when the value of one datum has no influence on the value of another. Thus, the outcome of a "head" for a coin toss does not increase the probability that the next toss produces a "tail". Local independence specifies that the value of one datum has no influence on another once the underlying variable has been accounted for (conditioned out). Local independence includes, but goes beyond, unidimensionality. Including the same math item twice in a math test would not effect its dimensionality. Yet an examinee would be expected to either succeed or fail on both items, so responses to the two identical items would not be locally independent.

Most commonly encountered Rasch quality-control fit statistics are designed to flag statistically unexpected responses or response patterns by considering the performance of one respondent or one test item at a time. These statistics are very powerful for detecting guessing, carelessness, response sets, social conformity, miskeyed items, data entry errors and the like. They also often hint at problems with local independence, but leave the analyst guessing at exactly where to look.

Smith & Miao (1994) demonstrate that factor analysis is a useful method for identifying multidimensionality in data that has been constructed to be unidimensional. They analyze the matrix of raw responses. But this has drawbacks:

(i) the raw responses are non-linear, so that the first factor only approximates the Rasch dimension.

(ii) the second factor, which might identify the nature of any general lack of local independence, is based on the non-linear residuals left after the extraction of the first factor from non-linear data. These residuals are doubly non-linear.

This suggests an improved method of factor analysis: I. Perform a conventional Rasch analysis. The resulting Rasch dimension is analogous to the first principal component, but now in a linear framework.

II. For each observation, X, compute its expected value according to the measurement model, E, and the model variance of the observed about expected Q. (For dichotomous data, Q is Jacob Bernoulli's binomial variance). The part of the observation not directly explained by the measurement model, the residual, is X-E. This is still in the raw score metric.

III. Linearize the residuals. Each residual indicates how much locally easier or harder that item was than expected (or how much more or less locally competent the examinee). A linear estimate of this local discrepancy is obtained by converting the residual into a logit value. An approximation to linearity is obtained by using the local slope of the logistic ogive (ICC) to convert from the raw score to the logit metric. This slope is the model variance, so that Linear (logit) residual = (X-E)/Q

IV. For each pair of items, compute the Pearson correlation between the linear residuals across all examinees who responded to both items. Potentially locally dependent pairs of items will have high positive correlations (e.g., items which cue each other) or high negative correlations (e.g., items which probe alternative therapeutic techniques).

Largest Logit Residual Correlations
used to identify dependent items
Corr. Item Item
.73
.64
.62
.56
.55
Comprehension
Bed transfer
Problem solving
Upper body dressing
Bowel
Expression
Toilet transfer
Memory
Lower body dressing
-.33 Upper body dressing Expression

The most extreme logit residual correlations for Functional Independence Measure (FIM-SM) data gathered for a sample of right-body (left-brain) stroke victims are shown in the Table. It is reassuring that, in retrospect, these local dependencies are so obvious. Only the negative correlation between upper-body dressing and expression causes us a second thought. Perhaps this is diagnostic of the manner in which left-brain stroke affects functioning. For right-brain stroke, the biggest negative correlation is:

 Corr. Item Item -.40 Toileting Expression

V. Perform a principal components factor analysis of the item correlation matrix. The first factor reported here is really the second factor, because the Rasch dimension is the first factor overall. This second factor identifies the strongest pattern in local dependency among the items as reflected in their correlations. For left-brain stroke, the second factor is:

 Factor Loading Item 2 2 2 2 2 .67 .62 .60 .59 .49 Expression Comprehension Problem solving Memory Social Interaction 2 2 2 2 -.56 -.64 -.53 -.49 Toilet transfer Lower body dressing Bed transfer Upper body dressing

This clearly indicates that there is a pattern among the five cognitive items on the FIM that is in opposition to that among the motor-oriented transfer and dressing items. In the case of the FIM, this second factor is so strong that, for many purposes, it is more clinically useful to obtain two measures from the FIM (a motor and a cognitive one) than a single composite measure.

Subsequent principal components may be useful diagnostically, but reflect weaker patterns in the data. For left-brain stroke, the third factor is indicative of the strong residual correlation between bladder and bowel already reported in the table of correlations.

VI. Tables of numbers are informative, but difficult to synthesize mentally. A picture is worth a thousand words, and a million numbers. A plot of the second factor (first residual factor) against the Rasch dimension depicts clearly the relationship among the FIM items (see Figure). The Rasch dimension (X-axis) is from easy to perform to difficult. The Residual factor (Y-axis) is from motor to cognitive.

Benjamin D. Wright

Smith RM, Miao CY (1994) Assessing unidimensionality for Rasch measurement. Chapter 18 in M. Wilson (Ed.) Objective Measurement: Theory into Practice. Vol. 2. Norwood NJ: Ablex.

Local dependency, correlations and principal components. Wright B.D. … Rasch Measurement Transactions, 1996, 10:3 p. 509-511.

Rasch Publications
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
in Spanish: Análisis de Rasch para todos, Agustín Tristán Mediciones, Posicionamientos y Diagnósticos Competitivos, Juan Ramón Oreja Rodríguez

 Forum Rasch Measurement Forum to discuss any Rasch-related topic

Go to Top of Page
Go to index of all Rasch Measurement Transactions
AERA members: Join the Rasch Measurement SIG and receive the printed version of RMT
Some back issues of RMT are available as bound volumes
Subscribe to Journal of Applied Measurement

Go to Institute for Objective Measurement Home Page. The Rasch Measurement SIG (AERA) thanks the Institute for Objective Measurement for inviting the publication of Rasch Measurement Transactions on the Institute's website, www.rasch.org.

Coming Rasch-related Events
March 21, 2019, Thur. 13th annual meeting of the UK Rasch user group, Cambridge, UK, http://www.cambridgeassessment.org.uk/events/uk-rasch-user-group-2019
April 4 - 8, 2019, Thur.-Mon. NCME annual meeting, Toronto, Canada,https://ncme.connectedcommunity.org/meetings/annual
April 5 - 9, 2019, Fri.-Tue. AERA annual meeting, Toronto, Canada,www.aera.net/Events-Meetings/Annual-Meeting
April 12, 2019, Fri. On-line course: Understanding Rasch Measurement Theory - Master's Level (G. Masters), https://www.acer.org/au/professional-learning/postgraduate/rasch
May 24 - June 21, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - 30, 2019, Wed.-Thu. Measuring and scale construction (with the Rasch Model), University of Manchester, England, https://www.cmist.manchester.ac.uk/study/short/intermediate/measurement-with-the-rasch-model/
June 4 - 7, 2019, Tue.-Fri.In-Person Italian Rasch Analysis Workshop based on RUMM (Fabio La Porta and Serena Caselli; entirely in Italian). Prof David Andrich from Western Australia University will be hosted by the workshop. For enquiries and registration email to workshop.rasch@gmail.com
June 17-19, 2019, Mon.-Wed. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Introduction to Rasch measurement (Trevor Bond, Winsteps), Announcement
June 20-21, 2019, Thurs.-Fri. In-person workshop, Melbourne, Australia: Applying the Rasch Model in the Human Sciences: Advanced Rasch measurement with Facets (Trevor Bond, Facets), Announcement
June 28 - July 26, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
July 2-5, 2019, Tue.-Fri. 2019 International Measurement Confederation (IMEKO) Joint Symposium, St. Petersburg, Russia,https://imeko19-spb.org
July 11-12 & 15-19, 2019, Thu.-Fri. A Course in Rasch Measurement Theory (D.Andrich), University of Western Australia, Perth, Australia, flyer - http://www.education.uwa.edu.au/ppl/courses
Aug 5 - 10, 2019, Mon.-Sat. 6th International Summer School "Applied Psychometrics in Psychology and Education", Institute of Education at HSE University Moscow, Russia.https://ioe.hse.ru/en/announcements/248134963.html
Aug. 9 - Sept. 6, 2019, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Aug. 14 - 16, 2019. Wed.-Fri. An Introduction to Rasch Measurement: Theory and Applications (workshop led by Richard M. Smith) https://www.hkr.se/pmhealth2019rs
August 25-30, 2019, Sun.-Fri. Pacific Rim Objective Measurement Society (PROMS) 2019, Surabaya, Indonesia https://proms.promsociety.org/2019/
Oct. 11 - Nov. 8, 2019, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Nov. 3 - Nov. 4, 2019, Sun.-Mon. International Outcome Measurement Conference, Chicago, IL,http://jampress.org/iomc2019.htm
Jan. 24 - Feb. 21, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 22 - June 19, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 26 - July 24, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 7 - Sept. 4, 2020, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 9 - Nov. 6, 2020, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 25 - July 23, 2021, Fri.-Fri. On-line workshop: Practical Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com