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 +                      |                                            |
2  .1 +             BowBladder                                            |
   .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).

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:

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:

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

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