Dependency and unidimensionality - Repeated measures - Longitudinal studies

 

If one time-point is crucial, that estimate that time-point and anchor the other. For instance, time-point 1 is usually crucial in medical applications when treatement decisions are made. Time-point 2 is crucial in educational situations when pass-fail decisions are made. In survey situations, 1 and 2 are probably equally important. Analyze them altogether. If time-point dependency is a concern than www.rasch.org/rmt/rmt251b.htm - Repeated measures, Mallinson.

 


 

There are three approaches to constructing repetition-bias-free Rasch measures of persons being re-measured.

 

1. The situations are such that the person being re-measured is substantively different from the original person. Any dependency between the pairs of measures of the persons is below the noise level caused by other activities. For instance, when measuring a child entering the educational system at age 6 and then measuring the child again at age 18, any specific dependency between the two measures will be at the noise level. All person records can be analyzed together.

 

2. The first of each person measurement is the benchmark. The persons are measured, and the item difficulties and responses structures estimated. For all subsequent time-points, the items are anchored (IAFILE=) at their values for the first time point, and similarly for the response structures (SAFILE=). Since the measurement framework is anchored, no dependency between the measurements biases the measurements. Since the analysis is anchored, all time-points can be analyzed together in one analysis.

 

3. All measurements of each person are equally important, but it is thought that local dependency between the measurements may bias the measurements or the findings. Then, randomly select the observations at one time-point for each person. Construct a data file with only these observations. Analyze this data set. The random set of person records are measured, and the item difficulties and responses structures estimated. For all other time-points, the items are anchored (IAFILE=) at these "random" values, and similarly for the response structures (SAFILE=). Since the measurement framework is anchored, no dependency between the measurements biases the measurements.  Since the analysis is anchored, all time-points can be analyzed together in one analysis.

 


 

Question: To calibrate item difficulty, I am using data from 75 subjects. Most of the subjects have been tested repeatedly, between two and 9 times each. The reason for this was that I assumed that by training and time (with natural development) the subjects ability was different between different testing situations. Now the referee has asked me to verify that "the requirement of local independence is not breached". How can I check this?

 

Unidimensionality can be violated in many different ways. If you run all known statistical tests to check for violations (even with your subjects tested only once), your data would undoubtedly fail some of them - (for technical details of some of these tests see Fischer & Molenaar, "Rasch Models", chapter 5.) Consequently, the question is not "are my data perfectly unidimensional" - because they aren't. The question becomes "Is the lack of unidimensionality in my data sufficiently large to threaten the validity of my results?"

 


 

Pre-Test - Post-Test dependency (or any two tests with the same persons)

 

1. Stack the data: all pre-test data records. Below them, all post-test data records, in the same sequence.

2. Rasch-analyze these data.

3. Output the IPMATRIX= of standardized residuals to Excel
4. For each item, correlate the pre-test standardized residuals with the post-test standardized residuals.

5. Noticeable positive correlations indicate dependency for those items between pre-test and post-test.

 


 

Imagine that you accidentally entered all your data twice. Then you know there is a lack of local independence. What would happen? Here is what happened when I did this with the dataset exam12lo.txt:

 

Data in once:

 

     SUMMARY OF 35 MEASURED PERSONS

+-----------------------------------------------------------------------------+

|           RAW                          MODEL         INFIT        OUTFIT    |

|          SCORE     COUNT     MEASURE   ERROR      MNSQ   ZSTD   MNSQ   ZSTD |

|-----------------------------------------------------------------------------|

| MEAN      38.2      13.0        -.18     .32      1.01    -.1   1.02     .0 |

| P.SD      10.1        .0         .99     .06       .56    1.4    .57    1.3 |

| MAX.      54.0      13.0        1.44     .59      2.36    2.9   2.28    2.5 |

| MIN.      16.0      13.0       -2.92     .29       .23   -2.9    .24   -2.3 |

|-----------------------------------------------------------------------------|

| REAL RMSE    .36 TRUE SD     .92  SEPARATION  2.55  PERSON RELIABILITY  .87 |

|MODEL RMSE    .33 TRUE SD     .94  SEPARATION  2.85  PERSON RELIABILITY  .89 |

| S.E. OF PERSON MEAN = .17                                                   |

+-----------------------------------------------------------------------------+

PERSON RAW SCORE-TO-MEASURE CORRELATION = .99

CRONBACH ALPHA (KR-20) PERSON RAW SCORE RELIABILITY = .89

 

     SUMMARY OF 13 MEASURED ITEMS

+-----------------------------------------------------------------------------+

|           RAW                          MODEL         INFIT        OUTFIT    |

|          SCORE     COUNT     MEASURE   ERROR      MNSQ   ZSTD   MNSQ   ZSTD |

|-----------------------------------------------------------------------------|

| MEAN     102.9      35.0         .00     .20      1.08    -.2   1.02    -.2 |

| P.SD      23.6        .0         .93     .03       .58    2.3    .53    2.0 |

| MAX.     145.0      35.0        2.45     .31      2.16    3.9   2.42    4.3 |

| MIN.      46.0      35.0       -1.65     .18       .31   -4.2    .39   -3.3 |

|-----------------------------------------------------------------------------|

| REAL RMSE    .24 TRUE SD     .90  SEPARATION  3.81  ITEM   RELIABILITY  .94 |

|MODEL RMSE    .20 TRUE SD     .91  SEPARATION  4.53  ITEM   RELIABILITY  .95 |

| S.E. OF ITEM MEAN = .27                                                     |

+-----------------------------------------------------------------------------+

 

Data in twice:

 

     SUMMARY OF 70 MEASURED PERSONS

+-----------------------------------------------------------------------------+

|           RAW                          MODEL         INFIT        OUTFIT    |

|          SCORE     COUNT     MEASURE   ERROR      MNSQ   ZSTD   MNSQ   ZSTD |

|-----------------------------------------------------------------------------|

| MEAN      38.2      13.0        -.18     .32      1.01    -.1   1.02     .0 |

| P.SD      10.1        .0         .99     .06       .56    1.4    .57    1.3 |

| MAX.      54.0      13.0        1.44     .59      2.36    2.9   2.28    2.5 |

| MIN.      16.0      13.0       -2.92     .29       .23   -2.9    .24   -2.3 |

|-----------------------------------------------------------------------------|

| REAL RMSE    .36 TRUE SD     .92  SEPARATION  2.55  PERSON RELIABILITY  .87 |

|MODEL RMSE    .33 TRUE SD     .94  SEPARATION  2.85  PERSON RELIABILITY  .89 |

| S.E. OF PERSON MEAN = .12                                                   |

+-----------------------------------------------------------------------------+

PERSON RAW SCORE-TO-MEASURE CORRELATION = .99

CRONBACH ALPHA (KR-20) PERSON RAW SCORE RELIABILITY = .89

 

     SUMMARY OF 13 MEASURED ITEMS

+-----------------------------------------------------------------------------+

|           RAW                          MODEL         INFIT        OUTFIT    |

|          SCORE     COUNT     MEASURE   ERROR      MNSQ   ZSTD   MNSQ   ZSTD |

|-----------------------------------------------------------------------------|

| MEAN     205.8      70.0         .00     .14      1.08    -.3   1.02    -.4 |

| P.SD      47.2        .0         .93     .02       .58    3.2    .53    2.9 |

| MAX.     290.0      70.0        2.45     .22      2.16    5.4   2.42    6.1 |

| MIN.      92.0      70.0       -1.65     .13       .31   -6.0    .39   -4.7 |

|-----------------------------------------------------------------------------|

| REAL RMSE    .17 TRUE SD     .92  SEPARATION  5.48  ITEM   RELIABILITY  .97 |

|MODEL RMSE    .14 TRUE SD     .92  SEPARATION  6.48  ITEM   RELIABILITY  .98 |

| S.E. OF ITEM MEAN = .27                                                     |

+-----------------------------------------------------------------------------+

 

There is almost no difference in the person report. Person reliability does not change merely because the sample size becomes larger. Person reliability changes when the person distribution changes.

 

The biggest impact the lack of local independence has in this situation is to make the item standard errors too small. Consequently you might report item results as statistically significant that aren't.

 

So, with your current data, you could adjust the size of the item standard errors to their biggest "worst case" size:

 

Compute k = number of observations in your data / number of observations if each person had only been tested once

 

Adjusted item standard error = reported item standard error * sqrt (k).

 

This would also affect item Reliability computations:

Adjusted item separation coefficient = reported item separation coefficient / sqrt(k)

Adjusted item Reliability (separation index) = Rel. / ( k + Rel. - Rel.*k) =TRUE Sep**2 + Adj. + Adj. Sep.**2)

 

The size of the item mean-square fit statistics does not change, but you would also need to adjust the size of the item t standardized fit statistics (if you use them). This is more complicated. It is probably easiest to read them off the plot from Rasch Measurement Transactions 17:1 shown below.

 

Look at your current item mean-square and significance. Find the point on the plot. Go down to the x-axis. Divide the value there by k. Go to the same mean-square value contour. The "worst case" lower statistical significance value is on the y-axis.

 

Another noticeable aspect of your current data could be that there are misfitting subjects who were tested 9 times, while fitting persons are tested only twice. This would introduce a small distortion into the measurement system. So, arrange all the Tables in fit order, and look at each end, do some subjects appear numerous times near the end of a Table? If so, drop out those subjects and compare item calibrations with and without those subjects. If there is no meaningful difference, then those subjects are merely at the ends of the probabilistic range predicted by the Rasch model.

 


Help for Winsteps Rasch Measurement and Rasch Analysis Software: www.winsteps.com. Author: John Michael Linacre

Facets Rasch measurement software. Buy for $149. & site licenses. Freeware student/evaluation Minifac download
Winsteps Rasch measurement software. Buy for $149. & site licenses. Freeware student/evaluation Ministep download

Rasch Books and Publications: Winsteps and Facets
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 Appliquer le modèle de Rasch: Défis et pistes de solution (Winsteps) E. Dionne, S. Béland
Introduction to Many-Facet Rasch Measurement (Facets), Thomas Eckes Rasch Models for Solving Measurement Problems (Facets), George Engelhard, Jr. & Jue Wang Statistical Analyses for Language Testers (Facets), Rita Green Invariant Measurement with Raters and Rating Scales: Rasch Models for Rater-Mediated Assessments (Facets), George Engelhard, Jr. & Stefanie Wind Aplicação do Modelo de Rasch (Português), de Bond, Trevor G., Fox, Christine M
Exploring Rating Scale Functioning for Survey Research (R, Facets), Stefanie Wind Rasch Measurement: Applications, Khine Winsteps Tutorials - free
Facets Tutorials - free
Many-Facet Rasch Measurement (Facets) - free, J.M. Linacre Fairness, Justice and Language Assessment (Winsteps, Facets), McNamara, Knoch, Fan
Other Rasch-Related Resources: Rasch Measurement YouTube Channel
Rasch Measurement Transactions & Rasch Measurement research papers - free An Introduction to the Rasch Model with Examples in R (eRm, etc.), Debelak, Strobl, Zeigenfuse Rasch Measurement Theory Analysis in R, Wind, Hua Applying the Rasch Model in Social Sciences Using R, Lamprianou Journal of Applied Measurement
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
As an Amazon Associate I earn from qualifying purchases. This does not change what you pay.

facebook Forum: Rasch Measurement Forum to discuss any Rasch-related topic

To receive News Emails about Winsteps and Facets by subscribing to the Winsteps.com email list,
enter your email address here:

I want to Subscribe: & click below
I want to Unsubscribe: & click below

Please set your SPAM filter to accept emails from Winsteps.com
The Winsteps.com email list is only used to email information about Winsteps, Facets and associated Rasch Measurement activities. Your email address is not shared with third-parties. Every email sent from the list includes the option to unsubscribe.

Questions, Suggestions? Want to update Winsteps or Facets? Please email Mike Linacre, author of Winsteps mike@winsteps.com


State-of-the-art : single-user and site licenses : free student/evaluation versions : download immediately : instructional PDFs : user forum : assistance by email : bugs fixed fast : free update eligibility : backwards compatible : money back if not satisfied
 
Rasch, Winsteps, Facets online Tutorials


 

 
Coming Rasch-related Events
May 17 - June 21, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 12 - 14, 2024, Wed.-Fri. 1st Scandinavian Applied Measurement Conference, Kristianstad University, Kristianstad, Sweden http://www.hkr.se/samc2024
June 21 - July 19, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Winsteps), www.statistics.com
Aug. 5 - Aug. 6, 2024, Fri.-Fri. 2024 Inaugural Conference of the Society for the Study of Measurement (Berkeley, CA), Call for Proposals
Aug. 9 - Sept. 6, 2024, Fri.-Fri. On-line workshop: Many-Facet Rasch Measurement (E. Smith, Facets), www.statistics.com
Oct. 4 - Nov. 8, 2024, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
Jan. 17 - Feb. 21, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
May 16 - June 20, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com
June 20 - July 18, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Further Topics (E. Smith, Facets), www.statistics.com
Oct. 3 - Nov. 7, 2025, Fri.-Fri. On-line workshop: Rasch Measurement - Core Topics (E. Smith, Winsteps), www.statistics.com

 

 

Our current URL is www.winsteps.com

Winsteps® is a registered trademark