Rasch Models in Health E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 565

Veröffentlichungsjahr: 2013

Beschreibung

The family of statistical models known as Rasch models started with a simple model for responses to questions in educational tests presented together with a number of related models that the Danish mathematician Georg Rasch referred to as models for measurement. Since the beginning of the 1950s the use of Rasch models has grown and has spread from education to the measurement of health status. This book contains a comprehensive overview of the statistical theory of Rasch models.

Part 1 contains the probabilistic definition of Rasch models, Part 2 describes the estimation of item and person parameters, Part 3 concerns the assessment of the data-model fit of Rasch models, Part 4 contains applications of Rasch models, Part 5 discusses how to develop health-related instruments for Rasch models, and Part 6 describes how to perform Rasch analysis and document results.

Leseprobe

Table of Contents

Preface

PART 1 Probabilistic Models

Chapter 1 The Rasch Model for Dichotomous Items

1.1. Introduction

1.2. Item characteristic curves

1.3. Guttman errors

1.4. Test characteristic curve

1.5. Implicit assumptions

1.6. Statistical properties

1.7. Inference frames

1.8. Specific objectivity

1.9. Rasch models as graphical models

1.10. Summary

1.11. Bibliography

Chapter 2 Rasch Models for Ordered Polytomous Items

2.1. Introduction

2.2. Derivation from the dichotomous model

2.3. Distributions derived from Rasch models

2.4. Bibliography

PART 2 Inference in the Rasch Model

Chapter 3 Estimation of Item Parameters

3.1. Introduction

3.2. Estimation of item parameters

3.3. Example

3.4. Bibliography

Chapter 4 Person Parameter Estimation and Measurement in Rasch Models

4.1. Introduction and notation

4.2. Maximum likelihood estimation of person parameters

4.3. Item and test information functions

4.4. Weighted likelihood estimation of person parameters

4.5. Example

4.6. Measurement quality

4.7. Bibliography

PART 3 Checking the Rasch Model

Chapter 5 Item Fit Statistics

5.1. Introduction

5.2. Rasch model residuals

5.3. Molenaar’s U

5.4. Analysis of item-restscore association

5.5. Group residuals and analysis of DIF

5.6. Kelderman’s conditional likelihood ratio test of no DIF

5.7. Test for conditional independence in three-way tables

5.8. Discussion and recommendations

5.9. Bibliography

Chapter 6 Overall Tests of the Rasch Model

6.1.Introduction

6.2. The conditional likelihood ratio test

6.3. Other overall tests of fit

6.4. Bibliography

Chapter 7 Local Dependence

7.1. Introduction

7.2. Local dependence in Rasch models

7.3. Effects of response dependence on measurement

7.4. Diagnosing and detecting response dependence

7.5. Summary

7.6. Bibliography

Chapter 8 Two Tests of Local Independence

8.1. Introduction

8.2. Kelderman’s conditional likelihood ratio test of local independence

8.3. Simple conditional independence tests

8.4. Discussion and recommendations

8.5. Bibliography

Chapter 9 Dimensionality

9.1. Introduction

9.2. Multidimensional models

9.3. Diagnostics for detection of multidimensionality

9.4. Tests of unidimensionality

9.5. Estimating the magnitude of multidimensionality

9.6. Implementation

9.7. Summary

9.8. Bibliography

PART 4 Applying the Rasch Model

Chapter 10 The Polytomous Rasch Model and the Equating of Two Instruments

10.1. Introduction

10.2. The Polytomous Rasch Model

10.3. Reparameterization ofthe thresholds

10.4. Tests of fit

10.5. Equating procedures

10.6. Example

10.7. Discussion

10.8. Bibliography

Chapter 11 A Multidimensional Latent Class Rasch Model for the Assessment of the Health-Related Quality of Life

11.1. Introduction

11.2. The data set

11.3. The multidimensional latent class Rasch model

11.4. Correlation between latent traits

11.5. Application results

11.6. Acknowledgments

11.7. Bibliography

Chapter 12 Analysis of Rater Agreement by Rasch and IRT Models

12.1. Introduction

12.2. An IRT model for modeling inter-rater agreement

12.3. Umbilical artery Doppler velocimetry and perinatal mortality

12.4. Quantifying the rater agreement in the Rasch model

12.5. Doppler velocimetry and perinatal mortality

12.6. Quantifying the rater agreement in the IRT model

12.7. Discussion

12.8. Bibliography

Chapter 13 From Measurement to Analysis

13.1. Introduction

13.2. Likelihood

13.3. First step: measurement models

13.4. Statistical validation of measurement instrument

13.5. Construction of scores

13.6. Two-step method to analyze change between groups

13.7. Latent regression to analyze change between groups

13.8. Conclusion

13.9. Bibliography

Chapter 14 Analysis with Repeatedly Measured Binary Item Response Data by Ad Hoc Rasch Scales

14.1. Introduction

14.2. The generalized multilevel Rasch model

14.3. The analysis of an ad hoc scale

14.4. Simulation study

14.5. Discussion

14.6. Bibliography

PART 5 Creating, Translating and Improving Rasch Scales

Chapter 15 Writing Health-Related Items for Rasch Models – Patient-Reported Outcome Scales for Health Sciences: From Medical Paternalism to Patient Autonomy

15.1. Introduction

15.2. The use of patient-reported outcome questionnaires

15.3. Writing new health-related items for new PRO scales

15.4. Selecting PROs for a clinical setting

15.5. Conclusions

15.6. Bibliography

Chapter 16 Adapting Patient-Reported Outcome Measures for Use in New Languages and Cultures

16.1. Introduction

16.2. Suitability for adaptation

16.3. Translation process

16.4. Translation methodology

16.5. Dual-panel translation

16.6. Assessment of psychometric and scaling properties

16.7. Bibliography

Chapter 17 Improving Items That Do Not Fit the Rasch Model

17.1. Introduction

17.2. The RM and the graphical log-linear RM

17.3. The scale improvement strategy

17.4. Application of the strategy to the Physical Functioning Scale

17.5. Closing remark

17.6. Bibliography

PART 6 Analyzing and Reporting Rasch Models

Chapter 18 Software for Rasch Analysis

18.1. Introduction

18.2. Stand alone softwares packages

18.3. Implementations in standard software

18.4. Fitting the Rasch model in SAS

18.5. Bibliography

Chapter 19 Reporting a Rasch Analysis

19.1. Introduction

19.2. Suggested elements

19.3. Bibliography

List of Authors

Index

First published 2013 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Karl Bang Christensen, Svend Kreiner and Mounir Mesbah to be identified as the author of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2012950096

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN: 978-1-84821-222-0

Preface

The family of statistical models known as Rasch models started with a simple model for responses to questions in educational tests presented together with a number of related models that the Danish mathematician Georg Rasch referred to as models for measurement. Since the beginning of the 1950s the use of Rasch models has grown and has spread from education to the measurement of health status. This book contains a comprehensive overview of the statistical theory of Rasch models.

Because of the seminal work of Georg Rasch [RAS 60] a large number of research papers discussing and using the model have been published. The views taken of the model are somewhat different. Some regard it as a measurement model and focus on the special features of measurement by items from Rasch models. Other publications see the Rasch model as a special case of the more general class of statistical models known as item response theory (IRT) models [VAN 97]. And, finally, some regard the Rasch model as a statistical model and focus on statistical inference using these models.

The statistical point of view is taken in this book, but it is important to stress that we see no real conflict between the different ways that the model is regarded. The Rasch model is one of the several measurement models defined by Rasch [RAS 60, RAS 61] and is, of course, also an IRT model. And even if measurement is the only concern, we need observed data and statistical estimates of person parameters to calculate the measures.

The statistical point of view is thus unavoidable. From this point of view, the sufficiency of the raw score is crucial and, following in the footsteps of Georg Rasch and his student Erling B. Andersen, we focus on methods depending on the conditional distribution of item responses given the raw score. The relationship between Rasch models and the family of multivariate models called graphical models [WHI 90, LAU 96] is also highlighted because this relationship enables analysis and modeling of properties like local dependence and non-differential item functioning in a very transparent way.

The book is structured as follows: Part I contains the probabilistic definition of Rasch models; Part II describes estimation of item and person parameters; Part III is about the assessment of the data-model fit of Rasch models; Part IV contains applications of Rasch models; Part V discusses how to develop health-related instruments for Rasch models; and Part VI describes how to perform Rasch analysis and document results.

The focus on the Rasch model as a statistical model with a latent variable means that little will be said about other IRT models, such as the two parameter logistic (2PL) model and the graded response model. This does not reflect a strong “religious” belief, that the Rasch model is the only interesting and useful IRT or measurement model, but only reflects our choice of a point of view for this book.

The book owes a lot to discussions at a series of workshops on Rasch models held in Stockholm (Sweden, 2001), Leeds (UK, 2002), Perth (Australia, 2003), Skagen (Denmark, 2005), Vannes (France, 2006), Bled (Slovenia, 2007), Perth (Australia, 2008 and 2012), Copenhagen (Denmark, 2010) and Dubrovnik (Croatia, 2011). Many of the authors have taken part and have helped create an atmosphere where topics relating to the Rasch model could be discussed in an open, friendly and productive manner.

The participants do not agree on everything and do not share all the points of views expressed. However, everyone agrees on the importance of Rasch’s contributions to measurement and statistics, and it is fair to say that this book would not exist if it had not been for these workshops.

Karl Bang CHRISTENSEN, Svend KREINER and Mounir MESBAHCopenhagen, November 2012

Bibliography

[LAU 96] LAURITZEN S. Graphical Models, Clarendon Press, 1996.

[RAS 60] RASCH G., Probabilistic Models for Some Intelligence and Attainment Tests, Danish National Institute for Educational Research, Copenhagen, 1960.

[RAS 61] RASCH G., “On general laws and the meaning of measurement in psychology”, Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, University of California Press, Berkeley, CA, pp. 321–334, 1961.

[VAN 97] VAN DER LINDEN W.J., HAMBLETON R.K., Handbook of Modern Item Response Theory, Springer-Verlag, New York, NY, 1997.

[WHI 90] WHITTAKER J., Graphical Models in Applied Multivariate Statistics, Wiley, Chichester, UK, 1990.

PART 1

Probabilistic Models

Introduction

This part introduces the models that are analyzed in the book. The Rasch model was originally formulated by Georg Rasch for dichotomous items [RAS 60]. This model is described in Chapter 1, where different parameterizations are also introduced. The sources of polytomous Rasch models are less clear. Georg Rasch formulated a quite general polytomous model where each item measures several latent variables [RAS 61]. However, this model has seen little use. Later, several authors [AND 77, AND 78, MAS 82] formulated models where items with more than two response categories measure a single underlying latent variable.

Bibliography

[AND 77] ANDERSEN E.B., “Sufficient statistics and latent trait models”, Psychometrika, vol. 42, pp. 69–81, 1977.

[AND 78] ANDRICH D., “A rating formulation for ordered response categories”, Psychometrika, vol. 43, pp. 561–573, 1978.

[MAS 82] MASTERS G.N., “A Rasch model for partial credit scoring”, Psychometrika, vol. 47, pp. 149–174, 1982.

[RAS 60] RASCH G., Probabilistic Models for Some Intelligence and Attainment Tests, Danish National Institute for Educational Research, Copenhagen, 1960.

[RAS 61] RASCH G., “On general laws and the meaning of measurement in psychology”, Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, IV, University of California Press, Berkeley, CA, pp. 321–334, 1961.

Chapter 1

The Rasch Model for Dichotomous Items

1.1. Introduction

The family of statistical models, which is known as Rasch models, was first introduced with a simple model for responses to dichotomous items (questions) in educational tests [RAS 60]. It was presented together with a number of related models that the Danish mathematician Georg Rasch called models for measurement. Since then, the family of Rasch models has grown to encompass a number of statistical models.

1.1.1. Original formulation of the model

All Rasch models share a number of fundamental properties, and we introduce this book with a brief recapitulation of the very first Rasch model: the Rasch model for dichotomous items. This model was developed during the 1950s when Georg Rasch got involved in educational research. The model describes responses to a number of items by a number of persons assuming that responses are stochastically independent, depending on unknown items and person parameters. In Rasch’s original conception of the model (see Figure 1.1), the structure of the model was multiplicative. In this model, the probability of a positive response to an item depends on a person parameterξ and an item parameterδ in such a way that the probability of a positive response to an item depends on the product of the person parameter and the item parameter.

Figure 1.1.The Rasch model 1952/1953

If we refer to the response of person v to item i as Xvi and code a positive response as 1 and a negative response as 0, the Rasch model asserts that

[1.1]

where both parameters are non-negative real numbers. It follows from [1.1] that

[1.2]

The interpretation of the parameters in this model is straightforward: the probability of a positive response increases as the parameters increase toward infinity. In educational testing, the person parameter represents the ability of the student and the item parameter represents the easiness of the item: the better the ability and the easier the item, the larger the probability of a correct response to the item. In health sciences, the person parameter could represent the level of depression whereas the item parameters could represent the risk of experiencing certain symptoms relating to depression.

EXAMPLE 1.1.– Consider the following dichotomous items intended to measure depression:

1) Did you have sleep disturbance every day for a period of two weeks or more?

2) Did you have a loss or decrease in activities every day for a period of two weeks or more?

3) Did you have low self-esteem every day for a period of two weeks or more?

4) Did you have decreased appetite every day for a period of two weeks or more?

Items like these appear in several questionnaires. According to the Rasch model, responses to these items depend on the level of depression measured by the ξ parameter and on four item parameters δ1–δ4. In a recent study, the item parameters were found to be 2.57, 1.57, 0.52 and 0.48, respectively [FRE 09, MES 09]. The interpretation of these numbers is that sleep disturbance is the most common and loss of appetite is the least common of the four symptoms. To better understand the role of the item parameters, we have to look at the relationships between the probabilities of positive responses to two questions. This is shown in Table 1.1, where it can be seen that the ratio between the two item parameters is the odds ratio (OR) comparing the odds of encountering the symptoms described by the items irrespective of the level of depression ξ of the persons. This interpretation should be familiar to persons with a working knowledge of epidemiological methods. According to the Rasch model, the level of depression does not modify the relative risk of the symptoms. In the theory of Rasch models, this is sometimes called no item-trait interaction.

Table 1.1.Response probabilities for two items when the person parameter is ξv