# Rasch Models in Health

By: Karl Bang Christensen (editor), Mounir Mesbah (editor), Svend Kreiner (editor)Hardback

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### Description

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.

### About Author

Karl Bang Christensen is Associate professor in the Department of Public Health at the University of Copenhagen, Denmark. His primary research areas include Biostatistics, IRT models, scale validation. Svend Kreiner is professor at the Department of Biostatistics, Institute of Public Health, University. He majored in mathematical statistics in 1971, and worked on and off at the Danish Institute of Educational Research. Mounir Mesbah is Professor of Statistics at Department of Mathematics and Statistics, University of Pierre et Marie Curie, Paris 6. Within the Department of Mathematics and Statistics, he is currently teaching at the Institut de Statistique de l'Universit Pierre et Marie Curie (ISUP) and in charge of Biostatistical options.

### Contents

I Probabilistic models 1 1 The Rasch model for dichotomous items 3 1.1 Introduction 4 1.1.1 original formulation of the model 4 1.1.2 Modern formulations of the model 7 1.2 Psychometric properties 8 1.2.1 Requirements of IRT models 9 1.2.2 Item Characteristic Curves 10 1.2.3 Guttman errors 10 1.2.4 Implicit assumptions 11 1.3 Statistical properties 11 1.3.1 The distribution of the total score 12 1.3.2 Symmetrical polynomials 13 1.3.3 Test characteristic curve (TCC) 14 1.3.4 Partial credit model parametrization of the score distribution 14 1.3.5 Rasch models for subscores 15 1.4 Inference frames 15 1.5 Specic objectivity 18 1.6 Rasch models as graphical models 19 1.7 Summary 20 2 Rasch models for ordered polytomous items 25 2.1 Introduction 26 2.1.1 Example 26 2.1.2 Ordered categories 26 2.1.3 Properties of the Polytomous Rasch model 30 2.1.4 Assumptions 32 2.2 Derivation from the dichotomous model 32 2.3 Distributions derived from Rasch models 37 2.3.1 The score distribution 37 2.3.2 Interpretation of thresholds in partial credit items and Rasch scores 39 2.3.3 Conditional distribution of item responses given the total score 39 2.4 Conclusion 39 2.4.1 Frames of inference for Rasch models 40 II Inference in the Rasch model 45 3 Estimation of item parameters 47 3.1 Introduction 48 3.2 Estimation of item parameters 50 3.2.1 Estimation using the conditional likelihood function 50 3.2.2 Pairwise conditional estimation 52 3.2.3 Marginal likelihood function 54 3.2.4 Extended likelihood function 55 3.2.5 Reduced rank parametrization 56 3.2.6 Parameter estimation in more general Rasch models 56 4 Person parameter estimation and measurement in Rasch models 59 4.1 Introduction and notation 60 4.2 Maximum likelihood estimation of person parameters 61 4.3 Item and test information functions 62 4.4 Weighted likelihood estimation of person parameters 63 4.5 Example 63 4.6 Measurement quality 65 4.6.1 Reliability in classical test theory 66 4.6.2 Reliability in Rasch models 67 4.6.3 Expected measurement precision 69 4.6.4 Targeting 69 III Checking the Rasch model 75 5 Itemt statistics 77 5.1 Introduction 78 5.2 Rasch model residuals 79 5.2.1 Notation 79 5.2.2 Individual response residuals: outts and ints 80 5.2.3 Group residuals 85 5.2.4 Group residuals for analysis of homogeneity 85 5.3 Molenaar's U 87 5.4 Analysis of item { restscore association 88 5.5 Group residuals and analysis of DIF 89 5.6 Kelderman's conditional likelihood ratio test of no DIF 90 5.7 Test for conditional independence in three-way tables 92 5.8 Discussion and recommendations 93 5.8.1 Technical issues 93 5.8.2 What to do when items do not agree with the Rasch model 95 6 Over-all tests of the Rasch model 99 6.1 Introduction 100 6.2 The conditional likelihood ratio test 100 6.3 Example: Diabetes and Eating habits 102 6.4 Other over-all tests of t 104 7 Local dependence 107 7.1 Introduction 108 7.1.1 Reduced rank parametrization model for sub tests 108 7.1.2 Reliability indexes 109 7.2 Local dependence in Rasch Models 109 7.2.1 Response dependence 110 7.3 E ects of response dependence on measurement 111 7.4 Diagnosing and detecting response dependence 114 7.4.1 Item t 114 7.4.2 Item residual correlations 116 7.4.3 Sub tests and reliability 118 7.4.4 Estimating the magnitude of response dependence 118 7.4.5 Illustration 119 7.5 Summary 124 8 Two tests of local independence 131 8.1 Introduction 132 8.2 Kelderman's conditional likelihood ratio test of local independence 132 8.3 Simple conditional independence tests 134 8.4 Discussion and recommendations 136 9 Dimensionality 139 9.1 Introduction 140 9.1.1 Background 140 9.1.2 Multidimensionality in health outcome scales 141 9.1.3 Consequences of multidimensionality 142 9.1.4 Motivating example: the HADS data 142 9.2 Multidimensional models 143 9.2.1 Marginal likelihood function 144 9.2.2 Conditional likelihood function 144 9.3 Diagnostics for detection of multidimensionality 144 9.3.1 Analysis of residuals 145 9.3.2 Observed and expected counts 145 9.3.3 Observed and expected correlations 147 9.3.4 The t-test approach 148 9.3.5 Using reliability estimates as diagnostics of multidimensionality 149 9.3.6 Tests of unidimensionality 150 9.4 Estimating the magnitude of multidimensionality 152 9.5 Implementation 153 9.6 Summary 153 IV Applying the Rasch model 161 10 The polytomous Rasch model and the equating of two instruments163 10.1 Introduction 164 10.2 The polytomous Rasch model 165 10.2.1 Conditional probabilities 166 10.2.2 Conditional estimates of the instrument parameters 167 10.2.3 An illustrative small example 169 10.3 Reparametrization of the thresholds 170 10.3.1 Thresholds reparametrized to two parameters for each instrument170 10.3.2 Thresholds reparametrized with more than two parameters 174 10.3.3 A reparametrization with four parameters 174 10.4 Tests of Fit 176 10.4.1 The conditional test of fit based on cell frequencies 176 10.4.2 The conditional test of fit based on class intervals 177 10.4.3 Graphical test of fit based on total scores 178 10.4.4 Graphical test of fit based on person estimates 179 10.5 Equating procedures 179 10.5.1 Equating using conditioning on total scores 180 10.5.2 Equating through person estimates 180 10.6 Example 180 10.6.1 Person threshold distribution 182 10.6.2 The test of t between the data and the model 182 10.6.3 Further analysis with the parametrization with two moments for each instrument 184 10.6.4 Equated scores based on the parametrization with two moments of the thresholds 190 10.7 Discussion 194 11 A multidimensional latent class Rasch model for the assessment of the Health-related Quality of Life 199 11.1 Introduction 200 11.2 The dataset 202 11.3 The multidimensional latent class Rasch model 205 11.3.1 Model assumptions 205 11.3.2 Maximum likelihood estimation and model selection 208 11.3.3 Software details 209 11.3.4 Concluding remarks about the model 210 11.4 Inference on the correlation between latent traits 211 11.5 Application results 214 12 Analysis of Rater Agreement by Rasch and IRT models 223 12.1 Introduction 224 12.2 An IRT model for modelling inter-rater agreement 224 12.3 Umbilical artery Doppler velocimetry and perinatal mortality 226 12.4 Quantifying the rater agreement in the Rasch model 227 12.4.1 Fixed Effects Approach 227 12.4.2 Random Effects approach and the median odds ratio 229 12.5 Doppler velocimetry and perinatal mortality 231 12.6 Quantifying the rater agreement in the IRT model 232 12.7 Discussion 233 13 From Measurement to Analysis: two steps or latent regression? 241 13.1 Introduction 242 13.2 Likelihood 243 13.2.1 Two-step model 244 13.2.2 Latent regression model 244 13.3 First step: Measurement models 245 13.4 Statistical Validation of Measurement Instrument 248 13.5 Construction of Scores 251 13.6 Two-step method to Analyze Change between Groups 253 13.6.1 Health related Quality of Life and Housing in Europe 253 13.6.2 Use of Surrogate in an Clinical Oncology trial 254 13.7 Latent Regression to Analyze Change between Groups 257 13.8 Conclusion 259 14 Analysis with repeatedly measured binary item response data byad hoc Rasch scales 265 14.1 Introduction 266 14.2 The generalized multilevel Rasch model 268 14.2.1 The multilevel form of the conventional Rasch model for binary items 268 14.2.2 Group comparison and repeated measurement 269 14.2.3 Differential item functioning and local dependence 270 14.3 The analysis of an ad hoc scale 272 14.4 Simulation study 277 14.5 Discussion 283 V Creating, translating, improving Rasch scales 287 15 Writing Health-Related Items for Rasch Models - Patient Reported Outcome Scales for Health Sciences: From Medical Paternalism to Patient Autonomy 289 15.1 Introduction 290 15.1.1 The emergence of the biopsychosocial model of illness 290 15.1.2 Changes in the consultation process in general medicine 291 15.2 The use of patient reported outcome questionnaires 292 15.2.1 Defining PRO constructs 293 15.2.2 Quality requirements for PRO questionnaires 298 15.3 Writing new Health-Related Items for new PRO scales 301 15.3.1 Consideration of measurement issues 302 15.3.2 Questionnaire Development 302 15.4 Selecting PROs for a clinical setting 305 15.5 Conclusions 305 16 Adapting patient-reported outcome measures for use in new lan- guages and cultures 313 16.1 Introduction 314 16.1.1 Background 314 16.1.2 Aim of the adaptation process 315 16.2 Suitability for adaptation 315 16.3 Translation Process 315 16.3.1 Linguistic Issues 316 16.3.2 Conceptual Issues 316 16.3.3 Technical Issues 316 16.4 Translation Methodology 317 16.4.1 Forward-backward translation 317 16.5 Dual-Panel translation 318 16.6 Assessment of psychometric and scaling properties 320 16.6.1 Cognitive debriefing interviews 320 16.6.2 Determining the psychometric properties of the new language version of the measure 322 16.6.3 Practice Guidelines 323 17 Improving items that do not fit the Rasch model 329 17.1 Introduction 330 17.2 The Rasch model and the graphical log linear Rasch model 330 17.3 The scale improvement strategy 332 17.3.1 Choice of modificational action 335 17.3.2 Result of applying the scale improvement strategy 339 17.4 Application of the strategy to the Physical Functioning Scale of the SF-36 340 17.4.1 Results of the GLLRM 340 17.4.2 Results of the subject matter analysis 341 17.4.3 Suggestions according to the strategy 342 17.5 Closing remark 345 VI Analyzing and reporting Rasch models 349 18 Software and program for Rasch Analysis 351 18.1 Introduction 352 18.2 Stand alone softwares packages 352 18.2.1 WINSTEPS 352 18.2.2 RUMM 353 18.2.3 Conquest 353 18.2.4 DIGRAM 354 18.3 Implementations in standard software 355 18.3.1 SAS macro for MML estimation: %ANAQOL 355 18.3.2 SAS Macros based on CML 356 18.3.3 eRm : an R Package 356 18.4 Fitting the Rasch model in SAS 356 18.4.1 Simulation of Rasch dichotomous items 356 18.4.2 MML Estimation of Rasch parameters using Proc NLMIXED 357 18.4.3 MML Estimation of Rasch parameters using Proc GLIMMIX 358 18.4.4 CML Estimation of Rasch parameters using Proc GENMOD 358 18.4.5 JML Estimation of Rasch parameters using Proc LOGISTIC 359 18.4.6 Loglinear Rasch model Estimation of Rasch parameters using Proc Logistic 360 18.4.7 Results 360 19 Reporting a Rasch analysis 363 19.1 Introduction 364 19.1.1 Objectives 364 19.1.2 Factors impacting a Rasch analysis report 364 19.1.3 The role of the substantive theory of the latent variable 366 19.1.4 The frame of reference 367 19.2 Suggested Elements 367 19.2.1 Construct: definition and operationalisation of the latent variable367 19.2.2 Response format and scoring 368 19.2.3 Sample and sampling design 368 19.2.4 Data 369 19.2.5 Measurement model and technical aspects 370 19.2.6 Fit analysis 370 19.2.7 Response scale suitability 371 19.2.8 Item fit assessment 372 19.2.9 Person fit assessment 372 19.2.10 Information 373 19.2.11Validated scale 374 19.2.12 Application and usefulness 375 19.2.13Further issues 376

### Product Details

• ISBN13: 9781848212220
• Format: Hardback
• Number Of Pages: 384
• ID: 9781848212220
• weight: 692
• ISBN10: 1848212224

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