# Rasch Models in Health

ISBN: 978-1-118-57454-6

Jan 2013, Wiley-ISTE

384 pages

Select type: E-Book

\$132.99

## 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.

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

16.3 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

""This book contains a comprehensive overview of the statistical theory of Rasch models."" (Zentralblatt MATH 2016)
This book contains a comprehensive
overview of the statistical theory of Rasch models.