Rasch Models in Health

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