Rasch Models in Health

Karl Bang Christensen is Associate Professor at the Department of Biostatistics at the University of Copenhagen in Denmark. With a background in mathematical statistics he has worked mainly within Biostatistics and Epidemiology. Inspired by the issue of measurement in social and health sciences he has published methodological work about Rasch models in journals such as Applied Psychological Measurement, the British Journal of Mathematical and Statistical Psychology and Psychometrika.Svend Kreiner is Professor at the Deptartment of Biostatistics, Institute of Public Health, University of Copenhagen, Denmark. He has for some years tried to combine his interest in Rasch models with his interest in graphical models for categorical data and has developed a family of Rasch-related models that he refers to as graphical loglinear Rasch models in which several of the problems with Rasch models for social and health science data have been resolved. Mounir Mesbah is Professor of Statistics at the Department of Mathematics and Statistics, University Pierre and Marie Curie, Paris, France. Within the Department of Mathematics and Statistics, he is currently teaching at the ISUP (UPMC Institute of Statistics) and is in charge of biostatistical options.

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

• I Probabilistic models 11 The Rasch model for dichotomous items 31.1 Introduction 41.1.1 original formulation of the model  41.1.2 Modern formulations of the model  71.2 Psychometric properties 81.2.1 Requirements of IRT models 91.2.2 Item Characteristic Curves 101.2.3 Guttman errors 101.2.4 Implicit assumptions  111.3 Statistical properties 111.3.1 The distribution of the total score  121.3.2 Symmetrical polynomials 131.3.3 Test characteristic curve (TCC)  141.3.4 Partial credit model parametrization of the score distribution 141.3.5 Rasch models for subscores 151.4 Inference frames  151.5 Specic objectivity 181.6 Rasch models as graphical models 191.7 Summary 202 Rasch models for ordered polytomous items 252.1 Introduction 262.1.1 Example  262.1.2 Ordered categories  262.1.3 Properties of the Polytomous Rasch model   302.1.4 Assumptions 322.2 Derivation from the dichotomous model  322.3 Distributions derived from Rasch models  372.3.1 The score distribution  372.3.2 Interpretation of thresholds in partial credit items and Raschscores  392.3.3 Conditional distribution of item responses given the total score 392.4 Conclusion 392.4.1 Frames of inference for Rasch models    40II Inference in the Rasch model 453 Estimation of item parameters 473.1 Introduction 483.2 Estimation of item parameters  503.2.1 Estimation using the conditional likelihood function  503.2.2 Pairwise conditional estimation  523.2.3 Marginal likelihood function 543.2.4 Extended likelihood function 553.2.5 Reduced rank parametrization 563.2.6 Parameter estimation in more general Rasch models  564 Person parameter estimation and measurement in Rasch models 594.1 Introduction and notation  604.2 Maximum likelihood estimation of person parameters   614.3 Item and test information functions 624.4 Weighted likelihood estimation of person parameters   634.5 Example 634.6 Measurement quality 654.6.1 Reliability in classical test theory  664.6.2 Reliability in Rasch models 674.6.3 Expected measurement precision  694.6.4 Targeting  69III Checking the Rasch model 755 Itemt statistics 775.1 Introduction 785.2 Rasch model residuals 795.2.1 Notation  795.2.2 Individual response residuals: outts and ints   805.2.3 Group residuals 855.2.4 Group residuals for analysis of homogeneity   855.3 Molenaar's U  875.4 Analysis of item { restscore association  885.5 Group residuals and analysis of DIF 895.6 Kelderman's conditional likelihood ratio test of no DIF   905.7 Test for conditional independence in three-way tables   925.8 Discussion and recommendations 935.8.1 Technical issues 935.8.2 What to do when items do not agree with the Rasch model 956 Over-all tests of the Rasch model 996.1 Introduction 1006.2 The conditional likelihood ratio test 1006.3 Example: Diabetes and Eating habits 1026.4 Other over-all tests of t 1047 Local dependence 1077.1 Introduction 1087.1.1 Reduced rank parametrization model for sub tests  1087.1.2 Reliability indexes  1097.2 Local dependence in Rasch Models 1097.2.1 Response dependence  1107.3 Eects of response dependence on measurement    1117.4 Diagnosing and detecting response dependence    1147.4.1 Item t  1147.4.2 Item residual correlations 1167.4.3 Sub tests and reliability 1187.4.4 Estimating the magnitude of response dependence  1187.4.5 Illustration 1197.5 Summary 1248 Two tests of local independence 1318.1 Introduction 1328.2 Kelderman's conditional likelihood ratio test of local independence 1328.3 Simple conditional independence tests 1348.4 Discussion and recommendations 1369 Dimensionality 1399.1 Introduction 1409.1.1 Background 1409.1.2 Multidimensionality in health outcome scales   1419.1.3 Consequences of multidimensionality    1429.1.4 Motivating example: the HADS data    1429.2 Multidimensional models 1439.2.1 Marginal likelihood function 1449.2.2 Conditional likelihood function  1449.3 Diagnostics for detection of multidimensionality    1449.3.1 Analysis of residuals  1459.3.2 Observed and expected counts 1459.3.3 Observed and expected correlations    1479.3.4 The t-test approach  1489.3.5 Using reliability estimates as diagnostics of multidimensionality 1499.3.6 Tests of unidimensionality 1509.4 Estimating the magnitude of multidimensionality   1529.5 Implementation  1539.6 Summary 153IV Applying the Rasch model 16110 The polytomous Rasch model and the equating of two instruments16310.1 Introduction 16410.2 The polytomous Rasch model  16510.2.1 Conditional probabilities 16610.2.2 Conditional estimates of the instrument parameters  16710.2.3 An illustrative small example 16910.3 Reparametrization of the thresholds 17010.3.1 Thresholds reparametrized to two parameters for each instrument17010.3.2 Thresholds reparametrized with more than two parameters 17410.3.3 A reparametrization with four parameters   17410.4 Tests of Fit 17610.4.1 The conditional test of fit based on cell frequencies  17610.4.2 The conditional test of fit based on class intervals  17710.4.3 Graphical test of fit based on total scores    17810.4.4 Graphical test of fit based on person estimates   17910.5 Equating procedures 17910.5.1 Equating using conditioning on total scores   18010.5.2 Equating through person estimates    18010.6 Example 18010.6.1 Person threshold distribution 18210.6.2 The test of t between the data and the model   18210.6.3 Further analysis with the parametrization with two momentsfor each instrument  18410.6.4 Equated scores based on the parametrization with two momentsof the thresholds 19010.7 Discussion 19411 A multidimensional latent class Rasch model for the assessment ofthe Health-related Quality of Life 19911.1 Introduction 20011.2 The dataset 20211.3 The multidimensional latent class Rasch model    20511.3.1 Model assumptions  20511.3.2 Maximum likelihood estimation and model selection  20811.3.3 Software details 20911.3.4 Concluding remarks about the model    21011.4 Inference on the correlation between latent traits   21111.5 Application results 21412 Analysis of Rater Agreement by Rasch and IRT models 22312.1 Introduction 22412.2 An IRT model for modelling inter-rater agreement   22412.3 Umbilical artery Doppler velocimetry and perinatal mortality  22612.4 Quantifying the rater agreement in the Rasch model   22712.4.1 Fixed Effects Approach  22712.4.2 Random Effects approach and the median odds ratio  22912.5 Doppler velocimetry and perinatal mortality    23112.6 Quantifying the rater agreement in the IRT model   23212.7 Discussion 23313 From Measurement to Analysis: two steps or latent regression? 24113.1 Introduction 24213.2 Likelihood 24313.2.1 Two-step model 24413.2.2 Latent regression model 24413.3 First step: Measurement models 24513.4 Statistical Validation of Measurement Instrument   24813.5 Construction of Scores 25113.6 Two-step method to Analyze Change between Groups   25313.6.1 Health related Quality of Life and Housing in Europe  25313.6.2 Use of Surrogate in an Clinical Oncology trial   25413.7 Latent Regression to Analyze Change between Groups   25713.8 Conclusion 25914 Analysis with repeatedly measured binary item response data byadhoc Rasch scales 26514.1 Introduction 26614.2 The generalized multilevel Rasch model  26814.2.1 The multilevel form of the conventional Rasch model for binaryitems 26814.2.2 Group comparison and repeated measurement   26914.2.3 Differential item functioning and local dependence  27014.3 The analysis of an ad hoc scale 27214.4 Simulation study  27714.5 Discussion 283V Creating, translating, improving Rasch scales 28715 Writing Health-Related Items for Rasch Models - Patient ReportedOutcome Scales for Health Sciences: From Medical Paternalism toPatient Autonomy 28915.1 Introduction 29015.1.1 The emergence of the biopsychosocial model of illness  29015.1.2 Changes in the consultation process in general medicine  29115.2 The use of patient reported outcome questionnaires   29215.2.1 Defining PRO constructs 29315.2.2 Quality requirements for PRO questionnaires   29815.3 Writing new Health-Related Items for new PRO scales   30115.3.1 Consideration of measurement issues    30215.3.2 Questionnaire Development 30215.4 Selecting PROs for a clinical setting 30515.5 Conclusions 30516 Adapting patient-reported outcome measures for use in new lan-guages and cultures 31316.1 Introduction 31416.1.1 Background 31416.1.2 Aim of the adaptation process 31516.2 Suitability for adaptation 31516.3 Translation Process 31516.3.1 Linguistic Issues 31616.3.2 Conceptual Issues 31616.3.3 Technical Issues 31616.4 Translation Methodology 31716.4.1 Forward-backward translation 31716.5 Dual-Panel translation 31816.6 Assessment of psychometric and scaling properties   32016.6.1 Cognitive debriefing interviews  32016.6.2 Determining the psychometric properties of the new languageversion of the measure  32216.6.3 Practice Guidelines  32317 Improving items that do not fit the Rasch model 32917.1 Introduction 33017.2 The Rasch model and the graphical log linear Rasch model  33017.3 The scale improvement strategy 33217.3.1 Choice of modificational action  33517.3.2 Result of applying the scale improvement strategy  33917.4 Application of the strategy to the Physical Functioning Scale of theSF-36 34017.4.1 Results of the GLLRM  34017.4.2 Results of the subject matter analysis    34117.4.3 Suggestions according to the strategy    34217.5 Closing remark  345VI Analyzing and reporting Rasch models 34918 Software and program for Rasch Analysis 35118.1 Introduction 35218.2 Stand alone softwares packages 35218.2.1 WINSTEPS 35218.2.2 RUMM  35318.2.3 Conquest  35318.2.4 DIGRAM  35418.3 Implementations in standard software 35518.3.1 SAS macro for MML estimation: %ANAQOL   35518.3.2 SAS Macros based on CML 35618.3.3 eRm : an R Package  35618.4 Fitting the Rasch model in SAS 35618.4.1 Simulation of Rasch dichotomous items    35618.4.2 MML Estimation of Rasch parameters using Proc NLMIXED 35718.4.3 MML Estimation of Rasch parameters using Proc GLIMMIX 35818.4.4 CML Estimation of Rasch parameters using Proc GENMOD 35818.4.5 JML Estimation of Rasch parameters using Proc LOGISTIC 35918.4.6 Loglinear Rasch model Estimation of Rasch parameters usingProc Logistic 36018.4.7 Results  36019 Reporting a Rasch analysis 36319.1 Introduction 36419.1.1 Objectives  36419.1.2 Factors impacting a Rasch analysis report   36419.1.3 The role of the substantive theory of the latent variable  36619.1.4 The frame of reference  36719.2 Suggested Elements 36719.2.1 Construct: definition and operationalisation of the latent variable36719.2.2 Response format and scoring 36819.2.3 Sample and sampling design 36819.2.4 Data 36919.2.5 Measurement model and technical aspects   37019.2.6 Fit analysis 37019.2.7 Response scale suitability 37119.2.8 Item fit assessment  37219.2.9 Person fit assessment  37219.2.10 Information 37319.2.11Validated scale 37419.2.12 Application and usefulness 37519.2.13Further issues 376