01.Invariant Measurement with Explanatory Rasch Models in the Human Sciences
10.What is Rasch Measurement Theory?
13.Explanatory Rasch Models
14.Discussion and Summary
15.2. Measurement, Explanation, and Invariance
16.Invariance and Measurement
21.Explanation and Invariance
24.Structural Equation Modeling
25.Explanatory Item Response Models
26.Guiding Principles for Measurement and Structural Analysis
27.Summary of Invariance and Measurement
28.Summary of Explanation and Invariance
29.Discussion and Summary
30.3. Rasch Measurement Theory and Generalized Linear Mixed Models
31.What Are Generalized Linear Mixed Models?
32.Linear Regression Model
33.Generalized Linear Model
35.Generalized Linear Mixed Models
36.Estimating the Dichotomous Model with a Generalized Linear Mixed Model
38.Steps for Conducting GLMM Analyses
39.Learning Stimulation Scale
40.Discussion and Summary
41.Appendix A: C3-Learning Stimulation Scale-Illustration.R
42.4. Evaluating Measurement Quality for Dichotomous Rasch Models
47.Overall Model-Data Fit
48.General Structural Aspects
52.Item Separation on Latent Variable
53.Structural Aspects: Item Covariates
57.Person Separation on the Latent Variable
58.Structural Aspects: Person Covariates
59.Illustration: Learning Stimulation Scale
63.Overall Model-Data Fit
66.Discussion and Summary
67.5. Explanatory Rasch Models: Linear Logistic Rasch Models and Latent Regression Rasch Models
68.Linear Logistic Rasch Model
70.Latent Regression Rasch Model
71.Discussion and Summary
72.6. Rater-Mediated Assessments with Explanatory Rasch Models
73.Explanatory Rasch Models for Rater-Mediated Assessments
77.Special Considerations for Measurement and Explanation in Rater-Mediated Assessments
78.Data Collection Designs for Rater-Mediated Assessments
79.Interpreting Explanatory Rasch Model Parameters in Rater-Mediated Assessments
80.Model-Data Fit for Explanatory Rasch Models with Raters
81.Specific Rater Errors and Biases
84.Differential Rater Functioning
85.Discussion and Summary
86.7. Instrument Development with Explanatory Rasch Models
87.Five Components for Instrument Development
98.Data Preparation for Transitive Reasoning
99.Summary of Transitive Reasoning Analyses
105.Data Preparation for Biology Exam
106.Summary of Biology Exam Analyses
107.Discussion and Summary
108.8. Rating Scale Analysis for Polytomous Explanatory Rasch Models
109.Estimating Polytomous Rasch Models in the GLMM Framework
112.Category Comparability
113.Rating Scale Analysis with Explanatory Rasch Models
114.Rating Scale Analysis with Item-Explanatory Rasch Models
115.Rating Scale Category Ordering
117.Category Comparability
118.Rating Scale Analysis with Person-Explanatory Rasch Models
119.Rating Scale Category Ordering
121.Category Comparability
122.Discussion and Summary
124.Summary and Implications of Each Chapter
125.Where Do we Go from Here?
134.Fig. 1: Overview of the chapters
135.Fig. 1.1: Generic variable map. Note Categories indicate subgroups of persons (e.g., male/female), while classifications indicate subsets of items (e.g., adult-/child-dominated activities)
136.Fig. 1.2: Three item response functions (IRFs) for Guttman and Rasch Scales with three persons (θ = −2, 0, 2)
137.Fig. 1.3: Variable map for learning stimulation scale. Note Home data (40 homes, 11 items, focus of activities: A = adult-dominated activities, and C = child-dominated activities)
138.Fig. 2.1: Illustration of latent structure analysis. Note Q is yule’s Q (Yule, 1912). This was used by lazarsfeld in his illustration
139.Fig. 2.2: Illustrative item response functions
140.Fig. 2.3: Path analysis (Wright, 1934, p. 161)
141.Fig. 2.4: Factor analysis (Thurstone, 1931, p. 100). Note Pitch discrimination (ability) is the latent variable
142.Fig. 2.5: Structural equation models with the LISREL model (Joreskog, 2007, p. 66). Note See Joreskog (2007) for a description of standard LISREL notation used for variables and arrows used in the model
143.Fig. 2.6: Graphical representations of the Rasch model. Note Adapted from wilson et al. (2008, p. 92)
144.Fig. 2.7: Graphical representation for three explanatory Rasch models. Note Dotted circle indicates random variable, Beta is fixed item effect, Z is random design effect, X is fixed design effect, and theta is random person effect. See text within the chapter for a deeper description
145.Fig. 2.8: Connections between invariance, measurement, and explanation
146.Fig. 3.1: Scatterplots for illustrative data with different dependent variables
147.Fig. 3.2: Simple linear model
148.Fig. 3.3: Wright Map. Note This version of a Wright Map was obtained in R using the EIRM package (Bulut, 2021)
149.Fig. 4.1: Definition of residuals
150.Fig. 4.2: Distribution of residual correlations
151.Fig. 4.3: Principal component analysis of residual correlations. Note Recommended critical value is 2.00—eigenvalues greater than 2.00 suggest additional potential dimensions in the residual correlations
152.Fig. 4.4: Item fit plots for learning stimulation scale
153.Fig. 4.5: Scatterplots of standardized residuals for selected items
154.Fig. 4.6: Scatterplots of standardized residuals for home 24 (misfitting, Outfit MSE = 2.00)
155.Fig. 5.1: Concept map for descriptive and explanatory models (dichotomous and polytomous data)
156.Fig. 5.2: Learning Stimulation Scale: Predicted probabilities for Item-Explanatory Rasch Model (Content Focus: Adult and Child Activities)
157.Fig. 5.3: Variable Map for ATC Scale. Note Variable Map was created using the Facets Program (Linacre, 1989)
158.Fig. 5.4: ATC Scale: Category Response Function (Rating Scale Model). Note The category probability functions were created using the Facets Program (Linacre, 1989)
159.Fig. 5.5: ATC Scale: Predicted probabilities for Item-Explanatory Rasch Model (Focus: Valence). Note −1 = Negative valence, 0 = Neutral valence, 1 = Positive valence
160.Fig. 5.6: ATC Scale: Predicted probabilities for Person-Explanatory Rasch Models (College and Gender)
161.Fig. 6.1: Principles for Invariant Rater-Mediated Assessments
162.Fig. 6.2: Illustrations of scoring designs for rater-mediated assessments
163.Fig. 6.3: Rater-mediated wright map for a descriptive RS-MFRM
164.Fig. 6.4: Examples of constant and non-constant adjustments for rater severity
165.Fig. 6.5: Scatterplots of standardized residuals for individual raters across domains
166.Fig. 6.6: Rater-mediated wright map with rater-specific threshold parameters
167.Fig. 6.7: Rater-mediated wright map with domain-specific threshold parameters
168.Fig. 6.8: Rater-mediated wright map for a person-explanatory PC-MFRM
169.Fig. 6.9: Illustrative category probability curves for a normal rater, central rater, and extreme rater
170.Fig. 6.10: Bivariate relationship between rater threshold SD and raters’ use of central and extreme categories
171.Fig. 6.11: Contrast estimates for rater*subgroup interaction
172.Fig. 7.1: Five components for instrument development
173.Fig. 7.2: Sample construct map for transitive reasoning assessment
174.Fig. 7.3: Scatterplot of item difficulty parameters for transitive reasoning items from the Dichotomous Rasch model
175.Fig. 7.4: Histogram of student achievement parameters from the transitive reasoning assessment based on the Dichotomous Rasch model
176.Fig. 7.5: Wright map showing item and student parameter estimates from the transitive reasoning assessment based on the Dichotomous Rasch model
177.Fig. 7.6: Item covariate parameter estimates from the transitive reasoning assessment based on the LLRM
178.Fig. 7.7: Histogram of student achievement parameter estimates from the transitive reasoning assessment based on the LLRM
179.Fig. 7.8: Wright map showing item and student achievement parameter estimates from the transitive reasoning assessment based on the LLRM
180.Fig. 7.9: Construct map for biology exam
181.Fig. 7.10: Wright map showing item and student parameter estimates from the biology exam based on the Dichotomous Rasch model
182.Fig. 7.11: Effects of cognitive demand and content for biology exam
183.Fig. 8.1: Illustration of recoded polytomous data using the EIRM package
184.Fig. 8.2: Plots of category probabilities for selected CESD items
185.Fig. 9.1: Principles for invariance
186.Table 1.1: Five principles of invariant measurement
187.Table 1.2: Rasch illustration of an ideal-type scale (Rasch, 1960/1980, pp. 65–66)
188.Table 1.3: Illustration of Guttman and Rasch item response patterns
189.Table 1.4: Summary of item calibrations for the learning stimulation scale
190.Table 2.1: Possible response patterns for four dichotomous items
191.Table 2.2: Guttman on the scaling of qualitative data
192.Table 2.3: Lazarsfeld on the translation of concepts into indices or scales
193.Table 2.4: Assumptions underlying mokken models for scaling
194.Table 2.5: Invariance and measurement
195.Table 2.6: Explanation and invariance
196.Table 3.1: Illustrative data for 10 persons
197.Table 3.2: Person by item matrix for learning stimulation scale (40 persons, 11 items)
198.Table 3.3: Wide to long format (5 persons, 4 items)
199.Table 3.4: Item calibrations for the learning stimulation scale
200.Table 4.1: Aspects of model-data fit
201.Table 4.2: Construct focus for learning stimulation scale
202.Table 4.3: Inter-item correlations of residuals: (aQ3)
203.Table 4.4: Summary statistics for dichotomous rasch model: Learning stimulation scale
204.Table 4.5: Summary statistics for items (Learning Stimulation Scale)
205.Table 4.6: Summary statistics for homes (Learning Stimulation Scale)
206.Table 5.2: Estimates of item covariates (Learning Stimulation Scale)
207.Table 5.3: Attitude Toward Censorship scale
208.Table 5.4: ATC scale: Description of the participants
209.Table 5.6: ATC scale: Model-data fit summary (Linear Logistic Rasch Models)
210.Table 5.7: Estimates of item covariates (ATC Scale)
211.Table 5.8: ATC scale: Model-data fit summary (Latent Regression Rasch Models)
212.Table 5.9: Estimates of person covariates (ATC Scale)
213.Table 6.1: Descriptive and explanatory many-facet Rasch Models
214.Table 6.2: Summary of parameter estimates and fit from descriptive MFRM-RS
215.Table 6.3: Summary of rater location estimates and fit from descriptive MFRM-RS
216.Table 6.4: Rater category use for Georgia writing data
217.Table 6.5: Differential rater functioning results for Georgia writing data
218.Table 7.1: Five components for instrument development
219.Table 7.2: Components for instrument development applied to the transitive reasoning assessment
220.Table 7.3: Transitive reasoning item classifications
221.Table 7.4: Components for instrument development applied to the biology exam
222.Table 7.5: Classification of biology Items by content domain and cognitive demand
223.Table 8.1: CESD item stems and classifications
224.Table 8.2: Illustration of polytomous recoding for GLMM estimation
225.Table 8.3: Indicators of rating scale functioning
226.Table 8.4: Average person location estimate (θ) within response categories for the CESD dataset
227.Table 8.5: Threshold ordering tests for the CESD items
228.Table 8.6: Average person location estimates within rating scale categories
229.Table 8.7: Rating scale threshold estimates for item classifications
230.Table 8.8: Average person location estimates within rating scale categories for education level groups
231.Table 8.9: Rating scale threshold estimates for item classifications
