Linganisha mbinu

	Mfumo wa Rasch wa Ngazi Nyingi	Uthabiti wa Upimaji wa Ngazi Nyingi
Nyanja	Saikometriki	Saikometriki
Familia	Latent structure	Latent structure
Mwaka wa asili	1997	2000s
Mwanzilishi	Adams, Wilson & Wu	Muthén, Asparouhov, and colleagues
Aina	Hierarchical item response model	Measurement model evaluation
Chanzo asilia	Adams, R. J., Wilson, M. & Wu, M. (1997). Multilevel item response models: An approach to errors in variables regression. Journal of Educational and Behavioral Statistics, 22(1), 47–76.	Muthén, B. O., & Asparouhov, T. (2009). Multilevel factor analysis of class and student achievement components. Journal of Educational and Behavioral Statistics, 34(2), 250–270.
Majina mbadala	hierarchical Rasch model, random-effects Rasch model, multilevel IRT Rasch, MRCML model	MLMI, multilevel factorial invariance, cross-level measurement invariance, multilevel CFA invariance
Zinazohusiana	5	3
Muhtasari	The multilevel Rasch model extends the standard Rasch model to data with a nested structure — for example, students within classrooms within schools — by embedding person ability parameters inside a hierarchical linear model. It yields item difficulty estimates on a logit scale while simultaneously partitioning person-ability variance across cluster levels and correcting standard errors for non-independence.	Multilevel measurement invariance testing evaluates whether a latent construct is measured equivalently both within clusters (e.g., individuals within teams) and between clusters (e.g., team-level aggregates). It extends standard measurement invariance procedures to nested data structures commonly encountered in organisational, educational, and cross-cultural research.