The achievement gap in reading competence: the effect of measurement non-invariance across school types

Rohm, Theresa; Carstensen, Claus H.; Fischer, Luise; Gnambs, Timo

doi:10.1186/s40536-021-00116-2

Large-scale Assessments in Education

An IERI – International Educational Research Institute Journal

Table 3 Results of multilevel structural equation models for longitudinal competence measurement (N = 7276)

From: The achievement gap in reading competence: the effect of measurement non-invariance across school types

	Model 1a			Model 1b				Model 2a			Model 2b
	M	SD	95% PPI	M	SD		95% PPI	M	SD	95% PPI	M	SD	95% PPI
Fixed effects: School level covariates
Grade 5
HS				0.865	0.031		(0.802, 0.920)				0.851	0.032	(0.784, 0.909)
MS				− 0.174	0.039		(− 0.252, − 0.099)				− 0.158	0.040	(− 0.238, − 0.080)
CS				− 0.207	0.039		(− 0.283, − 0.129)				− 0.201	0.040	(− 0.279, − 0.123)
AT				− 0.206	0.041		(− 0.286, − 0.124)				− 0.210	0.041	(− 0.291, − 0.130)
LS				− 0.689	0.030		(− 0.743, − 0.624)				− 0.692	0.030	(− 0.746, − 0.627)
Grade 7
HS				0.844	0.029		(0.783, 0.897)				0.848	0.029	(0.788, 0.902)
MS				− 0.206	0.036		(− 0.278, − 0.137)				− 0.209	0.036	(− 0.277, − 0.138)
CS				− 0.119	0.035		(− 0.188, − 0.049)				− 0.120	0.035	(− 0.187, − 0.050)
AT				− 0.174	0.038		(− 0.247, − 0.100)				− 0.179	0.038	(− 0.252, − 0.102)
LS				− 0.712	0.027		(− 0.762, − 0.656)				− 0.710	0.027	(− 0.762, − 0.655)
Grade 9
HS				0.861	0.031		(0.797, 0.917)				0.856	0.031	(0.790, 0.912)
MS				− 0.214	0.039		(− 0.288, − 0.138)				− 0.206	0.038	(− 0.282, − 0.134)
CS				− 0.111	0.038		(− 0.185, − 0.037)				− 0.110	0.038	(− 0.183, − 0.037)
AT				− 0.221	0.039		(− 0.298, − 0.142)				− 0.232	0.040	(− 0.308, − 0.152)
LS				− 0.682	0.030		(− 0.735, − 0.617)				− 0.676	0.032	(− 0.734, − 0.608)
Variance components of random effects Student level
Grade 5	0.383	0.047	(0.301, 0.491)	0.404	0.056		(0.319, 0.551)	0.353	0.042	(0.286, 0.445)	0.359	0.044	(0.289, 0.458)
Grade 7	0.154	0.019	(0.120, 0.192)	0.160	0.016		(0.135, 0.196)	0.151	0.016	(0.125, 0.187)	0.153	0.012	(0.131, 0.176)
Grade 9	0.277	0.030	(0.223, 0.340)	0.281	0.029		(0.227, 0.338)	0.265	0.025	(0.217, 0.313)	0.275	0.021	(0.237, 0.320)
School Level
Grade 5	0.289	0.044	(0.217, 0.388)	0.078	0.014		(0.057, 0.111)	0.260	0.039	(0.196, 0.349)	0.071	0.012	(0.052, 0.097)
Grade 7	0.104	0.016	(0.079, 0.139)	0.030	0.004		(0.023, 0.040)	0.103	0.014	(0.080, 0.135)	0.029	0.004	(0.023, 0.037)
Grade 9	0.212	0.030	(0.162, 0.280)	0.060	0.009		(0.045, 0.080)	0.202	0.027	(0.157, 0.262)	0.061	0.009	(0.047, 0.080)
Correlations and Covariances between latent factors Student Level	Correlation		Covariance	Correlation		Covariance		Correlation		Covariance	Correlation		Covariance
Grade 5 with Grade 7	0.644		0.172	0.632		0.179		0.631		0.139	0.634		0.139
Grade 5 with Grade 9	0.649		0.223	0.652		0.232		0.662		0.210	0.650		0.185
Grade 7 with Grade 9	0.723		0.163	0.722		0.160		0.722		0.156	0.721		0.131
School Level
Grade 5 with Grade 7	0.000		0.000	0.001		0.000		0.002		0.000	0.002		0.000
Grade 5 with Grade 9	0.000		0.000	0.001		0.000		0.002		0.000	0.003		0.001
Grade 7 with Grade 9	0.001		0.000	0.002		0.000		0.003		0.001	0.000		0.000
PPP	0.027			0.019				0.179			0.196

Note. Standardized results are presented for fixed effects. M = posterior mean. SD = posterior standard deviation, PPI posterior probability interval (2.5 and 97.5 percentile of the posterior distribution). School level covariates: HS high schools, MS middle secondary schools, CS comprehensive schools, AT schools offering all school tracks except high school, LS lower secondary schools, PPP posterior predictive p value. In Models 1a and 1b, no school-type specific item discrimination or item difficulty estimates were accounted for. In Models 2a and 2b, school-type specific item discrimination and item difficulty estimates were taken into account for items exhibiting DIF. Correlations and covariances were reported based on sample statistics

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