Assuming measurement invariance of background indicators in international comparative educational achievement studies: a challenge for the interpretation of achievement differences

Wendt, Heike; Kasper, Daniel; Trendtel, Matthias

doi:10.1186/s40536-017-0043-9

Large-scale Assessments in Education

An IERI – International Educational Research Institute Journal

Table 9 Distribution of \(\hat{\beta }_4\) across scaling models and countries, \(\chi ^2\)-value and asymmetric confidence interval (\(CI_l\) lower bound, \(CI_u\) upper bound; countries ordered in descending order of the conditional variance of \(\hat{\beta }_4\) across scaling models given country g)

From: Assuming measurement invariance of background indicators in international comparative educational achievement studies: a challenge for the interpretation of achievement differences

Country	\(\hat{\beta }_4\)				\(s^2_{\hat{\beta }_{4gz.g}}\)	\(\chi ^2\)	\(CI_l\)	\(CI_u\)
Country	1	2	3	4	\(s^2_{\hat{\beta }_{4gz.g}}\)	\(\chi ^2\)	\(CI_l\)	\(CI_u\)
Morocco	58.63	81.35	63.61	95.99	292.81	219,608.17	93.97	4070.68
Honduras	59.33	84.15	61.26	90.95	255.90	191,922.86	82.12	3557.50
Iran, Islamic Rep. of	67.99	89.82	67.52	90.78	169.56	127,170.62	54.41	2357.25
Qatar	158.42	138.73	150.80	138.54	94.71	71031.94	30.39	1316.65
Malta	73.36	63.28	67.27	55.85	53.95	40,465.24	17.31	750.07
Czech Republic	79.55	69.36	78.16	64.74	50.34	37,756.23	16.16	699.85
Romania	59.73	69.87	60.93	72.69	41.54	31,153.02	13.33	577.46
Abu Dhabi, UAE	92.34	82.41	93.06	94.07	29.40	22,047.72	9.43	408.68	\(\blacktriangle\)
Croatia	55.14	47.33	52.95	44.27	25.01	18,753.96	8.02	347.62	\(\blacktriangle\)
Austria	61.39	54.21	59.05	50.74	22.92	17,191.50	7.36	318.66	\(\blacktriangle\)
Slovenia	57.10	49.43	56.50	47.98	22.23	16,669.13	7.13	308.98	\(\blacktriangle\)
Hungary	79.65	84.35	77.74	87.68	20.32	15,242.15	6.52	282.53	\(\blacktriangle\)
Germany	71.93	66.12	70.09	62.12	19.05	14,287.74	6.11	264.84	\(\blacktriangle\)
Singapore	58.35	53.59	56.26	50.03	12.90	9674.75	4.14	179.33	\(\blacktriangle \triangle\)
Italy	51.74	45.44	50.29	44.61	12.43	9322.94	3.99	172.81	\(\blacktriangle \triangle\)
Azerbaijan	42.30	45.40	43.83	49.96	10.96	8218.86	3.52	152.35	\(\blacktriangle \triangle\)
Lithuania	48.05	43.30	46.25	40.65	10.65	7988.31	3.42	148.07	\(\blacktriangle \triangle\)
Slovak Republic	53.89	49.75	54.01	48.02	9.04	6781.11	2.90	125.70	\(\blacktriangle \triangle \bullet\)
Oman	53.24	57.78	52.77	53.87	5.23	3925.58	1.68	72.76	\(\blacktriangle \triangle \bullet \circ\)
Spain	47.14	43.82	44.89	41.87	4.83	3622.23	1.55	67.14	\(\blacktriangle \triangle \bullet \circ\)
Ireland	66.86	64.70	65.21	61.93	4.20	3148.12	1.35	58.35	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft\)
Georgia	47.75	44.71	49.56	48.01	4.12	3086.43	1.32	57.21	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft\)
Saudi Arabia	32.06	32.66	35.73	34.77	3.00	2253.27	0.96	41.77	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Australia	99.67	96.50	99.84	97.20	2.90	2176.08	0.93	40.34	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Poland	49.70	46.89	47.99	46.38	2.16	1619.93	0.69	30.03	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Chinese Taipei	50.96	48.85	49.95	48.21	1.47	1102.95	0.47	20.44	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Sweden	56.36	58.75	56.76	57.87	1.17	880.25	0.38	16.32	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Finland	38.91	38.08	38.31	36.58	0.98	736.06	0.31	13.64	\(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
\(s^2_{\hat{\beta }_{4gz.z}}\)	577.60	511.62	518.80	597.73
\(CL_l\)	361.05	319.80	324.29	373.63
\(CL_u\)	1070.11	947.88	961.18	1107.41

1, scaling model 1; 2, scaling model 2; 3, scaling model 3; 4, scaling model 4; \(\blacktriangle\), country variance is statistically different from Morocco and Honduras; \(\triangle\), country variance is statistically different from Iran, Islamic Rep. of; \(\bullet\), country variance is statistically different from Qatar; \(\circ\), country variance is statistically different from Malta and Czech Republic; \(\blacktriangleleft\), country variance is statistically different from Romania; \(\triangleleft\), country variance is statistically different from Abu Dhabi, UAE

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