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Table 11 Distribution of random effects \(\varvec{G}\) across scaling models and countries (part II)

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

Country Effect 1 2 3 4
\(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\) \(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\) \(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\) \(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\)
Morocco \(\hat{\alpha }_{0}\) 4358.44 85.63 4433.22 104.05 4314.43 53.49 4311.60 −97.25
\(\hat{\alpha }_{1}\) 85.63 432.09 104.05 959.00 53.49 392.12 −97.25 825.05
Oman \(\hat{\alpha }_{0}\) 2187.73 17.59 2186.26 15.53 2176.12 −38.90 2217.85 −108.31
\(\hat{\alpha }_{1}\) 17.59 319.17 15.53 377.71 −38.90 294.01 −108.31 349.03
Poland \(\hat{\alpha }_{0}\) 322.04 −40.50 322.36 −37.82 321.52 −44.02 317.37 −36.29
\(\hat{\alpha }_{1}\) −40.50 42.59 −37.82 39.27 −44.02 40.80 −36.29 37.58
Qatar \(\hat{\alpha }_{0}\) 1823.03 124.61 1824.27 107.88 1843.10 124.99 2290.76 154.27
\(\hat{\alpha }_{1}\) 124.61 217.05 107.88 169.92 124.99 186.00 154.27 181.57
Romania \(\hat{\alpha }_{0}\) 3132.69 −574.41 3133.08 −652.34 3119.28 −558.05 3138.69 −577.43
\(\hat{\alpha }_{1}\) −574.41 863.19 −652.34 1159.58 −558.05 818.38 −577.43 1098.26
Saudi Arabia \(\hat{\alpha }_{0}\) 3929.55 −146.25 3930.06 −147.93 3891.32 −174.19 3900.55 −180.56
\(\hat{\alpha }_{1}\) −146.25 379.72 −147.93 392.95 −174.19 340.87 −180.56 290.76
Singapore \(\hat{\alpha }_{0}\) 275.71 −9.82 273.85 −10.80 275.53 −16.42 281.66 −19.49
\(\hat{\alpha }_{1}\) −9.82 134.89 −10.80 112.66 −16.42 123.45 −19.49 86.69
Slovak Republic \(\hat{\alpha }_{0}\) 1359.48 −314.15 1353.27 −287.64 1355.48 −318.11 1381.23 −298.18
\(\hat{\alpha }_{1}\) −314.15 308.61 −287.64 256.59 −318.11 304.41 −298.18 250.48
Slovenia \(\hat{\alpha }_{0}\) 234.97 −33.21 234.17 −32.55 232.33 −43.19 234.77 −61.34
\(\hat{\alpha }_{1}\) −33.21 123.54 −32.55 91.74 −43.19 126.89 −61.34 98.47
Spain \(\hat{\alpha }_{0}\) 454.26 −8.45 453.99 −7.85 459.55 −9.76 494.53 −8.72
\(\hat{\alpha }_{1}\) −8.45 94.17 −7.85 80.19 −9.76 81.59 −8.72 74.55
Sweden \(\hat{\alpha }_{0}\) 134.78 −11.58 134.80 −11.66 135.76 −14.16 139.87 −15.65
\(\hat{\alpha }_{1}\) −11.58 159.41 −11.66 172.39 −14.16 167.74 −15.65 163.90
Abu Dhabi, UAE \(\hat{\alpha }_{0}\) 2138.29 46.38 2136.88 43.58 2006.65 28.57 1795.27 −51.16
\(\hat{\alpha }_{1}\) 46.38 210.85 43.58 170.91 28.57 219.04 −51.16 221.58
  1. 1, scaling model 1; 2, scaling model 2; 3, scaling model 3; 4, scaling model 4; \(\hat{\alpha }_{0}\), random intercept; \(\hat{\alpha }_{1}\), random slope of home resource for learning index