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Table 10 Distribution of random effects \(\varvec{G}\) across scaling models and countries (Part I)

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
\(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\) \(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\) \(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\) \(\hat{\alpha }_{0}\) \(\hat{\alpha }_{1}\)
Azerbaijan \(\hat{\alpha }_{0}\) 5248.25 153.86 5244.57 162.60 5226.53 74.15 5248.85 291.87
\(\hat{\alpha }_{1}\) 153.86 443.62 162.60 549.16 74.15 417.92 291.87 650.26
Australia \(\hat{\alpha }_{0}\) 590.11 24.08 588.47 25.68 592.42 −4.31 587.72 10.40
\(\hat{\alpha }_{1}\) 24.08 475.07 25.68 592.42 −4.31 428.68 10.40 415.55
Austria \(\hat{\alpha }_{0}\) 407.73 −6.63 402.97 −7.05 423.97 −3.26 415.20 −7.77
\(\hat{\alpha }_{1}\) −6.63 51.65 −7.05 35.44 −3.26 47.70 −7.77 30.53
Chinese Taipei \(\hat{\alpha }_{0}\) 176.44 −47.49 176.16 −46.11 171.42 −47.08 166.76 −42.88
\(\hat{\alpha }_{1}\) −47.49 74.59 −46.11 69.47 −47.08 77.28 −42.88 71.36
Croatia \(\hat{\alpha }_{0}\) 212.37 −3.58 210.72 −3.33 211.57 −5.62 211.39 −6.56
\(\hat{\alpha }_{1}\) −3.58 55.29 −3.33 43.16 −5.62 55.03 −6.56 40.84
Czech Republic \(\hat{\alpha }_{0}\) 244.49 −74.17 244.84 −63.68 244.47 −82.20 251.91 −86.89
\(\hat{\alpha }_{1}\) −74.17 344.00 −63.68 260.64 −82.20 361.47 −86.89 280.36
Finland \(\hat{\alpha }_{0}\) 350.47 −49.36 350.52 −48.31 347.46 −41.26 344.18 −34.64
\(\hat{\alpha }_{1}\) −49.36 66.14 −48.31 63.19 −41.26 62.60 −34.64 64.96
Georgia \(\hat{\alpha }_{0}\) 2429.86 −279.28 2432.33 −259.49 2428.40 −270.16 2420.54 −264.22
\(\hat{\alpha }_{1}\) −279.28 376.21 −259.49 329.27 −270.16 370.75 −264.22 312.49
Germany \(\hat{\alpha }_{0}\) 416.96 47.33 414.62 43.03 423.58 36.03 475.08 15.70
\(\hat{\alpha }_{1}\) 47.33 73.62 43.03 60.61 36.03 68.99 15.70 45.10
Honduras \(\hat{\alpha }_{0}\) 2131.60 293.87 2181.93 413.57 2096.88 239.06 2164.52 261.29
\(\hat{\alpha }_{1}\) 293.87 372.70 413.57 852.45 239.06 297.90 261.29 716.51
Hungary \(\hat{\alpha }_{0}\) 578.70 −190.88 578.38 −198.93 584.08 −184.64 672.32 −168.77
\(\hat{\alpha }_{1}\) −190.88 183.69 −198.93 206.38 −184.64 174.82 −168.77 257.79
Iran, Islamic Rep. of \(\hat{\alpha }_{0}\) 1536.11 73.79 1559.85 109.15 1564.47 47.99 1572.80 37.30
\(\hat{\alpha }_{1}\) 73.79 275.83 109.15 514.43 47.99 242.12 37.30 446.03
Ireland \(\hat{\alpha }_{0}\) 643.26 −56.05 643.50 −55.20 650.27 −55.71 664.92 −50.11
\(\hat{\alpha }_{1}\) −56.05 51.56 −55.20 48.60 −55.71 47.17 −50.11 45.44
Italy \(\hat{\alpha }_{0}\) 1346.42 −112.52 1349.74 −98.70 1339.77 −114.57 1339.30 −108.21
\(\hat{\alpha }_{1}\) −112.52 267.16 −98.70 206.35 −114.57 245.99 −108.21 170.05
Lithuania \(\hat{\alpha }_{0}\) 334.13 −63.92 333.83 −56.78 337.52 −69.82 346.11 −72.32
\(\hat{\alpha }_{1}\) −63.92 278.92 −56.78 226.04 −69.82 277.76 −72.32 222.45
Malta \(\hat{\alpha }_{0}\) 468.51 41.21 472.58 30.92 470.84 34.23 518.49 2.89
\(\hat{\alpha }_{1}\) 41.21 29.89 30.92 19.98 34.23 30.08 2.89 8.49
  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