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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 \(\hat{\beta }_4\) \(s^2_{\hat{\beta }_{4gz.g}}\) \(\chi ^2\) \(CI_l\) \(CI_u\)  
1 2 3 4
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. 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