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Table 8 Distribution of \(\hat{\beta }_3\) 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 \(\hat{\beta }_3\) 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 \(\hat{\beta }_3\) \(s^2_{\hat{\beta }_{3gz.g}}\) \(\chi ^2\) \(CI_l\) \(CI_u\)  
1 2 3 4
Iran, Islamic Rep. of 20.77 26.94 22.35 30.60 19.96 14,972.88 6.41 277.54  
Malta 37.08 32.20 35.90 27.47 18.72 14,038.53 6.01 260.23  
Slovenia 42.95 36.89 43.05 35.31 16.29 12,218.83 5.23 226.49  
Czech Republic 38.89 33.79 38.75 31.75 12.90 9676.91 4.14 179.37  
Abu Dhabi, UAE 18.66 16.57 21.90 24.43 12.07 9050.66 3.87 167.76  
Qatar 29.36 25.81 30.85 33.90 11.29 8467.50 3.62 156.95  
Romania 35.52 41.32 36.55 42.18 11.18 8383.35 3.59 155.39  
Austria 39.15 34.52 39.11 33.51 8.89 6669.48 2.85 123.63  
Germany 32.59 30.02 31.43 25.93 8.42 6313.93 2.70 117.03  
Ireland 43.67 42.27 41.90 37.96 5.99 4492.46 1.92 83.27  
Morocco 1.60 2.08 3.83 7.00 5.98 4482.99 1.92 83.10  
Slovak Republic 40.91 37.52 41.50 36.85 5.52 4142.23 1.77 76.78  
Oman 33.44 36.40 34.36 38.34 4.78 3583.85 1.53 66.43  
Croatia 25.08 21.49 25.84 22.19 4.54 3406.78 1.46 63.15  
Italy 27.06 23.71 27.00 23.40 4.04 3029.52 1.30 56.16  
Georgia 25.31 23.78 26.93 22.26 4.03 3020.26 1.29 55.98  
Lithuania 26.17 23.48 27.10 24.04 2.95 2210.88 0.95 40.98  
Hungary 44.69 47.21 44.15 47.28 2.71 2033.31 0.87 37.69  
Honduras −2.39 −3.81 −0.59 −0.36 2.65 1984.88 0.85 36.79  
Singapore 29.05 26.54 29.88 27.58 2.22 1664.81 0.71 30.86  
Azerbaijan 22.50 24.54 22.70 25.43 2.04 1530.08 0.65 28.36 \(\blacktriangle\)
Poland 37.16 35.03 36.58 34.20 1.86 1397.34 0.60 25.90 \(\blacktriangle \triangle\)
Spain 26.46 24.59 26.00 23.60 1.71 1285.60 0.55 23.83 \(\blacktriangle \triangle \bullet\)
Australia 40.50 39.20 40.92 38.52 1.25 935.29 0.40 17.34 \(\blacktriangle \triangle \bullet \circ\)
Saudi Arabia 11.24 11.27 12.91 13.28 1.15 865.93 0.37 16.05 \(\blacktriangle \triangle \bullet \circ \blacktriangleleft\)
Chinese Taipei 28.56 27.29 29.20 28.07 0.65 487.64 0.21 9.04 \(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Finland 25.44 24.89 25.37 23.78 0.59 440.63 0.19 8.17 \(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
Sweden 28.58 29.69 28.26 28.55 0.40 299.94 0.13 5.56 \(\blacktriangle \triangle \bullet \circ \blacktriangleleft \triangleleft\)
\(s^2_{\hat{\beta }_{3gz.z}}\) 135.82 129.56 119.69 102.71      
\(CL_l\) 84.90 80.99 74.82 64.20      
\(CL_u\) 251.64 240.04 221.75 190.28      
  1. 1, scaling model 1; 2, scaling model 2; 3, scaling model 3; 4, scaling model 4; \(\blacktriangle\), country variance is statistically different from Iran, Islamic Rep. of; \(\triangle\), country variance is statistically different from Malta; \(\bullet\), country variance is statistically different from Slovenia; \(\circ\), country variance is statistically different from Czech Republic and Abu Dhabi, UAE; \(\blacktriangleleft\), country variance is statistically different from Qatar and Romania; \(\triangleleft\), country variance is statistically different from Austria, Germany, Ireland and Morocco