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Table 3 Results of multilevel structural equation models for longitudinal competence measurement (N = 7276)

From: The achievement gap in reading competence: the effect of measurement non-invariance across school types

  Model 1a   Model 1b   Model 2a   Model 2b  
  M SD 95% PPI M SD 95% PPI M SD 95% PPI M SD 95% PPI
Fixed effects:
School level covariates
 Grade 5             
  HS     0.865 0.031 (0.802, 0.920)     0.851 0.032 (0.784, 0.909)
   MS     − 0.174 0.039 (− 0.252, − 0.099)     − 0.158 0.040 (− 0.238, − 0.080)
  CS     − 0.207 0.039 (− 0.283, − 0.129)     − 0.201 0.040 (− 0.279, − 0.123)
  AT     − 0.206 0.041 (− 0.286, − 0.124)     − 0.210 0.041 (− 0.291, − 0.130)
  LS     − 0.689 0.030 (− 0.743, − 0.624)     − 0.692 0.030 (− 0.746, − 0.627)
 Grade 7             
  HS     0.844 0.029 (0.783, 0.897)     0.848 0.029 (0.788, 0.902)
  MS     − 0.206 0.036 (− 0.278, − 0.137)     − 0.209 0.036 (− 0.277, − 0.138)
  CS     − 0.119 0.035 (− 0.188, − 0.049)     − 0.120 0.035 (− 0.187, − 0.050)
  AT     − 0.174 0.038 (− 0.247, − 0.100)     − 0.179 0.038 (− 0.252, − 0.102)
  LS     − 0.712 0.027 (− 0.762, − 0.656)     − 0.710 0.027 (− 0.762, − 0.655)
 Grade 9             
  HS     0.861 0.031 (0.797, 0.917)     0.856 0.031 (0.790, 0.912)
  MS     − 0.214 0.039 (− 0.288, − 0.138)     − 0.206 0.038 (− 0.282, − 0.134)
  CS     − 0.111 0.038 (− 0.185, − 0.037)     − 0.110 0.038 (− 0.183, − 0.037)
  AT     − 0.221 0.039 (− 0.298, − 0.142)     − 0.232 0.040 (− 0.308, − 0.152)
  LS     − 0.682 0.030 (− 0.735, − 0.617)     − 0.676 0.032 (− 0.734, − 0.608)
Variance components of random effects
Student level
 Grade 5 0.383 0.047 (0.301, 0.491) 0.404 0.056 (0.319, 0.551) 0.353 0.042 (0.286, 0.445) 0.359 0.044 (0.289, 0.458)
 Grade 7 0.154 0.019 (0.120, 0.192) 0.160 0.016 (0.135, 0.196) 0.151 0.016 (0.125, 0.187) 0.153 0.012 (0.131, 0.176)
 Grade 9 0.277 0.030 (0.223, 0.340) 0.281 0.029 (0.227, 0.338) 0.265 0.025 (0.217, 0.313) 0.275 0.021 (0.237, 0.320)
School Level
 Grade 5 0.289 0.044 (0.217, 0.388) 0.078 0.014 (0.057, 0.111) 0.260 0.039 (0.196, 0.349) 0.071 0.012 (0.052, 0.097)
 Grade 7 0.104 0.016 (0.079, 0.139) 0.030 0.004 (0.023, 0.040) 0.103 0.014 (0.080, 0.135) 0.029 0.004 (0.023, 0.037)
 Grade 9 0.212 0.030 (0.162, 0.280) 0.060 0.009 (0.045, 0.080) 0.202 0.027 (0.157, 0.262) 0.061 0.009 (0.047, 0.080)
Correlations and Covariances between latent factors
Student Level
Correlation Covariance Correlation Covariance Correlation Covariance Correlation Covariance
 Grade 5 with Grade 7 0.644 0.172 0.632 0.179 0.631 0.139 0.634 0.139
 Grade 5 with Grade 9 0.649 0.223 0.652 0.232 0.662 0.210 0.650 0.185
 Grade 7 with Grade 9 0.723 0.163 0.722 0.160 0.722 0.156 0.721 0.131
School Level
 Grade 5 with Grade 7 0.000 0.000 0.001 0.000 0.002 0.000 0.002 0.000
 Grade 5 with Grade 9 0.000 0.000 0.001 0.000 0.002 0.000 0.003 0.001
 Grade 7 with Grade 9 0.001 0.000 0.002 0.000 0.003 0.001 0.000 0.000
 PPP 0.027    0.019    0.179    0.196   
  1. Note. Standardized results are presented for fixed effects. M = posterior mean. SD = posterior standard deviation, PPI posterior probability interval (2.5 and 97.5 percentile of the posterior distribution). School level covariates: HS high schools, MS  middle secondary schools, CS comprehensive schools, AT schools offering all school tracks except high school, LS lower secondary schools, PPP posterior predictive p value. In Models 1a and 1b, no school-type specific item discrimination or item difficulty estimates were accounted for. In Models 2a and 2b, school-type specific item discrimination and item difficulty estimates were taken into account for items exhibiting DIF. Correlations and covariances were reported based on sample statistics