Table 4.2 on page 64.
We will use SPSS file imm23.sav and imm23_lev2.sav for this example. Variable schid is the ID variable that links the level-1 data file with the level-2 data file. We assume that an SSM file has been created based on these two files. The intra-class correlation can be calculated as 26.123/(26.123+81.244) = .243.
Sigma_squared = 81.24393
Tau INTRCPT1,B0 26.12302
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.857 ----------------------------------------------------
The value of the likelihood function at iteration 4 = -1.899339E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 50.758849 1.151098 44.096 22 0.000
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 50.758849 1.125728 45.090 22 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 5.11107 26.12302 22 240.76440 0.000
level-1, R 9.01354 81.24393
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3798.678611 Number of estimated parameters = 2
Table 4.3 on page 65.
Variable HOMEWORK is added as a level-1 predictor. The intra-class correlation can be calculated as 21.339/(21.339+71.284) = .23.
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.355709 1.162623 39.872 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 2.399882 0.277197 8.658 517 0.000
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.355709 1.584797 29.250 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 2.399882 0.601609 3.989 517 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 4.61942 21.33907 22 193.43666 0.000
level-1, R 8.44298 71.28391
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3727.481647 Number of estimated parameters = 2
Table 4.4 on page 67.
The effect of variable HOMEWORK is modeled as random effect.
Sigma_squared = 53.29476
Tau INTRCPT1,B0 62.42394 -27.59218 HOMEWORK,B1 -27.59218 17.72475
Tau (as correlations) INTRCPT1,B0 1.000 -0.830 HOMEWORK,B1 -0.830 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.876 HOMEWORK, B1 0.891 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.816862E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.325566 1.758922 26.337 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.980224 0.928325 2.133 22 0.044
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.325566 1.720031 26.933 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.980224 0.907872 2.181 22 0.040
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.90088 62.42394 22 214.58739 0.000
HOMEWORK slope, U1 4.21008 17.72475 22 191.51840 0.000
level-1, R 7.30033 53.29476
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3633.724203 Number of estimated parameters = 4
Table 4.5 on page 69.
The effect of variable PARENTED is included as a fixed effect. The output of parameter estimates is shown below.
Sigma_squared = 50.77830
Tau INTRCPT1,B0 47.86733 -21.95114 HOMEWORK,B1 -21.95114 13.87876
Tau (as correlations) INTRCPT1,B0 1.000 -0.852 HOMEWORK,B1 -0.852 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.851 HOMEWORK, B1 0.871 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.800022E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 40.854605 1.790145 22.822 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.881686 0.830257 2.266 22 0.034
For PARENTED slope, B2
INTRCPT2, G20 1.841435 0.295917 6.223 516 0.000
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 40.854605 2.102911 19.428 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.881686 0.812474 2.316 22 0.030
For PARENTED slope, B2
INTRCPT2, G20 1.841435 0.368670 4.995 516 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 6.91862 47.86733 22 168.89325 0.000
HOMEWORK slope, U1 3.72542 13.87876 22 167.77753 0.000
level-1, R 7.12589 50.77830
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3600.043367 Number of estimated parameters = 4
Table 4.6 on page 71.
A simple traditional regression analysis with HOMEWORK and PARENTED as predictors.
sigma_squared = 76.11628
The outcome variable is MATH
Least-squares estimates of fixed effects
----------------------------------------------------------------------------
Standard
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 37.239150 0.996279 37.378 516 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 2.335361 0.268397 8.701 516 0.000
For PARENTED slope, B2
INTRCPT2, G20 3.003987 0.276506 10.864 516 0.000
----------------------------------------------------------------------------
Least-squares estimates of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 37.239150 1.322404 28.160 516 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 2.335361 0.497836 4.691 516 0.000
For PARENTED slope, B2
INTRCPT2, G20 3.003987 0.409981 7.327 516 0.000
----------------------------------------------------------------------------
The least-squares likelihood value = -1859.993177 Deviance = 3719.98635 Number of estimated parameters = 1
Table 4.7 on page 74.
Sigma_squared = 53.30584
Tau INTRCPT1,B0 66.42699 -28.75043 HOMEWORK,B1 -28.75043 17.78417
Tau (as correlations) INTRCPT1,B0 1.000 -0.836 HOMEWORK,B1 -0.836 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.882 HOMEWORK, B1 0.891 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.817115E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.971968 2.710897 16.589 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.981191 0.929799 2.131 22 0.044
For SCSIZE slope, B2
INTRCPT2, G20 0.425935 0.640978 0.665 516 0.506
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.971968 2.509292 17.922 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.981191 0.907564 2.183 22 0.040
For SCSIZE slope, B2
INTRCPT2, G20 0.425935 0.426487 0.999 516 0.318
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 8.15028 66.42699 22 221.65118 0.000
HOMEWORK slope, U1 4.21713 17.78417 22 191.46611 0.000
level-1, R 7.30108 53.30584
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3634.230670 Number of estimated parameters = 4
Table 4.8 on page 75.
Variable PUBLIC is added as fixed effect and variable SCSIZE is taken out of the model.
Sigma_squared = 53.34329
Tau INTRCPT1,B0 60.16977 -27.45952 HOMEWORK,B1 -27.45952 17.31147
Tau (as correlations) INTRCPT1,B0 1.000 -0.851 HOMEWORK,B1 -0.851 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.871 HOMEWORK, B1 0.888 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.814194E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 49.050339 2.184865 22.450 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.975250 0.918466 2.151 22 0.043
For PUBLIC slope, B2
INTRCPT2, G20 -4.060951 1.977883 -2.053 516 0.040
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 49.050339 2.676867 18.324 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.975250 0.898964 2.197 22 0.039
For PUBLIC slope, B2
INTRCPT2, G20 -4.060951 2.292226 -1.772 516 0.076
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.75692 60.16977 22 192.87153 0.000
HOMEWORK slope, U1 4.16071 17.31147 22 191.40891 0.000
level-1, R 7.30365 53.34329
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3628.388074 Number of estimated parameters = 4
Table 4.10 on page 77.
This model involves an interaction term of variable HOMEWORK and PUBLIC. The interaction term is created by including variable PUBLIC as a level-2 predictor for variable HOMEWORK.
Sigma_squared = 53.33262
Tau INTRCPT1,B0 62.55533 -28.95353 HOMEWORK,B1 -28.95353 18.25230
Tau (as correlations) INTRCPT1,B0 1.000 -0.857 HOMEWORK,B1 -0.857 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.876 HOMEWORK, B1 0.893 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.811656E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 48.528899 3.016041 16.090 21 0.000
MPUBLIC, G01 -3.261963 3.714613 -0.878 21 0.390
For HOMEWORK slope, B1
INTRCPT2, G10 2.292764 1.591420 1.441 21 0.164
MPUBLIC, G11 -0.495648 1.972838 -0.251 21 0.804
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 48.528899 3.202969 15.151 21 0.000
MPUBLIC, G01 -3.261963 3.761037 -0.867 21 0.396
For HOMEWORK slope, B1
INTRCPT2, G10 2.292764 1.286462 1.782 21 0.089
MPUBLIC, G11 -0.495648 1.757265 -0.282 21 0.781
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.90919 62.55533 21 195.23099 0.000
HOMEWORK slope, U1 4.27227 18.25230 21 193.21281 0.000
level-1, R 7.30292 53.33262
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3623.312669 Number of estimated parameters = 4
Table 4.11 on page 80. This model uses full NELS-88 data (we dont have this data, so this is omitted).
Table 4.12 on page 82.
Variable WHITE is included as level-1 predictor.
Sigma_squared = 52.72832
Tau INTRCPT1,B0 56.03156 -26.81364 HOMEWORK,B1 -26.81364 16.77465
Tau (as correlations) INTRCPT1,B0 1.000 -0.875 HOMEWORK,B1 -0.875 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.865 HOMEWORK, B1 0.886 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.806808E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.628341 2.192272 21.269 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.899256 0.904929 2.099 22 0.047
For WHITE slope, B2
INTRCPT2, G20 3.309399 0.969861 3.412 515 0.001
For PUBLIC slope, B3
INTRCPT2, G30 -3.882560 1.798787 -2.158 515 0.031
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.628341 2.754269 16.929 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.899256 0.882304 2.153 22 0.042
For WHITE slope, B2
INTRCPT2, G20 3.309399 1.274330 2.597 515 0.010
For PUBLIC slope, B3
INTRCPT2, G30 -3.882560 2.065162 -1.880 515 0.060
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.48542 56.03156 22 184.16851 0.000
HOMEWORK slope, U1 4.09569 16.77465 22 190.78991 0.000
level-1, R 7.26143 52.72832
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3613.615347 Number of estimated parameters = 4
Table 4.13 on page 83.
Variable WHITE is included as a random effect.
Sigma_squared = 51.15595
Tau
INTRCPT1,B0 69.14492 -28.52701 -22.09856
HOMEWORK,B1 -28.52701 16.63111 2.90674
WHITE,B2 -22.09856 2.90674 26.55034
Tau (as correlations)
INTRCPT1,B0 1.000 -0.841 -0.516
HOMEWORK,B1 -0.841 1.000 0.138
WHITE,B2 -0.516 0.138 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.749
HOMEWORK, B1 0.904
WHITE, B2 0.540
----------------------------------------------------
Note: The reliability estimates reported above are based on only 16 of 23 units that had sufficient data for computation. Fixed effects and variance components are based on all the data.
The value of the likelihood function at iteration 17 = -1.804332E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 48.229687 2.340050 20.611 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.940057 0.900454 2.155 22 0.042
For WHITE slope, B2
INTRCPT2, G20 2.597336 1.554904 1.670 22 0.109
For PUBLIC slope, B3
INTRCPT2, G30 -4.930506 1.650828 -2.987 515 0.003
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 48.229687 2.716287 17.756 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.940057 0.881182 2.202 22 0.038
For WHITE slope, B2
INTRCPT2, G20 2.597336 1.503784 1.727 22 0.098
For PUBLIC slope, B3
INTRCPT2, G30 -4.930506 1.833957 -2.688 515 0.008
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 8.31534 69.14492 15 60.56538 0.000
HOMEWORK slope, U1 4.07813 16.63111 15 115.22516 0.000
WHITE slope, U2 5.15270 26.55034 15 29.68260 0.013
level-1, R 7.15234 51.15595
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 16 of 23 units that had sufficient data for computation. Fixed effects and variance components are based on all the data.
Statistics for current covariance components model -------------------------------------------------- Deviance = 3608.663161 Number of estimated parameters = 7
Table 4.14 on page 85.
Variable WHITE is included as fixed and variable MEANSES is included as fixed.
Sigma_squared = 52.79408
Tau INTRCPT1,B0 54.13863 -27.03574 HOMEWORK,B2 -27.03574 16.44395
Tau (as correlations) INTRCPT1,B0 1.000 -0.906 HOMEWORK,B2 -0.906 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.861 HOMEWORK, B2 0.884 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.803260E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.615943 2.235300 19.960 22 0.000
For MEANSES slope, B1
INTRCPT2, G10 4.977089 1.951297 2.551 514 0.011
For HOMEWORK slope, B2
INTRCPT2, G20 1.924156 0.896425 2.146 22 0.043
For WHITE slope, B3
INTRCPT2, G30 3.096787 0.963117 3.215 514 0.002
For PUBLIC slope, B4
INTRCPT2, G40 0.160388 2.271429 0.071 514 0.944
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.615943 2.852724 15.640 22 0.000
For MEANSES slope, B1
INTRCPT2, G10 4.977089 2.140173 2.326 514 0.020
For HOMEWORK slope, B2
INTRCPT2, G20 1.924156 0.870047 2.212 22 0.038
For WHITE slope, B3
INTRCPT2, G30 3.096787 1.275468 2.428 514 0.015
For PUBLIC slope, B4
INTRCPT2, G40 0.160388 2.738577 0.059 514 0.954
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.35790 54.13863 22 168.77148 0.000
HOMEWORK slope, U2 4.05511 16.44395 22 190.44614 0.000
level-1, R 7.26595 52.79408
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3606.519823 Number of estimated parameters = 4
Table 4.15 on page 86.
Variable PUBLIC is deleted from the model.
Sigma_squared = 52.79022
Tau INTRCPT1,B0 53.57697 -26.99882 HOMEWORK,B2 -26.99882 16.39853
Tau (as correlations) INTRCPT1,B0 1.000 -0.911 HOMEWORK,B2 -0.911 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.860 HOMEWORK, B2 0.884 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.804072E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.702254 1.787302 25.011 22 0.000
For MEANSES slope, B1
INTRCPT2, G10 4.892436 1.340903 3.649 515 0.000
For HOMEWORK slope, B2
INTRCPT2, G20 1.925101 0.895116 2.151 22 0.043
For WHITE slope, B3
INTRCPT2, G30 3.114806 0.957043 3.255 515 0.002
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.702254 2.050411 21.802 22 0.000
For MEANSES slope, B1
INTRCPT2, G10 4.892436 1.424591 3.434 515 0.001
For HOMEWORK slope, B2
INTRCPT2, G20 1.925101 0.869461 2.214 22 0.037
For WHITE slope, B3
INTRCPT2, G30 3.114806 1.287769 2.419 515 0.016
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.31963 53.57697 22 168.73655 0.000
HOMEWORK slope, U2 4.04951 16.39853 22 190.42898 0.000
level-1, R 7.26569 52.79022
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3608.143584 Number of estimated parameters = 4
Table 4.16 on page 88.
Interaction between HOMEWORK and MEANSES is included.
Sigma_squared = 52.78077
Tau INTRCPT1,B0 55.89070 -28.39071 HOMEWORK,B2 -28.39071 17.23834
Tau (as correlations) INTRCPT1,B0 1.000 -0.915 HOMEWORK,B2 -0.915 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.864 HOMEWORK, B2 0.889 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.803527E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.611344 1.836270 24.295 22 0.000
For MEANSES slope, B1
INTRCPT2, G10 3.977516 3.023012 1.316 514 0.188
For HOMEWORK slope, B2
INTRCPT2, G20 1.974713 0.929138 2.125 21 0.045
MMEANSES, G21 0.553145 1.645743 0.336 21 0.740
For WHITE slope, B3
INTRCPT2, G30 3.114979 0.957136 3.254 514 0.002
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 44.611344 2.029445 21.982 22 0.000
For MEANSES slope, B1
INTRCPT2, G10 3.977516 2.914961 1.365 514 0.173
For HOMEWORK slope, B2
INTRCPT2, G20 1.974713 0.862444 2.290 21 0.032
MMEANSES, G21 0.553145 1.340866 0.413 21 0.684
For WHITE slope, B3
INTRCPT2, G30 3.114979 1.286842 2.421 514 0.016
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.47601 55.89070 22 171.36655 0.000
HOMEWORK slope, U2 4.15191 17.23834 21 193.10306 0.000
level-1, R 7.26504 52.78077
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3607.054239 Number of estimated parameters = 4
Table 4.17 on page 89.
Variable SES is included as level-1 predictor.
Sigma_squared = 51.29601
Tau INTRCPT1,B0 49.90014 -24.37911 HOMEWORK,B3 -24.37911 14.64878
Tau (as correlations) INTRCPT1,B0 1.000 -0.902 HOMEWORK,B3 -0.902 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.855 HOMEWORK, B3 0.876 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.796378E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 45.674852 1.752109 26.068 22 0.000
For SES slope, B1
INTRCPT2, G10 2.205769 0.535632 4.118 514 0.000
For MEANSES slope, B2
INTRCPT2, G20 2.945244 1.428222 2.062 514 0.039
For HOMEWORK slope, B3
INTRCPT2, G30 1.825656 0.850414 2.147 22 0.043
For WHITE slope, B4
INTRCPT2, G40 2.170702 0.973294 2.230 514 0.026
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 45.674852 1.993801 22.908 22 0.000
For SES slope, B1
INTRCPT2, G10 2.205769 0.642834 3.431 514 0.001
For MEANSES slope, B2
INTRCPT2, G20 2.945244 1.591438 1.851 514 0.064
For HOMEWORK slope, B3
INTRCPT2, G30 1.825656 0.824166 2.215 22 0.037
For WHITE slope, B4
INTRCPT2, G40 2.170702 1.377362 1.576 514 0.115
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.06400 49.90014 22 162.57774 0.000
HOMEWORK slope, U3 3.82737 14.64878 22 177.92008 0.000
level-1, R 7.16212 51.29601
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3592.755431 Number of estimated parameters = 4
Table 4.19 on page 91.
Variable SES is the only predictor.
Sigma_squared = 75.32795
Tau INTRCPT1,B0 12.63048
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.762 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.872170E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 51.200859 0.850662 60.189 22 0.000
For SES slope, B1
INTRCPT2, G10 4.332382 0.566265 7.651 517 0.000
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 51.200859 0.823988 62.138 22 0.000
For SES slope, B1
INTRCPT2, G10 4.332382 0.529547 8.181 517 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 3.55394 12.63048 22 119.55639 0.000
level-1, R 8.67917 75.32795
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3744.339553 Number of estimated parameters = 2
Table 4.20 on page 92.
Variable SES is included as a random effect. Notice this model takes quite a few iterations to converge.
******* ITERATION 581 *******
Sigma_squared = 75.16657
Tau
INTRCPT1,B0 12.93101 -1.06844
SES,B1 -1.06844 0.16763
Tau (as correlations)
INTRCPT1,B0 1.000 -0.726
SES,B1 -0.726 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.659
SES, B1 0.019
----------------------------------------------------
The value of the likelihood function at iteration 581 = -1.872093E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 51.249011 0.857867 59.740 22 0.000
For SES slope, B1
INTRCPT2, G10 4.326502 0.571343 7.573 22 0.000
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 51.249011 0.835284 61.355 22 0.000
For SES slope, B1
INTRCPT2, G10 4.326502 0.517031 8.368 22 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 3.59597 12.93101 22 93.95627 0.000
SES slope, U1 0.40943 0.16763 22 29.95493 0.119
level-1, R 8.66987 75.16657
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3744.185216 Number of estimated parameters = 4
Table 4.21 on page 93.
Sigma_squared = 75.17279
Tau INTRCPT1,B0 10.69521
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.733 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.870805E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 53.126608 1.177803 45.107 22 0.000
For SES slope, B1
INTRCPT2, G10 4.299001 0.561803 7.652 516 0.000
For PERCMIN slope, B2
INTRCPT2, G20 -0.809360 0.364643 -2.220 516 0.026
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 53.126608 1.146173 46.351 22 0.000
For SES slope, B1
INTRCPT2, G10 4.299001 0.555372 7.741 516 0.000
For PERCMIN slope, B2
INTRCPT2, G20 -0.809360 0.402989 -2.008 516 0.044
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 3.27035 10.69521 22 110.56857 0.000
level-1, R 8.67022 75.17279
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3741.610871 Number of estimated parameters = 2
Table 4.22 on page 95.
Sigma_squared = 75.24047
Tau INTRCPT1,B0 8.84704
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.696 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.866889E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 53.096907 1.102689 48.152 22 0.000
For SES slope, B1
INTRCPT2, G10 3.884760 0.609773 6.371 515 0.000
For MEANSES slope, B2
INTRCPT2, G20 2.805058 1.479396 1.896 515 0.058
For PERCMIN slope, B3
INTRCPT2, G30 -0.692191 0.345699 -2.002 515 0.045
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 53.096907 1.058582 50.159 22 0.000
For SES slope, B1
INTRCPT2, G10 3.884760 0.693364 5.603 515 0.000
For MEANSES slope, B2
INTRCPT2, G20 2.805058 1.837170 1.527 515 0.127
For PERCMIN slope, B3
INTRCPT2, G30 -0.692191 0.461854 -1.499 515 0.134
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.97440 8.84704 22 77.37302 0.000
level-1, R 8.67413 75.24047
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3733.777145 Number of estimated parameters = 2
Table 4.23 and Table 4.24 on page 97. Analyses with NELS-88, models 2 and 3 (we do not have this data, so these analyses are omitted).
Table 4.25 on page 99.
Sigma_squared = 53.30420
Tau INTRCPT1,B0 63.48035 -27.70473 HOMEWORK,B1 -27.70473 17.74312
Tau (as correlations) INTRCPT1,B0 1.000 -0.826 HOMEWORK,B1 -0.826 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.877 HOMEWORK, B1 0.891 ----------------------------------------------------
The value of the likelihood function at iteration 7 = -1.818319E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 47.961602 4.083183 11.746 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.979239 0.928948 2.131 22 0.044
For RATIO slope, B2
INTRCPT2, G20 -0.094420 0.212382 -0.445 516 0.656
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 47.961602 4.759461 10.077 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 1.979239 0.908219 2.179 22 0.040
For RATIO slope, B2
INTRCPT2, G20 -0.094420 0.237873 -0.397 516 0.691
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.96746 63.48035 22 211.41465 0.000
HOMEWORK slope, U1 4.21226 17.74312 22 191.49725 0.000
level-1, R 7.30097 53.30420
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3636.637671 Number of estimated parameters = 4
Table 4.26 on page 100.
Sigma_squared = 53.29782
Tau INTRCPT1,B0 62.55451 -27.79939 HOMEWORK,B1 -27.79939 18.09403
Tau (as correlations) INTRCPT1,B0 1.000 -0.826 HOMEWORK,B1 -0.826 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.876 HOMEWORK, B1 0.893 ----------------------------------------------------
The value of the likelihood function at iteration 7 = -1.818945E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.330611 1.760574 26.316 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 2.884107 2.156618 1.337 21 0.196
MRATIO, G11 -0.052584 0.112271 -0.468 21 0.644
----------------------------------------------------------------------------
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 46.330611 1.721460 26.914 22 0.000
For HOMEWORK slope, B1
INTRCPT2, G10 2.884107 2.236130 1.290 21 0.211
MRATIO, G11 -0.052584 0.122894 -0.428 21 0.673
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 7.90914 62.55451 22 214.54663 0.000
HOMEWORK slope, U1 4.25371 18.09403 21 194.54422 0.000
level-1, R 7.30054 53.29782
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 3637.890963 Number of estimated parameters = 4
Table 4.27 on page 101. Analyses with NELS-88, (we do not have this data, so these analyses are omitted).