> print("Correlations in School-Level-2 Data")
[1] "Correlations in School-Level-2 Data"
> corrSummary(data = Lecture4Means, p.separate = TRUE, vars = c("homework", 
+ "mixgrd", "CMverb", "CMlang"))

Correlations:
 Variable homework mixgrd CMverb CMlang
 homework    1.000 -0.142  0.103  0.088
 mixgrd     -0.142  1.000 -0.158 -0.227
 CMverb      0.103 -0.158  1.000  0.721
 CMlang      0.088 -0.227  0.721  1.000

P-values:
 Variable homework mixgrd CMverb CMlang
 homework           0.050  0.158  0.224
 mixgrd      0.050         0.029  0.002
 CMverb      0.158  0.029         <.001
 CMlang      0.224  0.002  <.001       

> print("Descriptives in School-Level-2 Data")
[1] "Descriptives in School-Level-2 Data"
> univSummary(data = Lecture4Means, vars = c("homework", "mixgrd", 
+ "CMverb", "CMlang"))

 Variable   N   Mean    SD Variance    Min    Max
 homework 191  2.408 0.526    0.277  1.333  3.667
 mixgrd   191  0.366 0.483    0.233  0.000  1.000
 CMverb   191  9.948 0.931    0.867  5.776 12.477
 CMlang   191 40.932 4.877   23.786 23.714 50.333

> print("Correlations in Student-Level-1 Data")
[1] "Correlations in Student-Level-1 Data"
> corrSummary(data = Lecture4, p.separate = TRUE, vars = c("verbal", 
+ "langpost", "WCverb", "WClang", "CMverb", "CMlang"))

Correlations:
 Variable verbal langpost WCverb WClang CMverb CMlang
 verbal    1.000    0.603  0.918  0.541  0.397  0.268
 langpost  0.603    1.000  0.513  0.870  0.333  0.493
 WCverb    0.918    0.513  1.000  0.589  0.000  0.000
 WClang    0.541    0.870  0.589  1.000  0.000 -0.000
 CMverb    0.397    0.333  0.000  0.000  1.000  0.675
 CMlang    0.268    0.493  0.000 -0.000  0.675  1.000

P-values:
 Variable verbal langpost WCverb WClang CMverb CMlang
 verbal             <.001  <.001  <.001  <.001  <.001
 langpost  <.001           <.001  <.001  <.001  <.001
 WCverb    <.001    <.001         <.001  1.000  1.000
 WClang    <.001    <.001  <.001         1.000  1.000
 CMverb    <.001    <.001  1.000  1.000         <.001
 CMlang    <.001    <.001  1.000  1.000  <.001       

> print("Descriptives in Student-Level-1 Data")
[1] "Descriptives in Student-Level-1 Data"
> univSummary(data = Lecture4, vars = c("verbal", "langpost", "WCverb", 
+ "WClang", "CMverb", "CMlang"))

 Variable    N   Mean    SD Variance     Min    Max
 verbal   3566 10.036 2.033    4.132   2.133 16.633
 langpost 3566 41.464 8.814   77.691   9.000 58.000
 WCverb   3566  0.000 1.865    3.479  -7.400  6.826
 WClang   3566  0.000 7.667   58.777 -33.545 22.700
 CMverb   3566 10.036 0.808    0.653   5.776 12.477
 CMlang   3566 41.464 4.349   18.914  23.714 50.333

> print("Fixed Verbal Slopes Only")
[1] "Fixed Verbal Slopes Only"
> Model1 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + hw2 + mixgrd + CMverb10 + WCverb + (1 | schoolID))
> lmerSummary(Model1)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23629.074  23672.328 -11807.537  23615.074 

Model for the Variance:
 Groups    Name      Variance Std.Dev
 schoolID  Intercept    8.396   2.898
 Residual              40.550   6.368
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model1     4 253.499 498.717   <.001

Fixed Effects:
 Term          Est    SE       DF t.value p.value
 Intercept  41.579 0.362  172.098 114.721   <.001
 hw2        -0.053 0.459  178.450  -0.115   0.909
 mixgrd     -1.121 0.516  197.041  -2.173   0.031
 CMverb10    3.660 0.271  207.383  13.508   <.001
 WCverb      2.423 0.057 3372.684  42.370   <.001

> print("Add Random Cluster-MC Verbal Slopes")
[1] "Add Random Cluster-MC Verbal Slopes"
> Model2 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + hw2 + mixgrd + CMverb10 + WCverb + (1 + WCverb | schoolID))
> lmerSummary(Model2, LRT = TRUE, randCI = TRUE, slopeRel = TRUE)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23613.783  23669.396 -11797.891  23595.783 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.468   2.910       
           WCverb       0.224   0.473 -0.505
 Residual              39.758   6.305       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model2     4 182.249 374.445   <.001

Fixed Effects:
 Term          Est    SE      DF t.value p.value
 Intercept  41.528 0.358 176.855 116.127   <.001
 hw2        -0.095 0.446 177.653  -0.213   0.832
 mixgrd     -0.934 0.505 200.896  -1.848   0.066
 CMverb10    3.621 0.265 208.962  13.678   <.001
 WCverb      2.449 0.068 151.279  35.843   <.001

LRT for random effect(s):
                                  Npar     LL  -2LL   AIC   BIC    LRT DF p.value p.mix
<none>                               9 -11798 23596 23614 23669                        
WCverb in (1 + WCverb | schoolID)    7 -11808 23615 23629 23672 19.291  2   <.001 <.001

Random Slope Reliability (SR):
  SR = slope var / (slope var + res var / (mean L1n * var(X)))
 Term   SlopeVar ResVar Var.X    L1n    SR
 WCverb    0.224 39.758 3.479 18.670 0.268

Random Effect 95% CIs (Fixed Effect +/- 1.96*SD(random effect variance)):
 Term      FixedEffect Variance    SD  Lower  Upper
 Intercept      41.528    8.468 2.910 35.824 47.232
 WCverb          2.449    0.224 0.473  1.521  3.376

> print("Add 3 Cross-Level Interactions")
[1] "Add 3 Cross-Level Interactions"
> Model3 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + hw2 + mixgrd + CMverb10 + WCverb + hw2:WCverb + mixgrd:WCverb + 
+ 
+ CMverb10:WCverb + (1 + WCverb | schoolID))
> lmerSummary(Model3)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23622.028  23696.179 -11799.014  23598.028 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.471   2.911       
           WCverb       0.223   0.472 -0.517
 Residual              39.740   6.304       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model3     7 183.822 215.066   <.001

Fixed Effects:
 Term                Est    SE      DF t.value p.value
 Intercept        41.583 0.363 172.381 114.560   <.001
 hw2              -0.046 0.459 178.597  -0.100   0.920
 mixgrd           -1.137 0.516 196.788  -2.203   0.029
 CMverb10          3.644 0.271 206.890  13.448   <.001
 WCverb            2.390 0.100 128.552  23.861   <.001
 hw2:WCverb       -0.056 0.130 142.904  -0.429   0.668
 mixgrd:WCverb     0.321 0.159 228.287   2.021   0.044
 CMverb10:WCverb  -0.044 0.078 181.771  -0.559   0.577

> PseudoR2(baseModel = Model2, altModel1 = Model3)

Pseudo-R2
 Term       Model  Variance Pseudo.R2
 Intercept  Model2    8.468          
 Intercept  Model3    8.471    -0.000
 WCverb     Model2    0.224          
 WCverb     Model3    0.223     0.003
 Residual   Model2   39.758          
 Residual   Model3   39.740     0.000

> print("Add Random Constant-Centered Verbal Instead")
[1] "Add Random Constant-Centered Verbal Instead"
> Model4 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + verb10 + (1 + verb10 | schoolID))
> lmerSummary(Model4, LRT = TRUE, randCI = TRUE, slopeRel = TRUE)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23613.499  23662.933 -11798.750  23597.499 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.467   2.910       
           verb10       0.198   0.445 -0.533
 Residual              39.813   6.310       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model4     3 192.351 490.012   <.001

Fixed Effects:
 Term          Est    SE      DF t.value p.value
 Intercept  41.493 0.291 172.731 142.425   <.001
 mixgrd     -0.878 0.502 203.451  -1.750   0.082
 CMverb10    1.104 0.278 233.909   3.977   <.001
 verb10      2.446 0.067 159.764  36.465   <.001

LRT for random effect(s):
                                  Npar     LL  -2LL   AIC   BIC    LRT DF p.value p.mix
<none>                               8 -11799 23597 23613 23663                        
verb10 in (1 + verb10 | schoolID)    6 -11808 23615 23627 23664 17.864  2   <.001 <.001

Random Slope Reliability (SR):
  SR = slope var / (slope var + res var / (mean L1n * var(X)))
 Term   SlopeVar ResVar Var.X    L1n    SR
 verb10    0.198 39.813 4.132 18.670 0.277

Random Effect 95% CIs (Fixed Effect +/- 1.96*SD(random effect variance)):
 Term      FixedEffect Variance    SD  Lower  Upper
 Intercept      41.493    8.467 2.910 35.790 47.196
 verb10          2.446    0.198 0.445  1.574  3.319

> print("Hybrid Model for Verbal Slopes")
[1] "Hybrid Model for Verbal Slopes"
> Model5 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + verb10 + (1 + WCverb | schoolID))
> lmerSummary(Model5, LRT = TRUE, randCI = TRUE, slopeRel = TRUE)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23612.051  23661.485 -11798.025  23596.051 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.415   2.901       
           WCverb       0.224   0.473 -0.505
 Residual              39.757   6.305       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model5     3 204.049 499.535   <.001

Fixed Effects:
 Term          Est    SE      DF t.value p.value
 Intercept  41.484 0.290 171.027 142.918   <.001
 mixgrd     -0.920 0.500 202.309  -1.840   0.067
 CMverb10    1.167 0.271 236.228   4.301   <.001
 verb10      2.449 0.068 151.185  35.842   <.001

LRT for random effect(s):
                                  Npar     LL  -2LL   AIC   BIC    LRT DF p.value p.mix
<none>                               8 -11798 23596 23612 23661                        
WCverb in (1 + WCverb | schoolID)    6 -11808 23615 23627 23664 19.312  2   <.001 <.001

Random Slope Reliability (SR):
  SR = slope var / (slope var + res var / (mean L1n * var(X)))
 Term   SlopeVar ResVar Var.X    L1n    SR
 WCverb    0.224 39.757 3.479 18.670 0.268

Random Effect 95% CIs (Fixed Effect +/- 1.96*SD(random effect variance)):
 Term      FixedEffect Variance    SD  Lower  Upper
 Intercept      41.484    8.415 2.901 35.798 47.170

> print("Hybrid Model adding Smushed Cross-Level Interaction")
[1] "Hybrid Model adding Smushed Cross-Level Interaction"
> Model6 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + verb10 + verb10:mixgrd + (1 + WCverb | 
+ 
+ schoolID))
> lmerSummary(Model6)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23609.967  23665.580 -11795.984  23591.967 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.366   2.892       
           WCverb       0.217   0.466 -0.527
 Residual              39.733   6.303       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model6     4 224.144 379.780   <.001

Fixed Effects:
 Term              Est    SE      DF t.value p.value
 Intercept      41.585 0.292 167.679 142.201   <.001
 mixgrd         -1.146 0.507 199.835  -2.260   0.025
 CMverb10        1.043 0.275 246.384   3.791   <.001
 verb10          2.354 0.078 128.176  30.212   <.001
 mixgrd:verb10   0.370 0.150 281.186   2.463   0.014

> con1DTable(Model6, L = list(`L2 Between Verbal Main Effect` = c(0, 
+ 0, 1, 1, 0)))

Requested Fixed Effects:
 Name                            Est    SE      DF t.value p.value
 L2 Between Verbal Main Effect 3.396 0.277 245.882  12.279   <.001

> print("Hybrid Model with Unsmushed Cross-Level Interaction")
[1] "Hybrid Model with Unsmushed Cross-Level Interaction"
> Model7 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + verb10 + verb10:mixgrd + CMverb10:mixgrd + 
+ 
+ (1 + WCverb | schoolID))
> lmerSummary(Model7)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23610.751  23672.543 -11795.375  23590.751 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.412   2.900       
           WCverb       0.217   0.466 -0.532
 Residual              39.731   6.303       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model7     5 232.776 303.771   <.001

Fixed Effects:
 Term                Est    SE      DF t.value p.value
 Intercept        41.591 0.293 167.038 141.856   <.001
 mixgrd           -1.107 0.511 197.321  -2.166   0.031
 CMverb10          0.820 0.403 182.174   2.037   0.043
 verb10            2.361 0.079 123.558  30.055   <.001
 mixgrd:verb10     0.336 0.157 238.509   2.146   0.033
 mixgrd:CMverb10   0.417 0.552 236.541   0.756   0.451

> con1DTable(Model7, L = list(`L2 Between Verbal Main Effect` = c(0, 
+ 0, 1, 1, 0, 0), `L2 Between Verbal*mixgrd Interaction` = c(0, 
+ 0, 0, 0, 1, 1)))

Requested Fixed Effects:
 Name                                   Est    SE      DF t.value p.value
 L2 Between Verbal Main Effect        3.181 0.396 171.907   8.025   <.001
 L2 Between Verbal*mixgrd Interaction 0.753 0.529 200.554   1.424   0.156

> print("Cluster-MC Model with Both Interactions")
[1] "Cluster-MC Model with Both Interactions"
> Model8 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + WCverb + WCverb:mixgrd + CMverb10:mixgrd + 
+ 
+ (1 + WCverb | schoolID))
> lmerSummary(Model8)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23610.751  23672.543 -11795.375  23590.751 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.412   2.900       
           WCverb       0.217   0.466 -0.532
 Residual              39.731   6.303       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model8     5 197.988 303.771   <.001

Fixed Effects:
 Term                Est    SE      DF t.value p.value
 Intercept        41.591 0.293 167.038 141.856   <.001
 mixgrd           -1.107 0.511 197.321  -2.166   0.031
 CMverb10          3.181 0.396 171.907   8.025   <.001
 WCverb            2.361 0.079 123.558  30.055   <.001
 mixgrd:WCverb     0.336 0.157 238.509   2.146   0.033
 mixgrd:CMverb10   0.753 0.529 200.554   1.424   0.156

> con1DTable(Model8, L = list(`L2 Contextual Verbal Main Effect` = c(0, 
+ 0, 1, -1, 0, 0), `L2 Contextual Verbal*mixgrd Interaction` = c(0, 
+ 0, 0, 0, -1, 1)))

Requested Fixed Effects:
 Name                                      Est    SE      DF t.value p.value
 L2 Contextual Verbal Main Effect        0.820 0.403 182.174   2.037   0.043
 L2 Contextual Verbal*mixgrd Interaction 0.417 0.552 236.541   0.756   0.451

> print("Cluster-MC Model with Intra-Variable Interactions")
[1] "Cluster-MC Model with Intra-Variable Interactions"
> Model9 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + WCverb + I(CMverb10^2) + WCverb:mixgrd + 
+ 
+ CMverb10:mixgrd + WCverb:CMverb10 + (1 + WCverb | schoolID))
> lmerSummary(Model9)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23612.007  23686.157 -11794.003  23588.007 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.269   2.876       
           WCverb       0.229   0.478 -0.623
 Residual              39.717   6.302       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model  NumDF   DenDF F.value p.value
 Model9     7 204.137 217.209   <.001

Fixed Effects:
 Term                Est    SE      DF t.value p.value
 Intercept        41.807 0.301 167.169 138.999   <.001
 mixgrd           -0.832 0.516 196.971  -1.611   0.109
 CMverb10          3.212 0.398 181.255   8.077   <.001
 WCverb            2.359 0.079 122.803  29.832   <.001
 I(CMverb10^2)    -0.425 0.148 234.193  -2.868   0.005
 mixgrd:WCverb     0.340 0.158 230.599   2.151   0.032
 mixgrd:CMverb10   0.215 0.553 199.057   0.388   0.698
 CMverb10:WCverb  -0.043 0.078 179.153  -0.557   0.578

> con1DTable(Model9, L = list(`L2 Contextual Verbal Main Effect` = c(0, 
+ 0, 1, -1, 0, 0, 0, 0), `L2 Contextual Verbal*Verbal Interaction` = c(0, 
+ 0, 0, 0, 1, 0, 0, -1)))

Requested Fixed Effects:
 Name                                       Est    SE      DF t.value p.value
 L2 Contextual Verbal Main Effect         0.854 0.404 193.024   2.115   0.036
 L2 Contextual Verbal*Verbal Interaction -0.382 0.167 345.895  -2.290   0.023

> print("Hybrid Model with Intra-Variable Interactions")
[1] "Hybrid Model with Intra-Variable Interactions"
> Model10 = lmer(data = Lecture4, REML = TRUE, formula = langpost ~ 
+ 1 + mixgrd + CMverb10 + verb10 + I(CMverb10^2) + verb10:mixgrd + 
+ 
+ CMverb10:mixgrd + verb10:CMverb10 + (1 + WCverb | schoolID))
> lmerSummary(Model10)

Model Fit for Predicting langpost:
       AIC        BIC     logLik  -2*logLik 
 23612.007  23686.157 -11794.003  23588.007 

Model for the Variance:
 Groups    Name      Variance Std.Dev   Corr
 schoolID  Intercept    8.269   2.876       
           WCverb       0.229   0.478 -0.623
 Residual              39.717   6.302       
Number of obs: 3566 , groups: schoolID, 191 

Overall F-test for All Fixed Slopes:
 Model   NumDF   DenDF F.value p.value
 Model10     7 256.173 217.209   <.001

Fixed Effects:
 Term                Est    SE      DF t.value p.value
 Intercept        41.807 0.301 167.169 138.999   <.001
 mixgrd           -0.832 0.516 196.971  -1.611   0.109
 CMverb10          0.854 0.404 193.024   2.115   0.036
 verb10            2.359 0.079 122.803  29.832   <.001
 I(CMverb10^2)    -0.382 0.167 345.895  -2.290   0.023
 mixgrd:verb10     0.340 0.158 230.599   2.151   0.032
 mixgrd:CMverb10  -0.125 0.576 232.197  -0.217   0.828
 CMverb10:verb10  -0.043 0.078 179.153  -0.557   0.578

> con1DTable(Model10, L = list(`L2 Between Verbal Main Effect` = c(0, 
+ 0, 1, 1, 0, 0, 0, 0), `L2 Between Verbal*Verbal Interaction` = c(0, 
+ 0, 0, 0, 1, 0, 0, 1)))

Requested Fixed Effects:
 Name                                    Est    SE      DF t.value p.value
 L2 Between Verbal Main Effect         3.212 0.398 181.255   8.077   <.001
 L2 Between Verbal*Verbal Interaction -0.425 0.148 234.193  -2.868   0.005

