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HLM Textbook Examples
Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence
 by Judith D. Singer and John B. Willett
Chapter 4: Doing data analysis with the multilevel model for change

This chapter uses data file alcohol1_pp.sav.  The examples are created using HLM 6.06.  The method we choose for estimating these models is ML. Estimation method is selected from Other Settings -> Estimation Settings menu.


Figure 4.1, page 77, not quite, since we don't have a way of making the regression lines, but it gives us a way of looking at the data. Choose File -> Graph Data -> line plots, scatter plots from the pull-down menu to get the window below.


Table 4.1, pages 94-95

Model A:

Sigma_squared =      0.56175
 Standard Error of Sigma_squared =      0.06203
 Tau
 INTRCPT1,B0      0.56386 
 Standard Errors of Tau
 INTRCPT1,B0      0.11911 
Tau (as correlations)
 INTRCPT1,B0  1.000
 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.751
 ----------------------------------------------------
 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.921955   0.095707     9.633        81    0.000
 ----------------------------------------------------------------------------
 The outcome variable is   ALCUSE
 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           0.921955   0.095707     9.633        81    0.000
 ----------------------------------------------------------------------------
 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.75091       0.56386    81     328.92616    0.000
  level-1,       R         0.74950       0.56175
 -----------------------------------------------------------------------------
 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 670.155995
 Number of estimated parameters = 3

Model B:

Sigma_squared =      0.33729
 Standard Error of Sigma_squared =      0.05268
 Tau
 INTRCPT1,B0      0.62436      -0.06844 
   AGE_14,B1     -0.06844       0.15120 
 Standard Errors of Tau
 INTRCPT1,B0      0.14806       0.07008 
   AGE_14,B1      0.07008       0.05647 
Tau (as correlations)
 INTRCPT1,B0  1.000 -0.223
   AGE_14,B1 -0.223  1.000
 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.690
    AGE_14, B1                        0.473
 ----------------------------------------------------
 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.651304   0.105080     6.198        81    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.062455     4.334        81    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           0.651304   0.105080     6.198        81    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.062455     4.334        81    0.000
 ----------------------------------------------------------------------------
 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.79016       0.62436    81     264.14675    0.000
   AGE_14 slope, U1        0.38885       0.15120    81     155.51848    0.000
  level-1,       R         0.58077       0.33729
 -----------------------------------------------------------------------------
 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 636.611086
 Number of estimated parameters = 6

Model C:

 Sigma_squared =      0.33729
 Standard Error of Sigma_squared =      0.05268
 Tau
 INTRCPT1,B0      0.48758      -0.05934 
   AGE_14,B1     -0.05934       0.15060 
 Standard Errors of Tau
 INTRCPT1,B0      0.12782       0.06573 
   AGE_14,B1      0.06573       0.05639 
Tau (as correlations)
 INTRCPT1,B0  1.000 -0.219
   AGE_14,B1 -0.219  1.000
 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.634
    AGE_14, B1                        0.472
 ----------------------------------------------------
 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.315952   0.130695     2.417        80    0.016
         COA, G01           0.743212   0.194566     3.820        80    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.292955   0.084228     3.478        80    0.001
         COA, G11          -0.049430   0.125389    -0.394        80    0.693
 ----------------------------------------------------------------------------
 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           0.315952   0.092525     3.415        80    0.001
         COA, G01           0.743212   0.204789     3.629        80    0.001
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.292955   0.077620     3.774        80    0.000
         COA, G11          -0.049430   0.127416    -0.388        80    0.698
 ----------------------------------------------------------------------------
 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.69827       0.48758    80     224.24420    0.000
   AGE_14 slope, U1        0.38807       0.15060    80     155.22430    0.000
  level-1,       R         0.58077       0.33729
 -----------------------------------------------------------------------------
 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 621.202628
 Number of estimated parameters = 8

Model D:

 Sigma_squared =      0.33729
 Standard Error of Sigma_squared =      0.05268
 Tau
 INTRCPT1,B0      0.24091      -0.00612 
   AGE_14,B1     -0.00612       0.13911 
 Standard Errors of Tau
 INTRCPT1,B0      0.09259       0.05500 
   AGE_14,B1      0.05500       0.05481 
Tau (as correlations)
 INTRCPT1,B0  1.000 -0.033
   AGE_14,B1 -0.033  1.000
 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.462
    AGE_14, B1                        0.452
 ----------------------------------------------------
 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00          -0.316514   0.148062    -2.138        79    0.032
         COA, G01           0.579165   0.162486     3.564        79    0.001
        PEER, G02           0.694296   0.111533     6.225        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.429429   0.113689     3.777        79    0.000
         COA, G11          -0.014032   0.124765    -0.112        79    0.911
        PEER, G12          -0.149815   0.085641    -1.749        79    0.080
 ----------------------------------------------------------------------------
 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          -0.316514   0.117611    -2.691        79    0.008
         COA, G01           0.579165   0.168459     3.438        79    0.001
        PEER, G02           0.694296   0.131544     5.278        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.429429   0.115458     3.719        79    0.000
         COA, G11          -0.014032   0.127005    -0.110        79    0.912
        PEER, G12          -0.149815   0.095926    -1.562        79    0.118
 ----------------------------------------------------------------------------
 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.49082       0.24091    79     152.28049    0.000
   AGE_14 slope, U1        0.37298       0.13911    79     149.63981    0.000
  level-1,       R         0.58077       0.33729
 -----------------------------------------------------------------------------
 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 588.690655
 Number of estimated parameters = 10

Model E:

 Sigma_squared =      0.33729
 Standard Error of Sigma_squared =      0.05268
 Tau
 INTRCPT1,B0      0.24095      -0.00616 
   AGE_14,B1     -0.00616       0.13916 
 Standard Errors of Tau
 INTRCPT1,B0      0.09259       0.05501 
   AGE_14,B1      0.05501       0.05481 
Tau (as correlations)
 INTRCPT1,B0  1.000 -0.034
   AGE_14,B1 -0.034  1.000
 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.462
    AGE_14, B1                        0.452
 ----------------------------------------------------
 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00          -0.313821   0.146118    -2.148        79    0.032
         COA, G01           0.571196   0.146228     3.906        79    0.000
        PEER, G02           0.695183   0.111258     6.248        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.424687   0.105590     4.022        80    0.000
        PEER, G11          -0.151377   0.084513    -1.791        80    0.073
 ----------------------------------------------------------------------------
 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          -0.313821   0.117172    -2.678        79    0.008
         COA, G01           0.571196   0.151710     3.765        79    0.000
        PEER, G02           0.695183   0.131693     5.279        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.424687   0.111385     3.813        80    0.000
        PEER, G11          -0.151377   0.093888    -1.612        80    0.107
 ----------------------------------------------------------------------------
 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.49086       0.24095    79     152.28729    0.000
   AGE_14 slope, U1        0.37305       0.13916    80     149.66517    0.000
  level-1,       R         0.58076       0.33729
 -----------------------------------------------------------------------------
 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 588.703303
 Number of estimated parameters = 9

Model F:

 Sigma_squared =      0.33729

 Standard Error of Sigma_squared =      0.05268

 Tau
 INTRCPT1,B0      0.24095      -0.00616 
   AGE_14,B1     -0.00616       0.13916 


 Standard Errors of Tau
 INTRCPT1,B0      0.09259       0.05501 
   AGE_14,B1      0.05501       0.05481 


Tau (as correlations)
 INTRCPT1,B0  1.000 -0.034
   AGE_14,B1 -0.034  1.000

 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.462
    AGE_14, B1                        0.452
 ----------------------------------------------------

The value of the likelihood function at iteration 3 = -2.943516E+002
 The outcome variable is   ALCUSE

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.393569   0.103535     3.801        79    0.000
         COA, G01           0.571196   0.146228     3.906        79    0.000
        PEER, G02           0.695183   0.111258     6.248        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.061268     4.418        80    0.000
        PEER, G11          -0.151377   0.084513    -1.791        80    0.077
 ----------------------------------------------------------------------------

 The outcome variable is   ALCUSE

 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           0.393569   0.098960     3.977        79    0.000
         COA, G01           0.571196   0.151710     3.765        79    0.001
        PEER, G02           0.695183   0.131693     5.279        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.061268     4.418        80    0.000
        PEER, G11          -0.151377   0.093888    -1.612        80    0.111
 ----------------------------------------------------------------------------



 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.49086       0.24095    79     152.28731    0.000
   AGE_14 slope, U1        0.37305       0.13916    80     149.66519    0.000
  level-1,       R         0.58076       0.33729
 -----------------------------------------------------------------------------


 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 588.703296
 Number of estimated parameters = 9

Model G:

Sigma_squared =      0.33729

 Standard Error of Sigma_squared =      0.05268


 Tau
 INTRCPT1,B0      0.24095      -0.00616 
   AGE_14,B1     -0.00616       0.13916 


 Standard Errors of Tau
 INTRCPT1,B0      0.09259       0.05501 
   AGE_14,B1      0.05501       0.05481 


Tau (as correlations)
 INTRCPT1,B0  1.000 -0.034
   AGE_14,B1 -0.034  1.000

 ----------------------------------------------------
  Random level-1 coefficient   Reliability estimate
 ----------------------------------------------------
  INTRCPT1, B0                        0.462
    AGE_14, B1                        0.452
 ----------------------------------------------------

The value of the likelihood function at iteration 3 = -2.943516E+002
 The outcome variable is   ALCUSE

 Final estimation of fixed effects:
 ----------------------------------------------------------------------------
                                       Standard             Approx.
    Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
 ----------------------------------------------------------------------------
 For       INTRCPT1, B0
    INTRCPT2, G00           0.651303   0.079788     8.163        79    0.000
         COA, G01           0.571196   0.146228     3.906        79    0.000
        PEER, G02           0.695183   0.111258     6.248        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.061268     4.418        80    0.000
        PEER, G11          -0.151377   0.084513    -1.791        80    0.077
 ----------------------------------------------------------------------------

 The outcome variable is   ALCUSE

 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           0.651303   0.079786     8.163        79    0.000
         COA, G01           0.571196   0.151710     3.765        79    0.001
        PEER, G02           0.695183   0.131693     5.279        79    0.000
 For   AGE_14 slope, B1
    INTRCPT2, G10           0.270651   0.061268     4.418        80    0.000
        PEER, G11          -0.151377   0.093888    -1.612        80    0.111
 ----------------------------------------------------------------------------



 Final estimation of variance components:
 -----------------------------------------------------------------------------
 Random Effect           Standard      Variance     df    Chi-square  P-value
                         Deviation     Component
 -----------------------------------------------------------------------------
 INTRCPT1,       U0        0.49086       0.24095    79     152.28731    0.000
   AGE_14 slope, U1        0.37305       0.13916    80     149.66519    0.000
  level-1,       R         0.58076       0.33729
 -----------------------------------------------------------------------------


 Statistics for current covariance components model
 --------------------------------------------------
 Deviance                       = 588.703296
 Number of estimated parameters = 9

Figure 4.3, page 99

 


Graph for Model E:

 


Page 123

Test of equation 4.18 using Model F:

After building model F, we can further specify optional hypothesis tests through Optional Specifications menu. For example, to test equation 4.18, we can specify as shown in the sequence of screen shots below. After the hypothesis test specification, we can run the model and the result of the test will be at the end of the output.

               Results of General Linear Hypothesis Testing
 -----------------------------------------------------------------------------
                                   Coefficients      Contrast
 -----------------------------------------------------------------------------
 For       INTRCPT1, B0
     INTRCPT2, G00                   0.393569      1.000   0.000
          COA, G01                   0.571196      0.000   0.000
         PEER, G02                   0.695183      0.000   0.000
 For   AGE_14 slope, B1
     INTRCPT2, G10                   0.270651      0.000   1.000
         PEER, G11                  -0.151377      0.000   0.000

Chi-square statistic = 51.011646
Degrees of freedom   = 2
P-value              = 0.000000

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