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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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