HLM Textbook Examples
Multilevel Analysis by Tom Snijders and Roel Bosker
Chapter 5: The Hierarchical Linear Model
Example 5.1 on page 70-71 using https://stats.idre.ucla.edu/wp-content/uploads/2016/02/mlbook1.ssm. We use full maximum likelihood estimation method for this example. From Basic Model Specifications menu, choose Full maximum likelihood.
Level-1 variable IQ_VERB is entered as grand-mean centered. Level-2 variable IQ_VERB_ (which is the group mean of IQ_VERB) then is entered as grand-mean centered.
Tau
INTRCPT1,B0 7.91743 -0.82068
IQ_VERB,B1 -0.82068 0.20026
Standard Errors of Tau
INTRCPT1,B0 1.31508 0.26744
IQ_VERB,B1 0.26744 0.09832
Tau (as correlations)
INTRCPT1,B0 1.000 -0.652
IQ_VERB,B1 -0.652 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.698
IQ_VERB, B1 0.217
----------------------------------------------------
The value of the likelihood function at iteration 10 = -7.605849E+003
The outcome variable is LANGPOST
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 40.571236 0.287602 141.067 129 0.000
IQ_VERB_, G01 1.402053 0.321215 4.365 129 0.000
For IQ_VERB slope, B1
INTRCPT2, G10 2.458424 0.082982 29.626 130 0.000
----------------------------------------------------------------------------
The outcome variable is LANGPOST
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 40.571236 0.286613 141.554 129 0.000
IQ_VERB_, G01 1.402053 0.343986 4.076 129 0.000
For IQ_VERB slope, B1
INTRCPT2, G10 2.458424 0.086208 28.517 130 0.000
---------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.81379 7.91743 129 526.93537 0.000
IQ_VERB slope, U1 0.44751 0.20026 130 181.67794 0.002
level-1, R 6.43042 41.35027
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 15211.697947
Number of estimated parameters = 7
Example 5.2 on page 75. Based on the model above, level-2 variable GROUPSIZ is entered as grand-mean centered.
Sigma_squared = 41.31530
Standard Error of Sigma_squared = 1.28611
Tau
INTRCPT1,B0 7.66911 -0.77447
IQ_VERB,B1 -0.77447 0.17972
Standard Errors of Tau
INTRCPT1,B0 1.28283 0.25936
IQ_VERB,B1 0.25936 0.09512
Tau (as correlations)
INTRCPT1,B0 1.000 -0.660
IQ_VERB,B1 -0.660 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.692
IQ_VERB, B1 0.200
----------------------------------------------------
The value of the likelihood function at iteration 12 = -7.602153E+003
The outcome variable is LANGPOST
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 40.567658 0.286348 141.672 128 0.000
IQ_VERB_, G01 1.210282 0.326194 3.710 128 0.000
GROUPSIZ, G02 0.072898 0.038028 1.917 128 0.055
For IQ_VERB slope, B1
INTRCPT2, G10 2.513627 0.087337 28.781 129 0.000
GROUPSIZ, G11 -0.025716 0.011032 -2.331 129 0.020
----------------------------------------------------------------------------
The outcome variable is LANGPOST
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 40.567658 0.282597 143.553 128 0.000
IQ_VERB_, G01 1.210282 0.347659 3.481 128 0.001
GROUPSIZ, G02 0.072898 0.038695 1.884 128 0.059
For IQ_VERB slope, B1
INTRCPT2, G10 2.513627 0.094786 26.519 129 0.000
GROUPSIZ, G11 -0.025716 0.011874 -2.166 129 0.030
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.76932 7.66911 128 516.08521 0.000
IQ_VERB slope, U1 0.42394 0.17972 129 176.51886 0.004
level-1, R 6.42770 41.31530
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 15204.305563
Number of estimated parameters = 9
Example 5.3 on page 76, an extension of previous model. All variables except level-2 variable MIXEDGRD are entered as grand-mean centered. We didn’t constrain the covariance between u_1 and u_2.
Sigma_squared = 38.99286
Standard Error of Sigma_squared = 1.23538
Tau
INTRCPT1,B0 7.38326 -0.60748 -0.00614
IQ_VERB,B1 -0.60748 0.18584 -0.01634
SES,B2 -0.00614 -0.01634 0.00223
Standard Errors of Tau
INTRCPT1,B0 1.23567 0.25611 0.04530
IQ_VERB,B1 0.25611 0.09926 0.01393
SES,B2 0.04530 0.01393 0.00327
Tau (as correlations)
INTRCPT1,B0 1.000 -0.519 -0.048
IQ_VERB,B1 -0.519 1.000 -0.803
SES,B2 -0.048 -0.803 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.606
IQ_VERB, B1 0.190
SES, B2 0.062
----------------------------------------------------
Note: The reliability estimates reported above are based on only 130 of 131
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 840 = -7.540441E+003
The outcome variable is LANGPOST
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 41.601726 0.434270 95.797 127 0.000
IQ_VERB_, G01 0.770318 0.324693 2.372 127 0.018
MIXEDGRA, G02 -2.065145 0.798976 -2.585 127 0.010
GROUPSIZ, G03 -0.042985 0.051037 -0.842 127 0.400
For IQ_VERB slope, B1
INTRCPT2, G10 2.138082 0.132861 16.093 127 0.000
IQ_VERB_, G11 -0.031304 0.086105 -0.364 127 0.716
MIXEDGRA, G12 0.381452 0.253692 1.504 127 0.133
GROUPSIZ, G13 -0.003052 0.015920 -0.192 127 0.848
For SES slope, B2
INTRCPT2, G20 0.160610 0.026352 6.095 127 0.000
IQ_VERB_, G21 -0.022374 0.019494 -1.148 127 0.252
MIXEDGRA, G22 0.021394 0.047708 0.448 127 0.653
GROUPSIZ, G23 -0.001202 0.003006 -0.400 127 0.689
----------------------------------------------------------------------------
The outcome variable is LANGPOST
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 41.601726 0.398354 104.434 127 0.000
IQ_VERB_, G01 0.770318 0.317983 2.423 127 0.016
MIXEDGRA, G02 -2.065145 0.835415 -2.472 127 0.014
GROUPSIZ, G03 -0.042985 0.047773 -0.900 127 0.369
For IQ_VERB slope, B1
INTRCPT2, G10 2.138082 0.108641 19.680 127 0.000
IQ_VERB_, G11 -0.031304 0.104537 -0.299 127 0.764
MIXEDGRA, G12 0.381452 0.210430 1.813 127 0.069
GROUPSIZ, G13 -0.003052 0.013019 -0.234 127 0.815
For SES slope, B2
INTRCPT2, G20 0.160610 0.022928 7.005 127 0.000
IQ_VERB_, G21 -0.022374 0.018983 -1.179 127 0.239
MIXEDGRA, G22 0.021394 0.045037 0.475 127 0.634
GROUPSIZ, G23 -0.001202 0.002640 -0.455 127 0.648
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.71721 7.38326 126 442.91213 0.000
IQ_VERB slope, U1 0.43109 0.18584 126 174.18563 0.003
SES slope, U2 0.04721 0.00223 126 130.52808 0.373
level-1, R 6.24443 38.99286
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 130 of 131
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 = 15080.881187
Number of estimated parameters = 19