Welcome to the Institute for Digital Reseach and Education
Institute for Digital Reseach and Education Home
Help the Stat Consulting Group by giving a gift
HLM Paper Examples
Using SAS Proc Mixed to Fit Multilevel Models, Hierarchical Models, and
Individual Growth Models
by Judith Singer
This paper can be downloaded from Professor Singer's web site at
http://gseweb.harvard.edu/~faculty/singer/Papers/sasprocmixed.pdf .
The HLM SSM files used are hsb.SSM, willett.SSM
and willett_wave.MDM.
Page 329: Unconditional Means Model. The estimation method is REML (default).
Sigma_squared = 39.14831
Tau INTRCPT1,B0 8.61431
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.901 ----------------------------------------------------
The value of the likelihood function at iteration 4 = -2.355840E+004
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 12.636972 0.244412 51.704 159 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 12.636972 0.243628 51.870 159 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.93501 8.61431 159 1660.23264 0.000
level-1, R 6.25686 39.14831
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 47116.793475 Number of estimated parameters = 2
Page 331
Sigma_squared = 39.15708
Tau INTRCPT1,B0 2.63870
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.740 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.347972E+004
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 12.649436 0.149280 84.736 159 0.000
For MEANSES slope, B1
INTRCPT2, G10 5.863538 0.361457 16.222 7183 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 12.649436 0.148377 85.252 159 0.000
For MEANSES slope, B1
INTRCPT2, G10 5.863538 0.320211 18.311 7183 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 1.62441 2.63870 159 633.51740 0.000
level-1, R 6.25756 39.15708
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 46959.446955 Number of estimated parameters = 2
Page 335
Sigma_squared = 36.70356
Tau
INTRCPT1,B0 8.68087 0.04701
SES,B1 0.04701 0.68038
Tau (as correlations)
INTRCPT1,B0 1.000 0.019
SES,B1 0.019 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.908
SES, B1 0.260
----------------------------------------------------
The value of the likelihood function at iteration 18 = -2.335620E+004
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 12.636197 0.244503 51.681 159 0.000
For SES slope, B1
INTRCPT2, G10 2.193157 0.127879 17.150 159 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 12.636197 0.243738 51.843 159 0.000
For SES slope, B1
INTRCPT2, G10 2.193157 0.127846 17.155 159 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.94633 8.68087 159 1770.85120 0.000
SES slope, U1 0.82485 0.68038 159 213.43769 0.003
level-1, R 6.05835 36.70356
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 46712.398925 Number of estimated parameters = 4
Pages 337/338
Sigma_squared = 36.70313
Tau
INTRCPT1,B0 2.37996 0.19058
SES,B1 0.19058 0.14892
Tau (as correlations)
INTRCPT1,B0 1.000 0.320
SES,B1 0.320 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.733
SES, B1 0.073
----------------------------------------------------
The value of the likelihood function at iteration 61 = -2.325094E+004
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 12.096006 0.198734 60.865 157 0.000
SECTOR, G01 1.226384 0.306272 4.004 157 0.000
MEANSES, G02 5.333056 0.369161 14.446 157 0.000
For SES slope, B1
INTRCPT2, G10 2.937981 0.157135 18.697 157 0.000
SECTOR, G11 -1.640954 0.242905 -6.756 157 0.000
MEANSES, G12 1.034427 0.302566 3.419 157 0.001
----------------------------------------------------------------------------
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 12.096006 0.173699 69.638 157 0.000
SECTOR, G01 1.226384 0.308484 3.976 157 0.000
MEANSES, G02 5.333056 0.334600 15.939 157 0.000
For SES slope, B1
INTRCPT2, G10 2.937981 0.147620 19.902 157 0.000
SECTOR, G11 -1.640954 0.237401 -6.912 157 0.000
MEANSES, G12 1.034427 0.332785 3.108 157 0.002
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 1.54271 2.37996 157 605.29503 0.000
SES slope, U1 0.38590 0.14892 157 162.30867 0.369
level-1, R 6.05831 36.70313
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 46501.875635 Number of estimated parameters = 4
Page 339, model with random intercepts only, getting the -2LL value
Sigma_squared = 36.76611
Tau INTRCPT1,B0 2.37524
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.732 ----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.325148E+004
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 12.096251 0.198643 60.894 157 0.000
SECTOR, G01 1.224401 0.306117 4.000 157 0.000
MEANSES, G02 5.336698 0.368978 14.463 157 0.000
For SES slope, B1
INTRCPT2, G10 2.935860 0.150705 19.481 7179 0.000
SECTOR, G11 -1.642102 0.233097 -7.045 7179 0.000
MEANSES, G12 1.044120 0.291042 3.588 7179 0.001
----------------------------------------------------------------------------
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 12.096251 0.173691 69.642 157 0.000
SECTOR, G01 1.224401 0.308507 3.969 157 0.000
MEANSES, G02 5.336698 0.334617 15.949 157 0.000
For SES slope, B1
INTRCPT2, G10 2.935860 0.147580 19.893 7179 0.000
SECTOR, G11 -1.642102 0.237223 -6.922 7179 0.000
MEANSES, G12 1.044120 0.332897 3.136 7179 0.002
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 1.54118 2.37524 157 604.29893 0.000
level-1, R 6.06351 36.76611
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 46502.952734 Number of estimated parameters = 2
The difference in -2LL is 46502.952734 - 46501.875635 = 1.077099 on two degrees of freedom, which is not significant.
Second set of examples using willett.SSM file. To download the SSM file, you may right-click on the link and choose "Save Target As" option.
Page 342
Sigma_squared = 159.47717
Tau
INTRCPT1,B0 1198.77673 -179.25554
TIME,B1 -179.25554 132.40055
Tau (as correlations)
INTRCPT1,B0 1.000 -0.450
TIME,B1 -0.450 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.915
TIME, B1 0.806
----------------------------------------------------
The value of the likelihood function at iteration 2 = -6.334114E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 164.374286 6.118848 26.864 34 0.000
For TIME slope, B1
INTRCPT2, G10 26.960000 2.166603 12.443 34 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 164.374286 6.030803 27.256 34 0.000
For TIME slope, B1
INTRCPT2, G10 26.960000 2.135428 12.625 34 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 34.62336 1198.77673 34 399.10740 0.000
TIME slope, U1 11.50654 132.40055 34 175.13679 0.000
level-1, R 12.62843 159.47717
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 1266.822740 Number of estimated parameters = 4
Page 344
Sigma_squared = 159.47717
Tau
INTRCPT1,B0 1236.41265 -178.23322
TIME,B1 -178.23322 107.24917
Tau (as correlations)
INTRCPT1,B0 1.000 -0.489
TIME,B1 -0.489 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.917
TIME, B1 0.771
----------------------------------------------------
The value of the likelihood function at iteration 2 = -6.301424E+002 The outcome variable is Y
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 164.374285 6.206095 26.486 33 0.000
COVAR, G01 -0.113553 0.504012 -0.225 33 0.823
For TIME slope, B1
INTRCPT2, G10 26.960002 1.993880 13.521 33 0.000
COVAR, G11 0.432858 0.161928 2.673 33 0.012
----------------------------------------------------------------------------
The outcome variable is Y
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 164.374285 6.026170 27.277 33 0.000
COVAR, G01 -0.113553 0.415881 -0.273 33 0.787
For TIME slope, B1
INTRCPT2, G10 26.960002 1.936075 13.925 33 0.000
COVAR, G11 0.432858 0.121468 3.564 33 0.001
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 35.16266 1236.41265 33 398.49446 0.000
TIME slope, U1 10.35612 107.24917 33 143.96331 0.000
level-1, R 12.62843 159.47717
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 1260.284762 Number of estimated parameters = 4
Page 346-347 HLM takes multivariate approach for repeated measures and the files used for this type of analysis have extension as .MDM files. We converted willett data file into multivariate format first and created an MDM file, willett_wave.MDM for it.
Compound Symmetry: Notice that this model is different from the paper, since it has intercept as a random effect.
Parameter Standard Error
------------ ----------------
sigma_squared = 372.32114 51.385200
Tau INTRCPT1 877.16380
Tau (as correlations) INTRCPT1 1.000
Standard Errors of Tau INTRCPT1 232.28819
D _IND_1 1249.48494 877.16380 877.16380 877.16380 _IND_2 877.16380 1249.48494 877.16380 877.16380 _IND_3 877.16380 877.16380 1249.48494 877.16380 _IND_4 877.16380 877.16380 877.16380 1249.48494
D (as correlations) _IND_1 1.000 0.702 0.702 0.702 _IND_2 0.702 1.000 0.702 0.702 _IND_3 0.702 0.702 1.000 0.702 _IND_4 0.702 0.702 0.702 1.000
The value of the likelihood function at iteration 3 = -6.540559E+002
Final estimation of fixed effects:
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 164.374286 5.701600 28.830 34 0.000
For TIME slope, B1
INTRCPT2, G10 26.960000 1.458612 18.483 138 0.000
Statistics for current covariance components model -------------------------------------------------- Deviance = 1308.11172 Number of estimated parameters = 4
Summary of Model Fit
-------------------------------------------------------------------
Model Number of Deviance
Parameters
-------------------------------------------------------------------
1. Unrestricted 12 1271.86685
2. Homogeneous sigma_squared 4 1308.11172
------------------------------------------------------------------- Model Comparison Chi_square df P-value ------------------------------------------------------------------- Model 1 vs Model 2 36.24487 8 0.000
Unstructured. Notice that this model is different from the paper, since it has intercept as a random effect.
D _IND_1 1273.56708 948.57291 899.07508 546.55946 _IND_2 948.57291 1093.03078 992.98157 830.84891 _IND_3 899.07508 992.98157 1260.29185 1049.39588 _IND_4 546.55946 830.84891 1049.39588 1374.95666
D (as correlations) _IND_1 1.000 0.804 0.710 0.413 _IND_2 0.804 1.000 0.846 0.678 _IND_3 0.710 0.846 1.000 0.797 _IND_4 0.413 0.678 0.797 1.000
Standard Errors of D _IND_1 304.44076 255.89276 262.59160 242.00512 _IND_2 255.89276 261.28433 259.86536 250.32435 _IND_3 262.59160 259.86536 301.26738 284.55873 _IND_4 242.00512 250.32435 284.55873 328.67751
The value of the likelihood function at iteration 3 = -6.359334E+002
Final estimation of fixed effects:
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 165.832042 5.782419 28.679 34 0.000
For TIME slope, B1
INTRCPT2, G10 26.584650 2.090965 12.714 34 0.000
Statistics for current covariance components model -------------------------------------------------- Deviance = 1271.86685 Number of estimated parameters = 12
AR(1) HLM has trouble estimating this model, because HLM does not allow a model without any random effect.
Page 348.
Parameter Standard Error
------------ ----------------
rho = -0.14146 0.257333
sigma_squared = 140.89501 36.789326
Tau
INTRCPT1 1181.85384 -169.89054
TIME -169.89054 103.23221
Tau (as correlations)
INTRCPT1 1.000 -0.486
TIME -0.486 1.000
Standard Errors of Tau
INTRCPT1 306.17471 78.71459
TIME 78.71459 32.79528
D _IND_1 1322.74885 992.03261 844.89210 671.78339 _IND_2 992.03261 1086.19998 858.71595 814.80765 _IND_3 844.89210 858.71595 1056.11553 931.86371 _IND_4 671.78339 814.80765 931.86371 1232.49550
D (as correlations) _IND_1 1.000 0.828 0.715 0.526 _IND_2 0.828 1.000 0.802 0.704 _IND_3 0.715 0.802 1.000 0.817 _IND_4 0.526 0.704 0.817 1.000
The value of the likelihood function at iteration 6 = -6.333202E+002
Final estimation of fixed effects:
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 164.424411 6.019069 27.317 33 0.000
COVAR, G01 -0.123696 0.488823 -0.253 33 0.802
For TIME slope, B1
INTRCPT2, G10 26.906451 1.919652 14.016 33 0.000
COVAR, G11 0.435826 0.155900 2.796 33 0.009
Statistics for current covariance components model -------------------------------------------------- Deviance = 1266.64047 Number of estimated parameters = 9
Summary of Model Fit
-------------------------------------------------------------------
Model Number of Deviance
Parameters
-------------------------------------------------------------------
1. Unrestricted 14 1262.13753
2. Homogeneous sigma_squared 8 1266.90091
3. First-order Autoregressive 9 1266.64047
------------------------------------------------------------------- Model Comparison Chi_square df P-value ------------------------------------------------------------------- Model 1 vs Model 2 4.76338 6 >.500 Model 1 vs Model 3 4.50294 5 >.500 Model 2 vs Model 3 0.26044 1 >.500
The content of this web site should not be construed as an endorsement of any particular web site, book, or software product by the University of California.