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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.
Table 5.2, page 145
Table 5.2 uses data file reading_pp_with_centered_vars.sav.
Model A: Using AGEGRP-6.5 as a temporal predictor. This centered variable has been created in SPSS. It is called cagegrp.
Sigma_squared = 27.04307
Standard Error of Sigma_squared = 4.05393
Tau INTRCPT1,B0 11.04586 1.64665 CAGEGROP,B1 1.64665 4.39745
Standard Errors of Tau INTRCPT1,B0 6.06260 2.06295 CAGEGROP,B1 2.06295 1.27131
Tau (as correlations) INTRCPT1,B0 1.000 0.236 CAGEGROP,B1 0.236 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.329 CAGEGROP, B1 0.565 ----------------------------------------------------
The value of the likelihood function at iteration 2 = -9.090557E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 21.162921 0.614266 34.452 88 0.000
For CAGEGROP slope, B1
INTRCPT2, G10 5.030899 0.295620 17.018 88 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 21.162921 0.614266 34.452 88 0.000
For CAGEGROP slope, B1
INTRCPT2, G10 5.030899 0.295620 17.018 88 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 3.32353 11.04586 88 132.62294 0.002
CAGEGROP slope, U1 2.09701 4.39745 88 204.77777 0.000
level-1, R 5.20030 27.04307
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 1818.111416 Number of estimated parameters = 6
Model B: Using AGE-6.5 as a temporal predictor. This centered group has been created already and it is called cage.
Sigma_squared = 27.35397
Standard Error of Sigma_squared = 4.06532
Tau
INTRCPT1,B0 5.28326 2.32384
CAGE,B1 2.32384 3.31511
Standard Errors of Tau
INTRCPT1,B0 5.31505 1.68580
CAGE,B1 1.68580 0.99120
Tau (as correlations)
INTRCPT1,B0 1.000 0.555
CAGE,B1 0.555 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.189
CAGE, B1 0.546
----------------------------------------------------
The value of the likelihood function at iteration 13 = -9.010293E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 21.061431 0.560356 37.586 88 0.000
For CAGE slope, B1
INTRCPT2, G10 4.539983 0.260733 17.412 88 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 21.061431 0.559546 37.640 88 0.000
For CAGE slope, B1
INTRCPT2, G10 4.539983 0.260654 17.418 88 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 2.29853 5.28326 88 114.02426 0.032
CAGE slope, U1 1.82075 3.31511 88 201.45208 0.000
level-1, R 5.23010 27.35397
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 1802.058585 Number of estimated parameters = 6
Table 5.4, page 149 uses wages.sav file.
Model A:
Sigma_squared = 0.09510
Standard Error of Sigma_squared = 0.00192
Tau
INTRCPT1,B0 0.05428 -0.00292
EXPER,B1 -0.00292 0.00173
Standard Errors of Tau
INTRCPT1,B0 0.00487 0.00084
EXPER,B1 0.00084 0.00020
Tau (as correlations)
INTRCPT1,B0 1.000 -0.301
EXPER,B1 -0.301 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.477
EXPER, B1 0.353
----------------------------------------------------
Note: The reliability estimates reported above are based on only 850 of 888 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 17 = -2.460697E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.715603 0.010797 158.894 887 0.000
For EXPER slope, B1
INTRCPT2, G10 0.045681 0.002342 19.504 887 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 1.715603 0.010797 158.902 887 0.000
For EXPER slope, B1
INTRCPT2, G10 0.045681 0.002342 19.508 887 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.23298 0.05428 849 1787.79896 0.000
EXPER slope, U1 0.04156 0.00173 849 1647.11657 0.000
level-1, R 0.30838 0.09510
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 850 of 888 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 = 4921.394319 Number of estimated parameters = 6
Model B:
Sigma_squared = 0.09519
Standard Error of Sigma_squared = 0.00192
Tau
INTRCPT1,B0 0.05176 -0.00286
EXPER,B1 -0.00286 0.00164
Standard Errors of Tau
INTRCPT1,B0 0.00474 0.00082
EXPER,B1 0.00082 0.00020
Tau (as correlations)
INTRCPT1,B0 1.000 -0.310
EXPER,B1 -0.310 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.466
EXPER, B1 0.344
----------------------------------------------------
Note: The reliability estimates reported above are based on only 850 of 888 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 17 = -2.436876E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.717138 0.012543 136.899 885 0.000
HGC_9, G01 0.034922 0.007882 4.431 885 0.000
BLACK, G02 0.015396 0.023928 0.643 885 0.520
For EXPER slope, B1
INTRCPT2, G10 0.049343 0.002632 18.744 885 0.000
HGC_9, G11 0.001279 0.001724 0.742 885 0.458
BLACK, G12 -0.018213 0.005501 -3.311 885 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 1.717138 0.011954 143.645 885 0.000
HGC_9, G01 0.034922 0.007900 4.420 885 0.000
BLACK, G02 0.015396 0.025805 0.597 885 0.550
For EXPER slope, B1
INTRCPT2, G10 0.049343 0.002576 19.154 885 0.000
HGC_9, G11 0.001279 0.001830 0.699 885 0.485
BLACK, G12 -0.018213 0.005686 -3.203 885 0.002
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.22750 0.05176 847 1753.69961 0.000
EXPER slope, U1 0.04047 0.00164 847 1623.65666 0.000
level-1, R 0.30853 0.09519
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 850 of 888 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 = 4873.751523 Number of estimated parameters = 10
Model C:
Sigma_squared = 0.09517
Standard Error of Sigma_squared = 0.00192
Tau
INTRCPT1,B0 0.05184 -0.00288
EXPER,B1 -0.00288 0.00165
Standard Errors of Tau
INTRCPT1,B0 0.00474 0.00082
EXPER,B1 0.00082 0.00020
Tau (as correlations)
INTRCPT1,B0 1.000 -0.312
EXPER,B1 -0.312 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.466
EXPER, B1 0.345
----------------------------------------------------
Note: The reliability estimates reported above are based on only 850 of 888 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 17 = -2.437352E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.721474 0.010698 160.920 886 0.000
HGC_9, G01 0.038360 0.006433 5.963 886 0.000
For EXPER slope, B1
INTRCPT2, G10 0.048847 0.002514 19.430 886 0.000
BLACK, G11 -0.016115 0.004512 -3.571 886 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 1.721474 0.010777 159.736 886 0.000
HGC_9, G01 0.038360 0.006523 5.880 886 0.000
For EXPER slope, B1
INTRCPT2, G10 0.048847 0.002515 19.423 886 0.000
BLACK, G11 -0.016115 0.004326 -3.725 886 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.22768 0.05184 848 1755.87073 0.000
EXPER slope, U1 0.04060 0.00165 848 1627.08336 0.000
level-1, R 0.30850 0.09517
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 850 of 888 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 = 4874.703577 Number of estimated parameters = 8
Figure 5.2 on page 150 using Model C from previous example.
Table 5.5, page 154
Table 5.5: Severely unbalanced data using wages_small.sav, running Model C from pervious analysis.
Method 1: Using automatic fix-up from HLM 6.06. The automatic fix-up allows negative covariance.
******* ITERATION 6481 *******
Sigma_squared = 0.11494
Standard Error of Sigma_squared = 0.01452
Tau
INTRCPT1,B0 0.08230 0.00040
EXPER,B1 0.00040 0.00007
Standard Errors of Tau
INTRCPT1,B0 0.03492 0.01373
EXPER,B1 0.01373 0.00486
Tau (as correlations)
INTRCPT1,B0 1.000 0.167
EXPER,B1 0.167 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.196
EXPER, B1 0.001
----------------------------------------------------
Note: The reliability estimates reported above are based on only 86 of 124
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 6481 = -1.410252E+002
The outcome variable is LNW
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.737334 0.047617 36.486 122 0.000
HGC_9, G01 0.046080 0.024474 1.883 122 0.062
For EXPER slope, B1
INTRCPT2, G10 0.051562 0.021113 2.442 122 0.016
BLACK, G11 -0.059512 0.034786 -1.711 122 0.089
----------------------------------------------------------------------------
The outcome variable is LNW
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 1.737334 0.046911 37.035 122 0.000
HGC_9, G01 0.046080 0.025071 1.838 122 0.068
For EXPER slope, B1
INTRCPT2, G10 0.051562 0.019773 2.608 122 0.011
BLACK, G11 -0.059512 0.032093 -1.854 122 0.066
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.28689 0.08230 84 95.44909 0.185
EXPER slope, U1 0.00843 0.00007 84 93.23607 0.230
level-1, R 0.33902 0.11494
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 86 of 124
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 = 282.050499
Number of estimated parameters = 8
Method 2: Fixing rates of change
Sigma_squared = 0.11479
Standard Error of Sigma_squared = 0.01385
Tau INTRCPT1,B0 0.08428
Standard Errors of Tau INTRCPT1,B0 0.01945
Tau (as correlations) INTRCPT1,B0 1.000
---------------------------------------------------- Random level-1 coefficient Reliability estimate ---------------------------------------------------- INTRCPT1, B0 0.578 ----------------------------------------------------
The value of the likelihood function at iteration 13 = -1.410197E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.737349 0.047755 36.381 122 0.000
HGC_9, G01 0.045757 0.024499 1.868 122 0.061
For EXPER slope, B1
INTRCPT2, G10 0.051780 0.020935 2.473 253 0.014
BLACK, G11 -0.060078 0.034581 -1.737 253 0.082
----------------------------------------------------------------------------
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 1.737349 0.046911 37.035 122 0.000
HGC_9, G01 0.045757 0.025159 1.819 122 0.068
For EXPER slope, B1
INTRCPT2, G10 0.051780 0.019667 2.633 253 0.009
BLACK, G11 -0.060078 0.032014 -1.877 253 0.060
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.29031 0.08428 122 308.28132 0.000
level-1, R 0.33880 0.11479
-----------------------------------------------------------------------------
Statistics for current covariance components model -------------------------------------------------- Deviance = 282.039344 Number of estimated parameters = 6
Model A: Initial growth model.Table 5.7, page 163 using file unemp1sav. The interaction term of level-1 variable month with level-1 variable unemp has to be created in SPSS first.
Sigma_squared = 68.84760
Standard Error of Sigma_squared = 6.64275
Tau INTRCPT1,B0 86.85172 -3.05790 MONTHS,B1 -3.05790 0.35516
Standard Errors of Tau INTRCPT1,B0 14.87235 1.36915 MONTHS,B1 1.36915 0.18349
Tau (as correlations) INTRCPT1,B0 1.000 -0.551 MONTHS,B1 -0.551 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.562
MONTHS, B1 0.215
----------------------------------------------------
Note: The reliability estimates reported above are based on only 227 of 254 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 12 = -2.566569E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 17.669366 0.775565 22.783 253 0.000
For MONTHS slope, B1
INTRCPT2, G10 -0.421996 0.082981 -5.085 253 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 17.669366 0.775564 22.783 253 0.000
For MONTHS slope, B1
INTRCPT2, G10 -0.421996 0.082979 -5.086 253 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 9.31943 86.85172 226 501.97897 0.000
MONTHS slope, U1 0.59595 0.35516 226 288.21447 0.003
level-1, R 8.29745 68.84760
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 227 of 254 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 = 5133.137432 Number of estimated parameters = 6
Model B: Main effect of unemployment.
Sigma_squared = 62.38632
Standard Error of Sigma_squared = 6.03235
Tau INTRCPT1,B0 93.52127 -3.89455 MONTHS,B1 -3.89455 0.46478
Standard Errors of Tau INTRCPT1,B0 14.70550 1.35009 MONTHS,B1 1.35009 0.17804
Tau (as correlations) INTRCPT1,B0 1.000 -0.591 MONTHS,B1 -0.591 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.603
MONTHS, B1 0.282
----------------------------------------------------
Note: The reliability estimates reported above are based on only 227 of 254 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 12 = -2.553802E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 12.665553 1.242072 10.197 253 0.000
For MONTHS slope, B1
INTRCPT2, G10 -0.201983 0.093318 -2.164 253 0.030
For UNEMP slope, B2
INTRCPT2, G20 5.111351 0.988844 5.169 671 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 12.665553 1.140832 11.102 253 0.000
For MONTHS slope, B1
INTRCPT2, G10 -0.201983 0.093598 -2.158 253 0.031
For UNEMP slope, B2
INTRCPT2, G20 5.111351 0.919029 5.562 671 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 9.67064 93.52127 226 551.38649 0.000
MONTHS slope, U1 0.68175 0.46478 226 315.17536 0.000
level-1, R 7.89850 62.38632
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 227 of 254 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 = 5107.603283 Number of estimated parameters = 7
Model C: Effect of unemployment on initial status and growth rate.
Sigma_squared = 62.02980
Standard Error of Sigma_squared = 5.99668
Tau INTRCPT1,B0 93.71534 -3.87355 MONTHS,B1 -3.87355 0.45128
Standard Errors of Tau INTRCPT1,B0 14.68016 1.34112 MONTHS,B1 1.34112 0.17608
Tau (as correlations) INTRCPT1,B0 1.000 -0.596 MONTHS,B1 -0.596 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.605
MONTHS, B1 0.277
----------------------------------------------------
Note: The reliability estimates reported above are based on only 227 of 254 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 12 = -2.551523E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 9.616767 1.889295 5.090 253 0.000
For MONTHS slope, B1
INTRCPT2, G10 0.162032 0.193662 0.837 253 0.403
For UNEMP slope, B2
INTRCPT2, G20 8.529033 1.877857 4.542 670 0.000
For MXUNEMP slope, B3
INTRCPT2, G30 -0.465218 0.217215 -2.142 670 0.032
----------------------------------------------------------------------------
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 9.616767 1.596038 6.025 253 0.000
For MONTHS slope, B1
INTRCPT2, G10 0.162032 0.160972 1.007 253 0.315
For UNEMP slope, B2
INTRCPT2, G20 8.529033 1.655325 5.152 670 0.000
For MXUNEMP slope, B3
INTRCPT2, G30 -0.465218 0.198188 -2.347 670 0.019
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 9.68067 93.71534 226 554.39081 0.000
MONTHS slope, U1 0.67177 0.45128 226 313.16280 0.000
level-1, R 7.87590 62.02980
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 227 of 254 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 = 5103.046912 Number of estimated parameters = 8
Model D: Allowing unemployment to have both fix and random effects. The model does not run in HLM 6.06 and here is the message in the HLM output: "There are no degrees of freedom to estimate sigma_squared. Set sigma_squared to a constant, or delete one or more of the random effects from the model."
Figure 5.4, Model B on page 167. For Model C, since the interaction is between two level-1 variables, it is not possible to graph using HLM. Model D dose not run in HLM.
Table 5.8, page 175, High school dropout wage data using wages.sav --Effects of time-varying unemployment rate.
Model A: Unemployment rate centered around 7.
Sigma_squared = 0.09479
Standard Error of Sigma_squared = 0.00191
Tau
INTRCPT1,B0 0.05065 -0.00292
EXPER,B1 -0.00292 0.00163
Standard Errors of Tau
INTRCPT1,B0 0.00467 0.00081
EXPER,B1 0.00081 0.00019
Tau (as correlations)
INTRCPT1,B0 1.000 -0.321
EXPER,B1 -0.321 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.462
EXPER, B1 0.344
----------------------------------------------------
Note: The reliability estimates reported above are based on only 850 of 888
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 17 = -2.415260E+003
The outcome variable is LNW
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.748988 0.011400 153.421 886 0.000
HGC_9, G01 0.040010 0.006363 6.288 886 0.000
For EXPER slope, B1
INTRCPT2, G10 0.044054 0.002604 16.918 886 0.000
BLACK, G11 -0.018183 0.004485 -4.055 886 0.000
For UE_7 slope, B2
INTRCPT2, G20 -0.011950 0.001792 -6.670 6397 0.000
----------------------------------------------------------------------------
The outcome variable is LNW
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 1.748988 0.011429 153.033 886 0.000
HGC_9, G01 0.040010 0.006435 6.218 886 0.000
For EXPER slope, B1
INTRCPT2, G10 0.044054 0.002532 17.396 886 0.000
BLACK, G11 -0.018183 0.004307 -4.222 886 0.000
For UE_7 slope, B2
INTRCPT2, G20 -0.011950 0.002002 -5.969 6397 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.22505 0.05065 848 1740.10878 0.000
EXPER slope, U1 0.04041 0.00163 848 1630.45144 0.000
level-1, R 0.30788 0.09479
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 850 of 888
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 = 4830.519088
Number of estimated parameters = 9
Model B: Within context centering
Sigma_squared = 0.09480
Standard Error of Sigma_squared = 0.00191
Tau
INTRCPT1,B0 0.05102 -0.00303
EXPER,B1 -0.00303 0.00163
Standard Errors of Tau
INTRCPT1,B0 0.00469 0.00081
EXPER,B1 0.00081 0.00019
Tau (as correlations)
INTRCPT1,B0 1.000 -0.332
EXPER,B1 -0.332 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.463
EXPER, B1 0.344
----------------------------------------------------
Note: The reliability estimates reported above are based on only 850 of 888 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 17 = -2.413489E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.874252 0.029524 63.483 887 0.000
For EXPER slope, B1
INTRCPT2, G10 0.045057 0.002650 17.000 886 0.000
BLACK, G11 -0.018870 0.004478 -4.214 886 0.000
For HGC_9 slope, B2
INTRCPT2, G20 0.040169 0.006351 6.325 6396 0.000
For UE_MEAN slope, B3
INTRCPT2, G30 -0.017708 0.003520 -5.030 6396 0.000
For UE_PERSO slope, B4
INTRCPT2, G40 -0.009902 0.002097 -4.721 6396 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 1.874252 0.030807 60.838 887 0.000
For EXPER slope, B1
INTRCPT2, G10 0.045057 0.002568 17.548 886 0.000
BLACK, G11 -0.018870 0.004296 -4.393 886 0.000
For HGC_9 slope, B2
INTRCPT2, G20 0.040169 0.006425 6.252 6396 0.000
For UE_MEAN slope, B3
INTRCPT2, G30 -0.017708 0.003822 -4.633 6396 0.000
For UE_PERSO slope, B4
INTRCPT2, G40 -0.009902 0.002408 -4.111 6396 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.22588 0.05102 849 1746.61337 0.000
EXPER slope, U1 0.04038 0.00163 848 1629.32384 0.000
level-1, R 0.30789 0.09480
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 850 of 888 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 = 4826.978556 Number of estimated parameters = 10
Model C: Centering on time-1
Sigma_squared = 0.09476
Standard Error of Sigma_squared = 0.00191
Tau
INTRCPT1,B0 0.05029 -0.00291
EXPER,B1 -0.00291 0.00164
Standard Errors of Tau
INTRCPT1,B0 0.00466 0.00081
EXPER,B1 0.00081 0.00019
Tau (as correlations)
INTRCPT1,B0 1.000 -0.320
EXPER,B1 -0.320 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.460
EXPER, B1 0.344
----------------------------------------------------
Note: The reliability estimates reported above are based on only 850 of 888
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 17 = -2.412921E+003
The outcome variable is LNW
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 1.869345 0.026032 71.810 885 0.000
HGC_9, G01 0.039927 0.006349 6.288 885 0.000
UE1, G02 -0.016177 0.002648 -6.108 885 0.000
For EXPER slope, B1
INTRCPT2, G10 0.044765 0.002626 17.050 886 0.000
BLACK, G11 -0.018324 0.004486 -4.085 886 0.000
For UE_CENTE slope, B2
INTRCPT2, G20 -0.010309 0.001944 -5.302 6396 0.000
----------------------------------------------------------------------------
The outcome variable is LNW
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 1.869345 0.025514 73.268 885 0.000
HGC_9, G01 0.039927 0.006416 6.223 885 0.000
UE1, G02 -0.016177 0.002618 -6.180 885 0.000
For EXPER slope, B1
INTRCPT2, G10 0.044765 0.002551 17.549 886 0.000
BLACK, G11 -0.018324 0.004319 -4.243 886 0.000
For UE_CENTE slope, B2
INTRCPT2, G20 -0.010309 0.002278 -4.525 6396 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.22425 0.05029 847 1736.02864 0.000
EXPER slope, U1 0.04046 0.00164 848 1630.02882 0.000
level-1, R 0.30784 0.09476
-----------------------------------------------------------------------------
Note: The chi-square statistics reported above are based on only 850 of 888
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 = 4825.841693
Number of estimated parameters = 10
Table 5.10, page 184 using medication_pp.sav.
Model A: TIME
Sigma_squared = 1229.93078
Standard Error of Sigma_squared = 52.08495
Tau
INTRCPT1,B0 2111.36261 -121.61411
TIME,B1 -121.61411 63.73437
Standard Errors of Tau
INTRCPT1,B0 418.88535 58.95872
TIME,B1 58.95872 14.27753
Tau (as correlations)
INTRCPT1,B0 1.000 -0.332
TIME,B1 -0.332 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.883
TIME, B1 0.782
----------------------------------------------------
The value of the likelihood function at iteration 8 = -6.340226E+003
The outcome variable is POS
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 167.463443 9.326039 17.957 62 0.000
TREAT, G01 -3.109305 12.332266 -0.252 62 0.802
For TIME slope, B1
INTRCPT2, G10 -2.418118 1.730814 -1.397 62 0.167
TREAT, G11 5.536808 2.277795 2.431 62 0.018
----------------------------------------------------------------------------
The outcome variable is POS
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 167.463443 9.061094 18.482 62 0.000
TREAT, G01 -3.109305 12.240884 -0.254 62 0.800
For TIME slope, B1
INTRCPT2, G10 -2.418118 1.531620 -1.579 62 0.119
TREAT, G11 5.536808 2.207681 2.508 62 0.015
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 45.94957 2111.36261 62 629.96487 0.000
TIME slope, U1 7.98338 63.73437 62 315.00705 0.000
level-1, R 35.07037 1229.93078
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 12680.451557
Number of estimated parameters = 8
Model B: TIME-3.33
Sigma_squared = 1229.93106
Standard Error of Sigma_squared = 52.08496
Tau
INTRCPT1,B0 2008.75916 90.83527
TIME333,B1 90.83527 63.73439
Standard Errors of Tau
INTRCPT1,B0 367.89696 52.44551
TIME333,B1 52.44551 14.27753
Tau (as correlations)
INTRCPT1,B0 1.000 0.254
TIME333,B1 0.254 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.954
TIME333, B1 0.782
----------------------------------------------------
The value of the likelihood function at iteration 7 = -6.340226E+003
The outcome variable is POS
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 159.403049 8.764424 18.188 62 0.000
TREAT, G01 15.346713 11.544537 1.329 62 0.189
For TIME333 slope, B1
INTRCPT2, G10 -2.418119 1.730816 -1.397 62 0.167
TREAT, G11 5.536812 2.277798 2.431 62 0.018
----------------------------------------------------------------------------
The outcome variable is POS
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 159.403049 7.095204 22.466 62 0.000
TREAT, G01 15.346713 10.990788 1.396 62 0.168
For TIME333 slope, B1
INTRCPT2, G10 -2.418119 1.531619 -1.579 62 0.119
TREAT, G11 5.536812 2.207681 2.508 62 0.015
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 44.81918 2008.75916 62 2052.73726 0.000
TIME333 slope, U1 7.98338 63.73439 62 315.00698 0.000
level-1, R 35.07037 1229.93106
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 12680.451557
Number of estimated parameters = 8
Model C: TIME-6.67
Sigma_squared = 1229.92983
Standard Error of Sigma_squared = 52.08491
Tau
INTRCPT1,B0 3322.48179 303.28462
TIME667,B1 303.28462 63.73585
Standard Errors of Tau
INTRCPT1,B0 633.25472 80.96293
TIME667,B1 80.96293 14.27779
Tau (as correlations)
INTRCPT1,B0 1.000 0.659
TIME667,B1 0.659 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.925
TIME667, B1 0.782
----------------------------------------------------
The value of the likelihood function at iteration 10 = -6.340226E+003
The outcome variable is POS
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 151.342917 11.542088 13.112 62 0.000
TREAT, G01 33.802520 15.157593 2.230 62 0.029
For TIME667 slope, B1
INTRCPT2, G10 -2.418062 1.730569 -1.397 62 0.167
TREAT, G11 5.536760 2.277472 2.431 62 0.018
----------------------------------------------------------------------------
The outcome variable is POS
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 151.342917 8.408985 17.998 62 0.000
TREAT, G01 33.802520 14.144390 2.390 62 0.020
For TIME667 slope, B1
INTRCPT2, G10 -2.418062 1.531621 -1.579 62 0.119
TREAT, G11 5.536760 2.207686 2.508 62 0.015
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 57.64097 3322.48179 62 931.61037 0.000
TIME667 slope, U1 7.98347 63.73585 62 315.00719 0.000
level-1, R 35.07036 1229.92983
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 12680.451561
Number of estimated parameters = 8
page 188, using the variable initial and the variable final as the level-1 predictors. Click on INTRCPT1 to remove the intercept.
Initial and Final model in text
Sigma_squared = 1229.93033
Standard Error of Sigma_squared = 52.08493
Tau
INITIAL,B1 2111.37414 1300.59248
FINAL,B2 1300.59248 3322.48597
Standard Errors of Tau
INITIAL,B1 418.88740 393.74865
FINAL,B2 393.74865 633.25547
Tau (as correlations)
INITIAL,B1 1.000 0.491
FINAL,B2 0.491 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INITIAL, B1 0.883
FINAL, B2 0.925
----------------------------------------------------
The value of the likelihood function at iteration 7 = -6.340226E+003
The outcome variable is POS
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INITIAL slope, B1
INTRCPT2, G10 167.463454 9.326091 17.956 62 0.000
TREAT, G11 -3.109313 12.332335 -0.252 62 0.802
For FINAL slope, B2
INTRCPT2, G20 151.342629 11.542480 13.112 62 0.000
TREAT, G21 33.802766 15.158110 2.230 62 0.029
----------------------------------------------------------------------------
The outcome variable is POS
Final estimation of fixed effects
(with robust standard errors)
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INITIAL slope, B1
INTRCPT2, G10 167.463454 9.061097 18.482 62 0.000
TREAT, G11 -3.109313 12.240889 -0.254 62 0.800
For FINAL slope, B2
INTRCPT2, G20 151.342629 8.408955 17.998 62 0.000
TREAT, G21 33.802766 14.144363 2.390 62 0.020
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INITIAL slope, U1 45.94969 2111.37414 62 629.96510 0.000
FINAL slope, U2 57.64101 3322.48597 62 931.61012 0.000
level-1, R 35.07036 1229.93033
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 12680.451557
Number of estimated parameters = 8
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