#### HLM Textbook Examples Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence  by Judith D. Singer and John B. Willett Chapter 5: Treating time more flexibly

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

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.

Model A: Initial growth model.

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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