Table 5.1 on page 78, Table 5.2 on page 79 and Figure 5.3 using data file gpa.sav. These tables are produced using SPSS.
GLM gpa1 gpa2 gpa3 gpa4 gpa5 gpa6 BY sex WITH highgpa /WSFACTOR = gpa 6 Polynomial /METHOD = SSTYPE(3) /PLOT = PROFILE( gpa*sex ) /PRINT = DESCRIPTIVE /CRITERIA = ALPHA(.05) /WSDESIGN = gpa /DESIGN = highgpa sex .
Table 5.2:
Figure 5.3 on page 80.
GET FILE='E:hoxspssgpaflat.sav'. GRAPH /HISTOGRAM(NORMAL)=gpa .
Table 5.3 on page 81. Data sets we are going to use is https://stats.idre.ucla.edu/wp-content/uploads/2016/02/gpa4chp5.sav and gpa4ch5_lev2.sav and we assume that an SSM file has been created based on these data files. The variable student links the two data sets.
Part 1: Null model.
Sigma_squared = 0.09759
Tau
INTRCPT1,B0 0.05714
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.778
----------------------------------------------------
The value of the likelihood function at iteration 2 = -4.588466E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.865000 0.019157 149.551 199 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 2.865000 0.019109 149.927 199 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.23903 0.05714 199 898.04285 0.000
level-1, R 0.31239 0.09759
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 917.693135
Number of estimated parameters = 2
Part 2: Additional variable time is added.
Sigma_squared = 0.05809
Tau
INTRCPT1,B0 0.06372
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.868
----------------------------------------------------
The value of the likelihood function at iteration 4 = -2.044464E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.599214 0.021697 119.799 199 0.000
For TIME slope, B1
INTRCPT2, G10 0.106314 0.004074 26.096 1198 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 2.599214 0.018311 141.947 199 0.000
For TIME slope, B1
INTRCPT2, G10 0.106314 0.005870 18.111 1198 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.25243 0.06372 199 1508.73327 0.000
level-1, R 0.24102 0.05809
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 408.892880
Number of estimated parameters = 2
Part 3: The variable job is added to the model as a fixed effect.
Sigma_squared = 0.05572
Tau
INTRCPT1,B0 0.05368
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.853
----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.700095E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.945613 0.044520 66.164 199 0.000
For JOB slope, B1
INTRCPT2, G10 -0.160760 0.018375 -8.749 1197 0.000
For TIME slope, B2
INTRCPT2, G20 0.103168 0.004006 25.753 1197 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 2.945613 0.040529 72.680 199 0.000
For JOB slope, B1
INTRCPT2, G10 -0.160760 0.016312 -9.855 1197 0.000
For TIME slope, B2
INTRCPT2, G20 0.103168 0.005512 18.715 1197 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.23169 0.05368 199 1349.00558 0.000
level-1, R 0.23604 0.05572
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 340.019023
Number of estimated parameters = 2
Part 4: The variables sex and highgpa are added.
Sigma_squared = 0.05572
Tau
INTRCPT1,B0 0.04661
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.834
----------------------------------------------------
The value of the likelihood function at iteration 7 = -1.623520E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.464035 0.112080 21.985 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.148417 0.033577 4.420 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.085856 0.028198 3.045 1195 0.003
For JOB slope, B3
INTRCPT2, G30 -0.161340 0.018328 -8.803 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103157 0.004006 25.750 1195 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 2.464035 0.105358 23.387 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.148417 0.033188 4.472 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.085856 0.026782 3.206 1195 0.002
For JOB slope, B3
INTRCPT2, G30 -0.161340 0.016274 -9.914 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103157 0.005509 18.726 1195 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.21590 0.04661 199 1185.49373 0.000
level-1, R 0.23605 0.05572
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 324.704078
Number of estimated parameters = 2
Table 5.4 on page 83.
Part 1: The variable time is a random effect.
Sigma_squared = 0.04180
Tau
INTRCPT1,B0 0.03978 -0.00260
TIME,B4 -0.00260 0.00395
Tau (as correlations)
INTRCPT1,B0 1.000 -0.208
TIME,B4 -0.208 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.645
TIME, B4 0.623
----------------------------------------------------
The value of the likelihood function at iteration 13 = -1.050516E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.409738 0.105786 22.779 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116763 0.031778 3.674 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089846 0.026681 3.367 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119271 0.017477 -6.824 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103980 0.005639 18.440 199 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 2.409738 0.099731 24.162 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116763 0.031295 3.731 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089846 0.024903 3.608 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119271 0.015345 -7.773 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103980 0.005591 18.597 199 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.19945 0.03978 199 555.76390 0.000
TIME slope, U4 0.06282 0.00395 199 527.44573 0.000
level-1, R 0.20446 0.04180
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 210.103251
Number of estimated parameters = 4
Part 2: Cross level interaction of variable time and gender is included. The column with standardized coefficients can be calculated manually and is omitted here.
Sigma_squared = 0.04181
Tau
INTRCPT1,B0 0.03942 -0.00235
TIME,B4 -0.00235 0.00376
Tau (as correlations)
INTRCPT1,B0 1.000 -0.194
TIME,B4 -0.194 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.643
TIME, B4 0.611
----------------------------------------------------
The value of the likelihood function at iteration 14 = -1.060738E+002
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.472653 0.108253 22.841 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.076821 0.035154 2.185 1194 0.029
For HIGHGPA slope, B2
INTRCPT2, G20 0.089820 0.026674 3.367 1194 0.001
For JOB slope, B3
INTRCPT2, G30 -0.120165 0.017449 -6.887 1194 0.000
For TIME slope, B4
INTRCPT2, G40 0.059017 0.017819 3.312 198 0.001
SEX_1, G41 0.029472 0.011101 2.655 198 0.008
----------------------------------------------------------------------------
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 2.472653 0.102489 24.126 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.076821 0.034651 2.217 1194 0.027
For HIGHGPA slope, B2
INTRCPT2, G20 0.089820 0.024893 3.608 1194 0.001
For JOB slope, B3
INTRCPT2, G30 -0.120165 0.015436 -7.784 1194 0.000
For TIME slope, B4
INTRCPT2, G40 0.059017 0.017913 3.295 198 0.001
SEX_1, G41 0.029472 0.011062 2.664 198 0.008
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.19855 0.03942 199 551.91797 0.000
TIME slope, U4 0.06129 0.00376 198 508.91753 0.000
level-1, R 0.20449 0.04181
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 212.147638
Number of estimated parameters = 4
Figure 5.4 on page 84 based on Table 5.4 Part 2. Before the model is run, we have to request data for graphing equations. This is done through Basic Model Specifications by simply clicking on Graph equations.
Then we can run the model and we will be able to use the Graph Equations option from the File menu.
Select variables as shown from the above screen shot, we can produce the following plot. Notice that the title for the graph and the x-axis and y-axis can be modified by clicking on Range/Titles/Color. The dashed red line corresponds to the group of SEX = 2 and SEX_1 = 2 (level 2 variable of SEX), the female group and the solid blue line corresponds to the group of SEX = 1 and SEX_1 = 1, the male group.
Table 5.5 on page 85. This requires some recoding of the variable time. Since HLM does not provide any data management facility, we will do the recoding using SPSS. The new data file will be https://stats.idre.ucla.edu/wp-content/uploads/2016/02/gpa4chp5_time.sav and the level 2 data file is the same as before.
Part 1: 1st occasion = 0, same as the first part of Table 5.4. We omit it here.
Part 2: The variable time has been recoded as -5, …,-1, 0, …with the last occasion coded as zero as variable time1.
Sigma_squared = 0.04180
Tau
INTRCPT1,B0 0.11240 0.01713
TIME1,B4 0.01713 0.00395
Tau (as correlations)
INTRCPT1,B0 1.000 0.813
TIME1,B4 0.813 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.837
TIME1, B4 0.623
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.929631 0.106862 27.415 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116769 0.031779 3.674 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089845 0.026681 3.367 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119271 0.017477 -6.824 1195 0.000
For TIME1 slope, B4
INTRCPT2, G40 0.103980 0.005639 18.440 199 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 2.929631 0.102510 28.579 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116769 0.031295 3.731 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089845 0.024903 3.608 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119271 0.015345 -7.773 1195 0.000
For TIME1 slope, B4
INTRCPT2, G40 0.103980 0.005591 18.597 199 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.33525 0.11240 199 1215.34816 0.000
TIME1 slope, U4 0.06282 0.00395 199 527.45147 0.000
level-1, R 0.20446 0.04180
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 210.103292
Number of estimated parameters = 4
Part 3: The variable time is recoded centered around its mean and is included as a fixed effect as variable timec.
Sigma_squared = 0.04180
Tau
INTRCPT1,B0 0.05142 0.00726
TIMEC,B4 0.00726 0.00395
Tau (as correlations)
INTRCPT1,B0 1.000 0.510
TIMEC,B4 0.510 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.881
TIMEC, B4 0.623
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.669686 0.105387 25.332 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116766 0.031778 3.674 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089845 0.026681 3.367 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119272 0.017477 -6.824 1195 0.000
For TIMEC slope, B4
INTRCPT2, G40 0.103980 0.005639 18.441 199 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 2.669686 0.100159 26.654 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116766 0.031295 3.731 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089845 0.024903 3.608 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119272 0.015345 -7.773 1195 0.000
For TIMEC slope, B4
INTRCPT2, G40 0.103980 0.005591 18.597 199 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.22676 0.05142 199 1652.91589 0.000
TIMEC slope, U4 0.06282 0.00395 199 527.44545 0.000
level-1, R 0.20446 0.04180
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 210.103247
Number of estimated parameters = 4
Table 5.7 on page 89 using children’s vocabulary data. Variable study has been recoded into variable sdummy. The variable sdummy is -.5 for the first study and .5 for the second study. We have created a level-1 SPSS data file https://stats.idre.ucla.edu/wp-content/uploads/2016/02/vocab_sdummy.sav and a level-2 SPSS https://stats.idre.ucla.edu/wp-content/uploads/2016/02/vocab_l2.sav data file. We assume that an SSM has been created based on these two files. The linking variable is child.
Part 1: Intercept only
Sigma_squared = 31206.85925
Tau
INTRCPT1,B0 2999.57775
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.347
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 132.056729 20.225570 6.529 21 0.000
For SDUMMY slope, B1
INTRCPT2, G10 -110.886184 40.451140 -2.741 124 0.007
----------------------------------------------------------------------------
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 132.056729 16.147278 8.178 21 0.000
For SDUMMY slope, B1
INTRCPT2, G10 -110.886184 32.294556 -3.434 124 0.001
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 54.76840 2999.57775 21 28.88002 0.117
level-1, R 176.65463 31206.85925
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1651.929448
Number of estimated parameters = 2
Part 2: The variable age is grand mean centered and is included as a fixed effect.
Sigma_squared = 10498.58940
Tau
INTRCPT1,B0 5471.56387
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.735
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 137.771784 18.457888 7.464 21 0.000
For AGE slope, B1
INTRCPT2, G10 29.558847 2.025611 14.593 123 0.000
For SDUMMY slope, B2
INTRCPT2, G20 -69.767209 37.013631 -1.885 123 0.059
----------------------------------------------------------------------------
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 137.771784 17.151265 8.033 21 0.000
For AGE slope, B1
INTRCPT2, G10 29.558847 4.124667 7.166 123 0.000
For SDUMMY slope, B2
INTRCPT2, G20 -69.767209 28.676218 -2.433 123 0.015
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 73.97002 5471.56387 21 87.94757 0.000
level-1, R 102.46262 10498.58940
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1529.224765
Number of estimated parameters = 2
Part 3: The variable agec2 (age being centered and squared) is also included.
Sigma_squared = 7883.51225
Tau
INTRCPT1,B0 5749.18627
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.794
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 84.785740 20.215591 4.194 21 0.000
For AGE slope, B1
INTRCPT2, G10 30.629929 1.764394 17.360 122 0.000
For AGEC2 slope, B2
INTRCPT2, G20 2.557290 0.427127 5.987 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -68.233833 36.431132 -1.873 122 0.061
----------------------------------------------------------------------------
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 84.785740 11.942216 7.100 21 0.000
For AGE slope, B1
INTRCPT2, G10 30.629929 4.032159 7.596 122 0.000
For AGEC2 slope, B2
INTRCPT2, G20 2.557290 0.308919 8.278 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -68.233833 28.490300 -2.395 122 0.017
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 75.82339 5749.18627 21 117.06725 0.000
level-1, R 88.78914 7883.51225
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1499.835215
Number of estimated parameters = 2
Part 4: The variable age is centered and included as a random effect.
Sigma_squared = 888.15776
Tau
INTRCPT1,B0 6723.81032 1315.60698
AGE,B1 1315.60698 257.43050
Tau (as correlations)
INTRCPT1,B0 1.000 1.000
AGE,B1 1.000 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.974
AGE, B1 0.968
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 87.983071 17.952062 4.901 21 0.000
For AGE slope, B1
INTRCPT2, G10 28.135347 3.477898 8.090 21 0.000
For AGEC2 slope, B2
INTRCPT2, G20 2.174898 0.147545 14.741 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -4.825112 8.132635 -0.593 122 0.553
----------------------------------------------------------------------------
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 87.983071 14.209464 6.192 21 0.000
For AGE slope, B1
INTRCPT2, G10 28.135347 3.503365 8.031 21 0.000
For AGEC2 slope, B2
INTRCPT2, G20 2.174898 0.201680 10.784 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -4.825112 3.476148 -1.388 122 0.165
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 81.99884 6723.81032 21 1047.69383 0.000
AGE slope, U1 16.04464 257.43050 21 839.05987 0.000
level-1, R 29.80198 888.15776
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1289.907514
Number of estimated parameters = 4
Table 5.9 on page 92.
Part 1: Intercept only model. This is the first part of Table 5.7 and we omit it here.
Part 2: The variable age is centered on 12 months and is included as a fixed effect.
Sigma_squared = 10498.58940
Tau
INTRCPT1,B0 5471.56387
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.735
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 -65.855842 22.897709 -2.876 21 0.010
For AGE12 slope, B1
INTRCPT2, G10 29.558847 2.025611 14.593 123 0.000
For SDUMMY slope, B2
INTRCPT2, G20 -69.767209 37.013631 -1.885 123 0.059
----------------------------------------------------------------------------
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 -65.855842 15.130493 -4.353 21 0.000
For AGE12 slope, B1
INTRCPT2, G10 29.558847 4.124667 7.166 123 0.000
For SDUMMY slope, B2
INTRCPT2, G20 -69.767209 28.676218 -2.433 123 0.015
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 73.97002 5471.56387 21 87.94757 0.000
level-1, R 102.46262 10498.58940
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1529.224765
Number of estimated parameters = 2
Part 3: The variable age12sq is included as a fixed effect.
Sigma_squared = 7883.51227
Tau
INTRCPT1,B0 5749.18625
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.794
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 -4.820517 23.929343 -0.201 21 0.843
For AGE12 slope, B1
INTRCPT2, G10 -4.609531 5.972419 -0.772 122 0.440
For AGE12SQ slope, B2
INTRCPT2, G20 2.557290 0.427127 5.987 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -68.233833 36.431131 -1.873 122 0.061
----------------------------------------------------------------------------
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 -4.820517 9.937283 -0.485 21 0.632
For AGE12 slope, B1
INTRCPT2, G10 -4.609531 2.285767 -2.017 122 0.043
For AGE12SQ slope, B2
INTRCPT2, G20 2.557290 0.308919 8.278 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -68.233833 28.490300 -2.395 122 0.017
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 75.82339 5749.18625 21 117.06726 0.000
level-1, R 88.78915 7883.51227
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1499.835215
Number of estimated parameters = 2
Part 4: The variable age12 is a random effect.
Sigma_squared = 888.15784
Tau
INTRCPT1,B0 814.50204 -457.80535
AGE12,B1 -457.80535 257.43077
Tau (as correlations)
INTRCPT1,B0 1.000 -1.000
AGE12,B1 -1.000 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.608
AGE12, B1 0.968
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 -2.591218 8.483686 -0.305 21 0.763
For AGE12 slope, B1
INTRCPT2, G10 -1.834766 3.973751 -0.462 21 0.649
For AGE12SQ slope, B2
INTRCPT2, G20 2.174898 0.147545 14.741 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -4.825202 8.132632 -0.593 122 0.553
----------------------------------------------------------------------------
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 -2.591218 2.794276 -0.927 21 0.365
For AGE12 slope, B1
INTRCPT2, G10 -1.834766 1.862511 -0.985 21 0.336
For AGE12SQ slope, B2
INTRCPT2, G20 2.174898 0.201680 10.784 122 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -4.825202 3.476167 -1.388 122 0.165
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 28.53948 814.50204 21 40.18939 0.007
AGE12 slope, U1 16.04465 257.43077 21 839.06013 0.000
level-1, R 29.80198 888.15784
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1289.907521
Number of estimated parameters = 4
Part 5: The variable age12 and age12sq are random effects.
Sigma_squared = 679.38774
Tau
INTRCPT1,B0 128.31705 -84.78083 -7.03366
AGE12,B1 -84.78083 57.62219 4.32581
AGE12SQ,B2 -7.03366 4.32581 0.46216
Tau (as correlations)
INTRCPT1,B0 1.000 -0.986 -0.913
AGE12,B1 -0.986 1.000 0.838
AGE12SQ,B2 -0.913 0.838 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.178
AGE12, B1 0.434
AGE12SQ, B2 0.528
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 -3.734573 5.708652 -0.654 21 0.520
For AGE12 slope, B1
INTRCPT2, G10 -0.389827 2.415984 -0.161 21 0.874
For AGE12SQ slope, B2
INTRCPT2, G20 2.040081 0.196456 10.384 21 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -1.117249 7.327091 -0.152 122 0.879
----------------------------------------------------------------------------
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 -3.734573 2.763753 -1.351 21 0.191
For AGE12 slope, B1
INTRCPT2, G10 -0.389827 2.080441 -0.187 21 0.853
For AGE12SQ slope, B2
INTRCPT2, G20 2.040081 0.185602 10.992 21 0.000
For SDUMMY slope, B3
INTRCPT2, G30 -1.117249 3.330613 -0.335 122 0.737
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 11.32771 128.31705 21 6.52714 >.500
AGE12 slope, U1 7.59093 57.62219 21 33.17943 0.044
AGE12SQ slope, U2 0.67982 0.46216 21 48.75723 0.001
level-1, R 26.06507 679.38774
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 1265.180714
Number of estimated parameters = 7
Table 5.10 on page 101.
Part 1: The variable time is a fixed effect and compound symmetry is assumed.
Sigma_squared = 0.05562
Standard Error of Sigma_squared = 0.00249
Tau
INTRCPT1,B0 0.04576
Standard Errors of Tau
INTRCPT1,B0 0.00552
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.832
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.465053 0.111312 22.145 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.148397 0.033312 4.455 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.085838 0.027977 3.068 1195 0.003
For JOB slope, B3
INTRCPT2, G30 -0.161774 0.018304 -8.838 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103148 0.004002 25.772 1195 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 2.465053 0.105348 23.399 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.148397 0.033181 4.472 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.085838 0.026778 3.206 1195 0.002
For JOB slope, B3
INTRCPT2, G30 -0.161774 0.016276 -9.940 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103148 0.005508 18.726 1195 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.21391 0.04576 199 1187.22534 0.000
level-1, R 0.23583 0.05562
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 294.544930
Number of estimated parameters = 7
Part 2: The variable time is a random effect and compound symmetry is assumed.
Sigma_squared = 0.04176
Standard Error of Sigma_squared = 0.00209
Tau
INTRCPT1,B0 0.03897 -0.00256
TIME,B4 -0.00256 0.00391
Standard Errors of Tau
INTRCPT1,B0 0.00618 0.00154
TIME,B4 0.00154 0.00064
Tau (as correlations)
INTRCPT1,B0 1.000 -0.208
TIME,B4 -0.208 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.640
TIME, B4 0.621
----------------------------------------------------
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 2.410526 0.105075 22.941 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116761 0.031532 3.703 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089835 0.026475 3.393 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119622 0.017458 -6.852 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103973 0.005622 18.493 199 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 2.410526 0.099711 24.175 199 0.000
For SEX slope, B1
INTRCPT2, G10 0.116761 0.031292 3.731 1195 0.000
For HIGHGPA slope, B2
INTRCPT2, G20 0.089835 0.024900 3.608 1195 0.001
For JOB slope, B3
INTRCPT2, G30 -0.119622 0.015338 -7.799 1195 0.000
For TIME slope, B4
INTRCPT2, G40 0.103973 0.005590 18.599 199 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.19741 0.03897 199 556.26173 0.000
TIME slope, U4 0.06255 0.00391 199 527.84181 0.000
level-1, R 0.20436 0.04176
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 180.204948
Number of estimated parameters = 9
Part 3: The variable time is a fixed effect; MANOVA approach. HLM does not handle any model that does not have any random effect.