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HLM Textbook Examples
Multilevel Analysis Techniques and Applications by Joop Hox
Chapter 4: Some Important Methodological and Statistical Issues
The data sets used Chapter 4 is the same as in Chapter 2 and we assume that an SSM file has been created based on these data files.
Part 1: The variable sex is not centered and is a fixed effect.
Sigma_squared = 0.45992
Tau
INTRCPT1,B0 0.86222
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.974
----------------------------------------------------
The value of the likelihood function at iteration 3 = -2.246446E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 4.897221 0.095295 51.390 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843701 0.030963 27.248 1998 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 4.897221 0.098254 49.843 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843701 0.059144 14.265 1998 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.92856 0.86222 99 3788.54686 0.000
level-1, R 0.67817 0.45992
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4492.892477
Number of estimated parameters = 2
Part 2: The variable sex is grand-mean centered and is a fixed effect.
Sigma_squared = 0.45992
Tau
INTRCPT1,B0 0.86222
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.974
----------------------------------------------------
The value of the likelihood function at iteration 3 = -2.246446E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 5.308103 0.094098 56.410 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843701 0.030963 27.248 1998 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 5.308103 0.093626 56.695 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843701 0.059144 14.265 1998 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.92856 0.86222 99 3788.54685 0.000
level-1, R 0.67817 0.45992
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4492.892477
Number of estimated parameters = 2
Part 3: The variable sex is included as a random effect, not centered.
Sigma_squared = 0.39244
Tau
INTRCPT1,B0 0.94027 -0.14100
SEX,B1 -0.14100 0.27252
Tau (as correlations)
INTRCPT1,B0 1.000 -0.279
SEX,B1 -0.279 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.959
SEX, B1 0.766
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.168157E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 4.890086 0.099016 49.387 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843106 0.059634 14.138 99 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 4.890086 0.098519 49.636 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843106 0.059335 14.209 99 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.96968 0.94027 99 2416.40514 0.000
SEX slope, U1 0.52204 0.27252 99 426.17652 0.000
level-1, R 0.62645 0.39244
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4336.314651
Number of estimated parameters = 4
Part 4: The variable sex is included as a random effect and is also grand-mean centered.
Sigma_squared = 0.39244
Tau
INTRCPT1,B0 0.86757 -0.00829
SEX,B1 -0.00829 0.27253
Tau (as correlations)
INTRCPT1,B0 1.000 -0.017
SEX,B1 -0.017 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.976
SEX, B1 0.766
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.168157E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 5.300679 0.094243 56.245 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843106 0.059636 14.138 99 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 5.300679 0.093770 56.529 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843106 0.059335 14.209 99 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.93143 0.86757 99 4277.32851 0.000
SEX slope, U1 0.52204 0.27253 99 426.17686 0.000
level-1, R 0.62645 0.39244
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4336.314638
Number of estimated parameters = 4
Table 4.2 on page 60.
Part 1: No interaction, no centering.
Sigma_squared = 0.39248
Tau
INTRCPT1,B0 0.41155 0.02093
SEX,B1 0.02093 0.27329
Tau (as correlations)
INTRCPT1,B0 1.000 0.062
SEX,B1 0.062 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.911
SEX, B1 0.767
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.137029E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 3.339816 0.160775 20.773 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843146 0.059689 14.126 99 0.000
For TEXP slope, B2
INTRCPT2, G20 0.108363 0.010215 10.609 1997 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 3.339816 0.165505 20.180 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843146 0.059388 14.197 99 0.000
For TEXP slope, B2
INTRCPT2, G20 0.108363 0.011382 9.521 1997 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.64152 0.41155 99 1125.84022 0.000
SEX slope, U1 0.52277 0.27329 99 426.12953 0.000
level-1, R 0.62648 0.39248
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4274.057054
Number of estimated parameters = 4
Part 2: With interaction, but no centering.
Sigma_squared = 0.39241
Tau
INTRCPT1,B0 0.41198 0.02342
SEX,B1 0.02342 0.22641
Tau (as correlations)
INTRCPT1,B0 1.000 0.077
SEX,B1 0.077 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.911
SEX, B1 0.731
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.134216E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 3.313521 0.161014 20.579 98 0.000
MTEXP, G01 0.110235 0.010232 10.774 98 0.000
For SEX slope, B1
INTRCPT2, G10 1.329594 0.133051 9.993 98 0.000
MTEXP, G11 -0.034035 0.008457 -4.024 98 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 3.313521 0.167064 19.834 98 0.000
MTEXP, G01 0.110235 0.011466 9.614 98 0.000
For SEX slope, B1
INTRCPT2, G10 1.329594 0.108340 12.272 98 0.000
MTEXP, G11 -0.034035 0.007293 -4.667 98 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.64186 0.41198 98 1126.47092 0.000
SEX slope, U1 0.47583 0.22641 98 367.94122 0.000
level-1, R 0.62643 0.39241
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4268.431480
Number of estimated parameters = 4
Part 3: Centering, but no interaction.
Sigma_squared = 0.39248
Tau
INTRCPT1,B0 0.49674 0.15400
SEX,B1 0.15400 0.27331
Tau (as correlations)
INTRCPT1,B0 1.000 0.418
SEX,B1 0.418 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.960
SEX, B1 0.767
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.137029E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 5.296010 0.071925 73.633 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843146 0.059693 14.125 99 0.000
For TEXP slope, B2
INTRCPT2, G20 0.108358 0.010215 10.608 1997 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 5.296010 0.071189 74.394 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843146 0.059388 14.197 99 0.000
For TEXP slope, B2
INTRCPT2, G20 0.108358 0.011382 9.520 1997 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.70480 0.49674 99 2433.70809 0.000
SEX slope, U1 0.52279 0.27331 99 426.13103 0.000
level-1, R 0.62648 0.39248
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4274.057036
Number of estimated parameters = 4
Part 4: Centering and with interaction.
Sigma_squared = 0.39241
Tau
INTRCPT1,B0 0.48850 0.13369
SEX,B1 0.13369 0.22643
Tau (as correlations)
INTRCPT1,B0 1.000 0.402
SEX,B1 0.402 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.959
SEX, B1 0.731
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.134216E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 5.300373 0.071345 74.293 98 0.000
MTEXP, G01 0.093660 0.010851 8.631 98 0.000
For SEX slope, B1
INTRCPT2, G10 0.842895 0.055619 15.155 98 0.000
MTEXP, G11 -0.034035 0.008458 -4.024 98 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 5.300373 0.070631 75.043 98 0.000
MTEXP, G01 0.093660 0.011275 8.307 98 0.000
For SEX slope, B1
INTRCPT2, G10 0.842895 0.055066 15.307 98 0.000
MTEXP, G11 -0.034035 0.007293 -4.667 98 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.69892 0.48850 98 2394.46566 0.000
SEX slope, U1 0.47585 0.22643 98 367.94224 0.000
level-1, R 0.62643 0.39241
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4268.431481
Number of estimated parameters = 4
Figure 4.3 on page 61.
Use the model from Part 2 of Table 4.2, click on Graph Equations from File Menu and choose as follows. You can also add x-label and y-label and title for the graph by clicking the Range/Titles/Color option.
Table 4.3 on page 63.
Part 1: Intercept only
Sigma_squared = 0.63868
Tau
INTRCPT1,B0 0.87981
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.965
----------------------------------------------------
The value of the likelihood function at iteration 2 = -2.556874E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 5.307603 0.095504 55.575 99 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 5.307603 0.095023 55.856 99 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.93798 0.87981 99 2803.92595 0.000
level-1, R 0.79917 0.63868
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 5113.748909
Number of estimated parameters = 2
Part 2: The variable sex is included as a fixed effect.
Sigma_squared = 0.45992
Tau
INTRCPT1,B0 0.86222
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.974
----------------------------------------------------
The value of the likelihood function at iteration 3 = -2.246446E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 4.897221 0.095295 51.390 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843701 0.030963 27.248 1998 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 4.897221 0.098254 49.843 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843701 0.059144 14.265 1998 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.92856 0.86222 99 3788.54686 0.000
level-1, R 0.67817 0.45992
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4492.892477
Number of estimated parameters = 2
Part 3: The variable texp is also included.
Sigma_squared = 0.45992
Tau
INTRCPT1,B0 0.48596
Tau (as correlations)
INTRCPT1,B0 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.954
----------------------------------------------------
The value of the likelihood function at iteration 4 = -2.221260E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 3.560682 0.171477 20.765 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.844669 0.030951 27.291 1997 0.000
For TEXP slope, B2
INTRCPT2, G20 0.093445 0.010854 8.609 1997 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 3.560682 0.163292 21.806 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.844669 0.059140 14.283 1997 0.000
For TEXP slope, B2
INTRCPT2, G20 0.093445 0.011187 8.353 1997 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.69711 0.48596 99 2156.96630 0.000
level-1, R 0.67818 0.45992
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4442.520460
Number of estimated parameters = 2
Part 4: The variable sex is included as a random effect.
Sigma_squared = 0.39248
Tau
INTRCPT1,B0 0.41155 0.02093
SEX,B1 0.02093 0.27329
Tau (as correlations)
INTRCPT1,B0 1.000 0.062
SEX,B1 0.062 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.911
SEX, B1 0.767
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.137029E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 3.339816 0.160775 20.773 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843146 0.059689 14.126 99 0.000
For TEXP slope, B2
INTRCPT2, G20 0.108363 0.010215 10.609 1997 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 3.339816 0.165505 20.180 99 0.000
For SEX slope, B1
INTRCPT2, G10 0.843146 0.059388 14.197 99 0.000
For TEXP slope, B2
INTRCPT2, G20 0.108363 0.011382 9.521 1997 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.64152 0.41155 99 1125.84022 0.000
SEX slope, U1 0.52277 0.27329 99 426.12953 0.000
level-1, R 0.62648 0.39248
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4274.057054
Number of estimated parameters = 4
Part 5: Interaction of sex and texp is included.
Sigma_squared = 0.39242
Tau
INTRCPT1,B0 0.41197 0.02346
SEX,B1 0.02346 0.22634
Tau (as correlations)
INTRCPT1,B0 1.000 0.077
SEX,B1 0.077 1.000
----------------------------------------------------
Random level-1 coefficient Reliability estimate
----------------------------------------------------
INTRCPT1, B0 0.911
SEX, B1 0.731
----------------------------------------------------
The value of the likelihood function at iteration 6 = -2.134216E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard Approx.
Fixed Effect Coefficient Error T-ratio d.f. P-value
----------------------------------------------------------------------------
For INTRCPT1, B0
INTRCPT2, G00 3.313525 0.161010 20.580 99 0.000
For SEX slope, B1
INTRCPT2, G10 1.329591 0.133020 9.995 98 0.000
MTEXP, G11 -0.034035 0.008455 -4.025 98 0.000
For TEXP slope, B2
INTRCPT2, G20 0.110235 0.010232 10.774 1996 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 3.313525 0.167065 19.834 99 0.000
For SEX slope, B1
INTRCPT2, G10 1.329591 0.108340 12.272 98 0.000
MTEXP, G11 -0.034035 0.007293 -4.667 98 0.000
For TEXP slope, B2
INTRCPT2, G20 0.110235 0.011466 9.614 1996 0.000
----------------------------------------------------------------------------
Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect Standard Variance df Chi-square P-value
Deviation Component
-----------------------------------------------------------------------------
INTRCPT1, U0 0.64185 0.41197 99 1126.46051 0.000
SEX slope, U1 0.47575 0.22634 98 367.93782 0.000
level-1, R 0.62643 0.39242
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance = 4268.431491
Number of estimated parameters = 4
Table 4.4 can be produced manually based on the equations provided in this section. We omit the calculation here.
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