### HLM Textbook Examples Multilevel Analysis Techniques and Applications by Joop Hox Chapter 2: The Basic Two-Level Regression Model: Introduction

This chapter uses the data file on popularity of students. We use SPSS file format since HLM directly import SPSS data files. In order to perform the analyses in HLM, we need to get a data file for level-2. This is obtained by simply aggregating the level-1 data file by SCHOOL. Here are the two SPSS data files, popular.sav and popular_lev2.sav needed for this chapter. The variable that links these two files is SCHOOL. We assume that an SSM file has been created based on these two data files.
Table 2.1 on page 17.
Part1: Intercept only. (M0)
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 random effect and teacher experience (texp) as fixed effect (M1).
 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
Table 2.2 on page 20.
Part 1: M1, as shown above.
Part 2: Cross-level interaction with teacher experience (M2).

 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
Table 2.3 on page 21.
Part 1: M1 from Table 2.2.
Part 2: Standardized variables. Since HLM doesn't have any data management ability, we have to create these standardized variables somewhere else. We choose to do it in SPSS since we use SPSS data format when creating a SSM file. Here is the SPSS file with standardized variables. We assume that an SSM will be created based on this level-1 data file and the level-2 data file provided before.
 Sigma_squared =      0.26115
Tau
INTRCPT1,B0      0.33052       0.05123
ZSEX,B1      0.05123       0.04546

Tau (as correlations)
INTRCPT1,B0  1.000  0.418
ZSEX,B1  0.418  1.000
----------------------------------------------------
Random level-1 coefficient   Reliability estimate
----------------------------------------------------
INTRCPT1, B0                        0.960
ZSEX, B1                        0.767
----------------------------------------------------
The value of the likelihood function at iteration 6 = -1.729065E+003
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard             Approx.
Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
For       INTRCPT1, B0
INTRCPT2, G00          -0.009780   0.058670    -0.167        99    0.868
For     ZSEX slope, B1
INTRCPT2, G10           0.343852   0.024344    14.125        99    0.000
For    ZTEXP slope, B2
INTRCPT2, G20           0.579106   0.054592    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          -0.009780   0.058070    -0.168        99    0.867
For     ZSEX slope, B1
INTRCPT2, G10           0.343852   0.024220    14.197        99    0.000
For    ZTEXP slope, B2
INTRCPT2, G20           0.579106   0.060828     9.520      1997    0.000
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square  P-value
Deviation     Component
-----------------------------------------------------------------------------
INTRCPT1,       U0        0.57491       0.33052    99    2433.70795    0.000
ZSEX slope, U1        0.21321       0.04546    99     426.13100    0.000
level-1,       R         0.51103       0.26115
-----------------------------------------------------------------------------

Statistics for current covariance components model
--------------------------------------------------
Deviance                       = 3458.129869
Number of estimated parameters = 4
Figure 2.1 to Figure 2.6. HLM does not much plotting ability and no data management facility. In order to create these plots, we need to save the residual file in other statistical software, such as SAS, SPSS or SYSTAT and perform necessary data management steps before we can actually graph them. Combining with the model estimate, the residual file is enough, in theory at least, for creating all the plots in this chapter. But actually doing will take a lot of efforts. We therefore would refer anyone to the same chapter done in other packages.

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