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, https://stats.idre.ucla.edu/wp-content/uploads/2016/02/popular.sav and https://stats.idre.ucla.edu/wp-content/uploads/2016/02/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.