### HLM Textbook Examples Multilevel Analysis Techniques and Applications by Joop Hox Chapter 8: The Multilevel Approach to Meta-Analysis

This chapter uses data file on the effect of social training on socially anxious children. We created an ascii data file meta20.txt (shown below) for it. The data file contains six variables: study, d, varofd, rii, ntot and weeks. The variable study is the identification variable and has to be the first column. The second column is the effect size variable and the third is the variance. The rest of the columns will be the level-2 predictor variables.
 1      -.264       .086       .9        47       3
2      -.23        .106       .75       38       1
3       .166       .055       .75       74       2
4       .173       .084       .9        48       4
5       .225       .071       .75       57       3
6       .291       .078       .75       53       6
7       .309       .051       .9        80       7
8       .435       .093       .9        51       9
9       .476       .149       .75       32       3
10       .617       .095       .75       46       6
11       .651       .11        .75       56       6
12       .718       .054       .9        79       7
13       .74        .081       .75       55       9
14       .745       .084       .9        51       5
15       .758       .087       .9        59       6
16       .922       .103       .9        56       5
16       .938       .113       .75       45       5
18       .962       .083       .9        54       7
19      1.522       .100       .900      66       9
20      1.844       .141       .75       45       9
The meta-analysis is a V-known model and is done through the command interface. In order to issue HLM commands via the command line from any directory, we need to add HLM to the path. Click here for detailed instructions.

Table 8.3 on page 148.This type of analysis is done through command line interactively. We will show the detailed steps here. At the end of this interactive process, we name our .ssm file to be meta20.ssm.
Step 1: Creating the SSM File
HLM creates .ssm (sufficient statistics matrix) file from a raw data file.

E:\hox\ascii>hlm2
Will you be starting with raw data?  y
Is the input file a v-known file? y
How many level-1 statistics are there? 1
How many level-2 predictors are there? 3
Enter 8 character name for level-1 variable number 1: d
Enter 8 character name for level-2 variable number 1: rii
Enter 8 character name for level-2 variable number 2: ntot
Enter 8 character name for level-2 variable number 3: weeks
Input format of raw data file (the first field must be the character ID)
format: (a2, 3F11.3, 2F8.0)
What file contains the data?meta20.csv
Enter name of SSM file: meta20.ssm
20 groups have been processed
Step 2: Estimating a V-known model
After creating the .ssm file, we are ready to perform the analysis.
Part 1: Intercept only.
 E:\hox\ascii>hlm2 meta20.ssm
SPECIFYING AN HLM2 MODEL
Level-1 predictor variable specification
Which level-1 predictors do you wish to use?
The choices are:
For        D enter  1
level-1 predictor? (Enter 0 to end)  1

Level-2 predictor variable specification
Which level-2 variables do you wish to use?
The choices are:
For      RII enter  1    For     NTOT enter  2    For    WEEKS enter  3

Which level-2 predictors to model        D?
Level-2 predictor? (Enter 0 to end)  0

Select the level-2 variables that you might consider for
inclusion as predictors in subsequent models.
The choices are:
For      RII enter  1    For     NTOT enter  2    For    WEEKS enter  3

Which level-2 variables to model        D?
Level-2 variable? (Enter 0 to end)  0
Do you want to run this analysis with a heterogeneous sigma^2? n
Do you wish to use any of the optional hypothesis testing procedures? n
Do you want to do a latent variable regression? n
OUTPUT SPECIFICATION
Do you want a residual file? n
How many iterations do you want to do? 1000
Do you want to see OLS estimates for all of the level-2 units? n
Enter a problem title: Intercept Only Model
Enter name of output file: meta20_m1.txt
Computing . . ., please wait
The output is included below.
 Problem Title: Intercept Only Model
Summary of the model specified (in equation format)
---------------------------------------------------
Level-1 Model
Y1 = B1 + E1

Level-2 Model
B1 = G10 + U1

Tau
D,B1      0.14461

Tau (as correlations)
D,B1  1.000
----------------------------------------------------
Random level-1 coefficient   Reliability estimate
----------------------------------------------------
D, B1                        0.620
----------------------------------------------------
The value of the likelihood function at iteration 11 = -4.416426E+001
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard             Approx.
Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
INTRCPT2, G10           0.580146   0.108002     5.372        19    0.000
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square  P-value
Deviation     Component
-----------------------------------------------------------------------------
D,       U1        0.38027       0.14461    19      49.79738    0.000
-----------------------------------------------------------------------------
Statistics for current covariance components model
--------------------------------------------------
Deviance                       = 88.328524
Number of estimated parameters = 2
Table 8.3 Part 2: The variable time is included.
E:\hox\ascii>hlm2 meta20.ssm
SPECIFYING AN HLM2 MODEL
Level-1 predictor variable specification
Which level-1 predictors do you wish to use?
The choices are:
For        D enter  1
level-1 predictor? (Enter 0 to end)  1

Level-2 predictor variable specification
Which level-2 variables do you wish to use?
The choices are:
For      RII enter  1    For     NTOT enter  2    For    WEEKS enter  3

Which level-2 predictors to model        D?
Level-2 predictor? (Enter 0 to end)  3
Level-2 predictor? (Enter 0 to end)  0

Select the level-2 variables that you might consider for
inclusion as predictors in subsequent models.
The choices are:
For      RII enter  1    For     NTOT enter  2    For    WEEKS enter  3

Which level-2 variables to model        D?
Level-2 variable? (Enter 0 to end)  0
Do you want to run this analysis with a heterogeneous sigma^2? n
Do you wish to use any of the optional hypothesis testing procedures? n
Do you want to do a latent variable regression? n
OUTPUT SPECIFICATION
Do you want a residual file? n
How many iterations do you want to do? 10000
Do you want to see OLS estimates for all of the level-2 units? n
Enter a problem title: Variable Time is included
Enter name of output file: meta20_m2.txt
The major part of the output is shown below.
Problem Title: Variable Time is included
Summary of the model specified (in equation format)
---------------------------------------------------
Level-1 Model
Y1 = B1 + E1

Level-2 Model
B1 = G10 + G11*(WEEKS) + U1

Tau
D,B1      0.03663

Tau (as correlations)
D,B1  1.000
----------------------------------------------------
Random level-1 coefficient   Reliability estimate
----------------------------------------------------
D, B1                        0.297
----------------------------------------------------
The value of the likelihood function at iteration 11 = -3.777495E+001
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard             Approx.
Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
INTRCPT2, G10          -0.216959   0.204343    -1.062        18    0.303
WEEKS, G11           0.139929   0.033791     4.141        18    0.001
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square  P-value
Deviation     Component
-----------------------------------------------------------------------------
D,       U1        0.19139       0.03663    18      26.45735    0.090
-----------------------------------------------------------------------------

Statistics for current covariance components model
--------------------------------------------------
Deviance                       = 75.549906
Number of estimated parameters = 2
Table 8.4 on page 151. The first model and the fourth model are the models from Table 8.3 and we omit them here.
Model 1: The variable ntot is included in the model.
The setup of the model is similar to the example above and we omit it here. We only show the output below.
Problem Title: Ntot is included
Level-1 Model
Y1 = B1 + E1

Level-2 Model
B1 = G10 + G11*(NTOT) + U1

Tau
D,B1      0.15921

Tau (as correlations)
D,B1  1.000
----------------------------------------------------
Random level-1 coefficient   Reliability estimate
----------------------------------------------------
D, B1                        0.642
----------------------------------------------------
The value of the likelihood function at iteration 9 = -5.353864E+001
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard             Approx.
Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
INTRCPT2, G10           0.442971   0.512570     0.864        18    0.399
NTOT, G11           0.002489   0.009006     0.276        18    0.785
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square  P-value
Deviation     Component
-----------------------------------------------------------------------------
D,       U1        0.39901       0.15921    18      49.92053    0.000
-----------------------------------------------------------------------------

Statistics for current covariance components model
--------------------------------------------------
Deviance                       = 107.077277
Number of estimated parameters = 2
Model 2: The variable rii is included in the model.
The setup of the model is similar to the example above and we omit it here. We only show the output below.
 Problem Title: Variable RII is included
Level-1 Model
Y1 = B1 + E1

Level-2 Model
B1 = G10 + G11*(RII) + U1

Tau
D,B1      0.15648

Tau (as correlations)
D,B1  1.000
----------------------------------------------------
Random level-1 coefficient   Reliability estimate
----------------------------------------------------
D, B1                        0.638
----------------------------------------------------
The value of the likelihood function at iteration 11 = -4.328604E+001
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard             Approx.
Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
INTRCPT2, G10           0.160176   1.227267     0.131        18    0.898
RII, G11           0.508698   1.477294     0.344        18    0.734
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square  P-value
Deviation     Component
-----------------------------------------------------------------------------
D,       U1        0.39557       0.15648    18      49.23675    0.000
-----------------------------------------------------------------------------

Statistics for current covariance components model
--------------------------------------------------
Deviance                       = 86.572080
Number of estimated parameters = 2
Model 3: All of the variables are included.
The setup of the model is similar to the example above and we omit it here. We only show the output below.
Problem Title: All Variables Are Included
Level-1 Model
Y1 = B1 + E1

Level-2 Model
B1 = G10 + G11*(RII) + G12*(NTOT) + G13*(WEEKS) + U1

Tau
D,B1      0.04934

Tau (as correlations)
D,B1  1.000
----------------------------------------------------
Random level-1 coefficient   Reliability estimate
----------------------------------------------------
D, B1                        0.362
----------------------------------------------------
The value of the likelihood function at iteration 11 = -4.683295E+001
Final estimation of fixed effects:
----------------------------------------------------------------------------
Standard             Approx.
Fixed Effect         Coefficient   Error      T-ratio   d.f.     P-value
----------------------------------------------------------------------------
INTRCPT2, G10           0.384280   0.923448     0.416        16    0.682
RII, G11          -0.550999   1.201005    -0.459        16    0.652
NTOT, G12          -0.003571   0.007035    -0.508        16    0.618
WEEKS, G13           0.150605   0.037551     4.011        16    0.001
----------------------------------------------------------------------------

Final estimation of variance components:
-----------------------------------------------------------------------------
Random Effect           Standard      Variance     df    Chi-square  P-value
Deviation     Component
-----------------------------------------------------------------------------
D,       U1        0.22213       0.04934    16      25.43980    0.062
-----------------------------------------------------------------------------

Statistics for current covariance components model
--------------------------------------------------
Deviance                       = 93.665910
Number of estimated parameters = 2

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