Download Program: HLM 5 Hierarchical Linear and Nonlinear Modeling

Program:
Authors:
Publisher:
HLM 5 Hierarchical Linear and Nonlinear Modeling
Stephen Raudenbush, Tony Bryk, & Richard Congdon
Scientific Software International, Inc. (c) 2000
techsupport@ssicentral.com
www.ssicentral.com
------------------------------------------------------------------------------Module:
HLM2S.EXE (5.00.2045.1)
Date:
8 April 2000, Saturday
Time:
14:55:22
------------------------------------------------------------------------------SPECIFICATIONS FOR THIS HLM2 RUN
------------------------------------------------------------------------------Problem Title: NULL MODEL / ONE-WAY ANALYSIS OF VARIANCE
The data source for this run = SIOPET.SSM
The command file for this run = C:\HLM5S\siopet\null.hlm
Output file name
= C:\HLM5S\SIOPET\null.OUT
The maximum number of level-2 units = 50
The maximum number of iterations = 1000
Method of estimation: restricted maximum likelihood
Weighting Specification
-----------------------
Level 1
Level 2
Weighting?
no
no
Weight
Variable
Name
The outcome variable is
Normalized?
no
no
HELPING
The model specified for the fixed effects was:
---------------------------------------------------Level-1
Coefficients
---------------------INTRCPT1, B0
Level-2
Predictors
--------------INTRCPT2, G00
The model specified for the covariance components was:
--------------------------------------------------------Sigma squared (constant across level-2 units)
Tau dimensions
INTRCPT1
Summary of the model specified (in equation format)
--------------------------------------------------Level-1 Model
Y = B0 + R
Level-2 Model
B0 = G00 + U0
1
Least Squares Estimates
----------------------sigma_squared =
129.67348
Least-squares estimates of fixed effects
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.394346
0.360102
87.182
999
0.000
----------------------------------------------------------------------------
Least-squares estimates of fixed effects
(with robust standard errors)
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.394346
1.409786
22.269
999
0.000
---------------------------------------------------------------------------The least-squares likelihood value = -3851.050779
Deviance =
7702.10156
Number of estimated parameters =
1
STARTING VALUES
--------------sigma(0)_squared =
Tau(0)
INTRCPT1,B0
31.75676
99.81511
Estimation of fixed effects
(Based on starting values of covariance components)
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.394354
1.424099
22.045
49
0.000
---------------------------------------------------------------------------The value of the likelihood function at iteration 1 = -3.249218E+003
Iterations stopped due to small change in likelihood function
******* ITERATION 2 *******
Sigma_squared =
Tau
INTRCPT1,B0
2 = level-1 residual. Since there are no
level-1 predictors, it simply represents the
within group variance in helping.
31.75676
99.81511
Tau (as correlations)
INTRCPT1,B0 1.000
---------------------------------------------------Random level-1 coefficient
Reliability estimate
---------------------------------------------------INTRCPT1, B0
0.984
----------------------------------------------------
2
00 = level-2 residual. Since there are no
level-2 predictors, it simply represents the
between group variance in helping.
The reliability estimates represent the
proportion of the variance in the
OLS level-1 estimates that is
parameter variance. It is essentially
a ratio of “true” parameter variance
to “observed” parameter variance.
The t-test below assesses the fixed effects in the level-2
equations (i.e.,00 ). In this case, the test of 00 is whether the
grand mean helping is significantly different from zero.
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.394354
1.424099
22.045
49
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
31.394354
1.409786
22.269
49
0.000
----------------------------------------------------------------------------
The Chi-square below indicates whether the level-2 variance
component is significantly different from zero (i.e., 00 ). In
this case, this tests whether the between group variance in
helping is significantly different from zero.
Final estimation of variance components:
----------------------------------------------------------------------------Random Effect
Standard
Variance
df
Chi-square P-value
Deviation
Component
----------------------------------------------------------------------------INTRCPT1,
U0
9.99075
99.81511
49
3129.25201
0.000
level-1,
R
5.63531
31.75676
----------------------------------------------------------------------------Statistics for current covariance components model
-------------------------------------------------Deviance =
6498.43618
Number of estimated parameters =
2
One of the purposes of estimating this null model is to assess the degree of between group variance in
helping behavior. A common metric for this is the Intra-Class Correlation (ICC). Given that we have
partitioned the variance in helping into its within and between components, an ICC can be computed as
follows:
ICC =  / ( +2 )
ICC = 99.82 / (99.82 + 31.76)
ICC = .76
Thus, 76 percent of the variance in helping resides between groups. The Chi-square test above indicates
that this between group variance is significant; that is, the intercept term significantly varies across
groups.
3
Program:
Authors:
Publisher:
HLM 5 Hierarchical Linear and Nonlinear Modeling
Stephen Raudenbush, Tony Bryk, & Richard Congdon
Scientific Software International, Inc. (c) 2000
techsupport@ssicentral.com
www.ssicentral.com
------------------------------------------------------------------------------Module:
HLM2S.EXE (5.00.2045.1)
Date:
8 April 2000, Saturday
Time:
14:46: 8
------------------------------------------------------------------------------SPECIFICATIONS FOR THIS HLM2 RUN
------------------------------------------------------------------------------Problem Title: RANDOM COEFFICIENT REGRESSION MODEL
The data source for this run = siopet.ssm
The command file for this run = C:\HLM5S\siopet\r_reg.hlm
Output file name
= C:\HLM5S\siopet\r_reg.out
The maximum number of level-2 units = 50
The maximum number of iterations = 1000
Method of estimation: restricted maximum likelihood
Weighting Specification
-----------------------
Level 1
Level 2
Weighting?
no
no
Weight
Variable
Name
The outcome variable is
Normalized?
no
no
HELPING
The model specified for the fixed effects was:
---------------------------------------------------Level-1
Coefficients
---------------------INTRCPT1, B0
%
MOOD slope, B1
Level-2
Predictors
--------------INTRCPT2, G00
INTRCPT2, G10
'%' - This level-1 predictor has been centered around its grand mean.
The model specified for the covariance components was:
--------------------------------------------------------Sigma squared (constant across level-2 units)
Tau dimensions
INTRCPT1
MOOD slope
Summary of the model specified (in equation format)
--------------------------------------------------Level-1 Model
Y = B0 + B1*(MOOD) + R
Level-2 Model
B0 = G00 + U0
B1 = G10 + U1
4
Note here that I have centered
mood around its grand mean;
that is, the mood variable
represents the deviation of the
individual’s score from the
grand mean of the sample.
Given this, the variance in the
intercept term is equal to the
between group variance in
helping after controlling for
mood (i.e., the adjusted between
group variance in helping).
We will discuss later in the
session the implications of these
centering decisions for
organizational research.
The “least-squares” estimates of fixed effects
are the OLS regression results. Remember,
mood is grand mean centered (mood – grand
mean of mood).
Least Squares Estimates
----------------------sigma_squared =
46.34416
Least-squares estimates of fixed effects
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.394348
0.215277
145.832
998
0.000
For
MOOD slope, B1
INTRCPT2, G10
3.889450
0.091745
42.394
998
0.000
---------------------------------------------------------------------------Least-squares estimates of fixed effects
(with robust standard errors)
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.394348
0.876668
35.811
998
0.000
For
MOOD slope, B1
INTRCPT2, G10
3.889450
0.245918
15.816
998
0.000
---------------------------------------------------------------------------The least-squares likelihood value = -3338.072922
Deviance =
6676.14584
Number of estimated parameters =
1
STARTING VALUES
--------------sigma(0)_squared =
Tau(0)
INTRCPT1,B0
MOOD,B1
5.60718
46.31097
0.56111
0.56111
0.95020
Estimation of fixed effects
(Based on starting values of covariance components)
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.486284
0.968396
32.514
49
0.000
For
MOOD slope, B1
INTRCPT2, G10
3.008250
0.145820
20.630
49
0.000
---------------------------------------------------------------------------The value of the likelihood function at iteration 1 = -2.445746E+003
The value of the likelihood function at iteration 10 = -2.427653E+003
Iterations stopped due to small change in likelihood function
******* ITERATION 11 *******
Sigma_squared =
Tau
INTRCPT1,B0
MOOD,B1
2 = the level-1 residual variance. We can use this residual variance
to compute an R2 for mood as a level-1 predictor.
5.61081
45.63410
0.63550
0.63550
0.12917
R2 = (2 null - 2random reg. ) / 2 null
R2 = (31.76 - 5.61) / 31.76
R2 = .82
(see Snijders & Bosker (1994). Modeled variance in two-level
models. Sociological methods & research, 22, 342-363, for an
alternative R2.)
Tau (as correlations)
INTRCPT1,B0 1.000 0.262
MOOD,B1 0.262 1.000
5
 is now a variance-covariance matrix of the level2 residuals. Since there are still no level-2
predictors, the diagonal elements in  represent the
between group variance in the intercepts and slopes
respectively. The off-diagonal element represents
the covariance between the intercepts and slopes.
---------------------------------------------------Random level-1 coefficient
Reliability estimate
---------------------------------------------------INTRCPT1, B0
0.987
MOOD, B1
0.541
----------------------------------------------------
The reliability estimates once again
represent the proportion of the
variance in the OLS level-1 estimates
that are parameter variance. Note
that the intercept is considerably
more reliable than the slope.
The value of the likelihood function at iteration 11 = -2.427653E+003
The outcome variable is HELPING
The t-tests indicate that the pooled intercept and pooled
level-1 slope are both significantly different from zero.
It is the t-test associated with the pooled level-1 slope
that is used to test Hypothesis 1.
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
31.429052
0.960006
32.738
49
0.000
For
MOOD slope, B1
INTRCPT2, G10
3.012354
0.068961
43.682
49
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
31.429052
0.950326
33.072
49
0.000
For
MOOD slope, B1
INTRCPT2, G10
3.012354
0.068257
44.133
49
0.000
----------------------------------------------------------------------------
The Chi-square tests indicate that there is significant
between group variance in the intercept and slope
parameters across groups. The intercept term
represents the between group variance in helping
after controlling for mood. In the next model, we
will model this variance with proximity.
Final estimation of variance components:
----------------------------------------------------------------------------Random Effect
Standard
Variance
df
Chi-square P-value
Deviation
Component
----------------------------------------------------------------------------INTRCPT1,
U0
6.75530
45.63410
49
4605.68268
0.000
MOOD slope, U1
0.35941
0.12917
49
110.16628
0.000
level-1,
R
2.36871
5.61081
----------------------------------------------------------------------------Statistics for current covariance components model
-------------------------------------------------Deviance =
4855.30564
Number of estimated parameters =
4
6
Program:
Authors:
Publisher:
HLM 5 Hierarchical Linear and Nonlinear Modeling
Stephen Raudenbush, Tony Bryk, & Richard Congdon
Scientific Software International, Inc. (c) 2000
techsupport@ssicentral.com
www.ssicentral.com
------------------------------------------------------------------------------Module:
HLM2S.EXE (5.00.2045.1)
Date:
8 April 2000, Saturday
Time:
14:49: 2
------------------------------------------------------------------------------SPECIFICATIONS FOR THIS HLM2 RUN
------------------------------------------------------------------------------Problem Title: INTERCEPTS-AS-OUTCOMES
The data source for this run = siopet.ssm
The command file for this run = C:\HLM5S\siopet\inter.hlm
Output file name
= C:\HLM5S\siopet\inter.out
The maximum number of level-2 units = 50
The maximum number of iterations = 1000
Method of estimation: restricted maximum likelihood
Weighting Specification
-----------------------
Level 1
Level 2
Weighting?
no
no
Weight
Variable
Name
The outcome variable is
Normalized?
no
no
HELPING
The model specified for the fixed effects was:
---------------------------------------------------Level-1
Coefficients
---------------------INTRCPT1, B0
%
MOOD slope, B1
Level-2
Predictors
--------------INTRCPT2, G00
PROX, G01
INTRCPT2, G10
'%' - This level-1 predictor has been centered around its grand mean.
The model specified for the covariance components was:
--------------------------------------------------------Sigma squared (constant across level-2 units)
Tau dimensions
INTRCPT1
MOOD slope
Summary of the model specified (in equation format)
--------------------------------------------------Level-1 Model
Y = B0 + B1*(MOOD) + R
Level-2 Model
B0 = G00 + G01*(PROX) + U0
B1 = G10 + U1
7
Least Squares Estimates
----------------------sigma_squared =
Once again, the least-squares estimates represent the
OLS results where mood is grand mean centered and
proximity is “assigned” down to the individual level.
41.08266
Least-squares estimates of fixed effects
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
24.714388
0.622483
39.703
997
0.000
PROX, G01
1.261370
0.111137
11.350
997
0.000
For
MOOD slope, B1
INTRCPT2, G10
3.975459
0.086712
45.847
997
0.000
---------------------------------------------------------------------------Least-squares estimates of fixed effects
(with robust standard errors)
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
24.714388
2.523145
9.795
997
0.000
PROX, G01
1.261370
0.422335
2.987
997
0.003
For
MOOD slope, B1
INTRCPT2, G10
3.975459
0.222004
17.907
997
0.000
---------------------------------------------------------------------------The least-squares likelihood value = -3278.716808
Deviance =
6557.43362
Number of estimated parameters =
1
STARTING VALUES
--------------sigma(0)_squared =
Tau(0)
INTRCPT1,B0
MOOD,B1
5.60718
42.29461
-0.21053
-0.21053
1.11501
Estimation of fixed effects
(Based on starting values of covariance components)
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
25.028366
2.833095
8.834
48
0.000
PROX, G01
1.221252
0.505558
2.416
48
0.020
For
MOOD slope, B1
INTRCPT2, G10
3.009646
0.156721
19.204
49
0.000
---------------------------------------------------------------------------The value of the likelihood function at iteration 1 = -2.444995E+003
The value of the likelihood function at iteration 10 = -2.424730E+003
Iterations stopped due to small change in likelihood function
Since we now include a level-2 predictor of the intercept, the
elements in  no longer represent the var-covar. of the intercepts and
slopes. Specifically, the first diagonal element (i.e., 41.68) is the
residual variance in the intercept term. The second diagonal element
still represents the between group variance in level-1 slopes.
******* ITERATION 11 *******
Sigma_squared =
Tau
INTRCPT1,B0
MOOD,B1
5.61079
41.67890
-0.08356
Tau (as correlations)
INTRCPT1,B0 1.000 -0.036
MOOD,B1 -0.036 1.000
-0.08356
0.12924
We can use this residual variance in the intercepts to compute an R2
for Proximity as a level-2 predictor.
R2 = (00; random reg. - 00; proximity model ) / 00; random reg.
R2 = (45.63 – 41.68 ) / 45.63
R2 = .09
8
---------------------------------------------------Random level-1 coefficient
Reliability estimate
---------------------------------------------------INTRCPT1, B0
0.985
MOOD, B1
0.541
----------------------------------------------------
The reliability for the intercept term
is now computed using the residual
variance in the intercept. Thus, the
reliability has dropped to .985 from
.987.
The t-tests for the fixed effects now assess the significance
of the level-2 regression model for the intercept term (00
and 01 , respectively). The test of 01 is the test of
Hypothesis 2. The interpretation of 10 remains unchanged
(pooled level-1 slope).
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
24.922580
2.808705
8.873
48
0.000
PROX, G01
1.235956
0.501261
2.466
48
0.018
For
MOOD slope, B1
INTRCPT2, G10
3.013404
0.068981
43.684
49
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
24.922580
2.920104
8.535
48
0.000
PROX, G01
1.235956
0.476908
2.592
48
0.013
For
MOOD slope, B1
INTRCPT2, G10
3.013404
0.068234
44.163
49
0.000
----------------------------------------------------------------------------
The Chi-square tests indicate that after including Proximity, there
is still significant variance in the intercept term across groups.
This variance could be modeled by additional group-level
variables. The Chi-square tests also indicate that there is
significant variance in the slopes (related to Hypothesis 3). Our
next model will see if proximity is related to this variance.
Final estimation of variance components:
----------------------------------------------------------------------------Random Effect
Standard
Variance
df
Chi-square P-value
Deviation
Component
----------------------------------------------------------------------------INTRCPT1,
U0
6.45592
41.67890
48
4127.70255
0.000
MOOD slope, U1
0.35950
0.12924
49
110.18076
0.000
level-1,
R
2.36871
5.61079
----------------------------------------------------------------------------Statistics for current covariance components model
-------------------------------------------------Deviance =
4849.46082
Number of estimated parameters =
4
9
Program:
Authors:
Publisher:
HLM 5 Hierarchical Linear and Nonlinear Modeling
Stephen Raudenbush, Tony Bryk, & Richard Congdon
Scientific Software International, Inc. (c) 2000
techsupport@ssicentral.com
www.ssicentral.com
------------------------------------------------------------------------------Module:
HLM2S.EXE (5.00.2045.1)
Date:
8 April 2000, Saturday
Time:
14:50:59
------------------------------------------------------------------------------SPECIFICATIONS FOR THIS HLM2 RUN
------------------------------------------------------------------------------Problem Title: SLOPES-AS-OUTCOMES
The data source for this run = siopet.ssm
The command file for this run = C:\HLM5S\siopet\slopes.hlm
Output file name
= C:\HLM5S\siopet\slopes.out
The maximum number of level-2 units = 50
The maximum number of iterations = 1000
Method of estimation: restricted maximum likelihood
Weighting Specification
-----------------------
Level 1
Level 2
Weighting?
no
no
Weight
Variable
Name
The outcome variable is
Normalized?
no
no
HELPING
The model specified for the fixed effects was:
---------------------------------------------------Level-1
Coefficients
---------------------INTRCPT1, B0
%
MOOD slope, B1
Level-2
Predictors
--------------INTRCPT2, G00
PROX, G01
INTRCPT2, G10
PROX, G11
'%' - This level-1 predictor has been centered around its grand mean.
The model specified for the covariance components was:
--------------------------------------------------------Sigma squared (constant across level-2 units)
Tau dimensions
INTRCPT1
MOOD slope
Summary of the model specified (in equation format)
--------------------------------------------------Level-1 Model
Y = B0 + B1*(MOOD) + R
Level-2 Model
B0 = G00 + G01*(PROX) + U0
B1 = G10 + G11*(PROX) + U1
10
Least Squares Estimates
----------------------sigma_squared =
40.98631
The least-squares estimates represent the OLS results where mood is
grand mean centered, proximity is “assigned” down to individuals
and the proximity*mood (grand mean centered) interaction is entered
into the equation.
Least-squares estimates of fixed effects
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
24.784955
0.622949
39.786
996
0.000
PROX, G01
1.241624
0.111531
11.133
996
0.000
For
MOOD slope, B1
INTRCPT2, G10
4.453726
0.275517
16.165
996
0.000
PROX, G11
-0.090575
0.049533
-1.829
996
0.067
---------------------------------------------------------------------------Least-squares estimates of fixed effects
(with robust standard errors)
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
24.784955
2.493567
9.940
996
0.000
PROX, G01
1.241624
0.421545
2.945
996
0.004
For
MOOD slope, B1
INTRCPT2, G10
4.453726
0.756304
5.889
996
0.000
PROX, G11
-0.090575
0.118460
-0.765
996
0.445
---------------------------------------------------------------------------The least-squares likelihood value = -3279.132496
Deviance =
6558.26499
Number of estimated parameters =
1
STARTING VALUES
--------------sigma(0)_squared =
5.60718
Tau(0)
INTRCPT1,B0
42.29566
-0.25107
MOOD,B1
-0.25107
1.27823
Estimation of fixed effects
(Based on starting values of covariance components)
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
25.167937
2.834108
8.880
48
0.000
PROX, G01
1.195398
0.505750
2.364
48
0.022
For
MOOD slope, B1
INTRCPT2, G10
2.096086
0.512071
4.093
48
0.000
PROX, G11
0.172432
0.091396
1.887
48
0.065
---------------------------------------------------------------------------The value of the likelihood function at iteration 1 = -2.448101E+003
The value of the likelihood function at iteration 30 = -2.414026E+003
Iterations stopped due to small change in likelihood function
******* ITERATION 31 *******
Sigma_squared =
Tau
INTRCPT1,B0
MOOD,B1
Since we now include a level-2 predictor of the slope, both diagonal
elements in  represent residual variance. Specifically, the first diagonal
element (i.e., 42.95) is the residual variance in the intercept term. The
second diagonal element is the residual variance in the slopes (i.e., .02).
5.61465
42.94618
0.00716
Tau (as correlations)
INTRCPT1,B0 1.000 0.007
MOOD,B1 0.007 1.000
0.00716
0.02207
We can use this residual variance in the slopes to compute an R2 for
Proximity as a level-2 predictor of level-1 slopes.
R2 = (11; intercepts-as-outcomes. - 11; proximity model ) / 11; inter-as-outcome
R2 = (.129 - .022 ) / .129
R2 = .83
11
---------------------------------------------------Random level-1 coefficient
Reliability estimate
---------------------------------------------------INTRCPT1, B0
0.986
MOOD, B1
0.173
----------------------------------------------------
The reliability for the slope is now
computed using the residual variance
in the intercept. Thus, the reliability
has dropped to .173 from .541.
The t-tests for the fixed effects now assess the
significance of the level-2 regression model for the
slope term (10 and 11 , respectively). The test of 11
is the test of Hypothesis 3.
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
25.140965
2.847498
8.829
48
0.000
PROX, G01
1.194664
0.508198
2.351
48
0.023
For
MOOD slope, B1
INTRCPT2, G10
2.064738
0.158144
13.056
48
0.000
PROX, G11
0.179906
0.028447
6.324
48
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
25.140965
2.985770
8.420
48
0.000
PROX, G01
1.194664
0.483758
2.470
48
0.017
For
MOOD slope, B1
INTRCPT2, G10
2.064738
0.167640
12.317
48
0.000
PROX, G11
0.179906
0.033319
5.400
48
0.000
----------------------------------------------------------------------------
Final estimation of variance components:
The Chi-square tests indicate that after
including Proximity, there is no longer
significant variance in the slopes across groups.
----------------------------------------------------------------------------Random Effect
Standard
Variance
df
Chi-square P-value
Deviation
Component
----------------------------------------------------------------------------INTRCPT1,
U0
6.55333
42.94618
48
4122.21950
0.000
MOOD slope, U1
0.14856
0.02207
48
59.21764
0.129
level-1,
R
2.36953
5.61465
----------------------------------------------------------------------------Statistics for current covariance components model
-------------------------------------------------Deviance =
4828.05107
Number of estimated parameters =
4
12
Program:
HLM 5 Hierarchical Linear and Nonlinear Modeling
Authors:
Stephen Raudenbush, Tony Bryk, & Richard Congdon
Publisher:
Scientific Software International, Inc. (c) 2000
------------------------------------------------------------------------------Module:
HLM2S.EXE (5.00.2045.1)
Date:
8 April 2000, Saturday
Time:
14:53:33
------------------------------------------------------------------------------SPECIFICATIONS FOR THIS HLM2 RUN
------------------------------------------------------------------------------Problem Title: SLOPES-AS-OUTCOMES / GROUP MEAN CENTERED WITH AVE-MOOD ENTERED
The data source for this run = siopet.ssm
The command file for this run = C:\HLM5S\siopet\slopes2.hlm
Output file name
= C:\HLM5S\siopet\slopes2.out
The maximum number of level-2 units = 50
The maximum number of iterations = 1000
Method of estimation: restricted maximum likelihood
Weighting Specification
-----------------------
Level 1
Level 2
Weighting?
no
no
Weight
Variable
Name
The outcome variable is
Normalized?
no
no
HELPING
The model specified for the fixed effects was:
---------------------------------------------------Level-1
Level-2
Coefficients
Predictors
-----------------------------------INTRCPT1, B0
INTRCPT2, G00
AVE_MOOD, G01
PROX, G02
*
MOOD slope, B1
INTRCPT2, G10
PROX, G11
'*' - This level-1 predictor has been centered around its group mean.
The model specified for the covariance components was:
--------------------------------------------------------Sigma squared (constant across level-2 units)
Tau dimensions
INTRCPT1
MOOD slope
Summary of the model specified (in equation format)
--------------------------------------------------Level-1 Model
Y = B0 + B1*(MOOD) + R
Level-2 Model
B0 = G00 + G01*(AVE_MOOD) + G02*(PROX) + U0
B1 = G10 + G11*(PROX) + U1
13
In this test of the cross-level
moderation, I have centered
mood around its group mean.
There are some arguments in
the literature that group mean
centering is a better choice of
scaling for the level-1
variables when investigating
cross-level interactions (i.e.,
moderation). More
specifically, this allows one to
separate out the cross-level
interaction from a between
group-interaction.
Raudenbush (1989a,b) as well
as Hofmann and Gavin
(1998) have recently
discussed this in greater
detail. We will demonstrate
this notion later in the session.
Note that, in addition to group
mean centering, this model
also includes an additional
predictor of the intercept;
namely, ave_mood.
Ave_mood is the average
mood within each group (i.e.,
the mean mood for each
group). We will discuss why
this variable is included as a
predictor of the intercept later
in the session.
Least Squares Estimates
----------------------sigma_squared =
35.67115
Here the least-squares estimates represent the OLS results where mood is
group-mean centered, ave_mood is entered in as a group level variable and
proximity*mood (group-mean centered) is entered as the interaction. Thus,
there are 4 predictors: mood (grp mean centered), ave_mood, proximity,
and the interaction between mood and proximity (mood*proximity).
Least-squares estimates of
fixed effects
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
-4.744154
0.934278
-5.078
995
0.000
AVE_MOOD, G01
4.946582
0.114113
43.348
995
0.000
PROX, G02
1.370148
0.103952
13.181
995
0.000
For
MOOD slope, B1
INTRCPT2, G10
2.028703
0.361477
5.612
995
0.000
PROX, G11
0.184708
0.065270
2.830
995
0.005
----------------------------------------------------------------------------
Least-squares estimates of fixed effects
(with robust standard errors)
---------------------------------------------------------------------------Standard
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
-4.744154
3.413768
-1.390
995
0.165
AVE_MOOD, G01
4.946582
0.438139
11.290
995
0.000
PROX, G02
1.370148
0.403980
3.392
995
0.001
For
MOOD slope, B1
INTRCPT2, G10
2.028703
0.166272
12.201
995
0.000
PROX, G11
0.184708
0.032923
5.610
995
0.000
---------------------------------------------------------------------------The least-squares likelihood value = -3210.020772
Deviance =
6420.04154
Number of estimated parameters =
1
STARTING VALUES
--------------sigma(0)_squared =
5.60718
Tau(0)
INTRCPT1,B0
31.75619
0.02801
MOOD,B1
0.02801
0.02752
Estimation of fixed effects
(Based on starting values of covariance components)
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
-4.760707
3.959427
-1.202
47
0.236
AVE_MOOD, G01
4.949141
0.483588
10.234
47
0.000
PROX, G02
1.370446
0.440566
3.111
47
0.004
For
MOOD slope, B1
INTRCPT2, G10
2.033388
0.161959
12.555
48
0.000
PROX, G11
0.183345
0.029116
6.297
48
0.000
---------------------------------------------------------------------------The value of the likelihood function at iteration 1 = -2.405170E+003
The value of the likelihood function at iteration 25 = -2.405129E+003
Iterations stopped due to small change in likelihood function
******* ITERATION 26 *******
Sigma_squared =
5.61932
Tau
INTRCPT1,B0
31.75516
MOOD,B1
0.05813
0.05813
0.01987
Tau (as correlations)
INTRCPT1,B0 1.000 0.073
MOOD,B1 0.073 1.000
14
---------------------------------------------------Random level-1 coefficient
Reliability estimate
---------------------------------------------------INTRCPT1, B0
0.991
MOOD, B1
0.159
----------------------------------------------------
Now that we have chosen group mean centering, note that the intercept model has changed
considerably. This is because under group mean centering, the intercept is simply equal to the
between group variance in helping; thus, the level-2 intercept model is simply the between group
regression between group level helping and the group variables -- in the preceding model the variance
in the intercept was equal to the adjusted between group variance in helping (i.e., after controlling for
mood). In this model, the cross-level interaction results (i.e., 10 and 11 ) are virtually identical. This,
however, will not always be the case.
Final estimation of fixed effects:
---------------------------------------------------------------------------Standard
Approx.
Fixed Effect
Coefficient
Error
T-ratio
d.f.
P-value
---------------------------------------------------------------------------For
INTRCPT1, B0
INTRCPT2, G00
-4.780527
3.958677
-1.208
47
0.234
AVE_MOOD, G01
4.952222
0.483442
10.244
47
0.000
PROX, G02
1.370791
0.440561
3.111
47
0.004
For
MOOD slope, B1
INTRCPT2, G10
2.032463
0.157216
12.928
48
0.000
PROX, G11
0.183645
0.028289
6.492
48
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
-4.780527
3.426374
-1.395
47
0.170
AVE_MOOD, G01
4.952222
0.439851
11.259
47
0.000
PROX, G02
1.370791
0.404053
3.393
47
0.002
For
MOOD slope, B1
INTRCPT2, G10
2.032463
0.168370
12.071
48
0.000
PROX, G11
0.183645
0.033537
5.476
48
0.000
----------------------------------------------------------------------------
Final estimation of variance components:
----------------------------------------------------------------------------Random Effect
Standard
Variance
df
Chi-square P-value
Deviation
Component
----------------------------------------------------------------------------INTRCPT1,
U0
5.63517
31.75516
47
5359.09498
0.000
MOOD slope, U1
0.14095
0.01987
48
59.08077
0.131
level-1,
R
2.37051
5.61932
----------------------------------------------------------------------------Statistics for current covariance components model
-------------------------------------------------Deviance =
4810.25736
Number of estimated parameters =
4
15