Transcript Class 24 Lecture

Multilevel Models 2
Sociology 8811, Class 24
Copyright © 2007 by Evan Schofer
Do not copy or distribute without permission
Announcements
• Paper #2 due in 2 weeks
• Come see me ASAP if you don’t have a plan
• Unfortunately, I’m unavailable during office hours today
– Please send me an email to make an appointment at some
other time.
Multilevel Data
• Simple example: 2-level data
Class
Class
Class
Class
Class
Class
• Which can be shown as:
Level 2
Level 1
Class 1
S1
S2
Class 2
S3
S1
S2
Class 3
S3
S1
S2
S3
Review: Multilevel Strategies
• Problems of multilevel models
• Non-independence; correlated error
• Standard errors = underestimated
• Solutions:
– Each has benefits, disadvantages…
•
•
•
•
•
•
1.
2.
3.
4.
5.
6.
OLS regression
Aggregation (between effects model)
Robust Standard Errors
Robust Cluster Standard Errors
Dummy variables (Fixed Effects Model)
Random effects models
Example: Pro-environmental values
• Source: World Values Survey (27 countries)
• Let’s simply try OLS regression
. reg supportenv age male dmar demp educ incomerel ses
Source |
SS
df
MS
-------------+-----------------------------Model | 2761.86228
7 394.551755
Residual | 105404.878 27799 3.79167876
-------------+-----------------------------Total |
108166.74 27806 3.89005036
Number of obs
F( 7, 27799)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
27807
104.06
0.0000
0.0255
0.0253
1.9472
-----------------------------------------------------------------------------supportenv |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age | -.0021927
.000803
-2.73
0.006
-.0037666
-.0006187
male |
.0960975
.0236758
4.06
0.000
.0496918
.1425032
dmar |
.0959759
.02527
3.80
0.000
.0464455
.1455063
demp | -.1226363
.0254293
-4.82
0.000
-.172479
-.0727937
educ |
.1117587
.0058261
19.18
0.000
.1003393
.1231781
incomerel |
.0131716
.0056011
2.35
0.019
.0021931
.0241501
ses |
.0922855
.0134349
6.87
0.000
.0659525
.1186186
_cons |
5.742023
.0518026
110.84
0.000
5.640487
5.843559
Dummy Variables
• Another solution to correlated error within
groups/clusters: Add dummy variables
• Include a dummy variable for each Level-2 group, to
explicitly model variance in means
• A simple version of a “fixed effects” model (see below)
• Ex: Student achievement; data from 3 classes
• Level 1: students; Level 2: classroom
• Create dummy variables for each class
– Include all but one dummy variable in the model
– Or include all dummies and suppress the intercept
Yi    DClass 2 X i  DClass 3 X i  X i   i
Dummy Variables
• What is the consequence of adding group
dummy variables?
• A separate intercept is estimated for each group
• Correlated error is absorbed into intercept
– Groups won’t systematically fall above or below the regression
line
• In fact, all “between group” variation (not just error) is
absorbed into the intercept
– Thus, other variables are really just looking at within group
effects
– This can be good or bad, depending on your goals.
Example: Pro-environmental values
• Dummy variable model
. reg supportenv age male dmar demp educ incomerel ses _Icountry*
Source |
SS
df
MS
-------------+-----------------------------Model | 11024.1401
32 344.504377
Residual | 97142.6001 27774 3.49760928
-------------+-----------------------------Total |
108166.74 27806 3.89005036
Number of obs
F( 32, 27774)
Prob > F
R-squared
Adj R-squared
Root MSE
=
=
=
=
=
=
27807
98.50
0.0000
0.1019
0.1009
1.8702
-----------------------------------------------------------------------------supportenv |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age | -.0038917
.0008158
-4.77
0.000
-.0054906
-.0022927
male |
.0979514
.0229672
4.26
0.000
.0529346
.1429683
dmar |
.0024493
.0252179
0.10
0.923
-.046979
.0518777
demp | -.0733992
.0252937
-2.90
0.004
-.1229761
-.0238223
educ |
.0856092
.0061574
13.90
0.000
.0735404
.097678
incomerel |
.0088841
.0059384
1.50
0.135
-.0027554
.0205237
ses |
.1318295
.0134313
9.82
0.000
.1055036
.1581554
_Icountry_32 | -.4775214
.085175
-5.61
0.000
-.6444687
-.3105742
_Icountry_50 |
.3943565
.0844248
4.67
0.000
.2288798
.5598332
_Icountry_70 |
.1696262
.0865254
1.96
0.050
.0000321
.3392203
… dummies omitted …
_Icountr~891 |
.243995
.0802556
3.04
0.002
.08669
.4012999
_cons |
5.848789
.082609
70.80
0.000
5.686872
6.010707
Dummy Variables
• Benefits of the dummy variable approach
• It is simple
– Just estimate a different intercept for each group
• sometimes the dummy interpretations can be of interest
• Weaknesses
• Cumbersome if you have many groups
• Uses up lots of degrees of freedom (not parsimonious)
• Makes it hard to look at other kinds of group dummies
– Non-varying group variables = collinear with dummies
• Can be problematic if your main interest is to study effects of
variables across groups
– Dummies purge that variation… focus on within-group variation
– If you don’t have much within group variation, there isn’t much left to
analyze.
Dummy Variables
• Note: Dummy variables are a simple example
of a “fixed effects” model (FEM)
• Effect of each group is modeled as a “fixed effect”
rather than a random variable
• Also can be thought of as the “within-group” estimator
– Looks purely at variation within groups
– Stata can do a Fixed Effects Model without the
effort of using all the dummy variables
• Simply request the “fixed effects” estimator in xtreg.
Fixed Effects Model (FEM)
• Fixed effects model:
Yij   j   X ij   ij
• For i cases within j groups
• Therefore j is a separate intercept for each group
• It is equivalent to solely at within-group variation:
Yij  Y j   ( X ij  X j )   ij   j
• X-bar-sub-j is mean of X for group j, etc
• Model is “within group” because all variables are
centered around mean of each group.
Fixed Effects Model (FEM)
. xtreg supportenv age male dmar demp educ incomerel ses, i(country) fe
Fixed-effects (within) regression
Group variable (i): country
Number of obs
Number of groups
=
=
27807
26
R-sq:
Obs per group: min =
avg =
max =
511
1069.5
2154
within = 0.0220
between = 0.0368
overall = 0.0239
F(7,27774)
=
89.23
corr(u_i, Xb) = 0.0213
Prob > F
=
0.0000
-----------------------------------------------------------------------------supportenv |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age | -.0038917
.0008158
-4.77
0.000
-.0054906
-.0022927
male |
.0979514
.0229672
4.26
0.000
.0529346
.1429683
dmar |
.0024493
.0252179
0.10
0.923
-.046979
.0518777
demp | -.0733992
.0252937
-2.90
0.004
-.1229761
-.0238223
educ |
.0856092
.0061574
13.90
0.000
.0735404
.097678
incomerel |
.0088841
.0059384
1.50
0.135
-.0027554
.0205237
ses |
.1318295
.0134313
9.82
0.000
.1055036
.1581554
_cons |
5.878524
.052746
111.45
0.000
5.775139
5.981908
-------------+---------------------------------------------------------------sigma_u | .55408807
Identical to dummy variable model!
sigma_e | 1.8701896
rho | .08069488
(fraction of variance due to u_i)
-----------------------------------------------------------------------------F test that all u_i=0:
F(25, 27774) =
94.49
Prob > F = 0.0000
ANOVA: A Digression
• Suppose you wish to model variable Y for j
groups (clusters)
• Ex: Wages for different racial groups
• Definitions:
• The grand mean is the mean of all groups
– Y-bar
• The group mean is the mean of a particular sub-group
of the population
– Y-bar-sub-j
ANOVA: Concepts & Definitions
• Y is the dependent variable
• We are looking to see if Y depends upon the particular
group a person is in
• The effect of a group is the difference
between a group’s mean & the grand mean
• Effect is denoted by alpha (a)
• If Y-bar = $8.75, YGroup 1 = $8.90, then Group 1= $0.15
• Effect of being in group j is:
α j  Yj  Y
• It is like a deviation, but for a group.
ANOVA: Concepts & Definitions
• ANOVA is based on partitioning deviation
• We initially calculated deviation as the
distance of a point from the grand mean:
d i  Yi  Y
• But, you can also think of deviation from a
group mean (called “e”):
ei ,Group1  Yi ,Group1  YGroup1
• Or, for any case i in group j:
eij  Yij  Y j
ANOVA: Concepts & Definitions
• The location of any case is determined by:
• The Grand Mean, m, common to all cases
• The group “effect” , common to members
• The distance between a group and the grand mean
• “Between group” variation
• The within-group deviation (e): called “error”
• The distance from group mean to an case’s value
The ANOVA Model
• This is the basis for a formal model:
• For any population with mean m
• Comprised of J subgroups, Nj in each group
• Each with a group effect 
• The location of any individual can be
expressed as follows: Y  μ  α
ij
j
 eij
• Yij refers to the value of case i in group j
• eij refers to the “error” (i.e., deviation from
group mean) for case i in group j
Sum of Squared Deviation
• We are most interested in two parts of model
• The group effects:
j
• Deviation of the group from the grand mean
• Individual case error:
eij
• Deviation of the individual from the group mean
• Each are deviations that can be summed up
• Remember, we square deviations when summing
• Otherwise, they add up to zero
• Remember variance is just squared deviation
Sum of Squared Deviation
• The total deviation can partitioned into j and
eij components:
• That is, j + eij = total deviation:
α j  Yj  Y
eij  Yij  Yj
eij  α j  (Yj  Y)  (Yij  Yj )  Yij  Y
Sum of Squared Deviation
• The total deviation can partitioned into j and
eij components:
• The total variance (SStotal) is made up of:
–
–
–
j : between group variance (SSbetween)
eij : within group variance (SSwithin)
SStotal = SSbetween + SSwithin
ANOVA & Fixed Effects
• Note that the ANOVA model is similar to the
fixed effects model
• But FEM also includes a X term to model linear trend
ANOVA
Yij  μ  α j  eij
Fixed Effects Model
Yij   j   X ij   ij
• In fact, if you don’t specify any X variables, they are
pretty much the same
Within Group & Between Group
Models
• Group-effect dummy variables in regression
model creates a specific estimate of group
effects for all cases
• Bs & error are based on remaining “within group”
variation
• We could do the opposite: ignore within-group
variation and just look at differences between
• Stata’s xtreg command can do this, too
• This is essentially just modeling group means!
Between Group Model
. xtreg supportenv age male dmar demp educ incomerel ses, i(country) be
Between regression (regression on group means)
Group variable (i): country
Number of obs
Number of groups
=
=
27
27
R-sq:
Obs per group: min =
avg =
max =
1
1.0
1
within =
.
between = 0.2505
overall = 0.2505
sd(u_i + avg(e_i.))=
.6378002
F(7,19)
Prob > F
=
=
0.91
0.5216
-----------------------------------------------------------------------------supportenv |
Coef.
Std. Err.
t
P>|t|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age |
.0211517
.0391649
0.54
0.595
-.0608215
.1031248
male |
3.966173
4.479358
0.89
0.387
-5.409232
13.34158
dmar |
.8001333
1.127099
0.71
0.486
-1.558913
3.15918
demp | -.0571511
1.165915
-0.05
0.961
-2.497439
2.383137
educ |
.3743473
.2098779
1.78
0.090
-.0649321
.8136268
incomerel |
.148134
.1687438
0.88
0.391
-.2050508
.5013188
ses | -.4126738
.4916416
-0.84
0.412
-1.441691
.6163439
_cons |
2.031181
3.370978
0.60
0.554
-5.024358
9.08672
Note: Results are identical to the aggregated
analysis… Note that N is reduced to 27
Fixed vs. Random Effects
• Dummy variables produce a “fixed” estimate
of the intercept for each group
• But, models don’t need to be based on fixed effects
• Example: The error term (ei)
• We could estimate a fixed value for all cases
– This would use up lots of degrees of freedom – even more
than using group dummies
• In fact, we would use up ALL degrees of freedom
– Stata output would simply report back the raw data (expressed
as deviations from the constant)
• Instead, we model e as a random variable
– We assume it is normal, with standard deviation sigma.
Random Effects
• Issue: The dummy variable approach
(ANOVA, FEM) treats group differences as a
fixed effect
• Alternatively, we can treat it as a random effect
• Don’t estimate values for each case, but model it
• This requires making assumptions
– e.g., that group differences are normally distributed with a
standard deviation that can be estimated from data
Random Effects
• A simple random intercept model
– Notation from Rabe-Hesketh & Skrondal 2005, p. 4-5
Random Intercept Model
Yij   0   j   ij
• Where  is the main intercept
• Zeta () is a random effect for each group
– Allowing each of j groups to have its own intercept
– Assumed to be independent & normally distributed
• Error (e) is the error term for each case
– Also assumed to be independent & normally distributed
• Note: Other texts refer to random intercepts as uj or nj.
Linear Random Intercepts Model
• The random intercept idea can be applied to
linear regression
•
•
•
•
Often called a “random effects” model…
Result is similar to FEM, BUT:
FEM looks only at within group effects
Aggregate models (“between effects”) looks across
groups
– Random effects models yield a weighted average
of between & within group effects
• It exploits between & within information, and thus can
be more efficient than FEM & aggregate models.
– IF distributional assumptions are correct.
Linear Random Intercepts Model
. xtreg supportenv age male dmar demp educ incomerel ses, i(country) re
Random-effects GLS regression
Group variable (i): country
R-sq:
within = 0.0220
between = 0.0371
overall = 0.0240
Random effects u_i ~ Gaussian
corr(u_i, X)
= 0 (assumed)
Assumes
normal uj,
uncorrelated
with X vars
Number of obs
Number of groups
=
=
27807
26
Obs per group: min =
avg =
max =
511
1069.5
2154
Wald chi2(7)
Prob > chi2
625.50
0.0000
=
=
-----------------------------------------------------------------------------supportenv |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------age | -.0038709
.0008152
-4.75
0.000
-.0054688
-.0022731
male |
.0978732
.0229632
4.26
0.000
.0528661
.1428802
dmar |
.0030441
.0252075
0.12
0.904
-.0463618
.05245
demp | -.0737466
.0252831
-2.92
0.004
-.1233007
-.0241926
educ |
.0857407
.0061501
13.94
0.000
.0736867
.0977947
incomerel |
.0090308
.0059314
1.52
0.128
-.0025945
.0206561
ses |
.131528
.0134248
9.80
0.000
.1052158
.1578402
_cons |
5.924611
.1287468
46.02
0.000
5.672272
6.17695
-------------+---------------------------------------------------------------sigma_u | .59876138
SD of u (intercepts); SD of e; intra-class correlation
sigma_e | 1.8701896
rho | .09297293
(fraction of variance due to u_i)
Linear Random Intercepts Model
• Notes: Model can also be estimated with
maximum likelihood estimation (MLE)
• Stata:
xtreg y x1 x2 x3, i(groupid) mle
– Versus “re”, which specifies weighted least squares estimator
• Results tend to be similar
• But, MLE results include a formal test to see whether
intercepts really vary across groups
– Significant p-value indicates that intercepts vary
. xtreg supportenv age male dmar demp educ incomerel ses, i(country) mle
Random-effects ML regression
Number of obs
=
27807
Group variable (i): country
Number of groups
=
26
… MODEL RESULTS OMITTED …
/sigma_u |
.5397755
.0758087
.4098891
.7108206
/sigma_e |
1.869954
.0079331
1.85447
1.885568
rho |
.0769142
.019952
.0448349
.1240176
-----------------------------------------------------------------------------Likelihood-ratio test of sigma_u=0: chibar2(01)= 2128.07 Prob>=chibar2 = 0.000
Choosing Models
• Which model is best?
• There is much discussion (e.g, Halaby 2004)
• Fixed effects are most consistent under a
wide range of circumstances
• Consistent: Estimates approach true parameter values
as N grows very large
• But, they are less efficient than random effects
– In cases with low within-group variation (big between group
variation) and small sample size, results can be very poor
– Random Effects = more efficient
• But, runs into problems if specification is poor
– Esp. if X variables correlate with random group effects.
Hausman Specification Test
• Hausman Specification Test: A tool to help
evaluate fit of fixed vs. random effects
• Logic: Both fixed & random effects models are
consistent if models are properly specified
• However, some model violations cause random effects
models to be inconsistent
– Ex: if X variables are correlated to random error
• In short: Models should give the same results… If not,
random effects may be biased
– If results are similar, use the most efficient model: random
effects
– If results diverge, odds are that the random effects model is
biased. In that case use fixed effects…
Hausman Specification Test
• Strategy: Estimate both fixed & random
effects models
• Save the estimates each time
• Finally invoke Hausman test
– Ex:
•
•
•
•
•
streg var1 var2 var3, i(groupid) fe
estimates store fixed
streg var1 var2 var3, i(groupid) fe
estimates store fixed
hausman fixed random
Hausman Specification Test
• Example: Environmental attitudes fe vs re
. hausman fixed random
Direct comparison of coefficients…
---- Coefficients ---|
(b)
(B)
(b-B)
sqrt(diag(V_b-V_B))
|
fixed
random
Difference
S.E.
-------------+---------------------------------------------------------------age |
-.0038917
-.0038709
-.0000207
.0000297
male |
.0979514
.0978732
.0000783
.0004277
dmar |
.0024493
.0030441
-.0005948
.0007222
demp |
-.0733992
-.0737466
.0003475
.0007303
educ |
.0856092
.0857407
-.0001314
.0002993
incomerel |
.0088841
.0090308
-.0001467
.0002885
ses |
.1318295
.131528
.0003015
.0004153
-----------------------------------------------------------------------------b = consistent under Ho and Ha; obtained from xtreg
B = inconsistent under Ha, efficient under Ho; obtained from xtreg
Test:
Ho:
difference in coefficients not systematic
chi2(7) = (b-B)'[(V_b-V_B)^(-1)](b-B)
=
2.70
Prob>chi2 =
0.9116
Non-significant pvalue indicates
that models yield
similar results…