Transcript I wish I had learned HLM in this simple way.

Doing HLM using SAS PROC
MIXED
Kaz
www.estat.us
Feb 2005
My points
(1) Easy to compare HLM and other models
that are not HLM; thus, helpful. This is
because PROC MIXED lets you run models
that are not HLM.
(2) Easy to understand what makes HLM
HLM. In SAS, what is not essential to
HLM is done outside PROC MIXED (e.g.,
centering)
OLS vs. HLM in PROC MIXED.
(1)
The difference is a RANDOM statement.
OLS
Y_jk = b0 + error_jk
(2)
HLM
Level1: Y_jk=b0 + error_jk
Level2: b0=g0 + error_k
or
Y_jk=b0 + error_jk + error_k
• OLS regression syntax
PROC MIXED ;
MODEL Y= X;
Run;
• HLM syntax
PROC MIXED;
Model Y=X;
random intercept X/ subject=school;
Run;
Again, turning a simple linear
model into HLM
(1)
PROC MIXED;
Model Y=X W;
Run;
(3)
Random statement below reads:
I request that the intercept, as well
as the effects of X and W be
Estimated for each subject which
can be identified by GroupID.
(2)
PROC MIXED;
Model Y=X W;
random intercept X W/subject=GroupID;
Run;
How to write SAS PROC MIXED syntax:
Intuitive way
(1) Write all the variable names at the model
statement.
model Y=X W;
(2) Decide which variables’ effect you want to
estimate by schools
random intercept X W/subject=school;
More careful way
1.
Start from level-specific specification.
2.
3.
4.
Insert level-2 equations into level-1 equations.
Write the variable names involved in model statement.
Find “random components” (written in Roman alphabets)
RULE1: Put “intercept” in the random statement to accommodate
higher level errors.
RULE2: If the name of any variables sits right next to level-2
error with an asterisk (e.g., X*level-2 error), put those variable
names in the random statement.
(RULE3:No worry about residual. It is set by default.)
e.g., level1:y=b0 + b1*X + error_ij
level2: b0=g00 + g01*W + error_0j
level2: b1=g10 + g11*W + error_1j
Example 1 Anova Model
Level1: Y_ij=b0j + Residual_ij
Level2: b0j= g00 + U_0j

Y= g00 + U_0j + Residual
I said:
RULE1: Put “intercept” in the random statement to
accommodate higher level errors.
RULE2: If the name of any variables sits right next to
level-2 error with an asterisk (e.g., X*level-2 error), put
those variable names in the random statement.
ONLY RULE1 relevant in this model.
proc mixed ;
class group;
model Y= ;
random intercept/subject=school;
run;
Example 2 Slope as outcome models
Level1: Y_ij=b0j + b1j*X + Residual_ij
Level2: b0j= g00 + g01*W + U_0j
Level2: b1j=g10+ g11*W + U_1j

Y= g00 + g01*W + g10*X + g11*W*X
+ U_1j*X + U_0j + Residual
What were..
proc mixed ;
RULE1?
class group;
RULE2?
model Y= W X W*X;
random intercept X/subject=school;
run;
How to do substitution:
Cheating using HLM software!
PUSH
MIXED button
to get a little
window like this.
How to do substitution by hand
Level1: Y_ij=b0j + b1j*X + Residual_ij
Level2: b0j= g00 + t g01*W + U_0j
Level2: b1j=gt10+ g11*W + U_1j
1. Insert higher level equations into the level-1 equation.
 Y=[g00 + g01*W + U_0j] + [g10+g11*W + U_1j]*X + Residual_ij
2. Take out the brackets
--> Y=g00 + g01*W + U_0j + g10*X +g11*W*X + U_1j *X + Residual_ij
3. Notice which parts are structural part and which parts are random components.
 Y=g00 + g01*W + g10*X +g11*W*X + U_1j *X + U_0j + Residual_ij
What were rule1 and rule 2?
proc mixed ;
model W X W*X;
random intercept X /subject=school;
run;
Fixed Effects or Random Effects
OLS regression is a fixed effect model
PROC MIXED;
Model Y=X;
Run;
OLS regression is a model with fixed effects. So in a way OLS
is a special case of HLM. This is an awfully inflexible model
that does not consider the existence of various sources of
errors.
HLM
PROC MIXED;
Model Y=X;
random intercept X/ subject=groupID;
Run;
If a researcher thinks the effect of X (and the intercept) is
different by groups, so we should treat these coefficients
as random effects.
Benefit 1 of using random effect
Conceptual one
Useful to think about Micro-Macro problems
(1)
Student: Math score=b0 + b1*parents’ education level +
…. + error
Country:
b1=g00 + g01*SELECTION + error
(2)
Classroom: teacher perception of math ability of class
=b0 + b1*average parents’ education level
+b2*average math score
+b3*noise + error
Country:
b1=g10 + g11*National Exam + error
b2=g20 + g21*National Exam + error
b3=g30 + g31*National Exam + error
Benefit 2: Statistical benefit
•
Statistical Benefits
–
In deriving a grand mean (re: the effect of X or an intercept) HLM does
“shrinkage.” This pulls inaccurate group means towards the grand mean, so we
can reduce the influence of outliners if their estimates are inaccurate (i.e.,
having large error variance and/or coming from a small number of observations
within each group unit)
Shrunk School mean=reliability*school mean
where reliability is a function of N of observation in a group unit and variance.
(R&B HLM book, p. 48)
Quiz: 1) what happens to a school whose reliability is 1?
2) What happens if all schools are 1 on reliability?
3) What happens if all schools are .5 on reliability?
Quick decision rule
Random or fix
• Do I open the door at 11PM?
– Literature
– Theory
– Exploratory analysis (let’s see what
happens.)
Moore complicated: Two step decisions
regarding random effects
(I need your help in phrasing this.)
• Step 1: Effect different by school?
• Step 2: Random or Fixed?
– Fixed: Use a series of dummy variables
(in reality too tedious)
– Random: Shrinkage applies and get a
precision guided grand mean
Example
Student Engagement study
using ESM
by Uekawa, Borman, and Lee
Engagement Level (Rasch
model composite)
•
•
•
•
•
•
•
•
•
When you were signaled the first time today,
•I was paying attention………………………..
•I did not feel like listening……………………
•My motivation level was high………………..
•I was bored…………………………………..
•I was enjoying class…………………………
•I was focused more on class than anything else
•I wished the class would end soon………… .
•I was completely into class …………………
SD
O
O
O
O
O
O
O
O
The MEANS Procedure
Analysis Variable : engagement engagement
N
Mean
Std Dev
Minimum
Maximum
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2316
0.1167889
10.0106694 -31.5283511
26.4164991
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D A
OO
OO
OO
OO
OO
OO
OO
OO
SA
O
O
O
O
O
O
O
O
3-level HLM
• Level1: Repeated Measures (10 beeps)
• Level2: Students (10 kids from a class)
• Level3: courses (34 courses, Monday to Friday)
3-level HLM
Libname here "C:\";
/*This is three level model*/
proc mixed data=here.esm covtest noclprint;
class IDclass IDstudent;
model engagement= /solution ddfm=kr;
random intercept /sub=IDstudent(IDclass);
random intercept /sub=IDclass ;
run;
Quiz: how can we make this a 2-level hlm?
PROC MIXED statement
proc mixed data=here.esm covtest noclprint;
•
•
“covtest” does a test for covariance components (whether variances are
significantly larger than zero.). The reason why you have to request such a
simple thing is that COVTEST is not based on chi-square test that one
would use for a test of variance. It uses instead t-test or something that
is not really appropriate. Shockingly, SAS has not corrected this problem
for a while. Anyways, because SAS feels bad about it, it does not want to
make it into a default option, which is why you have to request this. Not
many people know this and I myself could not believe this. So I guess that
means that we cannot really believe in the result of COVTEST and must
use it with caution.
When there are lots of group units, use NOCLPRINT to suppress the
printing of group names.
CLASS statement
class IDclass IDstudent Hisp;
•
•
We throw in the variables that we want SAS to treat as categorical
variables. Variables that are characters (e.g., city names) must be on this
line (it won’t run otherwise). Group IDs, such as IDclass in my example
data, must be also in these lines; otherwise, it won’t run. Variables that
are numeric but dummy-coded (e.g., black=1 if black;else 0) don’t have to be
in this line, but the outputs will look easier if you do.
One thing that is a pain in the neck with CLASS statement is that it
chooses a reference category by alphabetical order. Whatever group in a
classification variable that comes the last when alphabetically ordered will
be used as a reference group. We can control this by data manipulation.
For example, if gender=BOY or GIRL, then I tend to create a new variable
to make it explicit that I get girl to be a reference group:
If gender=”Boy” then gender2=”(1) Boy”;
If gender=”Girl” then gender2=”(2) Girl”;
MODEL statement
model engagement= /solution ddfm=kr;
ddfm=kr specifics the ways in which the degree of
freedom is calculated. It seems most close to the
degree of freedom option used by Bryk, Raudenbush,
and Congdon’s HLM program.
Could be computationally very heavy if a model is
complicated. ddfm=bw would run faster, though DF
would be wrong.
Random statement
random intercept X/sub=IDstudent(IDclass);
random intercept X/sub=IDclass ;
We can estimate variance of slopes for categorical variables
using “group=“ option --- without necessarily making them
into dummy variables.
random intercept race /sub=IDclass group=race;
(instead of “random intercept black white hispanic/sub=IDclass;”)
MODEL 1
Libname here "G:\SAS";
proc mixed data=here.esm covtest noclprint;
weight precision_weight;
class IDclass IDstudent;
model engagement= /solution ddfm=kr;
random intercept /sub=IDstudent(IDclass);
random intercept /sub=IDclass ;
run;
•
RQ: How
does the
engagement
score varies
by students
and class?
The Mixed Procedure
•
Covariance Parameter Estimates
•
Standard
•
Cov Parm
Subject
•
•
•
Intercept
Intercept
Residual
IDstudent(IDclass)
IDclass
Z
Estimate
Error
Value
Pr Z
23.3556
5.3168
31.9932
2.5061
2.0947
1.0282
9.32
2.54
31.12
<.0001
0.0056
<.0001
Solution for Fixed Effects
Effect
Intercept
Estimate
Standard
Error
DF
t Value
Pr > |t|
-1.2315
0.5081
30.2
-2.42
0.0216
Engagement=b0 + residual + e_student + e_course
Residual Student Course
Course
9%
Student
38%
Residual
53%
MODEL 2
Libname here "G:\SAS";
proc mixed data=here.esm covtest noclprint;
weight precision_weight;
class IDclass IDstudent subject;
model engagement= hisp /solution ddfm=kr;
random intercept /sub=IDstudent(IDclass);
random intercept hisp /sub=IDclass ;
run;
RQ: What is the effect
of being Hispanic students?
And is the effect vary by
class?
Solution for Fixed Effects
Effect
Intercept
hisp
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.6287
-2.2113
0.5101
1.0031
33.1
17.7
-1.23
-2.20
0.2265
0.0410
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
Intercept
hisp
Residual
IDstudent(IDclass)
IDclass
IDclass
Estimate
Standard
Error
Z
Value
Pr Z
22.4743
3.6944
5.9656
31.9871
2.4712
1.7791
5.3290
1.0275
9.09
2.08
1.12
31.13
<.0001
0.0189
0.1315
<.0001
MODEL 3
proc mixed data=here.esm covtest noclprint;
weight precision_weight;
class IDclass IDstudent subject;
model engagement= hisp math hisp*math /solution ddfm=kr;
random intercept /sub=IDstudent(IDclass);
random intercept hisp /sub=IDclass ;
run;
Solution for Fixed Effects
Effect
Intercept
hisp
math
hisp*math
Hispanic
alienation
phenomenon—
related to
subject?
Math versus
Science
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.3249
-4.1236
-0.6081
3.3305
0.7061
1.4562
1.0108
1.9233
34.2
14.5
33.7
15.2
-0.46
-2.83
-0.60
1.73
0.6484
0.0129
0.5515
0.1035
The Mixed Procedure
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
Intercept
hisp
Residual
IDstudent(IDclass)
IDclass
IDclass
Estimate
Standard
Error
Z
Value
Pr Z
22.6987
3.5328
4.2309
31.9761
2.5020
1.7254
5.0161
1.0269
9.07
2.05
0.84
31.14
<.0001
0.0203
0.1995
<.0001
MODEL 3
Solution for Fixed Effects
Effect
Intercept
hisp
math
hisp*math
Estimate
Standard
Error
DF
t Value
Pr > |t|
-0.3249
-4.1236
-0.6081
3.3305
0.7061
1.4562
1.0108
1.9233
34.2
14.5
33.7
15.2
-0.46
-2.83
-0.60
1.73
0.6484
0.0129
0.5515
0.1035
The Mixed Procedure
Covariance Parameter Estimates
Cov Parm
Subject
Intercept
Intercept
hisp
Residual
IDstudent(IDclass)
IDclass
IDclass
Estimate
Standard
Error
Z
Value
Pr Z
22.6987
3.5328
4.2309
31.9761
2.5020
1.7254
5.0161
1.0269
9.07
2.05
0.84
31.14
<.0001
0.0203
0.1995
<.0001
Level1:engagement=b0+ b1*Hispanic +
residual
Level2: b0=g00 + A
Level2: b1=g10
Level3: g00=t_000 + t_100*Math + B
Level3: g10=t_100 + t_101*Math + C
VERSUS
 A
 B
-> C
Level1:engagement=
t_000
+ t_100*Math
+ t_100*Hispanic
+ t_101*Math*Hispanic
+ C*Hispanic
+ B + A
+ residual
Which is easy to understand?
HLM way
PROC MIXED way
Level1:engagement=b0+ b1*Hispanic +
residual
Level2: b0=g00 + A
Level2: b1=g10
Level3: g00=t_000 + t_100*Math + B
Level3: g10=t_100 + t_101*Math + C
In HLM software
Level-1 Intercept
Level-2 Intercept
Level-3 Intercept
Level-1 Error
Level-2 Error
Level-3 Error
Level1:engagement=
t_000
+ t_100*Math
+ t_100*Hispanic
+ t_101*Math*Hispanic
+ C*Hispanic
+ B + A
+ residual
In SAS PROC MIXED
 Disappears
 Disappears
 Intercept
 Residual
 Random effects
 Random effects
Why do we center variables?
Level1:engagement=
t_000
+ t_100*Math
+ t_100*Hispanic
+ t_101*Math*Hispanic
+ C*Hispanic
+ B + A
+ residual Imagine we have to report to teachers
their students’ average engagement score.
We want to use B + t_000. To be clear
about
Meaning of t_000 part, we “could” center
variables,
if it makes sense.
What about Centering?
In SAS, we use PROC STANDARD to do centering and this is outside
of PROC MIXED. When I learned this, I thought, “I have done it
before!” because centering is similar to the concept of Z-scores.
This is GROUP MEAN Centering
Proc standard data=X mean=0;
by GroupID;
var X;
Run;
This is GRAND MEAN Centering
proc standard data=X mean=0;
var X;
Run;
By the way, just for your information, this is to create Z-scores
proc standard data=X mean=0 STD=1;
var X;
Run;
When you use
SAS PROC
MIXED, you
notice Centering
is not really a
topic that is
specific to
HLM—because
it is done
outside PROC
MIXED.
What does it mean to center
dummy variable, like gender?
1.To adjust for gender composition.
2.
-Without it, the intercept = either male or
female
-With it, the intercept is adjusted for gender
composition.
3. See my Excel Presentation if we have
time. www.estat.us/sas/centering.xls
END
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