1. Doing HLM using SAS PROC MIXED
Kaz
Feb 2005

2. 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)

3. OLS vs. HLM in PROC MIXED. The difference is a RANDOM statement.
(1)
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;

4. 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;

5. 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;

6. More careful way Start from level-specific specification.
e.g., level1:y=b0 + b1*X + error_ij
level2: b0=g00 + g01*W + error_0j
level2: b1=g10 + g11*W + error_1j
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.)

7. Level1: Y_ij=b0j + Residual_ij Level2: b0j= g00 + U_0j 
Example 1 Anova Model
Level1: Y_ij=b0j + Residual_ij
Level2: b0j= g00 + U_0j 
Y= g00 + U_0j + Residual
proc mixed ;
class group;
model Y= ;
random intercept/subject=school;
run;
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.

8. 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
proc mixed ;
class group;
model Y= W   X   W*X;
random intercept X/subject=school;
run;
What were..
RULE1?
RULE2?

9. How to do substitution: Cheating using HLM software!
PUSH
MIXED button
to get a little
window like this.

10. 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;

11. 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
random intercept X/ subject=groupID;
If a researcher thinks the effect of X (and the intercept) is different by groups, so we should treat these coefficients as random effects.

12. Classroom: teacher perception of math ability of class
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

13. 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?

14. Quick decision rule Random or fix
Do I open the door at 11PM?
Literature
Theory
Exploratory analysis (let’s see what happens.)

15. Step 1: Effect different by school? Step 2: Random or Fixed?
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

16. Example Student Engagement study using ESM by Uekawa, Borman, and Lee

17. Engagement Level (Rasch model composite)
When you were signaled the first time today, SD D A SA
•I was paying attention……………………….. O O O O
•I did not feel like listening…………………… O O O O
•My motivation level was high……………….. O O O O
•I was bored………………………………….. O O O O
•I was enjoying class………………………… O O O O
•I was focused more on class than anything else O O O O
•I wished the class would end soon………… . O O O O
•I was completely into class ………………… O O O O
The MEANS Procedure
Analysis Variable : engagement engagement
N Mean Std Dev Minimum Maximum
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18. 3-level HLM Level1: Repeated Measures (10 beeps)
Level2: Students (10 kids from a class)
Level3: courses (34 courses, Monday to Friday)

19. 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?

20. proc mixed data=here.esm covtest noclprint;
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.

21. class IDclass IDstudent Hisp;
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”;

22. model engagement= /solution ddfm=kr;
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.

23. 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;”)

24. RQ: How does the engagement score varies by students and class?
MODEL 1
RQ: How does the engagement score varies by students and class?
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;
The Mixed Procedure
Covariance Parameter Estimates
Standard Z
Cov Parm Subject Estimate Error Value Pr Z
Intercept IDstudent(IDclass) <.0001
Intercept IDclass
Residual <.0001
Solution for Fixed Effects
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept
Engagement=b0 + residual + e_student + e_course

25. of being Hispanic students? And is the effect vary by class?
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
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept
hisp
Covariance Parameter Estimates
Standard Z
Cov Parm Subject Estimate Error Value Pr Z
Intercept IDstudent(IDclass) <.0001
Intercept IDclass
hisp IDclass
Residual <.0001

26. Solution for Fixed Effects
MODEL 3
Hispanic alienation phenomenon—related to subject? Math versus Science
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
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept
hisp
math
hisp*math
The Mixed Procedure
Covariance Parameter Estimates
Standard Z
Cov Parm Subject Estimate Error Value Pr Z
Intercept IDstudent(IDclass) <.0001
Intercept IDclass
hisp IDclass
Residual <.0001

27. Solution for Fixed Effects
MODEL 3
Solution for Fixed Effects
Standard
Effect Estimate Error DF t Value Pr > |t|
Intercept
hisp
math
hisp*math
The Mixed Procedure
Covariance Parameter Estimates
Standard Z
Cov Parm Subject Estimate Error Value Pr Z
Intercept IDstudent(IDclass) <  A
Intercept IDclass  B
hisp IDclass > C
Residual <.0001
Level1:engagement=
t_000
+ t_100*Math
+ t_100*Hispanic
+ t_101*Math*Hispanic
+ C*Hispanic
+ B + A
+ residual
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

28. Which is easy to understand?
HLM 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
PROC MIXED way
Level1:engagement=
t_000
+ t_100*Math
+ t_100*Hispanic
+ t_101*Math*Hispanic
+ C*Hispanic
+ B + A
+ residual
In HLM software In SAS PROC MIXED
Level-1 Intercept  Disappears
Level-2 Intercept  Disappears
Level-3 Intercept  Intercept
Level-1 Error  Residual
Level-2 Error  Random effects
Level-3 Error  Random effects

29. 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.

30. 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;
By the way, just for your information, this is to create Z-scores
proc standard data=X mean=0 STD=1;
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.

31. 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.

32. END
To go back to my HLM page