1. Chapter 5 Multilevel Models
5.1 Cross-sectional multilevel models
5.1.1 Two-level models
5.1.2 Multiple level models
5.1.3 Multiple level modeling in other fields
5.2 Longitudinal multilevel models
5.2.1 Two-level models
5.2.2 Multiple level models
5.3 Prediction
5.4 Testing variance components
Appendix 5A – High order multilevel models

2. Multilevel Models
Multilevel models - a conditional modeling framework that takes into account hierarchical and clustered data structures.
Used extensively in educational science and other disciplines in the social and behavioral sciences.
A multilevel model can be viewed as a linear mixed effects model and hence, the statistical inference techniques introduced in Chapter 3 are readily applicable.
By considering multilevel data and models as a separate unit, we expand the breadth of applications that linear mixed effects models enjoy.
Also known as hierarchical models

3. 5.1 Cross-sectional multilevel models
Two-level model example
Level 2 (Schools), Level 1(Students within a school)
Level 1 Model (student replications – j)
yij = β0i + β1i zij + ij
yij - student’s performance on an achievement test
zij - total family income
Level 2 Model
Thinking of the schools as a random sample, we model {β0i, β1i} as random quantities.
β0i = β0 + α0i and β1i = β1 + α1i ,
where α0i, α1i are mean zero random variables.

4. Two-level model example
The combined level 1 and level 2 models form:
yij = (β0 + α0i ) + (β1 + α1i) zij + ij
= α0i + α1i zij + β0 + β1 zij + ij .
The two-level model may be written as a single linear mixed effects model.
Specifically, we define αi = (α0i , α1i)´, zij = (1, zij)´, β = ( β0, β1)´ and xij = zij, to write
yij = zij´ αi + xij´ β + ij .
We model and interpret behavior through the succession of level models.
We estimate the combined levels through a single (linear mixed effects) model.

5. Two-level model example - variation
Modify the level-2 model so that
β0i = β0 + β01 xi + α0i and β1i = β1 + β11 xi + α1i ,
where xi indicates whether the school was a Catholic based or a public school.
The combined level 1 and (new) level 2 models form:
yij = α0i + α1i zij + β0 + β01 xi + β1 zij + β11 xi zij + ij .
That we can again write as
yij = zij´ αi + xij´ β + ij .
by defining αi = (α0i , α1i)´, zij = (1, zij)´,
β = ( β0, β01, β1, β11)´ and xij = (1, xi, zij, xi zij)´.
The term β11 xi zij , interacting between the level-1 variable zij and the level-2 variable xi, is known as a cross-level interaction.
Many researchers argue that understanding cross-level interactions is a major motivation for analyzing multilevel data.

6. yi,j,k = z1,i,j,k´ βi,j + x1,i,j,k´ β1 + ε1,i,j,k ,
Three Level Models
Level 1 model (students)
yi,j,k = z1,i,j,k´ βi,j + x1,i,j,k´ β1 + ε1,i,j,k ,
The predictors z1,i,j,k and x1,i,j,k may depend on the student (gender, family income and so on), classroom (teacher characteristics, classroom facilities and so on) or school (organization, structure, location and so on).
Level 2 model (classroom)
βi,j = Z2,i,j γi + X2,i,j β2 + ε2,i,j.
The predictors Z2,i,j and X2,i,j may depend on the classroom or school, but not students.
Level 3 model (School)
γi = Z3,i β3 + ε3,i .
The predictors Z3,i may depend on the school, but not students or classroom.

7. = xi,j,k´ β + zi,j,k´ αi,j+ ε1,i,j,k ,
Combined Model
The combined level 1, 2 and 3 models form:
yi,j,k = z1,i,j,k´ ( Z2,i,j (Z3,i β3 + ε3,i) + X2,i,j β2 + ε2,i,j)
+ x1,i,j,k´ β1 + ε1,i,j,k
= xi,j,k´ β + zi,j,k´ αi,j+ ε1,i,j,k ,
where

8. Motivation for multilevel models
Multilevel modeling provides a structure for hypothesizing relationships in a complex system.
The ability to estimate cross-level effects is one advantage of multilevel modeling when compared to an alternate research strategy calling for the analysis of each level in isolation of the others.
Second and higher levels of multilevel models also provide use with an opportunity to estimate the variance structure using a parsimonious, parametric structure.
One typically assumes that disturbance terms from different levels are uncorrelated.

9. 5.2 Longitudinal multilevel models
Similar to cross-sectional multilevel models, except:
Use a t subscript to denote the Level 1 replication, for time
Allow for correlation among Level 1 observations to represent serial patterns.
Possibly include functions of time as Level 1 predictors.
Typical example – use students as Level 2 unit of analysis and time as Level 1 unit of analysis.
Growth curve models are a classic example:
we seek to monitor the natural development or aging of an individual.
This development is typically monitored without intervention and the goal is to assess differences among groups.
In growth curve modeling, one uses a polynomial function of age or time to track growth.

10. Example – Dental Data
Originally due to Potthoff and Roy (1964); see also Rao (1987).
y is the distance, measured in millimeters, from the center of the pituitary to the pterygomaxillary fissure.
Measurements were taken on 11 girls and 16 boys at ages 8, 10, 12, and 14.
The interest is in
how the distance grows with age and
whether there is a difference between males and females.

12. Measure 16 18 20 22 24 26 28 30 32
Age 8 10 12 14
Figure 5.1 Multiple Time Series Plot of Dental Measurements. Open circles are girls, solid circles are boys

13. Dental Model Level 1 model yit = β0i + β1i z1,it + it , z1,it is age.
β0i = β00 + β01 GENDERi + α0i and
β1i = β10 + β11 GENDERi + α1i.
GENDER - 1 for females and 0 for males.

14. Three Level Example
Children Mental Health Assessment by Guo and Hussey (1999)
The Level 1 model (time replications) is
yi,j,t = z1,i,j,t´ βi,j + x1,i,j,t´ β1 + ε1,i,j,t ,
Assessment y is the “Deveroux Scale of Mental Disorders,” a score made up of 111 items.
x1,i,j,t = PROGRAMi,j,t -1 if the child was in program residence at the time of the assessment and 0 if the child was in day treatment or day treatment combined with treatment foster care.
z1,i,j,t = (1 TIMEi,j,t)´. TIMEi,j,t is measured in days since the inception of the study.
Thus, the level-1 model can be written as
yi,j,t = β0,i,j + β1,i,j TIMEi,j,t + β1 PROGRAMi,j,t + ε1,i,j,t .

15. Children Mental Health Assessment
The level 2 model (child replications) is
βi,j = Z2,i,j γi + X2,i,j β2 + ε2,i,j,
where there are i =1 ,…, n children and j = 1, …, Ji raters.
The level 2 model of Guo and Hussey can be written as
β0,i,j = β0,i,0 + β0,0,1 RATERi,j + ε2,i,j
and
β1,i,j = β2,0 + β2,1 RATERi,j .
The variable RATERi,j = 1 if rater was a teacher and = 0 if the rater was a caretaker.
The level 3 model (rater replications) is
γi = Z3,i β3 + ε3,i
β0,i,0 = β0,0,0 + β0,1,0 GENDERi + ε3,i .

16. 5.3 Prediction
Recall that we estimate model parameters and predict random variables.
Consider a two-level longitudinal model
Level 1 model (replication on time)
yi,t = z1,i,t´ βi + x1,i,t´ β1 + ε1,i,t ,
Level 2 model - βi = Z2,i β2 + i.
Linear mixed model is
yi,t = z1,i,t´ (Z2,i β2 + i) + x1,i,t´ β1 + ε1,i,t ,
The best linear unbiased predictor (BLUP) of βi is
bi,BLUP = ai,BLUP + Z2,i b2,GLS ,
where ai,BLUP = D Zi Vi-1 (yi - Xi bGLS ).

17. Three-Level Model Prediction
Estimate model parameters
Next, compute BLUP residuals
Use the formula, ai,BLUP = D Zi Vi-1 (yi - Xi bGLS ).
This yields the BLUPs for αi,j = (ε2,i,j´ ε3,i´)´, say,
ai,j,BLUP = (e2,i,j,BLUP ´ e3,i,BLUP ´)´.
Then, compute BLUP predictors of γi and βi,j
gi,BLUP = Z3,i b3,GLS + e3,i,BLUP
bi,j,BLUP = Z2,i,j gi,BLUP + X2,i,j b2, GLS + e2,i,j,BLUP .
Forecasts are also straightforward: for AR(1) level-1 disturbances, this simplifies to

18. 5.4 Testing variance components
For the error components model, do we wish to pool? We can express this as an hypothesis of the form H0: σα2 = 0.
For a two-level model, do the data provide evidence that our 2nd level model is viable? We may wish to test H0: Var α = 0.
Unfortunately, the usual likelihood ratio testing procedure is not valid for testing many variance components of interest.
In particular, the concern is for testing parameters where the null hypothesis is on the boundary of possible values.
That is,
σα2 = 0 is on the boundary.
Usual approximations are not valid when we use the boundary restriction in our definition of the estimator.
As a general rule, the standard hypothesis testing procedures favors the simpler null hypothesis more often than it should.

19. Alternative Testing Procedures
Suppose that we wish to test the null hypothesis H0: σ 2 = σ02, where σ02 is a known positive constant.
Standard likelihood ratio test is okay.
This procedure is not available when σ02 = 0 because the log-likelihood under H0 is not well defined.
However, H0: σ 2 = 0 is still a testable hypothesis
A simple test is to reject H0 if the maximum likelihood estimator,
exceeds zero.
This test procedure has power 1 versus all alternatives and a significance level of zero, a good test!!!

20. Error Components Model
Consider the likelihood ratio test statistic for assessing H0: σα2 = 0.
The asymptotic distribution turns out to be
Typically, the asymptotic distribution of the likelihood ratio test statistic for one parameter is
This means that using nominal values, we will accept the null hypothesis more often than we should; thus, we will sometimes use a simpler model than suggested by the data.
Suppose that one allows for negative estimates. Then, the asymptotic distributions turns out to be the usual

21. Recommendations No general theory is available.
Some additional theoretical results are available.
This can be important for some applications.
Simulation methods are always possible.
You have to know what your software package is doing
It may be giving you the appropriate test statistics or it may ignore the boundary issue.
If it incorrectly ignores the boundary issue, the test procedures are biased towards simpler models.