Multilevel Modeling
Soc 543
Fall 2004
Presentation overview
What is multilevel modeling?
Problems with not using multilevel models
Benefits of using multilevel models
Basic multilevel model
Variation one: person and time
Variation two: person, time, and space
Multilevel models
Units of analysis are nested within higher-level
units of analysis
Students within schools
Observations with person
Problems without MLM
If we ignore higher-level units of analysis => cannot
account for context (individualistic approach)
If we ignore individual-level observation and rely on
higher-level units of analysis, we may commit ecological
fallacy (aggregated data approach)
Without explicit modeling, sampling errors at second
level may be large =>unreliable slopes
Homoscedasticity and no serial correlation assumptions
of OLS are violated (an efficiency problem).
No distinction between parameter and sampling
variances
Advantages of MLM
Cross-level comparisons
Controls for level differences
General MLM
Example: Raudenbush and Bryk, 1986
Dependent variable:
Continuous
Observed
General MLM
High school and beyond (HSB) survey
10,231 students from 82 Catholic and 94
public schools
Dependent variable: standardized math
achievement score
Independent variable: SES
General MLM
Variability among schools
Level one: within schools
mathij = b0j + b1j (SESij - SES•j) + rij
General MLM
Variability among schools
Level two: between schools
b0j = g00 + u0j
b1j = g10 + u1j
General MLM
Variability among schools
Combined model
= g00 + u0j
+ g10(SESij - SES•j)
+ u1j(SESij - SES•j) + rij
= g00 + g10(SESij - SES•j)
+ u0j + vij
(Easy interpretation given the “centering”
parameterization)
mathij
General MLM
Variability among schools
Combined model
mathij = 100.74
+ 4.52(SESij - SES•j)
+ u0j + vij
There is a positive relation between SES and
math score
General MLM
Variability among schools
Results: math score means
school means are different
90% of the variance is parameter variance
10% is sampling variance
Results: math score-SES relation
school relations are different
35% is parameter variance (this requires
additional assumption and analysis)
65% is sampling variance
General MLM
Covariates at level 2
Level one: within schools
mathij = b0j + b1j (SESij - SES•j) + rij
General MLM
Covariates at level 2
Level two: between schools
b0j = g00 + g01sectj+ u0j
b1j = g10 + g11sectj+ u1j
General MLM
Covariates at level 2
Combined model:
mathij = g00 + g01sectj
+ g10 (SESij - SES•j)
+ g11sectj(SESij - SES•j)
+ rij + vj
General MLM
Combined model:
mathij = 98.37 + 5.06sectj
+ 6.23(SESij - SES•j)
- 3.86sectj(SESij - SES•j)
+ rij + vj
General MLM
Variability as a function of sector
Results: math score means
80.7% is parameter variance
differences in school means is not entirely
accounted for by sector
Results: SES-math score relation
9.7% is parameter variance
differences in school SES-math score
relation may be accounted for by sector
General MLM
Sector effects
Cannot say that previous relations are
causal – may be selection effects
Use example of homework to explain
sector differences
General MLM
Sector effects
Results:
school SES is strongly related to mean
math score, but SES composition accounts
for Catholic difference
schools with lower SES had weaker SESmath score relation than higher SES
schools
General MLM
Sector effects
Results:
variation in SES-math score relation may
be accounted for by school SES
variation in mean math score is not entirely
accounted for by school SES
MLM with person and time
When observations are repeated for the
same units, we also have a nested
structure.
Examining within-person changes over
time – growth curve analysis.
Growth curves may be similar across
persons within a class.
Example: Muthén and Muthén
Dependent variable: categorical, latent
Muthen and Muthen
NLSY
N=7326 (part 1); N=924 (part 2); N=922
(part 3); N=1225 (part 4)
Dependent variables: antisocial behavior
(excluding alcohol use) during past year, in
17 dichotomous items; alcohol use during
past year, in 22 dichotomous items
MLM with person and time
Part 1: latent class determination by latent
class analysis and factor analysis
It’s a cross-sectional analysis of baseline
data in 1980.
It found 4 latent classes.
MLM with person and time
Part 2: growth curve determination by latent
class growth curve analysis and growth mixture
modeling
It uses longitudinal information.
Different growth curves are allowed and
estimated for different latent classes.
Growth mixture modeling is a generalization of
latent class growth analysis, in allowing growth
variance within class
GMM yields a 4-class solution.
MLM with person and time
Part 3: latent class relation to growth
curve model by general growth mixture
modeling (GGMM)
What’s new is to the ability to predict a
categorical outcome variable from latent
classes.
The example also illustrates how
covariates that predict membership in
classes (Table 4).
MLM with person and time
Part 4: latent class relation to growth
curve model by GGMM
Multiple (2) latent class variables.
The first one comes from Part 1; the
second one comes from Part 2.
It bridges the two component parts, asking
how the first class membership affects
membership in the second class scheme.
MLM with person, time & space
Example: Axinn and Yabiku
Dependent variable: dichotomous,
observed
Hazard model with event history
MLM with person, time & space
Chitwan Valley Family Study (CVFS)
171 neighborhoods (5-15 household
cluster)
Dependent variable: initiated
contraception to terminate childbearing
MLM with person, time & space
Age 0
Age 12
Time-invariant childhood
community context
Time-invariant early life
nonfamily experiences
Birth of
1st child
Contraceptive use or
end of observation
Time-varying contemporary
community context
Time-varying contemporary
nonfamily experiences
MLM with person, time & space
Level one:
Logit(ptij) = b0j + b1Cj+ b2Xij +b3Djt + b4Zijt
C: time-invariant community var.
D: time-variant community var.
X: time-invariant personal var.
Z: time-variant personal var.
(Note that there is no interaction across
levels)
Multi-level Hazard Models
There is a general problem with non-linear
multi-level models.
Unbiasedness breaks down.
Special attention needs to be paid to
estimation of hazard models in a multilevel setting.
See Barber et al (2000).