Multilevel Modeling

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Multilevel Modeling

• Soc 543
• Fall 2004

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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

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Multilevel models

• Units of analysis are nested within higher-level
  units of analysis
• Students within schools
• Observations with person

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Problems without MLM

• If we ignore higher-level units of analysis gt
  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 gtunreliable slopes
• Homoscedasticity and no serial correlation
  assumptions of OLS are violated (an efficiency
  problem).
• No distinction between parameter and sampling
  variances

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Advantages of MLM

• Cross-level comparisons
• Controls for level differences

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General MLM

• Example Raudenbush and Bryk, 1986
• Dependent variable
• Continuous
• Observed

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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

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General MLM

• Variability among schools
• Level one within schools
• mathij b0j b1j (SESij - SESj) rij

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General MLM

• Variability among schools
• Level two between schools
• b0j g00 u0j
• b1j g10 u1j

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General MLM

• Variability among schools
• Combined model
• mathij g00 u0j
• g10(SESij - SESj)
• u1j(SESij - SESj) rij
• g00 g10(SESij - SESj)
• u0j vij
• (Easy interpretation given the centering
  parameterization)

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General MLM

• Variability among schools
• Combined model
• mathij 100.74
• 4.52(SESij - SESj)
• u0j vij
• There is a positive relation between SES and math
  score

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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

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General MLM

• Covariates at level 2
• Level one within schools
• mathij b0j b1j (SESij - SESj) rij

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General MLM

• Covariates at level 2
• Level two between schools
• b0j g00 g01sectj u0j
• b1j g10 g11sectj u1j

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General MLM

• Covariates at level 2
• Combined model
• mathij g00 g01sectj
• g10 (SESij - SESj)
• g11sectj(SESij - SESj)
• rij vj

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General MLM

• Combined model
• mathij 98.37 5.06sectj
• 6.23(SESij - SESj) - 3.86sectj(SESij -
  SESj)
• rij vj

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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

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General MLM

• Sector effects
• Cannot say that previous relations are causal
  may be selection effects
• Use example of homework to explain sector
  differences

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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 SES-math score
  relation than higher SES schools

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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

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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

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Muthen and Muthen

• NLSY
• N7326 (part 1) N924 (part 2) N922 (part 3)
  N1225 (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

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MLM with person and time

• Part 1 latent class determination by latent
  class analysis and factor analysis
• Its a cross-sectional analysis of baseline data
  in 1980.
• It found 4 latent classes.

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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.

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MLM with person and time

• Part 3 latent class relation to growth curve
  model by general growth mixture modeling (GGMM)
• Whats 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).

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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.

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MLM with person, time space

• Example Axinn and Yabiku
• Dependent variable dichotomous, observed
• Hazard model with event history

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MLM with person, time space

• Chitwan Valley Family Study (CVFS)
• 171 neighborhoods (5-15 household cluster)
• Dependent variable initiated contraception to
  terminate childbearing

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MLM with person, time space
Age 0
Age 12
Birth of 1st child
Contraceptive use or end of observation
Time-invariant childhood community context
Time-varying contemporary community context
Time-invariant early life nonfamily experiences
Time-varying contemporary nonfamily experiences
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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)

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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 multi-level setting.
• See Barber et al (2000).