Chapter 5 Multilevel Models

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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

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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

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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 ß0i ß1i zij ?ij
• yij - students performance on an achievement
  test
• zij - total family income
• Level 2 Model
• Thinking of the schools as a random sample, we
  model ß0i, ß1i as random quantities.
• ß0i ß0 a0i and ß1i ß1 a1i ,
• where a0i, a1i are mean zero random variables.

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Two-level model example

• The combined level 1 and level 2 models form
• yij (ß0 a0i ) (ß1 a1i) zij ?ij
• a0i a1i zij ß0 ß1 zij ?ij .
• The two-level model may be written as a single
  linear mixed effects model.
• Specifically, we define ai (a0i , a1i), zij
  (1, zij), ß ( ß0, ß1) and xij zij, to write
  
• yij zij ai xij ß ?ij .
• We model and interpret behavior through the
  succession of level models.
• We estimate the combined levels through a single
  (linear mixed effects) model.

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Two-level model example - variation

• Modify the level-2 model so that
• ß0i ß0 ß01 xi a0i and ß1i ß1 ß11 xi
  a1i ,
• 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 a0i a1i zij ß0 ß01 xi ß1 zij
  ß11 xi zij ?ij .
• That we can again write as
• yij zij ai xij ß ?ij .
• by defining ai (a0i , a1i), zij (1, zij),
• ß ( ß0, ß01, ß1, ß11) and xij (1, xi, zij,
  xi zij).
• The term ß11 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.

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Three Level Models

• Level 1 model (students)
• yi,j,k z1,i,j,k ßi,j x1,i,j,k ß1 e1,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)
• ßi,j Z2,i,j ?i X2,i,j ß2 e2,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)
• ?i Z3,i ß3 e3,i .
• The predictors Z3,i may depend on the school, but
  not students or classroom.

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Combined Model

• The combined level 1, 2 and 3 models form
• yi,j,k z1,i,j,k ( Z2,i,j (Z3,i ß3 e3,i)
  X2,i,j ß2 e2,i,j)
• x1,i,j,k ß1 e1,i,j,k
• xi,j,k ß zi,j,k ai,j e1,i,j,k ,
• where

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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.

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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.

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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.

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Dental Model

• Level 1 model
• yit ß0i ß1i z1,it ?it ,
• z1,it is age.
• Level 2 model
• ß0i ß00 ß01 GENDERi a0i and
• ß1i ß10 ß11 GENDERi a1i.
• GENDER - 1 for females and 0 for males.

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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 ßi,j x1,i,j,t ß1 e1,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 ß0,i,j ß1,i,j TIMEi,j,t ß1
  PROGRAMi,j,t e1,i,j,t .

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Children Mental Health Assessment

• The level 2 model (child replications) is
• ßi,j Z2,i,j ?i X2,i,j ß2 e2,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
• ß0,i,j ß0,i,0 ß0,0,1 RATERi,j e2,i,j
• and
• ß1,i,j ß2,0 ß2,1 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
• ?i Z3,i ß3 e3,i
• ß0,i,0 ß0,0,0 ß0,1,0 GENDERi e3,i .

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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 ßi x1,i,t ß1 e1,i,t ,
• Level 2 model - ßi Z2,i ß2 ?i.
• Linear mixed model is
• yi,t z1,i,t (Z2,i ß2 ?i) x1,i,t ß1
  e1,i,t ,
• The best linear unbiased predictor (BLUP) of ßi
  is
• bi,BLUP ai,BLUP Z2,i b2,GLS ,
• where ai,BLUP D Zi? Vi-1 (yi - Xi bGLS ).

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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 ai,j (e2,i,j
  e3,i), say,
• ai,j,BLUP (e2,i,j,BLUP e3,i,BLUP ).
• Then, compute BLUP predictors of ?i and ßi,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

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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 sa2 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 a 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,
• sa2 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.

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Alternative Testing Procedures

• Suppose that we wish to test the null hypothesis
  H0 s 2 s02, where s02 is a known positive
  constant.
• Standard likelihood ratio test is okay.
• This procedure is not available when s02 0
  because the log-likelihood under H0 is not well
  defined.
• However, H0 s 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!!!

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Error Components Model

• Consider the likelihood ratio test statistic for
  assessing H0 sa2 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

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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.