Linear Models I: A TwoLevel Model

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Linear Models I A Two-Level Model
Session 2
Damon Berridge
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Random Effects ANOVA

• The simplest multilevel model is equivalent to a
  one-way analysis of variance with random effects
  in which there are no explanatory variables.
• This model is useful as a conceptual building
  block in multilevel modelling as it possesses
  only the explicit partition of the variability in
  the data between the two levels.

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The Intraclass Correlation
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• The intraclass correlation coefficient r
  measures the proportion of the variance in the
  outcome that is between the level-2 units.
• We note that the true correlation coefficient r
  is restricted to take non-negative values, i.e. r
  ³ 0.
• Note that conditional on being in a group,
• But across the population,

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Parameter Estimation by Maximum Likelihood
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• The overall mean,

where
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Regression with level-2 effects

• In multilevel analysis the level-2 unit means
  (group means for explanatory variables) can be
  considered as an explanatory variable.
• A level-2 unit mean for a given level-1
  explanatory variable is defined as the mean over
  all level-1 units, within the given level-2 unit.
  
• The level-2 unit mean of a level-1 explanatory
  variable allows us to express the difference
  between within-group and between-group
  regressions.
• The within-group regression coefficient
  expresses the effect of the explanatory variable
  within a given group the between-group
  regression coefficient expresses the effect of
  the group mean of the explanatory variable on the
  group mean of the response variable.

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Linear Model Example C1

• The data we use in this example are a sub-sample
  from the 1982 High School and Beyond Survey
  (Raudenbush, Bryk, 2002), and include information
  on 7,185 students nested within 160 schools 90
  public and 70 Catholic. Sample sizes vary from 14
  to 67 students per school.

Raudenbush, S.W., Bryk, A.S., 2002, Heirarchical
Linear Models, Thousand Oaks, CA. Sage
Number of observations (rows) 7185 Number of
variables (columns) 15 The variables include the
following schoolschool identifier studentstuden
t identifier minority 1 if student is from an
ethnic minority, 0 other) gender 1 if student
is female, 0 otherwise ses a standardized scale
constructed from variables measuring parental
education, occupation, and income, socio economic
status meanses mean of the SES values for the
students in this school mathach a measure of the
students mathematics achievement size school
enrolment sector 1 if school is from the
Catholic sector, 0 public pracad proportion
of students in the academic track disclim a
scale measuring disciplinary climate himnty 1 if
more than 40 minority enrolment, 0 if less than
40)
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• We take standardized measure of mathematics
  achievement (mathach) as the student-level
  outcome, yij.
• The student level (level-1) explanatory
  variables are the student socioeconomic status,
  sesij , which is a composite of parental
  education, occupation, and income an indicator
  for student minority (1 yes, 0 other), and
  an indicator for student gender ( 1 female, 0
  male).
• There are two school-level (level-2) variables
  a school-level variable sector, which is an
  indicator variable taking on a value of one for
  Catholic schools and zero for public schools, and
  an aggregate of school-level characteristics
  (meanses) j, the average of the student ses
  values within each school. Two variables ses and
  meanses are centred at the grand mean.

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Questions motivating these analyses include the
following

• How much do the high schools vary in their mean
  mathematics achievement?
• Do schools with high meanses also have high math
  achievement?
• Is the strength of association between student
  ses and mathach similar across schools?
• Or is ses a more important predictor of
  achievement in some schools than in others?
• How do public and Catholic schools compare in
  terms of mean mathach and in terms of the
  strength of the ses- relationship, after we
  control for meanses?

To obtain some preliminary information about how
much variation in the outcome lies within and
between schools, we may fit the one-way ANOVA to
the high school data.
The student-level model is
The combined model is given by
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Sabre commands
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Sabre log file
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Example Including Effects of School Level

• The simple model
  provides a baseline against which we can compare
  more complex models.

Each school's mean is now predicted by the
meanses of the school
Substituting the level-2 equation into the
level-1 yields
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The estimated regression equation is given by