HIERARCHICAL LINEAR MODELS

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HIERARCHICAL LINEARMODELS
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NESTED DESIGNS

• A factor A is said to be nested in factor B if
  the levels of A are divided among the levels of
  B. This is given the notation A(B). We have
  encountered nesting before, since Subjects are
  typically nested in Treatment, S(T), in the
  randomized two group experiment.

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NESTED DESIGNS
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NESTED DESIGNS
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ANOVA TABLE
Note no interactions can occur between nested
factors
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ESTIMATING VARIANCES

• ?2C (MSC MSP )/p
• Conceptually this is
• (?2? p?2c - ?2?)/p

?
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TESTING CONTRASTS

• Thus, if one wanted to compare School 1 to School
  2, the contrast would be
• C12 Xschool 1 - Xschool 2
• Since the school mean is equal to overall mean
  school 1 effect error of school
• Xschool 1 ? ... ?1. e1. ,

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TESTING CONTRASTS

• the variance of School 1 is
• VAR(Xschool 1 ) ?2? ?2S /s
• MS(P(C(S))) MS(S) -
  MS(C(S)/cp / s
• Then t C12 / 2MS(P(C(S)))MS(S)-MS(C(S)/cp/
  s
• which is t-distributed with 1, df Satterthwaite
  approximation

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Satterthwaite approximation

• df cpMS(P(C(S)))/s MS(S) - MS(C(S)/s 2
• cpMS(P(C(S)))/s 2 MS(S)2
  MS(C(S)/s 2
• (p-1)cs cp(s-1) p(c-1)

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HLM - GLM differences

• GLM uses incorrect error terms in HLM designs
• Multiple comparisons using GLM estimates will be
  incorrect in many designs
• HLM uses estimates of all variances associated
  with an effect to calculate error terms

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Repeated Measures

• Multiple measurements on the same individual
• Time series
• Identically scaled variables
• Measurements on related individuals or units
• Siblings (youngest to oldest among trios of
  brothers)
• Spatially ordered observations along a dimension

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WITHIN-GROUP DESIGNS

• Within group designs
• We encountered a repeated measures design in
  Chapter Six in the guise of the dependent t-test
  design.
• _ _
• t x1. x2. / sd
• where
• sd ( s21 s22 2 r12 s1s2 )/n 1/2

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WITHIN-GROUP DESIGNSMODEL

• y ij ? ?i ?j eij
• where y ij score of person i at time j,
• ? mean of all persons over all occasions,
• ?i effect of person i,
• ?j effect of occasion j,
• eij error or unpredictable part of score.

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EXPECTED MEAN SQUARESFOR WITHIN-GROUP DESIGN
Source
df
Expected mean square
s
2
s
2
P
P-1
O
p
e

s
2
s
2
s
2
O
O-1

P
t
p
t
e
s
2
s
2
PO
(P-1)(O-1)

t
p
e
2
s
error

0
e
Table 11.1 Expected mean square table for P x O
design
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EXPECTED MEAN SQUARESFOR WITHIN-GROUP DESIGN
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VENN DIAGRAM FOR WITHIN-GROUP DESIGN
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SPHERICITY ASSUMPTION

• ?ij ?ij? for all j, j? (equal covariances)
  and ?ij ?ij for all I and j (equal variances)
• By treating each occasion as a variable, we can
  represent this covariance matrix, called a
  compound symmetric matrix, as
• ?11 ?12 ?13
• ? ?21 ?22 ?23
• ?31 ?32 ?33
• .
• .
• with ?12 ?21 ?31 ?32

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Testing Sphericity

• GLM uses Huynh-Feldt or Greenhouse-Geisser
  corrections to the degrees of freedom as
  sphericity is violated
• reduces degrees of freedom and power
• HLM allows specifying the form of the covariance
  matrix
• Compound symmetry (sphericity)
• Autoregressive processes
• Unstructured covariance (no limitations)

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Factorial Within-Group Designs
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Between- and Within-group Designs

• BETWEEN
• SOURCE df SS MS F error term
• Treat 1 20 20 4.0 P(Treat)
• Person 18 90 5.0 -
• WITHIN
• Time 2 50 25 12.5 P(Treat) x Time
• Treat x Time 2 30 15 7.5 P(Treat) x Time
• P(Treat) x Time 36 72 2.0 -

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Venn Diagram for Between and Within Design
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Doubly Repeated (Time x Rep) Between and Within
Design
BETWEEN
WITHIN
Time
Treatment
Time x Treatment
Person (Treatment) x Time
Person (Treatment)
Person (Treatment) x Rep
Treatment x Rep x Time
Person (Treatment) x Time x Rep
Treatment x Rep
Rep
Time x Rep
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HLM-GLM distinctions

• HLM correctly estimates contrasts for any
  hierarchical between-factors
• HLM correctly estimates all within-subject
  contrasts
• GLM does not estimate within-subject contrasts
  correctly

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The corrections to the F-test should be made
given that the sphericity test was significant.
For Greenhouse-Geisser, the df for the F-test are
reduced to 1, N-1 or 1, 1658, so that the
F-statistic is still significant at p lt .001. For
the Huynh and Feldt epsilon statistic, the
degrees of freedom are adjusted by the amount
.732 dfnumerator 3 x .732 2.196
dfdenominator 4974 x .732 3640.968. The
fraction df can either be rounded down or a
program, such as available in SAS, can provide
the exact probability. For the df 2,3640 the
F-statistic is still significant. Kirk (1996)
discussed in detail various adjustments and
recommends one by Collier, Baker, Mandeville, and
Hayes (1967), but the computation is cumbersome
HLM analyses compute it.