ANOVA approaches to Repeated Measures • univariate repeated

63 Pages·2007·2.38 MB·English

ANOVA approaches to Repeated Measures
• univariate repeated-measures ANOVA (chapter 2)
• repeated measures MANOVA (chapter 3)
Assumptions
• Interval measurement and normally distributed errors
(homogeneous across groups) - transformation may help
• Group comparisons
– estimation and comparison of group means
– not informative about individual growth
• Fixed time points
– time treated as classification variable
– timepoints can be evenly or unevenly spaced
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• Better suited for short-term or moderate change
esp. true for trend or growth-curve analysis
• LS techniques adversely affected by outliers
• Missing data
– ANOVA can handle some - unbalanced ANOVA
– MANOVA can’t handle any
• Variance-covariance structure for y
i
– compound symmetry for ANOVA
– unstructured for MANOVA
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Univariate Repeated Measures ANOVA
One-sample case: randomized block ANOVA
y = µ + π + τ + e (i = 1, . . . , N ; j = 1, . . . , n)
ij i j ij
• µ = grand mean
• π = individual difference component for subject i
i
(constant over time)
• τ = effect of time - same for all subjects
j
• e = error for subject i and time j
ij
Random components
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• π ∼ N(0, σ ) subjects random; between-subjects variance
i π
2
• e ∼ N(0, σ ) within-subjects variance
ij e
3
Data layout
timepoint
subject 1 2 . . . n
1 y y . . . y
11 12 1n
2 y y . . . y
21 22 2n
. . . . . . .
. . . . . . .
N y y . . . y
N1 N2 Nn
• one observation per cell
• similar to randomized block design with subject as block
• identical to paired-t test when n = 2
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Model assumptions
n
τ = 0 (n − 1 df for Time)
X j
j=1
E(y ) = µ + τ
ij j
2 2
V (y ) = V (µ + τ + π + e ) = σ + σ
ij j i ij π e
0
C(y , y ) = 0 for i 6= i (different subjects)
ij i0j
2 0
C(y , y ) = σ for j 6= j (same subject)
ij ij0 π
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Intraclass correlation
2
σ
π
Corr(y , y ) =
ij ij0 2 2
σ + σ
π e
• same correlation for all longitudinal pairs
(average correlation of y from any two timepoints)
• ranges from 0 to 1, so more like a proportion of variance
• proportion of total variance that is attributable to subjects
(that is not accounted for by Time)
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Compound symmetry structure of
variance-covariance matrix
2 2 2 2 2
 σ + σ σ σ . . . σ 
e π π π π
 
 2 2 2 2 2 
 σ σ + σ σ . . . σ 
 
 π e π π π 
 
 
Σ =  . . . . . . . . . . . . . . . 
y  
 
i  
 
 2 2 2 2 2 
 σ . . . σ σ + σ σ 
 π π e π π 
 
 
 2 2 2 2 2 
 σ σ . . . σ σ + σ 
 π π π e π 
 
2 2
• variance homogeneous across time: σ + σ
e π
2
• covariances homogenous across time: σ
π
2 2 2
correlation = σ /(σ + σ )
π e π
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Not too realistic
• variances often change over time: increase is common in
clinical trials (subjects more similar at start of trial)
• covariances close in time usually greater than covariances
distil in time
⇒ test of variance-covariance structure is necessary for the
validity of significance tests (more on this later)
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ANOVA table: subjects random and time fixed (balanced)
Source df SS MS E(MS)
SS
Subjects N − 1 SS = n N (y¯ − y¯ )2 S σ2 + nσ2
S Pi=1 i. .. N−1 e π
SS
Time n − 1 SS = N n (y¯ − y¯ )2 T σ2 + N (τ − τ )2
T Pj=1 .j .. n−1 e P j .
SS
Residual (N − 1) SS = N n (y R σ2
R Pi=1 Pj=1 ij (N−1)(n−1) e
×(n − 1) −y¯ − y¯ + y¯ )2
i. .j ..
total Nn − 1 SS = N n (y − y¯ )2
y Pi=1 Pj=1 ij ..
• y¯ = grand mean (averaged over time and subjects)
..
• y¯ = subject mean (i = 1, . . . , N)
i.
• y¯ = timepoint mean (j = 1, . . . , n)
.j
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