Repeated Measures ANOVA

90 Pages·2015·5.92 MB·English

Repeated Measures ANOVA
Prof. Wei Zhu
Department of Applied Mathematics & Statistics
Stony Brook University
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The One-way ANOVA we have just learnt can
test the equality of several population means.
It is an extension of the pooled variance t-test
That is:

 H (null hypothesis) : µ = µ = µ =…….. = µ
0 1 2 3 k
H (alternative hypothesis): At least one of
a

means differs from the rest.

Assumptions:
 Unknown but equal population variances
 Normal populations
 Independent samples
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The F statistic
k
 
2

1 n x  x
k1 i i
F  i1
n
k
j  
  2
1 x  x
Nk ij i
i1 j1
where x = the jth observation in the i th sample.
ij
 
i 1,2,,k and j 1,2,,n
i
n
i

x
ij
x  j1  mean for ith sample i 1,2,,k
i
n
i
k n
i
 k
x

ij N  n  Total sample size
i
i1 j1
x   Overall mean
i1
N
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The ANOVA table
Source S.S d.f, M.S. F

k MS
Between SS  n x  x2 MS  1 k n x  x2 F  B
B i i k 1 MS
B k1 i i
i1 W
i1

k nj   k nj  
Within SS   x  x 2 MS  1  x  x 2
N k
W ij i W Nk ij i
i1 j1 i1 j1
The ANOVA table is a tool for displaying the
computations for the F test. It is very important
when the Between Sample variability is due to
two or more factors. We reject the ANOVA null
hypothesis of equal means if F > F
k-1,N-k, α
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Limitations of the one-way ANOVA:
The most distinct disadvantage to the
analysis of variance (ANOVA) method is
that it requires three assumptions to be
made:

☼ The samples must be independent to
each other.
☼ All variances from each data group,
though unknown, must be equal.The
normality assumption.

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☼ A repeated measures design is one in
which at least one of the factors consists of
repeated measurements on the same
subjects or experimental units, under
different conditions and/or at different time
points.

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 A repeated measures design often involves
measuring subjects at different points in
time – or subjects measured under different
experimental conditions
 It can be viewed as an extension of the
paired-samples t-test (which involved only
two related measures)
 Thus, the measures—unlike in the “regular”
ANOVA—are correlated, that is, the
observations are not independent
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 Example: Data collected in a sequence of
(often) evenly spaced points in time – this is
usually referred to as the ‘longitudinal data’

 Example: Different treatments are assigned
to the each experimental unit

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