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CLASS NOTES: Repeated Measures of Analysis of Variance (ANOVA)
CONCEPT
CALCULATION/EXAMPLES
APPLICATION
Null Hypothesis (Ho): There is no
difference b/t treatment effects &
any differences that exist occur due
to chance.
Remember here that for the null
hypothesis, it can simply be written
as A = B = C = D. They all are the
same, so no changes exist.
However for the alternative
hypothesis, we are comparing more
than 2 treatment methods, & there
are a number of different
alternatives that can exist (A > B,
but B = C; A = B = C; A = B, but B
< C, etc..). So, we are looking for
at least one difference b/t
treatments. It is too much to write
every possible outcome in the case
of the alternative hypothesis.
* Researchers usually have some
idea of what difference they are
looking for in their study.
ANOVA for Repeated-Measures
Design: This is where the same
sample is used multiple times, or
that the same study group of
individuals participates in all of
the different treatment conditions.
* The general purpose of a
repeated-measures ANOVA is to
determine whether the differences
that are found between treatment
conditions are significantly greater
than would be expected by chance.
Single factor is used here
meaning that only one factor or
one variable is manipulated on two
or more levels w/ the same set of
individuals.
Hypotheses for the RepeatedMeasures ANOVA
Alternative Hypothesis (H1): At
least one treatment mean (µ) is
different from another.
F-Ratio for the RepeatedMeasures ANOVA
F=
Variance/differences between
Treatments (without individual
_________differences________
Variance/differences expected
By chance (w/ individual
With this formula, you do not have
to account for individual
differences as you do in
independent measures.
* For repeated-measured ANOVA,
the denominator is called the
differences removed)
residual variance or the error of
variance & measures how much
variance is to be expected just by
chance after the individual
differences have been removed.
This is the same for independentmeasures ANOVA
The Logic of the RepeatedMeasures ANOVA: There are 2
primary sources for chance
differences:
1. Individual differences: There
are variances b/t scores for each
participant.
2. Experimental Error: There is
always potential for some degree
of error in general
Testing Hypotheses w/ the
Repeated-Measures ANOVA
There are 2 different variances that
we are looking at for this
ANOVA:
This is the same logic as in
independent- measures design.
1) Between-Treatments Variance:
The variance b/t treatment
conditions
2) Within-Treatment Variance:
The variance w/ the sample
ANOVA Notation & Formulas
Example:
S = the number of subjects
This is not a complete list of all
notations & formulas, but those that
are different from earlier notations
& also those that are new. There
are other notations & formulas used
in ANOVA that are not listed here,
but that you are already familiar w/.
6 participants = 6 subjects
k = Number of treatment
conditions
3 different treatments for
Alzheimer’s Disease
k = 3 (treatment conditions)
n = Number of scores in each
If there are 5 scores in each
Remember that w/ Repeatedmeasures ANOVA, you only have
one sample group w/ multiple
treatment methods
treatment
N = Total # of scores in the entire
study
C = Correction Factor
G = The sum of all the scores in
the research study (the grand total)
SS = Sum of Squares
MS = Mean square
df = Degrees of Freedom
P = Used to represent the total of
all the scores for each individual
in the study.
treatment condition, n = 5
If there are 9 scores altogether (3
treatment conditions, 3 scores in
each treatment condition), then N =
9
For Repeated-Measures ANOVA,
“N” represents the total number of
all scores as opposed to total
number of all research participants.
C = (∑X1 + ∑X2 + ∑X3…..)2
N
Add up all the N scores or add up
the treatment totals G = Σ (∑X1 +
∑X2 + ∑X3…..)
SS = ∑X2 – (∑X)2
N
MS = SS
df
Remember that these formulas in
this column are general. The more
specific formulas used in the
process are listed below.
N–1
(see full example for calculations of
each degrees of freedom)
∑X1 + ∑X2 + ∑X3…..
“P” can be seen as “Personal
totals” or “Participant totals.”
ANOVA Formula Step 1:
For each subject, calculate..
a) ∑Xm, (∑Xm)2
(symbol P may also be
used to represent
personal totals)
For each treatment condition,
calculate…
The sum of each subject’s scores &
this value squared
Review example in text
a) ΣX, ΣX2 & (ΣX)2
b) n
n = number of participants/ sources
in each sample
c) M
The mean for each treatment
condition
The subscript ‘m’ represents values
associated w/ each subject. Another
notation representing personal
totals is also P. ‘P’ does not exist in
independent measures as these
values are not used in calculation.
For the full or TOTOAL set of
values, calculate….
d) N
N = The total number of all scores
in the experiment
e) M
Mean of each treatment mean
f) ΣX, ΣX2 & (ΣX)2
For the total set of values (see
example for calculations)
ANOVA Formula Step 2:
By adding the mean values together
& dividing by the number of
treatment measures
ANOVA SS formula:
Calculate Sum of Squares for…
a) SSTOT total
SSTOT = ∑X2 – (∑X)2
N
Using your total values.
b) SSb Between treatments
∑
The sum of squares between
treatment groups. The Sigma (∑)
represents the sum. The ‘g’
beneath the Sigma indicates that
you should repeat the formula for
each treatment. The subscript ‘g’
represents the value associated w/
each
g
(∑Xg)2 - (∑X)2
ng
N
c) SSsubj Within Subjects
∑ (∑Xm)2 - (∑X)2
s
k
N
d) SSerror Error
SSerror =
SSTOT – SSb - SSsubj
ANOVA Formula Step 3:
Calculate the df for…
a) dftot total
N–1
b) df between treatments
k–1
The ‘s’ beneath the sigma here
works the same as the ‘g’ beneath
the sigma in the between treatments
SS formula. It still means that you
are repeated the formula, but for
each subject as opposed to each
treatment.
c) df Between subjects
d) df error
S–1
(k – 1) (S – 1)
ANOVA Formula Step 4:
Compute Mean Square for…
a) MSb Between Treatments
b) MSerror Error
SSb
dfb
SSerror
dferror
Sum of Square between treatments
divided by degrees of freedom
between treatments
Sum of Square error divided by
degrees of freedom error
ANOVA Formula Step 5:
Compute your F-Ratio
So that you can compute your
final F-Ratio, which is…
F = MSbetween treatments
MSerror
Mean Square between treatments
divided by Mean Square error
The Distribution of F-Ratios
Since there are 2 df’s, then it is
expressed as :
df= 2, 12
The first df listed as your between
treatments df & your second the
within treatments df.
Once you have computed your FRatio score…
1) Go to the F distribution
table with your dfbetween
treatments score (calculated in
the numerator portion of
your F-Ratio formula) &
dfwithin treatments (subjects) score
(found in the denominator
portion of your F-Ratio
formula), which in the case
of repeated measures, you
will use your dferror as your
denominator.
2) Locate these 2 df’s on the
table (numerator is listed in
the row above whereas the
denominator is listed in the
column on the right).
3) Connect these 2 scores
together in the middle.
Regular type scores give
you the critical value for
alpha level of .05. Bold will
give you the critical value
for alpha level .01.
Example of ANOVA Repeated-Measures Design
Visual Detection Scores
Participant
No
Distraction
Visual
Distraction
Auditory
Distraction
A
47
22
41
B
57
31
52
C
38
18
40
D
45
32
43