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Unformatted text preview: Quantitative Methods in
Psychology
Psychology
August 4th, 2009: ANOVA week!
Jeff Vietri
corresponding to Chapter 14 in the
corresponding
text
text Effect Size
Effect
SSbetween
η=
SSbetween + SS within
2 • Gives the percent variance explained by
Gives the treatment (just like r2)
the the variability accounted for by the treatment
the
differences
differences 0.01 < η2 < 0.09 small effect
0.01
0.09 < η2 < 0.25 medium effect
0.09 η2 > 0.25
large effect Repeated Measures ANOVA
Repeated
Up until now we’ve just covered oneway
Up
ANOVA (the most basic type)
ANOVA
– This ANOVA merely determines if there are
This
significant differences between samples
significant However, its possible that differences
However,
between samples could be attributed to
the people in each group
the RepeatedMeasures ANOVA
RepeatedMeasures
Just like repeatedsamples ttest,
Just
Repeatedmeasures ANOVA uses the
same people in different treatment
conditions
conditions
– This reduces or may even eliminate extra
This
error
error
– This is an extremely powerful test
– Often the same people are assessed at
Often
different points in time (or under different
conditions) Repeated Measures ANOVA
Repeated Recall: Independent measures
ANOVA
ANOVA In repeated measures, individual difference ONLY contributes to differences within
treatments, because same individuals are used in all treatments. Variancebetween Variancewithin Independent
measures Treatment effect +
Experimental error +
individual differences Experimental error +
individual differences Repeated
measures Treatment effect +
Experimental error Experimental error +
individual differences Now: Repeatedmeasures ANOVA
Now: Repeatedmeasures ANOVA F= Variance between treatments
Variance due to chance/error Now, we can eliminate individual
differences in repeated measures ANOVA
differences
F= Variance between treatments (without individual differences)
Variance due to error (without individual differences) = Treatment effect + error (excluding individual differences)
Error (excluding individual differences) = Variance between treatments
Variance within treatments – Variance between individuals Repeated Measures: SSTotal
Repeated
• SSTotal = sum of squares for the entire set of scores (without regard to group)
of
Same as with oneway ANOVA SSTotal = ΣX
SSTotal 2 ( ΣX )
−
N 2 G
= ΣX −
N
2 2 G = Grand Total
G = ΣX
dftotal = N  1 Repeated Measures: SSBetween
Repeated
• SSBetween is the sum of squares of group mean
differences
differences
Pretending that each mean of a group is an
Pretending
individual score you use the same process for
the sum of squares
the
• dfbetween = k – 1
Again, same formula as with oneway ANOVA SS Between T12 T2 2 T3 2 G 2
−
=
+
+
n
n2 n3 N
1 T = group total Repeated Measures: SSWithin
Repeated
Within Groups variance in Oneway
Within
ANOVA is attributed to both individual
differences and random error
differences
Repeatedmeasures ANOVA attempts to
Repeatedmeasures
separate out these two components
separate
• SSwithin = SSerror + SSsubjects
• SSsubjects = individual differences
– This analyses accounts for individual
This
differences which strengthens the test
differences Repeated Measures: SSWithin
Repeated
• SSwithin is the sum of squares for each group added together
group
• SSwithin = SS1 + SS2 + SS3 + …+ SSk
SS
within
• dfwithin = N – k Same as with regular ANOVA Repeated Measures: SSBetween Subjects
Repeated
Represents individual differences so they can be
Represents
removed
removed
2 SS BetweenSubjects 2 2 Pn1
P
P2
G2
=1+
+ ... +
−
k
k
k
N • dfbetween subjects = n1 – 1
P represents person totals
represents
For each person, square the person’s total and
For
divide by the number of samples
Add all these up
Add
• Subtract [G2/N] Repeated Measures: SSError
Repeated
• SSError = SSwithin – SSbetween subjects
• dferror = (N – k) – (n1 – 1)
error
– “N” is the number of people in study
– “n1” is the number of people in each group For a repeatedmeasures ANOVA, which of the
following is computed differently, compared to an
independentmeasures ANOVA?
independentmeasures
A. total SS
B. between treatment
between SS
SS
C. within treatment SS
D. the denominator of
the
the F ratio
the Example
Example Effect Size
Effect
Just like with other statistical tests,
Just
Repeated Measures ANOVA has a way to
calculate effect size
calculate
Effect size gives a way to get a sense of
Effect
how important your results are
how
Just because they are significant does not
Just
mean that your IV has a huge impact on
your DV
your Post Hoc Tests
Post
Tukey’s HSD can be used for Repeated
Tukey’s
Measures
Measures
• Again, same formula as Oneway ANOVA
Again,
except must use MSError instead of MSWithin
except
• Also, df for qtable is dferror not dfwithin
for MS error
HSD = q
n1 Twoway ANOVA
Twoway Twoway ANOVA
Twoway Up until now we have only discussed Oneway
Up
ANOVA
ANOVA Where one discrete IV (“factor”) predicts one DV However, often times in research we not just
However,
concerned with one variable, researchers may want
to study more than one at once
to
Remember confounding variables? We can include
Remember
them into our design
them Twoway ANOVA
Twoway Oneway ANOVA design: Does my drug affect cholesterol levels (control, 5mg,
Does
10mg)?
10mg)? Twoway ANOVA design: Does my drug affect cholesterol levels (control, 5mg,
Does
10mg)?
10mg)?
Does an exercise program affect cholesterol levels?
Does drug affect cholesterol levels differently for people
Does
on exercise program vs. those not on exercise program?
on Twoway ANOVA
Twoway Twoway ANOVA is looking at two IVs and
Twoway
their effect on the DV
their
Twoway ANOVA has three null hypotheses: H0a: μA1 = μA2 = μA3
H0b: μB1 = μB2 = μB3
H0c: There is no interaction between A and B.
There This last hypothesis tests what is called an
This
interaction
interaction Twoway ANOVA
Twoway Since we are looking at two variables at the
Since
same time, we need to know the effect of the
first variable, the second variable, and the
combined effects of both variables
combined Example
Example
makes you
feel good makes you
feel good + = ? Example
Example
makes you
feel good makes you
feel good + makes you
dead = Interaction
Interaction An interaction means that the relationship
An
between one variable and the dependent
variable is affected by another variable
variable Q: What is the relationship of X and Y?
If the answer is “it depends”, then you have
If
an interaction
an What is the relationship between
temperature and test scores?
(A) Higher temperatures are
associated with lower scores.
(B) IT DEPENDS! If there is high
humidity then higher temperatures
are associated with lower scores.
If there is low humidity, there is no
relationship between temperature
scores. Effect Size
Effect
SSbetween
η=
SSbetween + SS error
2 Same as formula for Oneway ANOVA except replace SSwithin
by SSerror
Formula give the percent of variance explained
Similar to the r2 of the ttest effect size
“How much variation in the DV is explained by the IV?” Assumptions
Assumptions
Observations within each treatment condition
Observations
must be independent
must
– Independent observations Distribution of scores within each group must be
Distribution
normal
normal
– Normality Variances of scores within each group must be
Variances
normal
normal
– Homogeneity of variance
– Test with Fmax test (just like ttest) but we won’t be
doing that in this course
doing Exercise
Treatments
Participant
A
B
C
D
_________________________________________
1
6
3
3
0
P=12
ΣX2 = 172 2
4
4
2
2
P=12
G = 44 3
4
2
0
2
P=8
4
6
3
3
0
P=12
_________________________________________
T = 20 T = 12 T = 8 T = 4
SS = 4 SS = 2 SS = 6 SS = 4
Are there significant differences among treatments? Alpha = .05
Compute the percentage of variance explained by the treatment (η2). ...
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This note was uploaded on 11/17/2011 for the course PSYCHOLOGY 830:200 taught by Professor Staff during the Fall '11 term at Rutgers.
 Fall '11
 Staff
 Psychology

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