Unformatted text preview: Quantitative Methods in
Psychology
Psychology
August 4th, 2009: ANOVA week!
Jeff Vietri
corresponding to Chapter 14 in the
corresponding
text
text 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
Or, the differences are masked by
Or,
between subjects variability
between 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. 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 Important stuff about ANOVAs What do you need to know to use ANOVAs?
What do you need to know for the
quiz/exam? Terms
When to use
Logic
When to posthoc
How to interpret reports
What patterns of results look like ANOVA terms Factor An independent (or quasiindependent) variable Statistics training
Podcast listening
sex Levels How many different levels of the factor do you
have? None, undergraduate stats, graduate stats
No listening, listening
Female, male ANOVA terms Main effect In multifactorial ANOVA, this is if the factor
affects the DV, ignoring the effects of the other
factors Interaction In mulitfactorial ANOVA, this is when the effect
of a factor depends on the level of another factor i.e., students do well in Jeff’s classes as long as it’s not
stats
Alcohol has a greater effect in women than men Types of ANOVA Oneway (a.k.a. singlefactor) One factor, multiple levels, btwSs designs Repeated measures One factor, multiple levels, withinSs designs A oneway ANOVA with 2 levels of the factor is
functionally equivalent to an independentsamples ttest A RM ANOVA with 2 measurements is functionally
equivalent to a pairedsamples ttest Multifactorial ANOVA >1 factor, >1 level/factor, can have btwSs,
withinSs, or mixed designs ANOVA jargon # by # ANOVA i.e., 2 by 2 ANOVA
This describes each factor by how many levels
each has
2 by 2 means there are 2 factors, each with 2
levels
5 by 2 means there are 2 factors, one with 5
levels, one with 2 levels
5 by 2 by 2 means there are 3 factors, one with 5
and two factors with 2 levels each When should you use which
ANOVA? Oneway (singlefactor) ANOVA: When you have a single factor with >=2 levels,
and each level is a separate sample RM ANOVA: Need posthoc tests if levels > 2 When you’re trying to conduct a more sensitive
test, single factor, multiple levels, each level
applied to the same sample at different times Multifactorial ANOVA When you’re measuring the effect of >1 IV (or QIV) on the DV. Especially when you think the
two might interact. ANOVA logic Not all that different from ttest logic Numerator: variability in scores due to
treatments + error
Denominator: variability in scores due to
error
If the numerator is much bigger than
denominator, much of the variance in scores
is due to our treatment ANOVA logic Numerator is variability between treatment
conditions Denominator is either withintreatment
differences or error, depending on test As your treatment means get more different, the
numerator, and therefore the ration, gets bigger As the scores within a given treatment get more
different, denominator gets bigger, Fratio gets
smaller, failure to reject null Bigger Fratios = more the total variance is
due to the treatment. More likely to reject null hypothesis Ratio logic The Fratio is the ratio of one variance (mean
squares) to another.
Variances are always positive F >= 0 If there’s no treatment effect, Numerator = error
Denominator = error
F=1 Hypotheses Null is that there are no differences Therefore, reject the null means >= 1 mean is
different from >= another mean
Don’t know which unless there are only 2 means
Posthoc tests! Posthoc tests Tukey’s HSD & Scheffe are pretty common
Only do when: Reject null AND
>2 groups How to interpret ANOVA in lit Reject null: F (df_btw, df_within/error) = Fvalue, p < alpha
F (1, 28) = 7.23, p < .05 Fail to reject null F (df_btw, df_within/error) = Fvalue, p < alpha
F (1, 28) = 2.10, p > .05 For multivariate Each effect will have it’s own Fratio: Main effect for each factor There was no main effect of sex, F(1, 480) = 1.9, p > .
05
There was a main effect of studying, F (1, 480) = 6.2, p
< .05 And for the interaction And studying had a stronger effect for women than for
men, F (1, 480) = 5.1, p < .05 How to read reports This is important!
Looking at the pattern of means can often be
informative even before you perform
calculations
You can make good guesses at main effects
& interactions group individual Main effect of studying technique?
Is the mean for individualstudy (averaged across sex) higher
than the mean for groupstudy (averaged across sex)? Men Women
Main effect of sex?
Is the mean for men (averaged across studying technique) higher
than the mean for women (averaged across studying technique)? Men Women
Interaction between the two?
Is the relationship between studying technique and quiz score the
same for men and women?
Are the lines not parallel? ...
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 Fall '11
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 Psychology

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