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Unformatted text preview: Quantitative Methods in
Psychology
Psychology
August 3rd, 2009: ANOVA week!
Jeff Vietri
corresponding to Chapter 13 in the
corresponding
text
text This week Cover ANOVA Logic of ANOVA Only 2 chapters!
How does it work?
Why would we use it? Types of ANOVA Oneway
Repeated
Multifactorial Today's agenda Go over quiz
Introduce ANOVA Logic
Terms
Calculations? Don't forget the homework The problem What if we want to compare more than 2
groups? Ztest: sample vs. population
Student's ttest: sample vs. population
Independentsamples ttest: two samples
Pairedsamples ttest: one sample, two treatments,
or two matched samples We haven't covered a test that can do that The problem The basis of the previous tests were mean
differences, and so we could only use two
values
If we soldier ahead with multiple ttests, we
run into problems with experimentwise alpha
error The risk of false alarms rises with each test, and
the risk of making a single alpha error somewhere
could get rather high The solution ANOVA, of course! Uses variance, rather than difference between the
means The logic of ANOVA The variance among treatment means is due to
effects of the treatments and chance/error/etc.
The variability among scores within each
treatment condition are due to chance/error/etc.
If the variance between treatments is much
bigger than the variance within treatments, the
treatments did something ANOVA
ANOVA Recall: Many Types of ANOVAs Between Ss = independent measures
Within Ss = dependent measures
Between subjects, within subjects, or mixedmodels (both
Between
between and within Ss)
between
Single factor or multiple factors For now we will only consider single factor,
For
between subjects designs (chapter 13)
between ANOVA: Hypotheses
ANOVA: Null hypothesis H0: μ1 = μ2 = μ3 Populations behind all groups are the same Alternative hypothesis H1: there is at least one difference Not all populations represented by the groups are the
Not
same
same ANOVA uses variance Recall: Variance is a measurement of difference
Recall:
among a set of numbers
among The Fratio The test statistic for the ANOVA is the Fratio
How much variability is between my
treatments / how much should I expect by
chance
For oneway ANOVA, this is variance btw /
variance within F= = Treatment effect + Differences due to chance
Difference due to chance
Variability between treatments
Variability within treatments F= Treatment effect + Differences due to chance
Difference due to chance If there is no treatment effect (treatment effect = 0; null
hypothesis is true):
F= Treatment effect + Differences due to chance
Differences due to chance F=1
If there is a treatment effect (treatment effect ≠ 0; null
hypothesis is false):
F= F>1 Treatment effect + Differences due to chance
Difference due to chance ANOVA
ANOVA The ANOVA uses an Fdistribution (rather than a tdistribution or a zdistribution)
Fdistribution, like a tdistribution, is based on the
Fdistribution,
degrees of freedom (df)
(df) Bigger samples will give us better estimates of the
Bigger
population’s variances
population’s Represents all the possible ratios (since the F is a
Represents
ratio of variances)
ratio
The tail end of the Fdistribution represents the
The
critical region
critical Fdistribution is always positive
Fcritical are different for each df Total Sum of Squares
Total SSTotal = sum of squares for the entire set of
scores (without regard to which sample scores
come from)
come SSTotal = ΣX
SSTotal 2 ( ΣX )
−
N G = Grand Total
G = ΣX 2 G
= ΣX −
N
2 2 dftotal = N – 1 Withingroups Sum of Squares
Sum SSwithin is the sum of squares for each group
added together
added
SSwithin = ∑SSk = SS1 + SS2 + …+ SSk
∑SS
within
dfwithin = N – k
– Remember “N” is total number of people in the
Remember
study and “k” in the number of samples
study Betweengroups Sum of Squares
Betweengroups SSBetween is the sum of squares of sample mean
differences
differences
dfbetween = k – 1 (k: the number of samples) SS Between T12 T2 2 T3 2 G 2
−
=
+
+
n
n2 n3 N
1 2 or , SSTotal 2 T
G
=Σ
−
n
N T = total scores of
a sample
T1 = total scores of
sample 1
n1 = number of
people in sample 1 ANOVA
ANOVA We now have SSTotal, SSWithin, SSBetween
We now have dfTotal, dfWithin, dfBetween
For variance, just as we are used to, we divide
For
SS by df to get variance
SS
So, for each type of SS, we divide the
So,
appropriate df to get a measure of variance
df
Except the result we call a mean square (or
Except
mean
MS for short) rather than variance
MS ANOVA Table
ANOVA
Source df SS MS F Between Groups (k – 1) SSB SSB/dfB MSB/MSW Within Groups (N – k) SSW SSW/dfW  Total (N – 1) SSB + SSW   F = MS between / MS within Official reporting
Official If significant result:
F(dfbetween, dfwithin) = Fvalue, p < α
Fvalue,
F(3, 12) = 6.25, p < .05 If nonsignificant result:
F(3, 12) = 1.25, ns Example
Example
Control
3
0
2
0
0
T=5
SS = 8 Drug 1
Drug 2
4
6
3
3
1
4
1
3
1
4
T=10 T=20 T=25
SS= 8
SS= 6 n = 20
G =60 ΣX2 = 262 Drug 3
7
6
5
4
3
SS=10 Exercise
Exercise
Soda A
0
4
0
1
0
T=5
SS = 12 Soda B
6
8
5
4
2
T=25
SS= 20 n = 15
G = 60 ΣX2 = 356 Soda C
9
5
6
6
4
T=30
SS= 14
SS= ANOVA
ANOVA If the Ftest is significant this tells us that
If
there are differences between our groups
there
However, we may want to know exactly
However,
which groups are different
which
We can only assume that the sample with the
We
largest mean is significantly different than the
group with the smallest mean
group Post Hoc Tests
Post Post hoc means “after this”
These tests are conducted after the initial ANOVA
These
to find out which groups are significantly different
from one another
from
These tests are only done when:
These
only The null hypothesis is rejected (a significant F ratio)
AND you have three or more groups (if you just had two
AND
groups, you wouldn’t need post hoc tests because you
would know the group with the highest mean is different
from the group with the lowest)
from Tukey’s HSD
Tukey’s
MS within
HSD = q
n1 MSwithin is the same value from conducting the
ANOVA
ANOVA
q is calculated from the Studentized Range Statistic
is
table (Page 696)
table Use number of groups (total samples; k) and dfwithin
The “n1” in the formula refers to the number of people in
in
each sample (it assumes there are equal numbers)
each Tukey’s HSD
Tukey’s The value of HSD indicates the amount two sample
The
means must differ by to be considered significantly
different
different
Once you have the HSD value, calculate the
Once
differences in sample means for each comparison
you are interested in (µ1 µ2; µ1 µ3 ; µ2 µ3)
you
Each difference is compared to the HSD value, if
Each
that difference is greater than the HSD value then
those groups are significantly different
those Effect Size
Effect A significant effect is not always a big effect
We use measures of effect size to determine
We
the relative impact of the IV on the DV
the
With ttests we used Cohen’s d and r2
With
Cohen’s
With ANOVA we use η2
With Called “etasquared” Effect Size
Effect
SSbetween
η=
SSbetween + SS within
2 or SSbetween
η=
SS total
2 Effect Size
Effect
SSbetween
η=
SSbetween + SS within
2 Gives the percent variance explained (just like
Gives
r2)
0.01 < η2 < 0.09 small effect
0.01
0.09 < η2 < 0.25 medium effect
0.09
η2 > 0.25
large effect ...
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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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