Transcript Intro to Stats - Heather Lench, Ph.D.

ANOVAs
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Analysis of Variance (ANOVA)
Difference in two or more average scores in
different groups
Simplest is one-way ANOVA (one variable as
predictor); but can include multiple predictors
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Differences between the groups are separated
into two sources of variance
◦ Variance from within the group
◦ Variance from between the groups
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The variance between groups is typically of
interest
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Use when:
◦ you are examining differences between groups on
one or more variables,
◦ the participants in the study were tested only once
and
◦ you are comparing more than two groups
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Factor: the variable that designates the
groups to be compared
Levels: the individual comparable parts of the
factor
Factorial designs have more than one variable
as a predictor of an outcome
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F is based on variance, not mean differences
Partial out the between condition variance
from the within condition variance
F = MSbetween
MSwithin
MSbetween = SSbetween/dfbetween
MSwithin = SSwithin/ dfwithin
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Therapist wants to examine the effectiveness
of 3 techniques for treating phobias. Subjects
are randomly assigned to one of three
treatment groups. Below are the rated fear of
spiders after therapy.
X 1: 5 2 5 4 2
X 2: 3 3 0 2 2
X 3: 1 0 1 2 1
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1. State hypotheses
Null hypothesis: spider phobia does not differ
among the three treatment groups
◦ μTreatement1 = μTreatment2 = μTreatment3
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Research hypothesis: spider phobia differs in
at least one treatment group compared to
others OR there is an effect of at least one
treatment on spider phobia
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XTreatment1≠ XTreatment2
XTreatment1 ≠ XTreatment3
XTreatment2 ≠ XTreatment3
XTreatment1 ≠ XTreatment2≠ XTreatment3 (just write this one for ease,
but all are made)
F = MSbetween
MSwithin
MSbetween = SSbetween/dfbetween
MSwithin = SSwithin/ dfwithin
SSbetween = Σ(ΣX)2/n – (ΣΣX)2/N
ΣX = sum of scores in each group
ΣΣX = sum of all the scores across groups
n = number of participants in each group
N = number of participants (total)
SSwithin = ΣΣ(X2) – Σ(ΣX)2/n
ΣΣ(X2) = sum of all the sums of squared scores
Σ(ΣX)2 = sum of the sum of each group’s scores
squared
n = number of participants in each group
Sstotal = ΣΣ(X2) – (ΣΣX)2/N
ΣΣ(X2) = sum of all the sums of squared scores
(ΣΣX)2 = sum of all the scores across groups
squared
N = total number of participants (in all groups)
F = MSbetween
MSwithin
MSbetween = SSbetween/dfbetween
 Dfbetween = k-1 (k=# of groups)
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6. Determine whether the statistic exceeds
the critical value
◦ 6.01 > 3.89
◦ So it does exceed the critical value
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7. If over the critical value, reject the null
& conclude that there is a significant
difference in at least one of the groups
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For an ANOVA, the test statistic only tells
you that there is a difference
It does not tell you which groups were
different from other groups
There are numerous post-hoc tests that you
can use to tell the difference
Here, we will use Bonferroni corrected posthoc tests because they are already familiar
(similar to t-tests, but with corrected
critical value levels to reduce Type 1 error
rates)
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In results
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With post-hoc tests
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If it had not been significant:
◦ There was a significant effect of type of treatment on
spider phobia, F(2, 12) = 6.01, p < .05.
◦ There was a significant effect of type of treatment on
spider phobia, F(2, 12) = 6.01, p < .05. Participants who
received treatment X3 were less afraid of spiders (M =
1.00, SD = 0.71) than participants who received
treatment X1 (M = 3.60, SD = 1.52), t(8) = 3.47, p =
.008, but did not differ from participants who received
treatment X2 (M = 2.00, SD = 1.22, t(8) = 1.58, n.s.
Participants who received treatments X1 and X2 did not
significantly differ, t(8) = 1.84, n.s.
◦ There was no significant effect of type of treatment on
spider phobia, F(2, 12) = 2.22, n.s.
Source
Corrected Model
Intercept
cond
Error
Total
Corrected Total
Type III Sum
of Squares
df
Mean Square
F
Sig.
17.200(a)
2
8.600
6.000
.016
72.600
1
72.600
50.651
.000
17.200
2
8.600
6.000
.016
17.200
12
1.433
107.000
15
34.400
14