9. basic concepts_of_one_way_analysis_of_variance_(anova)

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Basic Concepts of One-way
Analysis of Variance
(ANOVA)
Sporiš Goran, PhD.
http://kif.hr/predmet/mki
http://www.science4performance.com/

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Overview
• What is ANOVA?
• When is it useful?
• How does it work?
• Some Examples
• Limitations
• Conclusions

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Definitions
• ANOVA: analysis of variation in an
experimental outcome and
especially of a statistical variance in
order to determine the contributions
of given factors or variables to the
variance.
• Remember: Variance: the square of
the standard deviation
Remember: RA
Fischer, 1919-
Evolutionary Biology

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Introduction
• Any data set has variability
• Variability exists within groups…
• and between groups
• Question that ANOVA allows us to
answer : Is this variability significant, or
merely by chance?

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• The difference between variation
within a group and variation
between groups may help us
determine this. If both are equal it is
likely that it is due to chance and
not significant.
• H0: Variability w/i groups =
variability b/t groups, this means
that 1 = n
• Ha: Variability w/i groups does not =
variability b/t groups, or, 1 ≠ n

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Assumptions
• Normal distribution
• Variances of dependent
variable are equal in all
populations
• Random samples; independent
scores

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One-Way ANOVA
• One factor (manipulated
variable)
• One response variable
• Two or more groups to
compare

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Usefulness
• Similar to t-test
• More versatile than t-test
• Compare one parameter
(response variable) between
two or more groups

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For instance, ANOVA
Could be Used to:
• Compare heights of plants with and
without galls
• Compare birth weights of deer in
different geographical regions
• Compare responses of patients to
real medication vs. placebo
• Compare attention spans of
undergraduate students in different
programs at PC.

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Why Not Just Use t-
tests?
• Tedious when many groups are
present
• Using all data increases
stability
• Large number of
comparisons some may
appear significant by chance

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Remember that…
• Standard deviation (s)
n
s = √[(Σ (xi
– X)2
)/(n-1)]
i = 1
• In this case: Degrees of freedom (df)
df = Number of observations or groups - 1

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Notation
• k = # of groups
• n = # observations in each group
• xij = one observation in group i
• Y = mean over all groups
• Yi = mean for group i
• SS = Sum of Squares
• MS = Mean of Squares
• λ = Between MS/Within MS

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FYI this is how SS
Values are calculated
k ni
• Total SS = Σ Σ (xij
– )2 =
SStot
i=1 j=1
k ni
• Within SS = Σ Σ (xij
– i
)2 =
SSw
i=1 j=1
k ni
• Between SS = Σ Σ ( i
– )2 =
SSbet
i=1 j=1

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and
• SStot = SSw + SSbet

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Calculating MS Values
• MS = SS/df
• For between groups, df = k-1
• For within groups, df= n-k

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Hypothesis Testing &
Significance Levels

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F-Ratio = MSBet/MSw
• If:
– The ratio of Between-Groups MS:
Within-Groups MS is LARGE reject
H0 there is a difference between
groups
– The ratio of Between-Groups MS:
Within-Groups MS is SMALLdo not
reject H0 there is no difference
between groups

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p-values
• Use table in stats book to determine
p
• Use df for numerator and
denominator
• Choose level of significance
• If F > critical value, reject the null
hypothesis (for one-tail test)

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Example 1, pp. 400 of
your handout
• Three groups:
– Middle class sample
– Persons on welfare
– Lower-middle class sample
• Question: Are attitudes toward
welfare payments the same?

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and
From the table with  = 0.05 and df = 2 and 24, we see that if
F > 3.40 we can reject Ho. This is what we would conclude
in this case.

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Example 2
• Bat cave gates:
– Group 1 = No gate (NG)
– Group 2 = Straight entrance gate (SE)
– Group 3 = Angled entrance gate (AE)
– Group 4 = Straight dark zone gate (SD)
– Group 5 = Angled dark zone gate (AD)
• Question: Is variation in bat flight
speed greater within or between
groups? Or Ho = no difference
significant difference in means.

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Just leave me alone
Max! Go back to
your hockey!

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Example 2 (cont’d)
Group #,
i
Gate
Type
Mean FS (m/s) sd FS (m/s) ni
1 NG 5.6 0.93 150
2 SE 3.8 1.05 150
3 AE 4.7 0.97 150
4 SD 4.2 1.23 137
5 AD 5.1 1.03 143
Hypothetical data for bat flight speed with various gate arrangements.
FS= Flight speed; sd = standard deviation

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Example 2  SSbet
Between SS = 300
Group
#, i
Gate
Type
Mean FS
(m/s)
sd FS (m/s) ni
1 NG 5.6 0.93 150
2 SE 3.8 1.05 150
3 AE 4.7 0.97 150
4 SD 4.2 1.23 137
5 AD 5.1 1.03 143

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Example 2  SSw
Within SS = 790
Group
#, i
Gate
Type
Mean FS
(m/s)
sd FS (m/s) ni
1 NG 5.6 0.93 150
2 SE 3.8 1.05 150
3 AE 4.7 0.97 150
4 SD 4.2 1.23 137
5 AD 5.1 1.03 143

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Example 2 (cont’d)
• Between MS = 300/4 = 75
• Within MS = 790/(730-5) = 1.09
• F Ratio = 75/1.09 = 68.8
• See Table find p-value based on
df= 4,
• Since F>value found on the table we
reject Ho.

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What ANOVA Cannot
Do
• Tell which groups are different
– Post-hoc test of mean differences
required
• Compare multiple parameters
for multiple groups (so it
cannot be used for multiple
response variables)

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Some Variations
• Two-Way, Three-Way, etc.
ANOVA (will talk about this next
class)
– 2+ factors
• MANOVA (Multiple analysis of
variance)
– multiple response variables

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Summary
• ANOVA:
– allows us to know if variability in a data
set is between groups or merely within
groups
– is more versatile than t-test
– can compare multiple groups at once
– cannot process multiple response
variables
– does not indicate which groups are
different

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Now, let’s go to our
SPSS manual
• Perform the sample problem on the effects
of attachment styles on the psychology of
sleep with the data set from the NAAGE
site called Delta Sleep.
• Pay attention to the procedure and the
post-hoc tests to determine which groups
are significantly different. Perform the
Tukey Test at a 5% significance level.
• Look at your output and interpret your
results.
• Tell me when you are done.

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When you are done
with this,
• Do practice exercises 1, 4, 6, 7
and 12 from the handout in
SPSS.
– Create the data sets.
– Run the one-way ANOVAS and
interpret your results.

9. basic concepts_of_one_way_analysis_of_variance_(anova)

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