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ANOVA
One-Way Analysis of Variance
Two-Way Analysis of Variance
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ONE WAY ANOVA
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Overview
Compares two or more populations of interval
data
Extension of independent T-Test
ANOVA (Analysis of Variance) determines
whether differences exist between population
means.
This procedure works by analyzing the sample
variance, hence the name.
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AIM
Whether means from several (>2)
independent groups differ
E.g. if a researcher is interested whether four
ethnic groups differ in their IQ scores.
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Checklist of Requirements for
ONE-WAY ANOVA
One IV (e.g., ethnicity) with more than two
levels
More than two levels for IV (e.g., Australian,
American, Chinese and African)
One DV...that is to be measured like IQ scores,
calories consumed, time taken to solve
problem.
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Assumptions
The populations from which the samples were
taken are normally distributed.
Homogeneity of variance
The observations are all independent of one
another.
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Example
A researcher is interested in finding out whether
the intensity of electric shock will affect the time
required to solve a set of difficult problems.
Eighteen subjects are randomly assigned to
three experimental conditions of low shock,
medium shock and high shock. The total time (in
minutes) required to solve all the problems isthe measure recorded for each subject.
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Dataset
Shock Intensity
Low Medium High
15 30 40
10 15 35
25 20 50
15 25 43
20 23 45
18 20 40
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State the Hypotheses
H0: The time taken to solve problems in each
shock level is same.
Ha
: The time taken to solve problems in each
shock level is not same., at least one is
different from the others.
H0: m1= m2= m3
Ha: Not all ms are equal
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Checking for the assumptions
INDEPENDENCE is judged through the problemstatement
NORMALITY: if sample size is large: the datatends to normal checked through Graphically (histograms, normality plots)
Numerically (Kolmogrov, Shapiro Wilk (when samplesize is less than 50))
HOMOGENEITY OF VARIANCES Checked through levenestest
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NORMALITY
The data is said to be normally distributed, as assessedby Shapiro-Wilk Test (p>.05), so the assumption ofnormality is satisfied.
Note: for sample size less than 50, Shapiro-Wilk test is displayed automatically and weinterpret through this test. For sample size larger than 50, Kolmogrov-Smirnov test isinterpreted.
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HOMOGENEITY OF VARIANCES
The significance value exceeds .05, suggesting
that the variances for the three shock levelsare equal, so the assumption of homogeneity
of variance is satisfied.
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Output and interpretation
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In one-way ANOVA, the total variation is partitioned into
two components. Between Groups represents variation of the group means
around the overall mean.
Within Groups represents variation of the individual scoresaround their respective group means.
Sigindicates the significance level of the F-test. Small significance values (
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Total Variability
Should besmall
Should belarge
Within group, response is
not exactly the same dueto;
1. Individual differences
2. Experimental error
different treatments
exposed to differentgroups and each
group responded
differently
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Interpretaion
The results from the analysis indicate that theintensity of the electric shock has a significanteffect on the time taken to solve the problem,
F(2,15)= 40.14,p
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Post Hoc Comparisons (when ANOVA
results are significant)
H1 supported, then researcher is interested to
know which of the two groups differ?
Post hoc comparisons provide the answer.
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Post Hoc Comparisons
Although the highly significant F-ratio (p
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Post Hoc Test
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Interpretation
The results indicate that the high shock level is
significantly different from both the low shock
and medium shock levels.
The low and medium shock levels do not differ
significantly.
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Graphical Representation
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Conclusion
These results show that overall difference in
the time taken to solve complex problems
between the three shock intensity levels is
because of significantly greater amount of
time taken by the subjects in high shock
conditions.
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