Transcript Statistics for Marketing and Consumer Research

Analysis of variance
(ANOVA)
(from Chapter 7)
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
1
Tests on multiple hypotheses
• Consider the situation where the means for
more than two groups are compared, e.g.
mean alcohol expenditure for: (a) students;
(b) unemployed; (c) employees
• One could run a set of two mean comparison
tests (students vs. unemployed, students vs.
employed, employed vs. unemployed)
• But.....too many results...
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
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Analysis of Variance
• It is an alternative approach to mean comparison
for multiple groups
• It is applicable to a sample of individuals that
differ for one or more given factors
• It allows tests where variability in a variable is
attributable to one (or more) factors
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
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Example
EFS: Total
Alcoholic
Beverages,
Tobacco
Economic position of Household Reference Person
Unoc Ret unoc
SelfFulltime
Pt
under
Unempl. over min
employed employee employee
min ni
ni age
age






Mean
18.56
14.64
12.39
19.48




St. Dev.
19.0
18.5
15.0
19.7
7.34

14.6
TOTAL

11.99
12.67


19.1
17.8
Are there significant difference across the means of
these groups?
Or do the differences depend on the different levels of
variability across the groups?
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
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Analysis of Variance
• Here:
the target variable is alcohol, bev., tobacco
expenditure, the factor is the economic
position of the HRP
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Copyright © 2008 - Mario Mazzocchi
5
One-way ANOVA
• Only one categorical variable (a single factor)
• Several levels (categories) for that factor
• The typical hypothesis tested through ANOVA is
that the factor is irrelevant to explain differences
in the target variable (i.e. the means are equal, as
in bivariate mean comparisons/t-tests)
• Apart from the tested factor(s), the groups should
be safely considered homogeneous between each
other
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
6
Null and alternative hypothesis for
ANOVA
• Null hypothesis (H0): all the means are equal
• Alternative hypothesis (H1): at least two
means are different
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Copyright © 2008 - Mario Mazzocchi
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Measuring and decomposing the total
variation
VARIATION BETWEEN THE GROUPS
+
VARIATION WITHIN EACH GROUP=
________________________________
TOTAL VARIATION
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Copyright © 2008 - Mario Mazzocchi
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The basic principle of the ANOVA:
If the variation explained by the different factor between the
groups is significantly more relevant than the variation
within the groups, then the factor is assumed to be
statistically relevant in explaining the differences
The test statistic:
• The test statistic is computed as:
2
B
2
W
s
Variance between groups
F

s
Variance within groups
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
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Distribution of the
F-statistic (one-tailed test)
if p<0,05 we refuse H0:
i.e. the means are not equal
Rejection area
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Copyright © 2008 - Mario Mazzocchi
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ANOVA in SPSS
Target variable
Factor
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SPSS output
ANOVA
Variance between
EFS: Total Alc oholic Beverages , Tobacc o
Between Groups
W ithin Groups
Total
Sum of
Squares
6171.784
151535.3
157707.1
df
5
494
499
Mean Square
1234.357
306.752
F
4.024
Sig.
.001
p-value < 0.05
Variance within
Variation decomposition
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
Degrees of
freedom
The null is
rejected
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Post-hoc tests
• They open the way to further explore the
sources of variability when the null
hypothesis of mean equality is rejected.
• It is usually relevant to understand which
particular means are different from each
other.
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Copyright © 2008 - Mario Mazzocchi
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Some post-hoc tests
•
•
•
•
•
•
LSD (least significant difference)
Duncan test
Tukey’s test
Scheffe test
Bonferroni post-hoc method
.......
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
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ANOVA assumptions
Two key assumptions are needed for
running analysis of variance without
risks
1)that the sub-samples defined by the
treatment are independent
2)that no big discrepancies exist in the
variances of the different sub-samples
Statistics for Marketing & Consumer Research
Copyright © 2008 - Mario Mazzocchi
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Multi-way (factorial) analysis of
variance
• This analysis measures the influence of two
or more factors
• Beside the influence of each individual
factor, it provides testing of interactions
between treatments belonging to different
factors
• ANOVA with more than two factors is rarely
employed, as interpretation of results
becomes quite complex
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Copyright © 2008 - Mario Mazzocchi
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