analysis of variance (anova)

Statistics for Marketing & Consumer ResearchCopyright © 2008 - Mario Mazzocchi

1

Analysis of variance(ANOVA)

(from Chapter 7)


2

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...


3

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


4

Example Economic position of Household Reference Person

Self-employed

Fulltime employee

Pt employee

Unempl. Ret unoc over min

ni age

Unoc - under min ni

age TOTAL

Mean 18.56 14.64 12.39 19.48 7.34 11.99 12.67

EFS: Total Alcoholic Beverages, Tobacco

St. Dev. 19.0 18.5 15.0 19.7 14.6 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?


5

Analysis of Variance

• Here: the target variable is alcohol, bev.,

tobacco expenditure, the factor is the economic position of the HRP


6

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


7

Null and alternative hypothesis for ANOVA

• Null hypothesis (H0): all the means are equal

• Alternative hypothesis (H1): at least two means are different


8

Measuring and decomposing the total variation

VARIATION BETWEEN THE GROUPS +VARIATION WITHIN EACH GROUP =________________________________

TOTAL VARIATION


9

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

2

Variance between groups

Variance within groupsB

W

sF

s


10

Distribution of theF-statistic (one-tailed test)

if p<0,05 we refuse H0:

i.e. the means are not equal

Rejection area


11

ANOVA in SPSS

Target variable

Factor


12

SPSS outputANOVA

EFS: Total Alcoholic Beverages, Tobacco

6171.784 5 1234.357 4.024 .001

151535.3 494 306.752

157707.1 499

Between Groups

Within Groups

Total

Sum ofSquares df Mean Square F Sig.

Variation decomposition Degrees of freedom

Variance between

Variance withinp-value < 0.05

The null is rejected


13

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.


14

Some post-hoc tests

• LSD (least significant difference)• Duncan test• Tukey’s test• Scheffe test• Bonferroni post-hoc method • .......


15

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


16

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

analysis of variance (anova)

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