1. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 1 Analysis of variance (ANOVA) (from Chapter 7)

2. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 4 Example 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. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 5 Analysis of Variance Here: the target variable is alcohol, bev., tobacco expenditure, the factor is the economic position of the HRP

6. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 7 Null and alternative hypothesis for ANOVA Null hypothesis (H 0 ): all the means are equal Alternative hypothesis (H 1 ): at least two means are different

8. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 8 Measuring and decomposing the total variation VARIATION BETWEEN THE GROUPS + VARIATION WITHIN EACH GROUP= ________________________________ TOTAL VARIATION

9. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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:

10. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 10 Distribution of the F-statistic (one-tailed test) if p<0,05 we refuse H0: i.e. the means are not equal Rejection area

11. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 11 ANOVA in SPSS Target variable Factor

12. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 12 SPSS output Variation decomposition Degrees of freedom Variance between Variance within p-value < 0.05 The null is rejected

13. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 14 Some post-hoc tests LSD (least significant difference) Duncan test Tukey’s test Scheffe test Bonferroni post-hoc method.......

15. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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. Statistics for Marketing & Consumer Research Copyright © 2008 - Mario Mazzocchi 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