1 Analysis of variance (ANOVA) LEZIONI IN LABORATORIO Corso di MARKETING L. Baldi Università degli Studi di Milano Estratto dal Cap. 7 di: “Statistics for Marketing and Consumer Research”, M. Mazzocchi, ed. SAGE, 2008.

2 Example: expenditure in alcohol by economic position Are there significant difference across the means of these groups? Or do the differences depend on the different levels of variability across the groups?

3 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...

4 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

5 Analysis of Variance Here: the target variable is alcohol expenditure the factor is the economic position of the consumers

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) 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 (H 0 ): all the means are equal Alternative hypothesis (H 1 ): 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

10 The test statistic The test statistic is computed as: Rejection area Distribution of the F-statistic (one-tailed test)

11 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.

12 Some post-hoc tests LSD (least significant difference) Duncan test Tukey’s test Scheffe test Bonferroni post-hoc method

13 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