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Analysis of variance (ANOVA)

Analysis of variance (ANOVA). (from Chapter 7). 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

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Analysis of variance (ANOVA)

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  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 tomean 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 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 groupsis significantly more relevant than the variationwithin 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. 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 output Variance between p-value < 0.05 The null is rejected Variance within Variation decomposition Degrees of freedom

  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 • that the sub-samples defined by the treatment are independent • that no big discrepancies exist in the variancesof 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

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