Analysis of variance

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ANOVA stands for Analysis of Variance. ANOVA is a family of multivariate statistical technique for helping to infer whether there are real differences between the means of three or more variables in a population, based on sample data.

[edit] Types

ANOVA models	Definitions
t-tests	Comparison of means of two groups.
One-way ANOVA	Comparison of means of three or more independent groups.
One-way repeated measures ANOVA	Comparison of means of three or more dependent groups.
Factorial ANOVA	Comparison of cell means for two or more categorical independent groups.
Mixed ANOVA (or Split-plot ANOVA)	Comparison of cells means for one or more independent groups and one or more dependent groups.
ANCOVA	Any ANOVA model with a covariate.
MANOVA	Any ANOVA model with multiple DVs.

[edit] Assumptions

ANOVA models are parametric, that is, they rely on assumptions about the distribution of the dependent variables.

Initially the array of assumptions for various types of ANOVA may seem bewildering. Let's keep it simple to start with. In practice, the first two assumptions here are the main ones to check. Note that the larger the sample size, the more robust ANOVA is to violation of normality and homoscedasticity (homogeneity of variance) assumptions.

Normality: check via histograms, skewness and kurtosis overall and for each group for each variable
Homogeneity of variance: check via Levene's test or other homogeneity of variance tests which are generally produced as part of the ANOVA statistical output
Sample size: per cell > 20 is preferred; aids robustness to violation of the first two assumptions, and a larger sample size increases power
Independent observations: scores on one variable or for one group should not be dependent on another variable or group

[edit] Interactions

Notes on understanding interaction

[edit] Post-hoc tests

Tukey’s HSD is particularly useful for comparing groups of unequal cell sizes
See also: ANOVA follow-up tests (Wikipedia)

[edit] Effect size

Effect size options for ANOVA, include:

Partial eta-squared for each of the main effects and interaction(s) (e.g., via SS formula or SPSS - ANOVA - Options)
(Total) eta-squared (e.g., via SS formula (SS between groups / Total SS); equivalent to R2 (total variance explained), i.e., provides % of variance in the dependent variable explained by the independent variables.
Cohen's d can be calculated, this is for the differences between two means; i.e., pairwise contrasts. So, you might just want to focus on some contrasts e.g., if there's a significant main effect for gender, then compute the Cohen's d for overall motivation for males and females. You can use the spreadsheet from Tutorial 5 or calculate yourself, using http://en.wikipedia.org/wiki/Effect_size#Cohen.27s_d

Recommended further reading: Measures of Effect Size (Strength of Association) for Analysis of Variance (Becker, 1999).

[edit] FAQ

Should I report effect sizes even when the F tests are not significant?

Effect size and statistical significance are two different, important pieces of information about an ANOVA. In a high power study, the results may be statistically significant but the size of the effect may be trivial. On the other hand, in a low power study, the results may not statistically significant, but the size of the effects may be small, medium, or even large. Thus, both are important.