Fundamental Concepts in Statistics and ANOVA

Fundamental Concepts in Statistics

• The goal of statistics is to estimate population characteristics.
• Statistics help us make inferences about characteristics in the population by calculating probabilities.
• We use samples to make inferences because we rarely have access to the whole population.
• Samples introduce error in our estimations, measured by standard error.
• Standard error indicates how much we expect a sample mean to vary across different samples of the population.
• A smaller sample size leads to a larger standard error.

Analysis of Variance (ANOVA)

• ANOVA analyzes the variance of data to determine if there are significant differences between groups.
• It is an omnibus test, meaning it performs all comparisons simultaneously.
• ANOVA checks for the presence of an effect, not the precise location of the difference like t-tests.

Assumptions of ANOVA

• Normality: The data should be normally distributed within each group.
• Homogeneity of Variance: The variance of the data should be roughly equal across all groups (Levene's test is used to check).
• Independence of Observations: Each observation should be independent of other observations.

Error Variance vs. Systematic Variance

• Error Variance: Random variance in sample means, captured by the sampling distribution of the mean, central limit theorem, and standard error.
• Systematic Variance: Variance in the sample that is attributed to the effect of the independent variable (IV).

ANOVA Terminology

• Variation: Spread or dispersion of scores around the mean.
• Variance: Average spread or dispersion.
• SSgroup: Sum of squared differences from the mean, representing variation between groups.
• SSerror: Sum of squared differences from the mean, representing variation within groups.
• MSgroup: Mean squares, representing variance between groups.
• MSerror: Mean squares, representing variance within groups.

Familywise Error Rate (FWER)

• The FWER is the probability of making at least one type I error (falsely rejecting the null hypothesis) in a family of comparisons.
• As the number of comparisons increases, the FWER increases.
• ANOVA provides a solution to the FWER issue by performing all comparisons simultaneously.

Factorial ANOVA

• Factorial ANOVA investigates the effects of multiple independent variables on a dependent variable.
• The main effect represents the overall impact of a single independent variable.
• The interaction effect detects how the effects of one independent variable change across different levels of another independent variable.
• An interaction effect is often more important than the main effects.
• Ordinal interactions occur when the simple effects of the independent variables are in the same direction (lines do not cross on the cell mean plot).
• Disordinal interactions occur when the simple effects are in opposite directions (lines cross on the cell mean plot).
• No interaction means the lines are parallel on the cell mean plot.
• Simple effects analysis is performed to understand where the significance of the interaction lies. It involves running one-way ANOVAs for each level of each independent variable.

Writing Up Factorial ANOVA Analysis

• State the statistical hypotheses.
• Report descriptive statistics for each group.
• Report on the checks of assumptions (normality, homogeneity of variance).
• Report the overall F test for the interaction effect.
• Interpret the interaction effect, focusing on its implications.
• Interpret the main effects, considering the interaction effect.