PSY3213 chap14

Chapter 14: The Comparison on More Than Two Conditions

What is Analysis of Variance (ANOVA)?

ANOVA is a statistical method used to test differences between two or more group means. It helps determine if the observed variance between group means is significantly greater than the variance within the groups.

• Key Concepts:
  • F-Test: Helps determine if the variance between groups is greater than the variance within groups.
  • t-Test Relation: While t-tests compare means between two groups, ANOVA can compare means across multiple groups.

Variability in ANOVA

One-way ANOVA

Variability in a one-way ANOVA is apportioned into:

1. Between-condition variability: Variance due to the differences among the group means.
2. Within-condition variability: Variance due to the differences within each group.

The total variability can be represented as:
\text{Total Variability} = \text{Between Condition Variability} + \text{Within Condition Variability}

Two-way ANOVA

• Involves two independent variables and assesses their individual effects as well as any interaction effects.
• Residuals: Differences between observed and predicted values, relevant for understanding interaction effects in ANOVA.

Setup and Interpretation of ANOVA Summary Tables

Degrees of Freedom (df)

• df-between: Number of conditions - 1 = k - 1
• df-within: Number of observations - number of conditions = N - k

Mean Squares Calculation

• Mean Squares (MS):
  • \text{MS-between} = \frac{\text{SS-between}}{df\text{-between}}
  • \text{MS-within} = \frac{\text{SS-within}}{df\text{-within}}

F-Ratio

• Ratio used to determine if there are significant differences between group means:
  • F = \frac{MS\text{-between}}{MS\text{-within}}

Understanding F-tests

The basic question addressed by F-tests in ANOVA is whether variation between conditions can be attributed to true differences in group means rather than sampling variability.

Sum of Squares (SS) Notation

• Total SS can be computed as:
  \text{Total SS} = \text{Between SS} + \text{Within SS}
• Example: 170 = 138 + 32.

Tests of Simple Effects

Two-way Designs

• Involves more than one independent variable (IV). The main effects and interaction effects are evaluated:
  • Main effect: The impact of one IV, while ignoring the other.
  • Interaction effect: The effect of one IV varies depending on the level of another IV, necessitating the use of residuals.

Calculation of Effects

• Grand mean (MG):
  MG = \frac{(19 + 15 + 12 + 10)}{4} = 14

• Row and Column Effects:

  • Row effect arises from the average of different conditions in rows relative to the grand mean.
  • Column effect arises from the averages of different conditions in columns relative to the grand mean.

Interaction Effects Calculation

• Interaction effect formula:
  \text{Group Mean} - \text{Grand Mean} - \text{Row Effect} - \text{Column Effect}

Conclusion

ANOVA is crucial in statistical analysis for comparing means across multiple groups. Understanding how to set up summary tables, interpret F-ratios, and evaluate simple and interaction effects is essential for robust research results.