Factorial ANOVA and Mixed Factorial Designs

Factorial ANOVA Overview

  • Factorial ANOVA is used to examine the interaction between multiple independent variables.

Split-Plot or Mixed Factorial Design

  • A design that includes one between-subjects variable and one within-subjects variable.
  • Whole plots refer to the entire range of a singular factor.

Design Structure

  • The levels of the between-subjects factor are randomly assigned to whole plots.
  • The levels of the within-subject factor are assigned at random to split-plots, ensuring each level is represented within each whole plot.

Application

  • Used frequently when multiple measurements are recorded over time.

    • Example: Subjects categorized by age (young vs old) are tested repeatedly across different conditions (e.g., three trials or drugs).

  • The design distinguishes between error terms for the between-subjects and within-subjects factors, allowing for more robust analysis.

ANOVA Table Structure

  • The ANOVA table typically includes:
    • Source
    • Sum of Squares (SS)
    • Degrees of Freedom (df)
    • Mean Squares (MS)
    • F-value
    • p-value

Example: 2 x 3 Mixed Factorial

  • Dependent Variable: Time (in seconds) to arrange objects by increasing weight (lower = better).
    • Factor 1: Social facilitation (alone vs. observed) - between-subjects factor.
    • Factor 2: Trial number (three trials) - within-subjects factor.

Hypotheses Formulation

  • Main Effect of Social Facilitation:

    • Null Hypothesis ($H_0$): ext{Mean for Alone} = ext{Mean for Observed}
    • Alternative Hypothesis ($HA$): H0 ext{ is not true}

  • Main Effect of Trial Number:

    • Null Hypothesis ($H_0$): ext{Mean for Trial 1} = ext{Mean for Trial 2} = ext{Mean for Trial 3}
    • Alternative Hypothesis ($HA$): H0 ext{ is not true}

  • Interaction Hypothesis:

    • Null Hypothesis ($H_0$): There is no interaction.
    • Alternative Hypothesis ($HA$): H0 ext{ is not true}

Data Organization

  • Results are organized by subject and trial number across different social facilitation conditions.
  • Statistical values such as means and standard deviations are computed for different categories of data.

ANOVA Table Calculation Steps

Step 1: Summarize the Data

  • Calculate Row and Column means for each condition based on the recorded times.

Step 2: Degrees of Freedom

  • Total degrees of freedom ($df_{total}$) = Total scores - 1 = 36 - 1 = 35.
  • For other factors:
    • df_{SF} = ext{levels of SF} - 1 = 2 - 1 = 1
    • df_{error(between)} = ext{levels of SF} imes (n - 1) = 2 imes (6 - 1) = 10

Step 3: Sum of Squares and Mean Squares

  • Calculate SS for each factor along with the Mean Square (MS) by dividing SS by df.

Step 4: Calculate F-statistics

  • For example,
    • F{SF} = rac{MS{SF}}{MS_{error(between)}} = rac{53.778}{0.644} = 83.45
  • Utilize the relationship for other factors similarly.

Final ANOVA Table Summary

  • Showcasing SS, df, MS, F, and p-values for Social facilitation, Trial number, and interaction effects.

Decision Criteria

  • Compare F-statistics against critical values to determine significance levels at specified degrees of freedom.
  • Response according to established significance thresholds (e.g., p < .05).

Effect Size Calculation

  • Calculate partial eta-squared (ηp²) for each factor:
    • Social facilitation: ext{Effect Size} = rac{SS{Social acil}}{SS{Social acil} + SS_{Error}}

Write-Up Guidelines

  • State results in an accessible format indicating whether main effects are significant including specific F-values, p-values, and effect sizes.
  • Describe the results for both main effects and any interaction.

Post Hoc Tests

  • Consider conducting post hoc analyses to further examine the main effects if significant differences are indicated.