Lecture 8 - Two-Factor Mixed and Within-Participants ANOVA

Course Overview

Two-Factor Mixed Design
- Raw Data
- Cell Means
- ANOVA Table
- Simple Main Effects
- Interaction Plot
- Calculating F Ratios
Two-Factor Fully Within-Participants Design
- Raw Data
- Cell Means
- ANOVA Table
- Simple Main Effects
- Interaction Plot
- Calculating F Ratios

Gain a comprehensive understanding of two-factor mixed and within-participants designs and their application in psychological research.
Focus on procedural aspects of statistical analysis rather than solely on calculation methods to enhance understanding of practical applications.
Learn to interpret complex ANOVA tables and graphical representations to draw meaningful conclusions from data.
Master handling significant main effects and simple main effects, especially when dealing with factors that contain three or more levels, which are common in psychological studies.

Between-participants Designs: Focuses on comparing different groups and splitting total variability into between-group and within-group variability to assess effects of treatments or conditions.
Within-participants Designs: Involves the same participants being measured across different conditions to split within-group variability into between-participant and residual variability, enhancing control over confounding variables.
Mixed Designs: Combines elements of both between-participants and within-participants designs, allowing researchers to analyze variability using main effects and interactions, while acknowledging the complexity of human behavior.
More complicated designs may not be beneficial to analyze by hand due to increased complexity and potential for errors in manual calculations.

Utilizes versatile ANOVAs that are frequently employed in psychological research, offering a robust framework for examining interactions between independent variables.
Comprises at least one between-participants factor and one within-participants factor, providing a comprehensive view of both individual and group-level effects.
Example Application:
- Stroop Task demonstrates cognitive interference, encapsulating how different cognitive processes interact under varying conditions.

Participants are instructed to name the ink colors of words presented on a screen.
- Congruent Trials: The ink color of the word matches the semantic meaning of the word.
- Incongruent Trials: The ink color conflicts with the semantic meaning of the word, creating cognitive interference.

Stroop Effect: Notable finding where longer response times (RTs) are observed on incongruent trials, showcasing the impact of competing cognitive processes on performance.

Investigates the impairment of response inhibition in patients with schizophrenia through the application of the Stroop task.
Design: 2 × 2 mixed design incorporating two distinct factors:
- Patient Group: Healthy vs. Schizophrenia (Between-participants)
- Trial Type: Congruent vs. Incongruent (Within-participants)
- Conditions: 4 unique combinations: Healthy-Congruent, Healthy-Incongruent, Schizophrenia-Congruent, and Schizophrenia-Incongruent.

Factor A: Patient Group
- Response times for Healthy and Schizophrenia participants measured across both trial types (congruent/incongruent) to identify differences in cognitive processing.
Data Table: Includes response times for multiple participants across conditions, facilitating the analysis of performance effects.

Mixed-design ANOVA produces distinct error terms that are crucial for accurate interpretation of results:
- Between-participants main effect error term.
- Within-participants main effect and interaction error term, essential for understanding the nuances of participant responses under different conditions.
Ensure correct degrees of freedom are reported for all ANOVA results, as this impacts the validity of conclusions drawn from the analysis.

Source listings must include detailed statistics for each factor:
- Group (A)
- Trial Type (B)
- Interaction (A × B)

With corresponding Sum of Squares, Degrees of Freedom, Mean Square, F, and P values to assess the significance of each factor.

Report significant means and their p-values across all design elements to provide a clear picture of the findings.

Use pooled error terms where the variance for each simple main effect is calculated akin to a fully between-participants design, enhancing clarity in significance testing.
all use the same error term
he other error term is the within-participants factor error term from theinitial ANOVA (Error B×S/A)• This is used to test the two within-participants simple main effects:• trial type at healthy• trial type at schizophrenia

Different approaches can be employed, including:
- T-tests for each pair of means compared (both independent-samples and repeated-measures) allow detailed pairwise comparisons.
- Generate interaction plots, which provide visual representation of interaction effects to complement statistical findings.

If the interaction is found not to be significant, conduct planned comparisons following initial main effect analysis to explore differences further.
If a significant interaction occurs, follow up with comparisons for all groups involved to understand the interplay of factors more intricately.
Ensure correct analytical methods are employed for each effect tested to maintain rigor in research conclusions.

Vital for within-participant factorial designs that utilize three or more levels.
Greenhouse and Geisser correction applied to adjust for violations to maintain the integrity of statistical inferences.

R code provided for generating plots related to the statistical analysis and visual representation of data insights.

Running a 2 × 3 mixed/within-participants ANOVA in R, which will involve coding and analysis of more extensive datasets.
Conducting follow-up tests specifically for factors exceeding two levels will deepen understanding of complex interactions.