ANCOVA and Repeated Measures ANOVA are powerful statistical tools that build on basic ANOVA concepts. They allow researchers to control for covariates and analyze data from repeated measurements, providing deeper insights into complex experimental designs.

These methods expand the analytical toolkit for comparing group means. ANCOVA adjusts for continuous variables, while Repeated Measures ANOVA handles multiple measurements from the same participants, enhancing the precision and scope of variance analysis.

Analysis of Covariance (ANCOVA)

Overview and Purpose

• Analysis of Covariance (ANCOVA) is a statistical method that combines ANOVA and regression to compare means of groups while controlling for the effect of a continuous variable (covariate)
• ANCOVA aims to remove the influence of the covariate on the dependent variable, allowing for a more accurate comparison of group means
• ANCOVA is useful when there is a variable (covariate) that may influence the dependent variable and is not of primary interest in the study

Key Components and Assumptions

• Covariate is a continuous variable that is measured along with the dependent variable and is believed to have an influence on it
• Between-subjects factor is a categorical independent variable that divides participants into different groups or conditions
• Assumptions of ANCOVA include linearity, homogeneity of regression slopes, independence of the covariate and treatment effect, and normality of residuals

Interpreting Results and Examples

• ANCOVA results indicate whether there are significant differences between group means after adjusting for the effect of the covariate
• If the ANCOVA is significant, it suggests that the between-subjects factor has an effect on the dependent variable, even after controlling for the covariate
• Example: Comparing the effectiveness of different teaching methods (between-subjects factor) on student performance (dependent variable) while controlling for students' prior knowledge (covariate)
• Example: Investigating the impact of different treatment options (between-subjects factor) on patient outcomes (dependent variable) while accounting for patients' age (covariate)

Repeated Measures ANOVA

Overview and Design

• Repeated measures design involves each participant being exposed to all levels of the independent variable, with measurements taken at each level
• Repeated measures ANOVA is used to analyze data from a repeated measures design, where the same participants are tested under different conditions or at different time points
• Repeated measures ANOVA is advantageous when dealing with small sample sizes or when individual differences between participants need to be controlled

Sphericity and Corrections

• Sphericity is an assumption of repeated measures ANOVA that requires the variances of the differences between all pairs of conditions to be equal
• Mauchly's test is used to assess whether the sphericity assumption is met in a repeated measures ANOVA
• If the sphericity assumption is violated, corrections such as the Greenhouse-Geisser correction can be applied to adjust the degrees of freedom and produce a more accurate F-statistic and p-value

Within-Subjects Factors and Examples

• Within-subjects factor is a variable that is manipulated within each participant, meaning that each participant experiences all levels of the variable
• Examples of within-subjects factors include time points in a longitudinal study, different treatment conditions, or various tasks performed by the same individuals
• Example: Comparing the effectiveness of three different memory techniques (within-subjects factor) on participants' recall performance, with each participant using all three techniques
• Example: Investigating the impact of different exercise intensities (within-subjects factor) on individuals' heart rate, with each participant performing exercises at all intensity levels

Mixed Design ANOVA

Overview and Combination of Factors

• Mixed design ANOVA, also known as split-plot ANOVA, combines both within-subjects and between-subjects factors in a single analysis
• Mixed design ANOVA allows researchers to examine the main effects of both within-subjects and between-subjects factors, as well as their interaction
• Mixed design ANOVA is useful when there are both repeated measures and independent groups in a study

Interpreting Results and Examples

• In a mixed design ANOVA, the main effects of both within-subjects and between-subjects factors are examined, as well as their interaction
• A significant main effect of the within-subjects factor indicates differences across the levels of that factor, while a significant main effect of the between-subjects factor suggests differences between the groups
• A significant interaction between the within-subjects and between-subjects factors indicates that the effect of one factor depends on the level of the other factor
• Example: Investigating the effectiveness of different teaching methods (between-subjects factor) on students' performance across multiple time points (within-subjects factor)
• Example: Comparing the impact of different drug treatments (between-subjects factor) on patients' symptoms over time (within-subjects factor), while also examining potential differences between male and female patients (between-subjects factor)