why can we use the same multiple comparison procedures for between subjects, repeated measures, and mixed designs without changing the formulas
The reason we can use the same multiple comparison procedures for between subjects, repeated measures, and mixed designs without changing the formulas is because the underlying principle and logic behind these procedures remain the same across these different designs.
Multiple comparison procedures are statistical techniques used to compare multiple groups or conditions while controlling for the overall probability of making a Type I error (i.e., falsely concluding there is a significant difference when there is none). These procedures aim to determine which specific group or condition differs significantly from others.
In all three designs (between subjects, repeated measures, and mixed designs), the goal is to compare multiple groups or conditions. However, the main difference lies in how the data is collected and the statistical models that are used to analyze the data.
In a between-subjects design, different groups of participants are assigned to different conditions, and a separate group of participants is observed for each condition. The analysis typically involves comparing means or proportions between the different groups.
In a repeated measures design, the same participants are tested under multiple conditions or at multiple time points. This design allows for within-subject comparisons, examining how participants change across conditions or over time. The analysis often involves comparing means or proportions within participants.
A mixed design combines both between-subjects and repeated measures factors. It involves having multiple groups of participants tested under multiple conditions, resulting in both between-subject and within-subject comparisons. The analysis requires accounting for both sources of variability.
While the data collection and statistical models differ between these designs, the core idea of multiple comparison procedures remains the same: comparing multiple groups or conditions. Therefore, the same multiple comparison procedures can be used across these designs without changing the underlying formulas. However, it is essential to choose the appropriate statistical model and make any necessary adjustments based on the design and assumptions of the analysis.