A contrast variable is fundamentally a linear combination of random variables (each variable representing random values in a group) instead of group means.
Understanding this definition requires breaking down the key components:
Deconstructing the Definition
- Linear Combination: This means the contrast variable is calculated by summing up the group values, each multiplied by a specific weight (or coefficient). For example,
c1*G1 + c2*G2 + ... + ck*Gk, where G represents the random variable for a group and c represents a coefficient. - Random Variables: Unlike traditional contrast analysis which often works directly with the fixed average (mean) of each group, a contrast variable is based on the random values observed within each group. This captures the variability within the data more directly at a lower level.
- Instead of Group Means: This highlights the distinction. While standard contrasts operate on
mean(G1)*c1 + mean(G2)*c2 + ..., a contrast variable is defined based on the raw random values before averaging.
Purpose and Application
Based on a contrast variable, researchers and analysts can compute the mean of a contrast. This "mean of a contrast" then serves the same function as a traditional contrast applied to group means – it allows for addressing specific questions about group comparisons.
- Group Comparisons: Contrast variables are used to test hypotheses that involve comparing specific groups or combinations of groups against others. For instance, comparing the average performance of Group A and B against Group C.
- First Level Analysis: The reference indicates that contrast variables can be used for comparisons "at the first level as in traditional contrast analysis". This implies its application early in the data analysis process, potentially before aggregating data to higher levels or conducting multi-level analyses.
In essence, a contrast variable provides an alternative perspective for defining comparisons between groups by focusing on the underlying random variability rather than just the aggregated group means.
