Contrast coefficients, often denoted as cL, are fundamental constants used in statistical analysis, particularly in ANOVA (Analysis of Variance) and related linear models.
Definition of Contrast Coefficients
Based on the provided reference, contrast coefficients (cL) are constants that sum to 0. This property is the defining characteristic of a linear contrast.
In mathematical terms, a linear contrast, often symbolized by Ψ (Psi), is formed using these coefficients in relation to group means (or other parameters). The defining equation for the coefficients themselves is:
∑ c ℓ = 0
This equation simply means that when you add up all the individual contrast coefficients used in a specific comparison, their total sum must equal zero.
Why the Sum-to-Zero Property is Important
While the reference doesn't explain why they sum to zero, this property is crucial because it ensures that the contrast is comparing differences between group means or effects, rather than combining them in a way that includes an overall mean effect. It allows for specific, targeted comparisons between subsets of groups or conditions within a study.
Application Example
The reference mentions an example context where contrast coefficients are used:
- Example: Section 10.1 describing a two-by-two design dealing with weight gain in rats.
This suggests that in a study examining, for instance, the effects of two different diets and two different exercise levels on rat weight gain, contrast coefficients would be used to compare specific group means (e.g., comparing the average weight gain of rats on diet A vs. diet B, or exercise group 1 vs. exercise group 2, or even interaction effects).
Key Takeaways
- Contrast coefficients are numerical constants.
- Their defining property is that they sum to zero (∑ c ℓ = 0).
- They are used to define linear contrasts (Ψ), which allow for specific comparisons between group means in statistical tests.
- An example application is seen in analyzing data from experimental designs, like a two-by-two study on weight gain in rats.
