Treatment and sum contrasts are two different ways to define how categorical variables are represented in statistical models, influencing how group means are compared.
Understanding these contrasts helps in interpreting the results of statistical analyses, particularly in regression models involving categorical predictors (like ANOVA or ANCOVA).
Treatment Contrasts
Treatment contrasts are designed to compare the mean of each level of a categorical variable against the mean of a designated reference or baseline level.
- Key Characteristic: They compare one or more means against a baseline condition.
- How they work: One level is chosen as the baseline (often the first or last level alphabetically or numerically). The contrasts then represent the difference between the mean of each other level and the mean of this baseline level.
- Interpretation: The coefficients in a model using treatment contrasts indicate the estimated difference in the outcome variable between each level and the baseline level, holding other variables constant.
- Use Cases: Commonly used when you have a natural control group, a standard treatment, or a baseline condition you want to compare all other conditions against.
Example: If you have a variable 'Treatment' with levels A, B, and C, and you set A as the baseline, treatment contrasts would test:
- Is Mean(B) different from Mean(A)?
- Is Mean(C) different from Mean(A)?
Sum Contrasts
Sum contrasts, also known as deviation contrasts, compare the mean of each level of a categorical variable against the overall grand mean (GM) of the dependent variable.
- Key Characteristic: They allow us to determine whether we can reject the null hypothesis that a condition's mean is the same as the GM.
- How they work: Each level's mean is compared to the grand mean across all levels of the variable. The sum of the contrast codes for each level typically sums to zero.
- Interpretation: The coefficients represent the estimated difference between the mean of each level (except potentially the last one, which is defined by the others) and the grand mean.
- Two-Group Case: In the specific case of only two groups, testing if a group mean is the same as the GM also implies a hypothesis test that the two group means are the same.
Example: If you have a variable 'Group' with levels 1 and 2, sum contrasts would test:
- Is Mean(Group 1) different from the Grand Mean?
- Is Mean(Group 2) different from the Grand Mean?
- Note: In a two-group case, if Mean(Group 1) != GM, and Mean(Group 2) != GM, this implies Mean(Group 1) != Mean(Group 2). Similarly, if Mean(Group 1) = GM, then Mean(Group 2) must also equal GM, implying Mean(Group 1) = Mean(Group 2). Thus, the comparison against the GM directly tests the equality of the two group means.
Key Differences Summarized
Here's a table highlighting the main distinctions:
| Feature | Treatment Contrasts | Sum Contrasts |
|---|---|---|
| Comparison Basis | Baseline/Reference Group Mean | Grand Mean (Overall Mean) |
| Hypothesis | Level Mean vs. Baseline Mean | Level Mean vs. Grand Mean |
| Interpretation | Difference from baseline | Difference from grand mean |
| Two Groups | Tests Mean(Group 2) vs Mean(Group 1) | Tests Mean(Group 1) vs GM and Mean(Group 2) vs GM (implicitly testing Mean(Group 1) vs Mean(Group 2)) |
Choosing between treatment and sum contrasts depends on the specific research question and the desired interpretation of the model coefficients.
