formula_7, where L is the weighted sum of group means, the cj coefficients represent the assigned weights of the means (these must sum to 0 for orthogonal contrasts), and formula_8j represents the group means. Coefficients can be positive or negative, and fractions or whole numbers, depending on the comparison of interest. Linear contrasts are very useful and can be used to test complex hypotheses when used in conjunction with ANOVA or multiple regression. In essence, each contrast defines and tests for a particular pattern of differences among the means.