Contrast (statistics)

In statistics, particularly in analysis of variance and linear regression, a contrast is a linear combination of variables (parameters or statistics) whose coefficients add up to zero, allowing comparison of different treatments.

Let ${\displaystyle \theta _{1},\ldots ,\theta _{t}}$ be a set of variables, either parameters or statistics, and ${\displaystyle a_{1},\ldots ,a_{t}}$ be known constants. The quantity ${\displaystyle \sum _{i=1}^{t}a_{i}\theta _{i}}$ is a linear combination. It is called a contrast if ${\displaystyle \sum _{i=1}^{t}a_{i}=0}$. Furthermore, two contrasts, ${\displaystyle \sum _{i=1}^{t}a_{i}\theta _{i}}$ and ${\displaystyle \sum _{i=1}^{t}b_{i}\theta _{i}}$, are orthogonal if ${\displaystyle \sum _{i=1}^{t}a_{i}b_{i}=0}$.

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