What is the formula for standard error of a contrast?

Answer 1

Denote: ai = contrast and ni = sample size for each level

Estimate of contrast:

sum( ai ybari )

note: sum is written as Sigma

Standard Error of contrast:

sqrt( sum( sigma2 ai2 / ni ) )

note: sum is written as Sigma, and lowercase sigma is usually estimated with MSE

Sums of Squares of contrast:

( sum( ai ybari ) )2 / ( sum( ai2 / ni ) )

note: sum is written as Sigma

Usually when one uses estimate divided by SE, the test statistic follows a t-distribution (unless he/she didn't estimate lowercase sigma). When one uses SS(contrast) divided by MSE, the test statistic follows a F-distribution. The formulas are similar because there's a strong relationship between the t-distribution and the F-distribution.

Hopes this helps and sorry I don't know how to write math equations here.