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How do I compare group means in a non-standard post-hoc contrast

There may be a specific group comparison you are interested in which is not routinely outputted with an analysis of variance.

Suppose we have a vector of contrast coefficients, c, with i-th term c(i) being the i-th group contrast coefficient.

Compare a group of controls (CO) with a group with general memory deficits (GM) and a group with delayed recall (DR) problems. control mean - 0.5(memory deficit group mean - delayed recall group mean) which therefore has contrast coefficients c(CO)=1, c(GM)=c(DR)=-0.5.

The variance of this contrast is

$$\frac{c(CO)^text{2}}{N(CO)} + \frac{c(GM)^text{2}}{N(GM)} + \frac{c(DR)^text{2}}{N(DR)} MSE

where N(.) is the number in each group and MSE is the mean square of the error term obtained from the full anova table.

For example if we are comparing the control mean with the average of the memory deficit and delayed recall groups the variance is $$\frac{1}{N(CO)} + \frac{1}{4N(GM)} + \frac{1}{4N(DR)} MSE}$$

If the control mean does not differ from the average of the two patient group means this difference in means divided by the square root of its variance approximately follows a t distribution with degrees of freedom equal to the error df from the anova table.

$$\frac{control mean - 0.5(memory deficit group mean - delayed recall group mean)}{frac{1}{N(CO)} + \frac{1}{4N(GM)} + \frac{1}{4N(DR)} MSE}