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A contrast is defined as the sum of each group mean multiplied by a coefficient for each group (i.e., a signed number, c j). [10] In equation form, = ¯ + ¯ + + ¯ ¯, where L is the weighted sum of group means, the c j coefficients represent the assigned weights of the means (these must sum to 0 for orthogonal contrasts), and ¯ j represents the group means. [8]
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If only a fixed number of pairwise comparisons are to be made, the Tukey–Kramer method will result in a more precise confidence interval. In the general case when many or all contrasts might be of interest, the Scheffé method is more appropriate and will give narrower confidence intervals in the case of a large number of comparisons.
The only difference between the confidence limits for simultaneous comparisons and those for a single comparison is the multiple of the estimated standard deviation. Also note that the sample sizes must be equal when using the studentized range approach.
The ATE measures the difference in mean (average) outcomes between units assigned to the treatment and units assigned to the control. In a randomized trial (i.e., an experimental study), the average treatment effect can be estimated from a sample using a comparison in mean outcomes for treated and untreated units.
The example discussed by Duncan in his 1955 paper is of a comparison of many means (i.e. 100), when one is interested only in two-mean and three-mean comparisons, and general p-mean comparisons (deciding whether there is some difference between p-means) are of no special interest (if p is 15 or more for example).
Dunnett's test's calculation is a procedure that is based on calculating confidence statements about the true or the expected values of the differences ¯ ¯, thus the differences between treatment groups' mean and control group's mean.
The F-test is computed by dividing the explained variance between groups (e.g., medical recovery differences) by the unexplained variance within the groups. Thus, = If this value is larger than a critical value, we conclude that there is a significant difference between groups.