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Estimating Differences of Means

Estimating Differences of Means

Usually the comparisons are presented in a table, like this one for the example with frying donuts:

i−x̅jCritical q
q(α,r,dfW)
Standardized
error
95% Conf Interval
for μi−μj
Signif
at 0.05?
Fat 1 − Fat 2−133.95974.1008−29.23.2
Fat 1 − Fat 3−43.95974.1008−20.212.2
Fat 1 − Fat 4103.95974.1008−6.226.2
Fat 2 − Fat 393.95974.1008−7.225.2
Fat 2 − Fat 4233.95974.10086.839.2YES
Fat 3 − Fat 4143.95974.1008−2.230.2

How do you read the table, and how was it constructed? Look first at the rows. Each row compares one pair of treatments.

If you have r treatments, there will be r(r−1)/2 pairs of means. The “/2” part comes because there’s no need to compare Fat 1 to Fat 2 and then Fat 2 to Fat 1. If Fat 1 is absorbed less than Fat 2, then Fat 2 is absorbed more than Fat 1 and by the same amount.

Now look at the columns. I’ll work through all the columns of the first row with you, and you can interpret the others in the same way.

1. The row heading tells you which treatments are being compared in this row, and the direction of comparison.
2. The next column gives the point estimate of difference, which is nothing more than the difference or the two sample means. The sample means of Fat 1 and Fat 2 were 72 and 85, so the difference is −13: the sample average of Fat 1 was 13 g less fat absorbed than the sample average of Fat 2.
3. Next is critical q, from the confidence interval formula. q(α,r,dfW) depends on the number of treatments and total number of data points, not on the individual treatments, so it’s the same for all rows in any given experiment.For this experiment, we had four treatments and dfW from the ANOVA table was 20, so we need q(0.05, 4, 20). Your textbook may have a table of critical values for the studentized range, or you can look up q in an online table such as the one at the end of Abdi and Williams 2010, or find it with an online calculator like Lowry 2001a. Most textbooks don’t have a table of q, and the TI calculators can’t compute it.)Different sources give slightly different critical values of q, I suspect because q is extremely difficult to compute. One value I found was q(0.05,4,20) = 3.9597.