# How to summarise ANOVA post-hoc Tukey test?

I am having trouble to interpret tukey's post hoc result from the following

They incorporated post-hoc test results into a table using the πβ€π<π style notation (π,π,π,... correspond to groups) but how to interpret it?

• Is Normal > osteopenic, osteoporotic means Normal has higher significant difference than osteopenic and osteoporotic ?
• In R, the results look like the following, is that mean Neutral has higher significant difference than Friends ? more specifically how to I write this result in πβ€π<π style notation?

Thanks!

• One thing you are going to need to answer the question is the means for each group --- Neutral, Friends, and so on. ... Some software will report a positive mean difference when the values in the first group are larger, and a negative mean difference when the values in the second group are larger. However, since this function in R is reporting all the mean differences as positive, I suspect this function is not following that convention. Commented Aug 4, 2022 at 0:44

Normal > osteopenic, osteoporotic means Normal has higher significant difference than osteopenic and osteoporotic?

That means that the outcome value for Normal is significantly higher than that for either osteopenic and osteoporotic, but osteopenic and osteoporotic aren't distinguishable statistically. In general, the > means that the outcome value for the category on the left is statistically distinguishable from and higher than that on the right. Categories that can't be distinguished statistically are separated by , (comma).

In R, the results look like the following, [does] that mean Neutral has higher significant difference than Friends? More specifically how do I write this result in πβ€π<π style notation?

What's displayed seems to be the differences in outcome between each pair of categories. In your case all differences except Alone-Neutral are statistically significant. You need to put the actual outcome values into order first. From what you show, that would seem to be Disliked highest, Alone and Neutral next highest but not distinguished from each other, and all of those higher than Friends. You could write that as Disliked > Alone, Neutral > Friends.

• sorry for late question, by outcome value are you referring to p-value, mean or something else? Commented Sep 1, 2022 at 1:17
• @jquery404 I refer to the mean value.
– EdM
Commented Sep 1, 2022 at 2:42
• I think I got it, but the only thing I'm struggling with is how much of a difference would be considered significant difference. In 1st row: Normal-Ost.Pen = 0.42, Normal-Ost.Por = 0.67 and Ost.Pen-Ost.Por=0.25 resulting in (N>OstPen,OstPor). but in the 5th row: the difference is clearly higher, but they put (-), which, I suppose indicates, difference is not significant. So is there a value/threshold above which I should report? Commented Sep 11, 2022 at 22:47
• @jquery404 statistical significance is typically evaluated by a ratio between an observed difference and an estimate of the error in determining the difference. Although the differences in the 5th row of your first table are numerically larger than those in the 1st row, those differences are very small compared against their errors, implicit in the large standard deviations. It's not a threshold in absolute difference that matters, it's a threshold in the reliability of judging that a difference is different from 0; p-values <0.05 provide a common threshold in that reliability.
– EdM
Commented Sep 12, 2022 at 11:43