Question about Pairwise significance and (in)equality

#1
Assume that I did a post-hoc test and I found that the means of Group X is not significantly different from Group Y and Group Z, and Group Y is significantly different from Group Z.

Does this imply that the mean of Group X = the mean of Group Y/Z and the mean of Group Y ≠ the mean of Group Z?

Perhaps, my question is more about the nature/semantics of the null/alternate hypotheses. Can one generalize the null/alternative hypotheses as simple mathematical equalities?
 
#2
It is likely that your group means are ordered
Y < X < Z (or the opposite).

The trick to understanding this is to picture a confidence interval around each group mean. You don't actually know the extact location of Y's mean with 100% certainty. You know the best guess of Y's mean and you know the interval around Y's mean with 95% confidence. This idea that you only really have a guess about an interval is what is going to inform your intuition.

It turns out that the intervals that you think might contain Ys mean and Xs means overlap. So you can't really be all that certain they are actually different means (or more precisely "this is insufficient evidence that they are statistically different". Likewise X and Z.

However there is no overlap of these intervals between Y and Z. So this is statistically significant evidence that Z and Y are different.

You essentially don't know whats up with X relative to Y and Z. X could be exactly Y. X could be exactly Z. X could be dead set in the middle. You don't have the statistical signficance to say really with these test and these data.
 
#3
It is likely that your group means are ordered
Y < X < Z (or the opposite).

The trick to understanding this is to picture a confidence interval around each group mean. You don't actually know the extact location of Y's mean with 100% certainty. You know the best guess of Y's mean and you know the interval around Y's mean with 95% confidence. This idea that you only really have a guess about an interval is what is going to inform your intuition.

It turns out that the intervals that you think might contain Ys mean and Xs means overlap. So you can't really be all that certain they are actually different means (or more precisely "this is insufficient evidence that they are statistically different". Likewise X and Z.

However there is no overlap of these intervals between Y and Z. So this is statistically significant evidence that Z and Y are different.

You essentially don't know whats up with X relative to Y and Z. X could be exactly Y. X could be exactly Z. X could be dead set in the middle. You don't have the statistical signficance to say really with these test and these data.
Alright, another question. If I were to assume that the CI.95 for the mean of group X is 1 <= mu <= 2, does that mean that the true mean can be 1.01 or 1.02 or 1.99 or 1.98, etc... all with an equivalent probability of happening? Or is there some sort of curve that dictates that the mean has a higher probability of being one number instead of another (such as the T distribution)?

I guess what I am asking is whether the confidence interval is actually an interval, where the true mean has exactly the same chance as being 1.01 as it does being 1.99.
 

Mean Joe

TS Contributor
#4
This is an interesting question. If you're using the sample mean (of the group) as the point estimator for the true mean, it has an approximately normal sampling distribution. This is often the case, so if your 95% CI is 1 <= mu <= 2, then you should have Xbar = 1.5 (right in the middle of the CI).

You could find other CIs, for example 90% CI may be 1.2 <= mu <= 1.8. Then there is a 5% chance that mu is in [1, 1.2] or [1.8, 2]--this is a total interval length of 0.2 + 0.2 = 0.4 (compare to 90% chance it's in the interval of length 0.6 from [1.2, 1.8]). These are just made-up numbers, but that's my idea.
 
#5
So, then, is there any stringent statistical method by which I can order my group means by rank, taking into account such things as confidence intervals, sample sizes, etc?