I have finally decided to reach out for help regarding the Mitchell-Olds & Shaw test. If you're not familiar, it is designed to help researchers better interpret quadratic regressions. That puts it very simply, but more information can be found here and the R source code here.
Please help me discern between seemingly conflicting information about the null hypothesis of this test. Within the "Details" section of ?MOStest, it states that: "The null hypothesis is that the extreme point u is located within the interval given by points p_1 and p_2" (p_1 is the min of x and p_2 is the max). That "extreme point" referred to is the peak or pit of the quadratic regression. This null makes sense to me, but when the code is run, the output states that the null is that the "hump of a quadratic linear predictor is at min or max". These two nulls are distinct from one another.
The output generally looks like this (I don't have the rep to embed a screenshot yet):
output of Mitchell-Olds & Shaw test screenshot
I don't understand the source code well enough to discern which null is accurate. That being said, here are a few more lines of evidence for one or the other null. The MOStest() outputs "isBracketed", which tells you if the pit or peak is within the bounds of x. And, as can be seen in the screenshot of the output, the test also tells us if the pit/peak takes place at the max or min, with p-values, though I am not sure what "combined" is.
What exactly is this test telling me? My guess is that both null hypotheses are being tested, one in the form of the min and max p-values, and one with the "combined" p-value.
What say you?
Thanks!