How Good Is The Test?
What can be the meaning of the question: "How good is the test?" We tested the Null hypothesis
against the Alternative hypothesis that the smallest interpoint distance between the two point sets
is smaller than expected. We used for our test statistic the smallest interpoint distance.
And indeed at the 1 % significance level we had to reject the Null hypothesis. -- OK. Just barely!
We did everything right. What can be the meaning of "How good is the test?"
Here goodness means, if we employ this test on data sets coming from the same population as
the initial moved point data set, so that the Null hypothesis is false, what is the probability
that we do not reject the Null hypothesis? This probability is misdetect rate of the test.
To determine the misdetect of the test, we will do a 10,000 trial Monte Carlo experiment in which
each trial we will generate 2 point sets, with one of the point sets moved closer to the other
in the same statistical manner as our first moved point data set. For each trial, we will compute
the test statistic, which is the closest interpoint distance between the sets.
We will find what fraction of trials had closest interpoint distances greater than the critical value .005513 . Recall that when the closest interpoint distance is greater than the critical value,
we do not reject the Null hypothesis. Rather we reject the Alternative hypothesis in favor of
the Null hypothesis.