The Best Fit number of joinpoints can decrease

Why does the number of joinpoints for the best fit sometimes decrease when the maximum number of joinpoints is increased?

Answer:

Users of joinpoint need to specify the minimum (denoted MIN) and maximum (denoted MAX) number of joinpoints to be considered.  To find the number of joinpoints for the best fit, a sequence of tests is performed, first testing H0: K0 = MIN vs. Ha: K1=MAX number of joinpoints.  If we reject the null hypothesis, we next test H0: K0=MIN+1 vs. Ha:K1= MAX.  If we fail to reject the null, we test H0: K0=MIN vs Ha:K1= MAX-1.  Testing continues in this manner until we determine the number of joinpoints by continuing to step the lower limit up by one if we reject the null hypothesis or by stepping the upper limit down by one if we fail to reject the null hypothesis.  To maintain an overall significance level α for this sequence of tests, each test is performed at the α/(MAX – K0) level.  Therefore, even when the overall significance level is kept constant, a change in the MAX or MIN number of joinpoints, which would change the significance levels of the individual tests, may result in a change to the number of joinpoints in the best fitting model.