We introduce formula-based approaches for statistical tests regarding peri-stimulus time histograms (PSTHs) in cases when a conditioning stimulus was triggered by a motor-unit discharge with a constant delay time. The individual bin test and the cumulative sum test have recently been reported as methods for the quantitative analysis of PSTHs in the evaluation of human neural projections. These tests calculate confidence intervals using simulated point processes on the discharge of a motor unit, but require a significant amount of calculation time for the point process simulation. In order to overcome such a disadvantage in practical use, we provide a statistical formulization of the two above-mentioned tests using a combination theory. In general, the time required for calculating confidence intervals using these formula-based approaches was 2-13 times faster than when using the previous approaches. Unlike the previous ones, these formula-based approaches do not need to judge that a simulation process converges to the substationary state, and calculate an ideal distribution of statistical noise on the histogram, thereby providing high accuracy. We conclude that the formula-based approaches increase reliability and are sufficiently sophisticated for practical use.
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