Why should we add/subtract 1 in the p-value calculation?

Question

I have seen this equation for computing the p-value after a Monte-Carlo test.

\begin{equation} P_{upper} = \frac{NGE+1}{N_{runs} + 1} \quad\quad P_{lower} = \frac{NLE+1}{N_{runs} + 1} \end{equation}

Source: https://www.biomedware.com/files/documentation/clusterseer/MCR/Monte_Carlo.htm

where Nruns is the total number of Monte Carlo simulations, NGE is the number of simulations for which the statistic was greater than or equal to the observed statistic, and NLE is the number of simulations for which the statistic was lower than or equal to the observed statistic.

In this equation, 1 is added to the numerator and the denominator because "the observed statistic is included in the reference distribution".

Questions:

1. What exactly does this mean and why should we add/subtract 1?

2. Even if I don't add/subtract a one, I still get a significant p-value. Therefore, which one is statistically correct?

Any thoughts on this? I'll appreciate any help!

Answer 1

In general, for p-value calculation, we first produce a distribution of the test statistic and integrate the distribution from the "observed" test statistic up to infinity (let's say for p-value upper). Consider the image below. There are two hypotheses being tested for some value $\mu=1$ and $\mu=0$. First, the integral from the observed value to infinity is calculated for both the blue and red histograms. Then p-value will be the integral of red histogram divided by the blue.

In your case, instead of the integral the total number of events (of MC) is taken directly and the +1 corresponds to the incorporation of the observed test statistic in that total number.