I am trying to implement the Pettitt test on R. I am coding by myself but with the reference of the built-in function pettitt.test of R.(https://www.rdocumentation.org/packages/trend/versions/1.1.4/topics/pettitt.test)
I could implement it successfully, but my question is about the p-value. I am calculating the p-value with this equation:
p=2e^((-6U ̂^2)/(n^3+n^2 ))
being U the value of the test statistic and n the length of my sample. (I have checked and I got the same than with the built-in function pettitt.test)
For a 5% confidence interval, the alternative hypothesis should be true if this p-value is lower than 0.5? or 0.05? I am confused by the documentation of R (weblink previously shared) that says lower or equal than 0.5, and for the Pettitt's paper (1979) page 5 that also indicates lower or equal than 0.5
Any help to clarify this question?
What the paper says is
... where the approximation holds good, accurate to two decimal places, for p(OA)<0.5
(emphasis added). That is, it's not about the critical level at which you want to reject the null hypothesis (typically p=0.05), but about what range of values the approximation holds for.
Most R functions and packages simply return the p-value and let the user decide what they want to do about interpreting it (i.e. reject or fail to reject the null hypothesis at a given alpha-level, treat it as a continuous ("Fisherian") strength of evidence against the null hypothesis, etc.).
If you wanted you could have your function issue a warning if the calculated value of p is >0.5 (e.g. "approximation of the p-value may be unreliable")
Pettitt, A. N. “A Non-Parametric Approach to the Change-Point Problem.” Journal of the Royal Statistical Society. Series C (Applied Statistics) 28, no. 2 (1979): 126–35. https://doi.org/10.2307/2346729.