The counting processes are features that increment by 1 every time a brand new occasion arrives. Clearly, there are fewer occasions occurring within the therapy than within the management. If these had been login occasions, this is able to recommend that the brand new code comprises a bug that stops some customers from having the ability to log in efficiently.
This can be a widespread state of affairs when coping with occasion timestamps. To offer one other instance, if occasions corresponded to errors or crashes, we wish to know if these are accruing quicker within the therapy than within the management. Furthermore, we wish to reply that query as shortly as potential to forestall any additional disruption to the service. This necessitates sequential testing methods which had been launched in part 1.
Time-Inhomogeneous Poisson Course of
Our knowledge for every therapy group is a realization of a one-dimensional level course of, that’s, a sequence of timestamps. As the speed at which the occasions arrive is time-varying (in each therapy and management), we mannequin the purpose course of as a time-inhomogeneous Poisson point process. This level course of is outlined by an depth operate λ: ℝ → [0, ∞). The variety of occasions within the interval [0,t), denoted N(t), has the next Poisson distribution
N(t) ~ Poisson(Λ(t)), the place Λ(t) = ∫₀ᵗ λ(s) ds.
We search to check the null speculation H₀: λᴬ(t) = λᴮ(t) for all t i.e. the depth features for management (A) and therapy (B) are the identical. This may be accomplished semiparametrically with out making any assumptions concerning the depth features λᴬ and λᴮ. Furthermore, the novelty of the analysis is that this may be accomplished sequentially, as described in section 4 of our paper. Conveniently, the one knowledge required to check this speculation at time t is Nᴬ(t) and Nᴮ(t), the overall variety of occasions noticed up to now in management and therapy. In different phrases, all you might want to check the null speculation is 2 integers, which may simply be up to date as new occasions arrive. Right here is an instance from a simulated A/A check, by which we all know by design that the depth operate is identical for the management (A) and the therapy (B), albeit nonstationary.
