April 19, 2024

The counting processes are capabilities that increment by 1 at any time when a brand new occasion arrives. Clearly, there are fewer occasions occurring within the therapy than within the management. If these have been login occasions, this might counsel that the brand new code comprises a bug that forestalls some customers from with the ability to log in efficiently.

This can be a widespread scenario when coping with occasion timestamps. To provide one other instance, if occasions corresponded to errors or crashes, we want to know if these are accruing sooner within the therapy than within the management. Furthermore, we wish to reply that query as shortly as doable to forestall any additional disruption to the service. This necessitates sequential testing strategies which have 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 perform 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 capabilities for management (A) and therapy (B) are the identical. This may be achieved semiparametrically with out making any assumptions concerning the depth capabilities λᴬ and λᴮ. Furthermore, the novelty of the analysis is that this may be achieved 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 thus far in management and therapy. In different phrases, all it’s essential take a look at 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 take a look at, wherein we all know by design that the depth perform is identical for the management (A) and the therapy (B), albeit nonstationary.