# Deformation (congruence) analysis - statistical tests

Dear Micha,

But please let me put forward a few thoughts yet.

- In my opinion, please correct me if I am wrong, we may not test observations against possible blunders and points against possible displacements at the same stage. We should first test observations against blunders, based on the unconstrained model given by me in Eq. (2). Only then, we may test datum points against displacements, based, e.g., on the congruence/constrained model (implicit hypothesis method). In other words, we should perform these two testing procedures separately. Otherwise, possible blunder(s) and displacement(s) will mix and it will be difficult to identify and adapt them.

- Please correct me if I am wrong, the global congruence test for the detection of datum point displacement(s) based on: vT*P*v - (viT*Pi*vi + vjT*Pj*vj) is more powerful, i.e., a given displacement will be detectable with a higher probability than by the test based on vT*P*v. It is due to the number of degrees of freedom of global test statistic. The test statistic which is based on vT*P*v - (viT*Pi*vi + vjT*Pj*vj) has fewer degrees of freedom and the test is more powerful.

- You have written: "But: Do you really test/check the object points without having a stable reference network (datum points)? I don't think so. Usually, one checks for a stable reference (sub-)network (a stable datum) first. If such a stable network is identified(!), object points or other deformation parameters are evaluated afterwards."
Yes, we have unstable datum points in practice and we accept this fact. Test (including detection test) does not have a success rate = 100% (it is not possible). It is always a lower rate and it is called test power. Please note that the probability of Type II error (beta) is, in fact, the probability that some point displacement(s) will not be detected by test. In other words, if we have some true value of displacement, we can calculate beta, or conversely, the power of test (1 - beta). Of course, the higher the true value of displacement is, the lower beta is.
For example, if the beta is 20% for some displacement value, it means that 20 times out of 100 times we won't detect this displacement by test. In consequence, this displacement will be in the congruent/datum part of a model.

Best regards,
Krzysztof

PS. Unfortunately, I do not have a work of Jäger et al. (2005) and, hence, I cannot refer to this book.