# Deformation (congruence) analysis - statistical tests

Hi Micha,

thank you very much for your fast and detailed response.

1. If I well understand, the global congruence test for the detection of datum point displacement(s) in JAG3D is based on the following quadratic form: omega = vT*P*v, where v is the residual vector from the congruence/constrained model presented by Eq. (1), i.e., v = [viT vjT]T :

Eq. (1)

While, in my opinion, the global congruence test for the detection of datum point displacement(s) should be based on the following quadratic form: vT*P*v - (viT*Pi*vi + vjT*Pj*vj), where vi, vj are the residual vectors from the unconstrained (single-epoch) models presented by Eq. (2).

Eq. (2)

The part vT*P*v from Eq. (1) is related to null hypothesis and the part (viT*Pi*vi + vjT*Pj*vj) from Eq. (2) is related to alternative hypothesis, according to the generalized likelihood ratio testing theory.

My point is, the global test for the detection of datum point displacement(s) based on omega = vT*P*v from Eq. (1) is less powerful than the one based on: vT*P*v - (viT*Pi*vi + vjT*Pj*vj), e.g., Caspary (2000, Eq. (10.13), (10.14)), Niemeier (2008, Eq. (13.3.21)), Heunecke et al. (2013, Eq. (11.11-11.13)) or Hekimoglu et al. (2010, Eq. (16) from the article ‘Increasing the Efficacy of the Conventional Deformation Analysis Methods: Alternative Strategy’).

2.1. Let me allow to give my second concern in a slightly different way. Object point displacements are estimated from congruence model in JAG3D, i.e., the model as it is presented in Eq. (1). This is clear and understandable. My concern was related to variance factor estimate (and, at the same time, its denominator), which is used in statistical significance testing of estimated object point displacements. If I well understand your explanation and user manual, the statistical significance testing of estimated object point displacements uses the variance factor estimate which is estimated from the congruence model, i.e., the same model, which is used for the estimation of object point displacements. And this raises my fears. Admittedly, the object points do not force the network. Nevertheless, the datum points do it. In other words, the observations are not here completely free as in the model from Eq. (2). In consequence, if the group of congruence/datum points included unstable (non-detected) point(s), the variance factor – which is estimated from the congruence model – would be biased.
So, in my opinion, this variance factor should be estimated from a completely free model as presented in Eq. (2). Such a solution is also presented in literature known to me.

Best regards,
Krzysztof