T-Test

As discussed in the topic on the W-Test, the W-Test is a 1-dimensional test which checks the conventional alternative hypothesis. This hypothesis assumes that there is just one erroneous observation at a time. This so-called data snooping works very well for single observations, for example for directions, distances, zenith angles, azimuths or height differences. However, for some observations such as GPS baselines, it is not enough to test the DX-, DY-, DZ-components of each vector separately. It is imperative to test the baseline as a whole as well.

For this purpose we introduce the T-Test. Depending on the dimension of the quantity to be tested, the T-Test is a 3- or 2-dimensional test. As with the W-Test, the T-Test is also interlinked with the F-Test by the B-Method of testing. The T-Test has the same power as both other tests, but has its own levels of significance and its own critical values (see tables below).

Significance level/critical value for 2-dimensional T-Test, based on α0 of the W-Test:

Significance level/critical value for 2-dimensional T-Test, based on α0 of the W-Test:

significance level α0	0.001	0.010	0.050
significance level α (2-dim)	0.003	0.022	0.089
critical value T-Test	5.91	3.81	2.42

significance level α0

significance level α (2-dim)

critical value T-Test

Significance level/critical value for 3-dimensional T-Test, based on a0 of the W-Test:

Significance level/critical value for 3-dimensional T-Test, based on a0 of the W-Test:

significance level α0	0.001	0.010	0.050
significance level α (3-dim)	0.005	0.037	0.129
critical value T-Test	4.24	2.83	1.89

significance level α0

significance level α (3-dim)

critical value T-Test

The T-Test is equally useful when you want to test for slight deformations on known stations. The data snooping successfully finds an outlier that might have occurred, for example, due to a typing error in either the Easting or the Northing or the height component causing a gross error in one of the coordinate components. But the deformation of a whole station might stay undetected by the data snooping when the shifts caused by the deformation in each coordinate component are relatively small. For testing a possible deformation influencing the whole coordinate triplet, like with the Easting and the Northing and the height, a different alternative hypothesis is needed. The 3-dimensional T-Test on the complete coordinate triplet is better equipped to trace the deformation, although its not able to trace the exact direction into which the station has moved.

A situation in which the W-Test is accepted but the associated T-Test of the observation is rejected, which is not unlikely in practice, does not necessarily imply a contradiction. It is simply a matter of having tested different hypotheses.