test 
V 
The power g(r) is an increasing function of A, so A should be as large as possible. 
A is maximal if N,, = 0 (Grayhill /9/), i.e. if dz is orthogonal to dx, dx", dt 
Thus a statistical advantage of the orthogonal additional parameter concept be- 
comes evident: It increases the power of this test. 
Beside this it is clear that the power of a test depends mainly on the alternative 
hypothesis and on the type I error size o. 
If He is rejected, i.e. Af a significant systematic influence is apparent, the 
detection of individual significant components becomes necessary. 
Therefore the set of hypotheses 
wii . dz, = dz, ; (A= 10505. 0p) (39) 
Oo 
is commonly used to test the individual parameters on significance. 
Hence we get the test criterions 
T "e —— : dziz4 7 variance of the ith (40) 
99 (0212 additional parameter 
Under pit) the TU). values are distributed as Student's t with r degrees of 
freedom. Thus the more-dimensional T-test (31) is reduced to an one-dimensional 
t-test. 
In the case of independence of the individual parameters the type I error size a 
of the individual test is related to the type I error size o of the global test 
as 
]el oi ei (411 e 8) ; (41) 
œ = 
"Ie 
The main problem in testing subsets or even single parameters estimated in multi- 
dimensional models arises with the dependence of these subsets (single parameters) 
on the other model parameters. In the case of significant correlations the pro- 
bability a = prse, | HO) of rejection of a true null-hypothesis Hit). 
dz; = dz; of a single event (parameter) is no further independent. It should be 
noticed that the application of the one-dimensional t-test with the usual limits 
and confidence intervals leads the more to wrong decisions the more the correla- 
tions do increase. Thus we have another important argument for using the concept 
of orthogonal additional parameter sets. 
Whereas the orthogonal concepts are regarded as very useful, the practical geo- 
metrical conditions however often do not provide for sufficient orthogonality. 
Then we have to set up simultaneous confidence intervals for the single events. 
This can be done by the a-posteriori orthogonalization of the additional parame- 
ter set, i.e. the transformation of the additional parameter vector (or, .if 
necessary, even the vector of all unknowns of the bundle system (1)) into ortho- 
gonal components, which can be tested independently (see Roy, Bose /19/, Pelzer 
/16/). 
Sometimes high correlations do appear only within a subset of the additional pa- 
rameters. Then these parameters can be tested together on common significance. 
The corresponding test criterion may analogously be derived from the general li-