Hi Paul,

I have to read A testing procedure for use in geodetic networks.

There are further publications dealing with the adaption mentioned by Baarda, e.g. Computation of Baarda’s lower bound of the non-centrality parameter.

I believed that there was a simplier implementation of this test but obviously, I need to study this method in more detail.

The main intention to adapt test statistics is to avoid contrary decisions, i.e., if the global test is rejected, at least one localization test should also be rejected. However, if many hypothesis tests are conducted using the same sample (e.g. to detect and identify outliers), the probability of at least on type-I error increases. This is known as alpha-error accumulation. There are some strategies to reduce this error. Popular methods are the Šidák correction, the Bonferroni correction, or the Holm–Bonferroni correction. The concept proposed by Baarda is - as far as I know - only known in the geodetic sciences. Baarda's idea was to equate the test power (type-II error) and the non-centrality of each test.

Let the type-I and type-II error, e.g. α and β, be specified by the user for the localization test, where the degrees of freedom are d1 = 1 and d2 = ∞ for the numerator and the denominator of the corresponding F-test, respectively. The non-centrality parameter λ is obtained as a function of both errors λ = λ(α, β). Thus, the parameters α, β, and λ are dependent to each other. If two of them are known, the missing value can be calculated. From that point of view, we are also able to obtain and to adapt α = α(β, λ) as a function of β and λ for test statistics having different degrees of freedom d1 and d2, for instance the global test.

An example in Matlab notation - perhaps you are familiar with this software package. Let the total redundancy of the network be four. Moreover, let d2 = ∞ for all tests under investigation.

dof = 4; % redundancy of the network (number of observation - number of unknowns)
% type-I and type-II error for hypothesis test (e.g. outlier in a single observation)
alpha_single = 0.001;                                     % --> 0.1 %
beta  = 0.2;                                              % --> 20 %
% quantile of localization test
q_single = norminv(1-alpha_single/2)^2                    % --> 10.83
% non-centrality parameter
lambda = (norminv(1-alpha_single/2) + norminv(1-beta))^2  % --> 17.07
% adapted quantile of global test
q_global = ncx2pdf(beta, dof, lambda)/dof                 % --> 3.38
% corresponding type-I error 
alpha_global = chi2cdf(q_global*dof, dof, 'upper')        % --> 0.89 %

You will find all theses value in attached screenshot of my last reply.

In html report, there is no tooltip for Baarda table.

Yes, you are right. However, the tooltip is available in the application; and, unusually, the data analysis takes place in the application itself and not with the report.

All the best
Micha