F-Test

F-Test

The F-Test is a commonly used multi-dimensional test for checking the null-hypothesis H0. The F-Test is often called the overall model test, because it tests the given mathematical and stochastic models in general.

The F-value is defined by the expression:

The F-value is defined by the expression:

F = s2/α2

F = s2/α2

with

s2 = a-posteriori variance factor derived from the computed residuals and the redundancy

s2

α2 = a-priori variance factor.

α2

The F-value is tested against a critical value of the F-distribution, which is a function of the redundancy and the significance level α. But the information provided by the F-Test, that is, the acceptance or rejection of the null-hypothesis, is not specific. Therefore, if H0 is rejected it makes sense to ask for the reasons why by tracing errors in observations or pre-assumptions.

There are three reasons for rejection:

There are three reasons for rejection:

• Gross errorsIf you suspect that H0 is rejected because of a gross error in one of the observations, then the so-called data snooping provides you with revealing information by applying the W-Test in order to seek for and find errors in individual observations. The F-Test and the W-Test are interlinked by a common value for the power of test β. The method is the called the B-Method of testing.
• Incorrect mathematical modelIf you suspect that H0 is rejected because of a mathematical model that is incorrect or not refined enough, then check the adjustment settings. For example, redefine and make the vertical refraction coefficient or the scale factor correction be used to improve the mathematical model.See also:Advanced Terrestrial Parameters
• Incorrect stochastic modelIf you suspect that H0 is rejected because of a stochastic model that is incorrect or not refined enough, then check the adjustment settings . If the a-priori variance-covariance matrix is too optimistic increase the input standard deviations and/or centering/height errors of the observations to improve the stochastic model.See also:TPS Accuracy Information

When you optimise the mathematical or stochastic model always bear in mind that the purpose of statistical testing is not to have all observations accepted, but to detect outliers and model errors.