STAR*NET V6 - Circe

Chapter 8 Analysis of Adjustment Output
In a STAR*NET survey adjustment, the number of degrees of freedom is the difference
between the number of observations and the number of unknowns, i.e. the redundancy in
the survey. Observations consist of the usual field measurements that are not declared
free, together with any coordinate values entered with partial fixity. The count of
unknowns includes all connected coordinates that are not fixed, one internal orientation
for each direction set, and the computed transformations applied to GPS vectors.
An alternate view of the Chi Square test is that it compares the Total Error Factor
obtained in the adjustment against its expected value of 1.0. The total error factor is a
function of the sum of the squares of the standardized residuals and the number of
degrees of freedom in the adjustment. When the sum of the squares of the standardized
residuals is exactly in the middle of its random distribution and as the number of degrees
of freedom increases, the total error factor approaches exactly 1.0. The assumption that
the adjustment residuals are due solely to random influences is rejected when the total
error factor is larger than 1.0 by some magnitude dependent on the number of degrees of
freedom and the confidence level desired.
Since STAR*NET uses a one-tail Chi Square test, it fails the test only if the total error
factor is greater than 1.0, how much greater depends on the number of degrees of
freedom. With four degrees of freedom, failure occurs at 1.54; with ten at 1.35; with 100
at 1.12; and with 1000 at 1.04.
If the total error factor is less than 1.0, the residuals are smaller than what is expected
relative to the applied standard errors. If the total error factor is much less than 1.0, you
have probably under estimated the quality of your observations, i.e. set your standard
errors too large. For example, if there are 100 degrees of freedom in your job and the
total error factor is less than 0.87, the sum of the squares of the standardized residuals
falls within the bottom 2.5 percent of its expected random distribution. If STAR*NET
were applying a two-tail Chi Square test, these “good” results would be rejected as
having a non-random cause which should be investigated.
In summary, the Chi Square test, often called the “goodness-of-fit test,” statistically tests
whether your residuals are due to normal random errors. If the adjustment fails the test,
first check the standard error values (weighting) you have assigned to your observations.
If you find no problems there, then next check for mistakes which can include blunders
in your actual observations, fieldbook recording errors, or data preparation errors such as
incorrectly entered measurements or misnamed stations in the input data file.
The failure of the Chi Square test is an alarm that indicates some problem and you then
need to identify the source of the problem. Locating the source of problems is discussed
later in this chapter.
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