Biology Reference
In-Depth Information
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B
Independent Variable
FIGURE 8-7.
The residuals for a straight line (panel A), or Eq. (8-15) (panel B), fit to the data in Figure 8-3.
The better fit gives more randomly distributed residuals.
this verification passes and the residuals do not follow a Gaussian
distribution, then it is likely that the fitting equation is incorrect.
One of the most useful tests is to simply plot the residuals as a function
of the independent and dependent variables. A visual inspection can
usually provide an indication of a problem with the analysis. For
example, Figure 8-7 presents the residuals for two different fits of the
data shown in Figure 8-3. Panel B is a fit of Eq. (8-15), and it appears the
residuals are random. By comparison, panel A corresponds to the fit
of a straight line, and it is clearly not random. Thus, by inspection we
can conclude this data cannot be described by a straight line.
There are many quantitative goodness-of-fit tests, such as the runs test,
autocorrelation, and the Kolmogorov-Smirnov test. The runs test has
proven to be very useful. A run is one or more residuals in a row with
the same sign. Panel A of Figure 8-7 contains five runs: the first seven
residuals are all negative; the 8th residual is positive; the 9th and 10th
residuals are negative; residuals 11 to 23 are all positive; and the 24th is
negative. Panel B contains 16 runs. This test is statistically based, and, if
the residuals follow a Gaussian distribution, the expected number of
runs and the variance of the expected number of runs can be computed.
A Z score and probability are then calculated, determining the likelihood
of the observed number of runs. More details for this and the other
quantitative methods mentioned above can be found in Straume and
Johnson (1992).
E. Uniqueness of the Least-Squares Values of the Parameters
For some types of fitting equations (i.e., linear models), it can be
algebraically demonstrated that only a single set of unique model
