Using the best fit model
Calculations and plots using the best fit model equation
You can use the best fit curve as a data smoothing curve, then
obtain areas and slopes by analytical mathematics. As this is
tedious and not always possible you can use numerical methods,
but you may have to experiment to get the step sizes right.
Estimating the area under a curve (AUC)
You can integrate over the range set by the data, or over any
chosen range. If the curve is complicated, you should increase
the number of Simpson rule points until the area stabilises.
Estimating derivatives (initial, maximum, minimum rates)
You can estimate the derivatives at chosen points, or plot over
the x-range set by your data, to get the initial, minimum and
maximum gradients. The step size used is a fraction of the x-
range and this can be adjusted (for complicated curves) until
the derivative estimate stabilises.
Calibration (predicting x given measured values of y)
This can be done after fitting any model, as long as the best
fit curve is free from turning points.
Extrapolation (Graph plotting)
After fitting, you can alter start and stop positions and number
of best-fit curve points in order to extrapolate, e.g. to the
origin, or the axes in a double-reciprocal, or Scatchard plot.
Graphical deconvolution (Graph plotting)
After fitting sums of terms you can deconvolute to see the best
fit curve dissected into components. Do this after fitting sums
of high-low affinity, or MM sites, exponentials, etc. since, if
a component is making a small contribution it can often be seen
graphically. Practise with the test file gauss3.tf1, a sum of 3
Gaussians plus error which can be used in the EXPERT mode.
Error bars (Graph plotting)
If you have replicates they can be used to calculate and plot
mean values and error bars with the best-fit curve.
Storing results
The WSSQ, no. points, no. parameters can be saved after fitting
for an F test, Akaike AIC or Schwarz criterion SC.
Using the best fit curve to calculate weights (w = 1/s^2)
You may believe that a suitable model for s-values is given by
s^2 = A + |B*exact-function|^C,
and wish to use such a formula
with best-fit-function replacing exact-function. For instance,
10% relative error would require A = 0, B = 0.1 and C = 2.
Before using such a technique you must be absolutely sure that
the best-fit model is the true model, and that the formula for
s is correct. Also you must have good data, excellent starting
estimates and reasonable values for A, B and C. It is unwise to
blindly use this approach and routinely estimate A, B and C from
the data set. A much better approach is to get suitable values
for A, B and C (by independent investigation) and then use the
formula after a solution has been found. The procedure used by
this program is now described.
First curve fitting is done with weighting taken from s-values
on the data file. After a satisfactory solution has been found
you re-enter from the current best-fit point, then calculate s
(using your own A, B and C values) as a final refinement.
Fitting a straight line by nonlinear regression
There are several circumstances when this program can be used
to fit straight lines using the model
y = mx + c
with p(1) = m and p(2) = c and nonlinear regression, rather
than using the
dedicated programs POLNOM or LINFIT. If you select this model
the following options are provided.
- y = mx + c will give the same result as POLNOM and LINFIT.
- y = mx is useful for calibration when you require a line
through the origin, i.e with zero background.
- Reduced major axis regression. This minimises the sum of
the areas of right angled triangles between the points and
best-fit line, and should be used when there is variation
in both x and y but x and y have widely differing scales.
- Major axis or orthogonal regression. This minimises the sum
of squares of perpendicular distances between points and
best-fit line and should be used where there is variation
in both x and y (as in correlation).
You can use weights for all these regressions but it is best to
set all s = 1 in cases 3) and 4) and do unweighted regression.)
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