# 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.

1. y = mx + c will give the same result as POLNOM and LINFIT.
2. y = mx is useful for calibration when you require a line through the origin, i.e with zero background.
3. 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.
4. 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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