If you don't know which of two equations is more appropriate for your data, fit both and compare the results. For example, compare models with one and two classes of binding sites. Or compare a one-phase exponential dissociation curve with a two-phase curve.

Approach to comparing two models

When comparing the fits of two models, the first step is to examine the best-fit values of each model to make sure they are scientifically reasonable. Also make sure that the confidence intervals are not extremely wide. If the best-fit values of one model make no sense (for example, if a rate constant is negative) or if the confidence intervals are very wide, then reject that model. You don't need statistical calculations to reject a model, if the best-fit parameters of that model make no scientific sense.

If both models fit the data with sensible values, compare goodness-of-fit as quantified by sum-of-squares. If the more complicated equation fits worse (has higher sum-of-squares) than the simpler equation, then you should clearly reject the more complicated equation and conclude that the simpler equation fits better. This will happen rarely, as the curve generated by the more complicated equation (the one with more variables) will nearly always have a lower sum-of-squares, simply because it has more inflection points (it wiggles more).

If both models fit the data with sensible values, and the more complicated model fits better, then you need to use statistical calculations to decide which model to accept. Do this with an F test to compare fits.

Comparing the fits of two equations using an F test

The F test compares the fit of two equations, where the more complicated equation (the one with more parameters) fits better (has a smaller sum-of-squares) than the simple equation. The question is whether this decrease in sum-of-squares is worth the "cost" of the additional variables (loss of degrees of freedom). The F test (detailed below) calculates a P value that answers this question: If the simpler model is really correct, what is the chance that you'd randomly obtain data that fits the more complicated model so much better? If the P value is low, conclude that the more complicated model is significantly better than the simpler model. Most investigators accept the more complicated model if the P value is less than 0.05.

This graph shows the results of fitting both a one-site and two-site competitive binding curve to some data. Both fits are scientifically plausible, so it makes sense to perform compare the sum-of-squares with the F test.

The F ratio, as well as the two degrees of freedom values, are explained in the next section. The P value is 0.0065. If the one-site model is correct, there is only a 0.07% chance that you'd randomly obtain data that fits the two-site model so much better. Since this is below the traditional threshold of 5%, conclude that the fit of the two-site model is significantly better, statistically, than the one-site model.

The F test is only strictly valid when the simpler equation is a special case of the more complicated equation. In other words, the two models should be nested. By setting some parameters to special values (often 0.0 or 1.0), you can turn the more complicated model into the simpler model.

How the F test works

To understand the F test, you need to compare the sum-of-squares and degrees of freedom for each fit. Here are the results for the example of the previous section.

	Two-site	One-site	% Increase
Degrees of freedom	7	9	28.57%
Sum-of-squares	52330	221100	322.51%

In going from the two-site to the one-site model, we gained two degrees of freedom because the one-site model has two fewer variables. Since the two-site model has 7 degrees of freedom (12 data points minus 5 variables), the degrees of freedom increased 28.6%. If the one-site model were correct, you'd expect the sum-of-squares to also increase about 28.6% just by chance. In fact, the one-site had a sum-of-squares 322%higher than the two-site model. The percent increase in sum-of-squares was 11.29 times higher than the increase in degrees of freedom (322.51/28.57). The F ratio is 11.29.

More generally, if the simpler model is correct you expect the relative increase in the sum of squares to equal the relative increase in degrees of freedom. In other words, if the simpler model is correct you expect that:

If the more complicated model is correct, then you expect the relative increase in sum-of-squares (going from complicated to simple model) to be greater than the relative increase in degrees of freedom:

The F ratio quantifies the relationship between the relative increase in sum-of-squares and the relative increase in degrees of freedom.

That equation is more commonly shown in an equivalent form:

F ratios are always associated with a certain number of degrees of freedom for the numerator and a certain number of degrees of freedom for the denominator. This F ratio has DF1-DF2 degrees of freedom for the numerator, and DF2 degrees of freedom for the denominator.

If the simpler model is correct you expect to get an F ratio near 1.0. If the ratio is much greater than 1.0, there are two possibilities:

	•	The more complicated model is correct.

	•	The simpler model is correct, but random scatter led the more complicated model to fit better. The P value tells you how rare this coincidence would be.

The P value answers this question: If model 1 is really correct, what is the chance that you'd randomly obtain data that fits model 2 so much better? If the P value is low, you conclude that model 2 is significantly better than model 1. Otherwise, conclude that there is no compelling evidence supporting model 2, so accept the simpler model (model 1).

Comparing two models with GraphPad Prism

Prism can compute the F test automatically. To compare fits, simply, check the option box on top of the nonlinear regression parameters dialog. Check the option button for equation 1, and choose that equation along with its initial values. Also set any variables to constants, if necessary. Then check the option button for equation 2, and choose an equation and adjust settings for initial values and constants.

Tip: It is not possible to automatically compare the fit of data to a built-in equation with a fit to a user-defined equation. Instead copy a built-in equation to the user-defined list. Then compare two user-defined equations.Prism places the results of the F test on the same resuts tables with the best-fit values. For the example data, here is how Prism reports the comparison: