# calculate goodness of fit

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Neha on 20 Feb 2014
Edited: Matt J on 27 Nov 2015
I have a data set which looks like this
I have around 7 equations like
y=atp*kcat*Et.*x ./ (kd_A*km_g + km_g*atp + km_a*x + atp*x.*(1+x./ksi));
y=atp*kcat*E.*x ./ (km_g*atp + km_a*x.*(1+x./ksi) + atp*x);
and more...
There are 4 - 5 parameters to fit for in each equation and there are 3 - 4 known variables.
One of these equations should be the best fit for the data and I want to find that. I tried using lsqcurvefit and got pretty good results. But similar values fpr the variables using all the equations and the resnorm is identical. Is there a way to find the goodness of fit? And am I doing it correct?

Star Strider on 20 Feb 2014
Edited: Star Strider on 20 Feb 2014
If you have the Statistics Toolbox, I suggest fitnlm. It produces a number of useful statistics. To the best of my knowledge, these aren’t available with lsqcurvefit and the Optimization Toolbox.
As a general rule, if all your functions produce similar residual errors, the model with the fewest parameters to estimate will have the best goodness-of-fit because it has the greatest degrees-of-freedom. The F-statistic may be the best measure, although the likelihood ratio may be best for for comparing two models with the same number of parameters.
That said, there are a number of other techniques to assess regression models. It is too broad a subject to go into here.
Pitch Mandava on 20 Mar 2014
Edited: Pitch Mandava on 20 Mar 2014
There is a webiste on sakai.duke.edu which deals with using F-test for comparing two different models with an example. Hope this link works for everybody. https://sakai.duke.edu/access/content/group/25e08a3d-9fc4-41b0-a7e9-815732c1c4ba/New%20folder/Stat%20Topic%20Files/Non-Linear%20Regression/FTestTutorial.pdf

jandas on 20 Feb 2014
Try to use the fit function
[fitobject,gof] = fit(x,y,fitType)
and have a look at the gof output argument

Matt J on 21 Feb 2014
Edited: Matt J on 21 Feb 2014
You could use curve fitters from other toolboxes as the others have been telling you. On the other hand, the goodness of fit measures that they return are rather simple functions of the output residual vector and Jacobian that lsqcurvefit does give you. You could easily calculate the goodness of fit metrics yourself.
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Matt J on 21 Nov 2015
Edited: Matt J on 27 Nov 2015
Hi Kate,
The usual formulas for goodness of fit do not apply to constrained problems. However, I assume your constraints are simple bounds LB<=x<=UB, since those are the only kind that lsqcurvefit supports. One can usually re-parametrize such problems as unconstrained e.g., by the change of variables x-->y,
x=(UB-LB)*(sin(y)+1)/2 - LB
Note that x, as given in terms of y above, is inherently bounded to [LB, UB] while y can vary freely.
I would recommend that you solve the problem first in terms of the natural variable x using lsqcurvefit and explicit bounds. Then, re-pose the problem as unconstrained and use nlinfit to obtain goodness of fit output. Because you already know the solution from lsqcurvefit, you can simply pass that solution as the beta0 input to nlinfit, and it will finish in 1 iteration or so, spitting out goodness of fit output with it. Of course, the goodness of fit will be in terms of the new unconstrained variables y. You will have to translate that through the change of variables formula to get things in terms of x.