You must calculate the Adjusted Coefficient Of Determination (R2-adj) and Prediction Coefficient Of Determination (R2-pred). If you use Minitab, these statistical factors are shown.

R2 adjusted is the percentage of response variable variation that is explained by its relationship with one or more predictor variables, adjusted for the number of predictors in the model. This adjustment is important because the R2 for any model will always increase when a new term is added. A model with more terms can seem to have a better fit because it has more terms. However, some increases in R2 might be because of chance alone.

Use R2 adjusted to determine how well the model fits your data when you want to adjust for the number of predictors in the model. The adjusted R2 increases only if the new term improves the model more than would be expected by chance. It decreases when a predictor improves the model less than expected by chance.

Use predicted R2 to determine how well the model predicts responses for new observations. Larger values of predicted R2 indicate models of greater predictive ability.Predicted R2 is calculated by systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. Predicted R2 ranges between 0 and 100% and is calculated from the PRESS statistic. Predicted R2 can prevent over-fitting the model and can be more useful than adjusted R2 for comparing models because it is calculated using observations not included in model estimation. Over-fitting refers to models that seem to explain the relationship between the predictor and response variables for the data set used for model calculation but fail to provide valid predictions for new observations

http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/goodness-of-fit-statistics/r-squared/

Assuming that you are fitting the non-linear model by Least Squares i.e. minimizing the sum of the squared residuals, then I would say that R2 is equally informative in the non-linear case as in the linear. After all, R2 describes the ratio between the model explained variation in the response (dependent) variable and the variation in the residuals.

@ Kenneth, thanks for the info. Is there any other parameter that I can use apart from R^2 ?

Please, access our papers about non linear models  and the best response you can find in the References like Bate and Watts, Ratkowsky and so on.

Good luck,

jwalter

Not sure what you mean by efficient. In statistics efficiency relates to the information matrix and tells one how capable an estimator is in exploiting the information provided by the data for estimating an unknown parameter. R2 is related to the F-statistic which can be used for simulateously test whether the regression coefficients all equal to zero (or some other value if you wish).

@ Jose Walter, thanks for the update. I would go through the mentioned papers.

@ Kenneth, by "efficient" i mean.... if for a set of data I fitted a polynomial which gives R^2 close to 0.99, but even then if a back substitute these data points into the polynomial and calculate the error. Error seams significant, this happened with me specially when I use Microsoft Excel for curve fitting. Since then I started using other software.

Thanks!

When the R2 is lower than 0.99, the most non linear parameter has bias and the estimate has no meaning for your data. The best temperature for a chemical reaction estimate in 26 is in fact 29 oC, for example. The linear parameter has goodness of fit with reliability.

Pay attention in the sample size, the number and the interval between the explanatory variables, and no more than 3 parameters in the model.

I suggest the confidence interval for the parameter when you do not have the routine for the goodness of fit, but you can fit these data using the SAS Software.

See the responses that I find in the papers after fitting the parameters of the Weibull distribution.

Good luck.

jwalter

A moment! We are talking about non-linearity in the regression model, not in the parameters, right? Like y=b*f(x)+e when it is not required that f(x)=x. If so I maintain my opinion on R2. Provided that f(x) are correctly specified, then LS is giving unbiased estimates, irrespective of the value of R2. If one uses some non-parametric approach to estimate f(x) then it will typically be biased as the scope usually is not minimize MSE in order to trade between biasedness and precision. But agian, this is unrelated to R2.

@Jose Walter, thanks for the info.

@ Kenneth, thanks for the info.

Yes we are taking about non-linearity in the regression model.

As for alternative goodness-of-fit measures, you could use some of the information criteria like AIC and BIC. Yet another alternative is cross-validation where you fit the model on a subset of your observations and then check how well the model predicts the values in the non-used subset of observations. However, in my view R2 is superior to these alternatives. 

@ Kenneth, thanks for the info.

It was really a great help to me, appreciated.

thanks.

Rajesh.

You may use Wilmott's index of agreement.

@ Naveen, depending on the implemtation of Wilmott's index it is possible to get the R2. If one argues that squared residuals would be inappropriate and therefore prefer to use absolute errors or the like, then one should also estimate the model according to the same principle.

@ Naveen, thanks for the info. I will dig deep on Wilmott's index.

I think the main issue here is that you aren't really talking about a non-linear model, or at least not unless we use "non-linear" in a very restricted sense. I know that this has been addressed somewhat in Kenneth's responses, but I think the terminological issue here is important.

A polynomial model such as Yhat = B0 + B1X + B2X2, is a linear model, at least in the sense that the term "linear model" is usually used in statistics. By that I mean that in this model, the predicted values of Y are a linear function of the parameters (B0, B1 and B2). We describe such a model as linear in the parameters.

A model like this can still be fit using OLS, R2 is still relevant, and from a statistical perspective it's no different than any other linear regression. If you want to communicate that the model includes non-linear relationships between the predictors and the response variable, you can describe it as non-linear in the variables.

The R^2 calculation for nonlinear regression models can very a lot depending on the software / application / etc. Also, they involve a lot of approximations in order to make any calculation and are not as easily interpretable as the linear case. I really think that in any application where you (or anyone else) may think that any of the approximations that go into developing the R^2 calculation are off, don't use it (and if you don't know how it's calculated, don't use it!). There are many other methods that will give you a good and very interpretable result directly from your data. For example, some kind of cross-validation like K-fold cross-validation or bootstrap methods give you a good sense of how trustworthy your estimates are directly from your data without relying on a bunch of normal approximations. Lastly, for most applications I would also recommend using some goodness of fit measure that takes into account parsimony like the AIC or BIC.

I also want to reiterate Matt Williams' comment. If it's a linear regression (linear in the parameters), then yes the R^2 is fine and is interpretable in the commonly thought of way.

@Matt, thanks for the info. So I got it wrong. Now it makes sense. 

What does non-linear model actually means, if polynomial is a linear model ?

Thanks !

@ Chris, thanks for the info.

@Rajesh, a non-linear model is one that cannot be linearized by any kind of transformations. Take for example next logistic model:

y~1/(1+Ae-Bx)

It seems nonlinear, OK? But look carefully:

y=1/(1+Ae-Bx) => 1/y=1+Ae-Bx 

=>1/y-1=Ae-Bx =>ln(1/y-1)=lnA -Bx

so if you take the new variable

z=ln(1/y-1) =ln((1-y)/y)

then you have the linearised version

z=(lnA)+(-B)*x

and you can find using OLS the b0=lnA, b1=-B.

Did you understand it?

@ Demetris, thanks for the info.

Ya, I understood. Now it makes sense. Thanks a lot...

Regards,

Rajesh.

You must calculate the Adjusted Coefficient Of Determination (R2-adj) and Prediction Coefficient Of Determination (R2-pred). If you use Minitab, these statistical factors are shown.

R2 adjusted is the percentage of response variable variation that is explained by its relationship with one or more predictor variables, adjusted for the number of predictors in the model. This adjustment is important because the R2 for any model will always increase when a new term is added. A model with more terms can seem to have a better fit because it has more terms. However, some increases in R2 might be because of chance alone.

Use R2 adjusted to determine how well the model fits your data when you want to adjust for the number of predictors in the model. The adjusted R2 increases only if the new term improves the model more than would be expected by chance. It decreases when a predictor improves the model less than expected by chance.

Use predicted R2 to determine how well the model predicts responses for new observations. Larger values of predicted R2 indicate models of greater predictive ability.Predicted R2 is calculated by systematically removing each observation from the data set, estimating the regression equation, and determining how well the model predicts the removed observation. Predicted R2 ranges between 0 and 100% and is calculated from the PRESS statistic. Predicted R2 can prevent over-fitting the model and can be more useful than adjusted R2 for comparing models because it is calculated using observations not included in model estimation. Over-fitting refers to models that seem to explain the relationship between the predictor and response variables for the data set used for model calculation but fail to provide valid predictions for new observations

http://support.minitab.com/en-us/minitab/17/topic-library/modeling-statistics/regression-and-correlation/goodness-of-fit-statistics/r-squared/

@ Arash, thanks for the info, it is really a great help. I go through the the link that you provided. 

Thanks, appreciated.

Regards,

Rajesh.

Check if  R^2  is greater than the Durban-Watson  (D-W) value .

@ Okafor, thanks for the info. I will do that and get back to you.

thanks & Regards,

Rajesh.

@ Lucky, thanks for the info.

R2 is good for cases where you have no model for the noise in your measurement. A good way to think of it is just as a measure of the fraction of the variance in the measurement that's covered by your fit model. But it's not necessarily what you want for cases where you're concerned you might be overfitting your data (i.e. using a fit function with more parameters than you can justify) or where the error bars on different data points are very different from one another (as happens, for example, when Poisson noise dominates). Chi2, and especially reduced chi2 (i.e. chi2 per degree of freedom) are probably more appropriate for these cases.

In chi2 curve fitting you're basically saying: Here's my data, here are the error bars on each data point, and here's a parametrized fit function. After fitting, are the residuals consistent with random noise?

If, after fitting, you've brought your reduced chi2 down to a value statistically consistent with 1 (and there are ways to calculate confidence intervals for that; read Bevington's "Data Reduction and Error Analysis for the Physical Sciences" for the full story), then you've extracted pretty much all the non-noise information you can. You can also use this for quantitatively comparing two fit functions, for example to see whether it's justified to add one or two more terms to your fit function. You want to use the smallest number of parameters that still reduces the residuals to random noise.

A common example of a nonlinear fit function is a Gaussian where one of the parameters is a constant background offset. Without the background, you can take the logarithm and it turns into a parabola, which as others have noted would be linear in the parameters. But when you include the background, then you lose that property. Warning: chi2 curve fitting in this case (as well as many others) is known to give a biased result, specifically in the amplitude of the peak. Again, Bevington's book covers this.