What type of non-linear regression model are you talking about?

Suppose that you have Y=aExp(cX), you can use non-linear regression modeling to determine 'a' and 'c' You can take the Ln of both sides and get Ln(Y) = Ln(a) + cX and use linear regression modeling on it. 

Andrew's answer should be applicable. Most biological phenomena are hypothesized to follow a simple linear trend.

thanks for the reply.. but i am afraid that will help me as my equation is of the form

Y=  b1* X1 ^ a1 + b2* X2 ^ a2 + b3* X3 ^ a3 + c

where c is a constant term.

I was trying to estimate the all parameters ai and bi

You must have used brute force to estimate the coefficients/parameters in that equation. In simple linear regression analysis using the least squares method, one way of estimating the regression coefficients, assumptions for y ~ niid apply for you test the significance of the coefficients. Therefore I see the t-test as valid in your case as long as the assumptions of niid hold for your y. I think.

For Luca: it is hard to say it estimates of the parameters would be biased or not by taking the log without knowing the model for the error, which is not given.

If the complete model is Y = a * exp( b X ) + epsilon, with epsilon Gaussien iid, you're right.

If the complete model is Y = a * exp( b X ) * epsilon with epsilon log-normal iif, you're wrong, taking the log is indeed a correct way to proceed.

If error model is anything else, it may be quite unclear if any of the two give unbiased estimates...

In addition, this is true if the minimization uses least-squares, but if other methods are used, things are again different. Maximum likelihood approachs, for instance, can give things quite different than least-squares, as robust methods

Both remarks also apply to the original model of Mithun.

So, in short, to test significance of your parameters, the most suited method will also depend on the choice of your error term. Nested models can be more suited than T-tests in your case, provided you use maximum likelyhood and probably do not use the khi-square law, but do some bootstrap.

I think it depends on which estimation methods you use. Usually, for nonlinear regression, maximum likelihood estimation (MLE) is usually chosen. MLE is asymptotical to follow a normal distribution. Based on this theory, you can use t-test to test the significance of each parameter if your sample size is big. AIC and BIC can also help you to choose which model is better.

@ Hai-Yan: Just take care that asymptot can be far away (not to say if your guess about the likelihood form is wrong)... Drawing contour plots or comparing profiled confidence intervals with Gaussian ones may help to detect such problems

There should be some software that can estimate your coefficients. I'm sure R has some function for this.

You should be able to use the "Optimization" function in Excel to get some estimates for your variables. You will need to create a Y(predicted) from you model. I would start off with  Y= 1* X1 ^ 1 + 1* X2 ^1 + 1* X3 ^1 + 1 and plug in your X1 to X3 values. Then set up a column with Y(actual) with your Y values for each X1 to X3 variables. Then create a third column (Y(pred) - Y(actual))^2. Sum this column and create your Sum of Squares value. Then, tell Excel to minimize this value by manipulating ai, bi and c. 

Thanks once again for the replies... many of which seems a bit beyond my understanding.

@ Andrew, I have already estimated the coefficients using SPSS that employs Levenberg-Marquardt algorithm, but the issue that I am facing is on how to ascertain that my coefficients are significant as the result of SPSS doesn't provide those. The result obtained is having less standard error, signs of coefficients seems logical and R-square is good enough (though I am not sure if it is a good measure for non-linear regression)

Oh. OK. Are you able to show the SPSS output? The software should give you the info you want. If not, you can create the excel program. The sum of squares will be your SSE. You can go back and set each ai, bi and c to zero, one at a time, and look at the change in the SSE. This change should be the SSR for each term. Then you will have to calculate MSR/MSE and find an F-value.    

Kindly have a look at the attached output file.

Mithun Mohan:

1. If you post the output in a format everybody can open/read (e.g., *.txt) or just copy and paste, you'll get more answers and, hence, increase the likelihood of getting a useful answer. Many here do not use SPSS but other software such as Stata, R, etc. or might use a device that doesn't allow them to open special file formats. 

2. I am note sure whether I understood your question: What exactly do you want to know? What regression model are you estimating (OLS/linear regression, logit/probit model, ...)? In case you want to learn about the relationship of your dependent variable and the independent variable (the one with the supposed non-linear effect), first look at the statistical significance of the quadratic polynomial (i.e., the coefficient for the squared variable). Interpretation of statistical significance in general and p-values in particular varies not only from model to model but also from field to field. However, with a random sample, the following should hold in many fields for quadratic polynomials in a linear regression model (OLS/ML): If the coefficient for the quadratic polynomial is significant, you have evidence for a non-linear relationship of this variable and your dependent variable in the population, if not, you have evidence against the non-linearity. The exact form of the relationship is, obviously, a different question.

3. Yes: This t-value can be used to assess the coefficient's significance. All you need to know is against what distribution and, of course, what critical value you want to assess it (common situation: you have >100 degrees of freedom and, hence, use the standard normal distribution with a critical value of 1.96 for a two-sided test with alpha=.05). SPSS should (but I don't know for sure), however, also give you p-values, which inform you about the statistical significance of the coefficients given the level of statistical significance and the characteristics of the test (e.g., one-sided vs two-sided) you asked for.

Best,

Sebastian

Personally I prefer the likelihood ratio test. Presumably you estimate the model by Maximum Likelihood. Then take the ratio (logLikelihood of the estimated full model) divided by (logLikelihood of restricted model) and multiply by two. If the restricted model is true, then this ratio is approximately chi-squared with r degrees of freedom, so you get the p-value as the probability that such a variable be grater than your computed ratio. Here r is the number of restrictions, i.e., r=1 if you test for a single parameter, and log is the natural log, i.e., ln.

Cheers -- Harald

@Lang: in such a model, with exponents and coefficients as parameters, likelihood ratio test may well not follow a khi-square distribution, or its number of degrees of freedom difficult to assess, because of degenracy of the null hypothesis: if you hypothesize that the coefficient is 0, what means the corresponding exponent? How much parameters are you indeed removing? And for the exponent, if it is null, what does it mean? In addition, there is degeneracy: if the right model is, say, Y = 1 + 3 * X^2, and you fit Y = a + b * X^c + d*X^e, you can have c=e=2 and any combination of b+d = 3 that will work... All of this makes that LRT may be much more difficult to perform than expected, because not following the khi-square law... (models are not even really nested, I guess)

The output file in pdf form is attached. I am trying to model a non-linear relation between the dependent and three independent variables. The relation which I would like to try out is of the form

Y= b1* X1 ^ a1 + b2* X2 ^ a2 + b3* X3 ^ a3 + c

c being a constant value. I want to know if the coefficients in the equation are statistically significant or not 

Correction of my previous answer:

The chi-square statistic is computed as 2*logLikelihood of full model minus 2*logLikelihood of restricted model. Sorry for the confusion!

@Emmanuel Curis.

I suppose that the null hypothesis for the exponents are that they equal one, not zero. If they are zero, then you add an extra intercept and the model suffer from multicollinearity. If you want to do away with a factor, you set the coefficient in front equal to zero, not the exponent.

@Emmanuel Curis again

If a b-coefficient is set to zero, i.e., if that factor is taken away, you have two less parameters in the model, so r=2 in that case. I don't see a problem with that?

Why two ? 0 * X^b can work with any value of b, so your question is only on the first coefficient... So you can also say that you remove one parameter...

And why 1 for the exponent, and not 2 or -1 or... ?

Anyway, this is sufficient (not to speak about other concerns) to make the results about the khi-square law inapplicable without great caution.

@Emmanuel Curis

The df (dimensionality) is the number of free parameters, i.e., the number of parameters you estimate. The df of the chi-square statistic is the difference between the df for the full model and the restricted (null) model. So if you take away a term bXa , then you estimate two less parameters (b and a).

Cheers -- Harald

@Emmanuel Curis

And of course, you may chose 2 in the exponent or whatever as the null to test for. But zero is a bad choice.

Mithun - 

I think Luca's first two postings might be particularly helpful, with regard to your original question about t.  The warning about relationships among the coefficients may be particularly important. But having worked with hypothesis tests, I think it more practical for you just to consider that if a coefficient is less than twice its standard error, that might be the most practical warning you need that you may have a substantial problem.  As for p-values, they are functions of sample size.  That is a very undesirable property when it comes to decision-making.  

Nice discussion, all. 

Jim 

When I read James R Knaub’s comment, I had the following thought.

In a way, significance testing and p-values — are they really relevant in this conext? After all, you are not looking for the “true” model (if there is such a one, it presumably has another functional shape,) you are fitting an ad hoc model to your data.

I would consider skipping the significance tests and CI:s altogether and concentrate on effect size. Compute the η2 or Cohens f2, and skip those factors that have low effect sizes. This seems also to ba a trend in applied statistical work.

Cheers -- Harald

Hi Mithun -  GraphPad Prism is a dedicated nonlinear regression application and it has an amazing help section which is online and available for anyone. Here is the page discussing whether the best fit parameters are plausible and ways of ascertaining that. http://www.graphpad.com/guides/prism/6/curve-fitting/

Also this book has been very helpful to me with nonlinear regression model fits. It presents a number of methods to ascertain the goodness of fit of nonlinear models (not only for biology). Motulsky and Christopoulos 'Fitting models to biological data using linear and nonlinear regression: a practical guide to curve fitting'. Many pages are available on google books for preview.

cheers - Kate