# Hypothesis Tests for Non-linear least squares coefficients

I have a set of results that take the following form:

σ       Standard error
0,06    0,50
0,91    >10
0,04    0,11
0,84    0,56
0,44    0,10
0,03    0,09
0,30    >10
0,19    0,07
0,04    1,15


Where σ is a coefficient in a non-linear model, which I've estimated directly using an iterative algorithm (BFGS).

If I want to test a hypothesis of the type σ=0, which test would be appropriate? We haven't covered non-linear optimization methods in my classes, and I'm really at loss.

Hypothesis testing and confidence intervals on non-linear regression models are based on asymptotic theory. To construct a test, you may extract the diagonal element of the covariance matrix that corresponds to your estimate and as usual form the t-ratio. You can then use the critical values of the standard normal distribution to reach a decision.

One warning is in order though. As these tests are based on large sample theory, it's not prudent to pay much attention to them when your sample is quite small. There is no clear guideline as to "how large is sufficiently large" but personally if it's less than 50-60 observations, I would be very suspicious.

Hope this helps.

• With 9 parameters, I would be very suspicious even w/ 30, eg. A rule of thumb w/ linear models is 10 residual df per parameter to be estimated to avoid approaching saturation (technically a different issue). Given that there are likely to be additional issues / complexity here, N=20-25 seems too small to rely on asymptotics. Commented May 25, 2015 at 19:10
• @gung You are right, I thought it was just one parameter for some reason. Commented May 25, 2015 at 19:12
• I probably misread half of the questions here. This is still a decent answer, you might just bump up the suggested minimum N. Commented May 25, 2015 at 19:18

It seems you have 9 estimates sigma, all of which are positive. The probability of this, calculated using the sign test is so low if its true value is zero, that you may deduce it is non-zero. Or did I misunderstand something? JOHN BIBBY

A likelihood ratio test is pretty typical for these kinds of problems. To test the hypothesis $H_0: \gamma = 0$ vs $H_1: \gamma \ne 0$, you would fit the model without $\gamma$, then fit it with $\gamma$. The test statistic is $\Lambda = -2\ln(L_0/L_1)$, where $L_1$ is the likelihood of the null model (the one without $\gamma$) and $L_0$ is the likelihood of the alternative model (the one with $\gamma$).

You reject $H_0$ if $\Lambda > \chi^2_{df=1}$ and fail to reject otherwise, assuming a reasonably large sample size. Note that this explanation of a likelihood ratio test is specifically for if you're testing a single coefficient. If you're doing it for multiple coefficients, your critical value is a different $\chi^2$ value.