As far as I can tell, curvilinear is defined vaguely but means the same as nonlinear. Is that correct? Or does curvilinear have a distinct definition?
Solved – What does “curvilinear” mean
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With the usual definitions of linear and nonlinear with regard to modelling, it's not linearity with respect to the predictors that's the critical aspect, but linearity with respect to the parameters. A nonlinear model is nonlinear because it's not linear in parameters.
For example, the first sentence here says:
In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables.
By contrast, Generalized Linear Models generally have a nonlinear relationship between response and predictors, but the link-transformed mean response (the linear predictor, $\eta$) is linear in the parameters.
[By that definition, I believe your model is nonlinear in the $\theta$s, though if the $\theta$s are specified (known) then that nonlinearity isn't relevant to estimation. If they're being fitted, then the model is nonlinear.]
The Hessian is the matrix of second derivatives of the objective function you are attempting to minimize (resp. maximize, depending on how SAS set this up). The Hessian is a square $k \times k$ matrix, where $k$ is the number of parameters in your model. In your case, the Hessian is singular, which means that your parameters are linear functions of each other (or almost collinear) ... either absolutely, or with respect to the data that you have.
A young colleague of mine once got the exact error you did in SAS. Turns out he had included an intercept term twice. So the first thing you want to do is to check the specification of the model in SAS.
Another source of difficulty will occur if one of the parameters is basically useless at the data points you have. With a large sample and a good model, the log likelihood should look like a paraboloid near the minimum. However, a useless parameter can produce a long, narrow valley. Any solution in the trough of the valley will produce much the same minimum in the objective function. Your algorithm will terminate normally (it will no longer be able to improve itself by taking additional steps), but the Hessian will be singular.
If you don't have loads of parameters, (and you shouldn't with a non-linear model) you could try plotting the likelihood in the vicinity of the "solution" that SAS came up with, to see where the problem lies.
The Hessian is used to estimate the standard errors of the maximum likelihood estimates, so if the Hessian is broken, you won't get complete entries in the correlation matrix. These are obtained from the inverse of the Hessian, which you can't have, since the Hessian is singular.
I'm not sure that you converged to a minimum. With a singular Hessian, you could have converged to a saddlepoint in the parameter space, or the optimizer may simply have become disorientated and quit in the naïve belief that it had accomplished its mission. Something about your model is not working: either the way you specified it to SAS, or the data you have, or the model you are attempting to fit. Trust nothing in the results you got and start over. Perhaps try a simpler, linear model and then add more complicated features to improve the fit.
Best Answer
"Nonlinear" has many meanings, only some of which are (directly) about curves. I would say that I have encountered "curvilinear" to mean smooth curves. So a parabola or a logarithmic curve are "curvilinear," but a bent line (e.g. from a simple threshold or saturation model, "broken stick" model, etc.) are not.
Caveat emptor: word use will vary by context. For example straight lines are themselves a kind of "curve" in some contexts. As always, if there is a specific usage of the word "curvilinear" that you are wondering about, a quote and citation or two would be helpful.