ANCOVA that specifies a curvilinear relationship between covariate and response?

Are you able to run ANCOVA's that specify a curvilinear/nonlinear relationship between the covariate and response variable? If so, what are these models called? Specifically, I'm interested in running a a model that involves a blocking term, a few covariates, and a response that has a negative binomial distribution.
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Not a robit
You can use polynomials in your model or spline terms when there is a non-linear relationship - depending on the its form.
After doing a bit more reading, I think I want to run a zero-inflated negative binomial GLM. Unfortunately, I have very little experience with these models and have not been able to find much online or in textbooks. Could anyone point me in the direction of resources that would help? Or would anyone be willing to walk me through the process? Interested in how to select appropriate estimation and validation methods as well as how to know I'm getting a good fit.
I have not unfortunately. I will be using JMP Pro 13 although I also have SPSS (with the downloaded STATS ZEROINFL extension installed).


Not a robit
I find it hard to believe you cannot find online resources - its the world wide web! You don't necessarily need to find software specific resources, any lists of assumptions should be of interest.

Do you code in JMP, is that an option. I regularly you SAS, so I may be able to help. Though, I don't run much count data, so my memory is somewhat hazy.

If I remember correctly, you first look at the distribution of the outcome and examine mean versus variance. There are some dispersion tests you can run. You can also overlay known distribution over distribution of outcome data to look for good fits. You can also take into account if data are say 0 truncated or zero inflated. Also if the mean is high, a linear regression can be used some times or if outcome rear you can sometimes use logistic regression.