non-linear-model-vs-non-linear-relationship
Table of Contents
1 non-linear-relationships-vs-non-linear-models
- I value precise language very highly
- this is because in multi-disciplinary teams it is easy to talk using the same words and mean different things
- in recent discussion about Distributed Lag Non-linear Models I started to reflect on something that has bothered me for a While
- back in 2005 my old mate Prof Keith Dear picked me up on using the term "non-linear model" incorrectly and explained the maths…
- I kind of understood but promptly forgot and found a lot of people use the term non-linear model a bit carelessly
- Yesterday I was in a discussion about comparing non-linear relationships between different studies in a meta-analysis
- I immediatly felt uncomfortable when we started to discuss these as "non-linear models"
- so here is a quick bit of google fu (with a session at the coffee shop with Steve and Mishka) to remind me about the difference between
1.1 Nonlinear Regression vs. Linear Regression
- the following comes from
- http://www.ats.ucla.edu/stat/sas/library/SASNLin_os.htm
- verbatim except for my attempt at mathjax notation in latex
A regression model is called nonlinear, if the derivatives of the model with respect to the model parameters depends on one or more parameters. This definition is essential to distinguish nonlinear from curvilinear regression. A regression model is not necessarily nonlinear if the graphed regression trend is curved. A polynomial model such as this:
1.1.1 Model 1
\(Y_{i} = \beta_{0} + \beta_{1} X_{i} + \beta_{2} X_{i}^2 + \epsilon_{i}\)
- appears curved when y is plotted against x. It is, however, not a nonlinear model. To see this, take derivatives of y with respect to the parameters b0, b1
- dy/db0 = 1
- dy/db1 = x
- dy/db2 = x2
- None of these derivatives depends on a model parameter, the model is linear. In contrast, consider the log-logistic model
1.1.2 Model 2
\(Y_{i} = d + (a - d)/(1 + e^{b \times log(x/g)}) + \epsilon\)
- Take derivatives with respect to d, for example:
- The derivative involves other parameters, hence the model is nonlinear.
1.2 Conclusions
- It is probably best to refer to the polynomial as a "non-linear relationship" in a linear model
- reserving "non-linear model" for things like Model 2
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