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# Non-linear Regression

Sometimes we have a mechanistic model for the relationship between y and x, and we want to estimate the parameters and standard errors of the parameters of a specific non-linear equation from data. Some frequently used non-linear models are shown in Table 20.1. What we mean in this case by ‘non-linear’ is not that the relationship is curved (it was curved in the case of polynomial regressions, but these were linear models), but that the relationship cannot be linearized by transformation of the response variable or the explanatory variable (or both). Here is an example: it shows jaw bone length as a function of age in deer. Theory indicates that the relationship is an asymptotic exponential with three parameters:

In R, the main difference between linear models and non-linear models is that we have to tell R the exact nature of the equation as part of the model formula when we use non-linear modelling. In place of lm we write nls (this stands for ‘non-linear least squares’). Then, instead of y~x, we write y~a-b*exp(-c*x) to spell out the precise nonlinear model we want R to fit to the data.

The slightly tedious thing is that R requires us to specify initial guesses for the values of the parameters a, b and c (note, however, that some common non-linear models have ‘self-starting’ versions in R which bypass this step; see p. 675). Let's plot the data to work ...

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