When to choose nls() over loess()?

Question

If I have some (x,y) data, I can easily draw straight-line through it, e.g.

f=glm(y~x)
plot(x,y)
lines(x,f$fitted.values)

But for curvy dat

Answer 1

loess() is non-parametric, meaning you don't get a set of coefficients you can use later - it's not a model, just a fit line. nls() will give you coefficients you could use to build an equation and predict values with a different but similar data set - you can create a model with nls().

Answer 2

Nonlinear-least squares is a means of fitting a model that is non-linear in the parameters. By fitting a model, I mean there is some a priori specified form for the relationship between the response and the covariates, with some unknown parameters that are to be estimated. As the model is non-linear in these parameters NLS is a means to estimate values for those coefficients by minimising a least-squares criterion in an iterative fashion.

LOESS was developed as a means of smoothing scatterplots. It has a very less well defined concept of a "model" that is fitted (IIRC there is no "model"). LOESS works by trying to identify pattern in the relationship between response and covariates without the user having to specify what form that relationship is. LOESS works out the relationship from the data themselves.

These are two fundamentally different ideas. If you know the data should follow a particular model then you should fit that model using NLS. You could always compare the two fits (NLS vs LOESS) to see if there is systematic variation from the presumed model etc - but that would show up in the NLS residuals.

Instead of LOESS, you might consider Generalized Additive Models (GAMs) fitted via gam() in recommended package mgcv. These models can be viewed as a penalised regression problem but allow for the fitted smooth functions to be estimated from the data like they are in LOESS. GAM extends GLM to allow smooth, arbitrary functions of covariates.