# Pseudo-$R^2$: what are the null models for linear and non-linear regressions?

I have data from an experiment. The independent variable is time, the dependent variable is mass loss of organic matter. Now I want to compare whether a linear or a non-linear model fits better. From my understanding the $R^2$ (adjusted or not) should not be used and instead I should go for pseudo-$R^2$ measures (e.g. McFadden, Nagelkerke). Is this true? If so I don't understand how to get the null models for the linear and non-linear model which are needed to calculate the McFadden Pseudo-$R^2$.

I am doing all statistics in R and I used the nls function for the non-linear and the lm function for the linear model. Now, I wanted to implement the Pseudo-$R^2$ by McFadden (1-(LL1/LL0)) but I am stuck in finding the null models.

Can someone please explain me the best way to compare a linear and a non-linear model? Any search on the web did not yield a solution.

Any direct help or linkage to other websites/posts is greatly appreciated.

Thanks in advance and have a nice weekend, Stefan

• It depends, what did you read specifically about $R^2$ that makes it a non-ideal indication of fit in your case? – Robert Kubrick May 11 '12 at 13:41

## 1 Answer

Perhaps I'm misunderstanding the situation, but I suspect much needs to be rethought here. First, metrics like McFadden's pseudo-$R^2$ are typically used with generalized linear models, so I'm guessing that's what you mean by "non-linear regression". But the choice between linear regression and a GLM (like logistic regression), is not based on model fit, but statistical theory. If your response variable is continuous and approximately normally distributed, then OLS (i.e., standard) regression is in order. On the other hand, if your response variable is the outcome of a Bernoulli trial (e.g., a success or failure), then logistic regression is appropriate. There are a variety of other GLMs, as well. Other GLMs include: multinomial logistic regression (when the response is one of a set of mutually-exclusive categories), Poisson regression (for counts), or beta regression (for continuous proportions, which may be appropriate here). In any case, the model is selected based on its appropriateness to the situation, not based on fit. A good introduction to these issues is Agresti's Introduction to Categorical Data Analysis, although it doesn't cover beta regression. This paper provides a basic introduction to BR and how it can be applied in R. Thus, my first point is that you don't compare linear and non-linear regression models in the manner you appear to be thinking of.

In addition, you should know that many people think that pseudo-$R^2$'s are not very good measures of model quality, and that this can even be true of regular $R^2$. It may be worth your while to read up on these issues and get a better sense of what they are and their pros and cons before you use them. Here is a good overview of pseudo-$R^2$. Here is a good discussion of their merits on CV, and here is another CV discussion of the merits of regular $R^2$. Moreover, statisticians tend to think these metrics should not be used for model comparison, but that information criteria, like the AIC, should be used instead.