Pitfalls in Fitting Nonlinear Models by Transforming to Linearity

Question

Some nonlinear models can be transform to linear models. My understanding is that there might be one-to-one relationship between the estimates of nonlinear model and its linear model form but their corresponding standard errors are not related to each other. Is this assertion true? Are there any pitfalls in fitting Nonlinear Models by transforming to linearity? Thanks in advance for your help.

Answer 1

Some nonlinear models can be transformed into linear models.

Usually only by ignoring the error term, unless by some amazing chance it enters in just such a way that the error term becomes additive after transforming.

My understanding is that there might be a one-to-one relationship between the estimates of the nonlinear model and its linear model form

Not in general. For example, consider:

$y = \exp(\alpha+\beta x)+e$

$\log(y) = \alpha+\beta x+\eta$

where in each case the model had the variance of the error term constant.

If you fit both models via least squares to the same data (nonlinear least squares and linear least squares), the parameter estimates will differ.

but their corresponding standard errors are not related to each other.

Yes, if we put "and" rather than "but".

Here's an example in R (the model isn't really suitable, but that doesn't change the issue):

> carsfit=lm(log(dist)~speed,cars)
> summary(carsfit)

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  1.67612    0.19614   8.546 3.34e-11 
speed        0.12077    0.01206  10.015 2.41e-13 

Residual standard error: 0.4463 on 48 degrees of freedom
Multiple R-squared:  0.6763,    Adjusted R-squared:  0.6696 
F-statistic: 100.3 on 1 and 48 DF,  p-value: 2.413e-13

(some unnecessary output removed)

vs:

> carsexp=nls(dist~exp(a+b*speed),data=cars,start=list(a=1.67,b=0.12))
> summary(carsexp)

Formula: dist ~ exp(a + b * speed)

Parameters:
  Estimate Std. Error t value Pr(>|t|)    
a  2.24119    0.20815  10.767 2.13e-14 ***
b  0.09168    0.01028   8.917 9.38e-12 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.07 on 48 degrees of freedom

Number of iterations to convergence: 4 
Achieved convergence tolerance: 3.598e-07

Sometimes it really matters which model you choose!

Are there any pitfalls in fitting Nonlinear Models by transforming to linearity?

Well, possibly quite a few, depending on how broad this question is -- answering it might fill a book. Once you're clear about what your error term is, it's usually clearer whether (and how) models should transform.

Clearly, at least, it potentially makes a large difference to parameter estimates.

Secondly, consider if the additive error were large enough that some of the smaller observations could be negative;

clearly we can't just take logs.