# Error analysis in “linearized” regression

I am currently taking an experimental chemistry course where we need to fit data to an equation of the form $y=a\exp(bx)$. They recommend "linearizing" this equation by taking the logarithms of both sides to get $\log(y)=\log(a)+bx$, and then perform a linear regression.

This is all fine, and I can do the regression no problem, but I am wondering about statistical properties of the parameter estimates of $a$ and $b$. Being chemists and not statisticians, most of them don't actually know how to perform a linear regression and just use excel's built in feature, and they also just care about $R^2$ instead of using confidence intervals, hypothesis testing, or some other more suitable statistitical test. I would like to use better statistics but I need to make some assumptions and I'm not sure which ones.

Originally I wanted to include error as $y=a\exp(bx)+\epsilon$, but then I get $\log(y)=\log(a\exp(bx)+\epsilon)$ and I can't write it as a linear regression anymore. Alternatively, I could just assume $\log(y)=\log(a)+bx+\epsilon$ which would then require me to incorporate error as $y=a\exp(bx+\epsilon)$, which seems better. Then I could assume $\epsilon$ is normally distributed and carry on with linear regression by least squares as usual. Is this a valid way of incorporating error to get meaningful statistical properties? If not, is there a better way, potentially using some non-normal distribution for $\epsilon$ or incorporating it differently into the model?

## 2 Answers

This is not a direct answer to your question but I would like to point out a few things.

Your model y=a Exp[b x] is nonlinear with respect to any of the involved parameters. So, nonlinear regression is required. However, the problem with nonlinear regression is that you must start with "good" estimates of model parameters.

You can obtain reasonable estimates by linearization of the model but what you minimize is the sum of the squares on Log[y] while your problem is to minimize the sum of the squares on y, which is very different.

So, if you go through the two consecutive steps, linearized model to get estimates and then nonlinear model, I think that your problem disappears taking you back to the classical assumptions.

One of your options is to use functions like nls in the S (or R) library. Given 'sufficient' data you can get asymptotic estimates of the variances of estimates of the parameters under assumptions about distributions on the error in the data. This is a reference.