# Choosing the right regression and validating it

I have two questions:

1. Which regression should I choose and why?

Based on the R squared value, exponential regression seems to be a better fit. But I am not really sure, if I should just go with it because it has a higher R squared value?

2. How do I validate it?

Lets say I decided to go with the linear regression (despite its smaller R squared value.) The only process I am following is to manually calculate the residuals and fit a histogram to check if they are normally distributed. And apparently they are OK in this case.

However, my concern is that some values (red circled ones) don't seem to be fitting well since their error is too high. But still, I suspect that they do not undermine R squared value, because are symmetric to each other. So it will probably add a bias?

So what is to best way to validate my choice of regression against such cases (besides high R squared value and normality check.) And are there other biases that I should look for?

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First of all, looking at the graph I don't think it is possible to distinguish exponential growth from linear reliably. Sure, you can run a battery of tests and come to some conclusion, but you'd be fooling yourself. Based solely on data it's impossible. Here, you need to bring in the information which is not in the data: e.g., if this bacteria growth, I'd imagine it has to be exponential.

Secondly, if you still want to get some hints from data itself, then I'd look at the variance - whether it's constant or changing with $x$. The distinguishing feature of exponential growth models is usually the variance increase: $\ln y(x) = \beta_o+\beta_x x+\varepsilon_x$, notice that ${\rm var}[y(x)]$ increases with $x$.

You can run a test like Engle's heteroscedasticity test, or plot the square of your regression residuals vs $x$.

UPDATE: The simplest and intuitive test I'd suggest is to regress the regress the squared residuals on $x$. The matter is that your linear regression residuals are guaranteed to have zero mean and conditional covariance with $x$. So if you regress the squared residuals with $x$ and see a significant model, then maybe variance is growing.

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In terms of validating the statistical tool I am using (regression,) I think your last paragraph addresses the question. The right test and its value? there are many tests listed through a google search, but the hard part is to select the right one. and why? – uha1 Apr 8 '14 at 18:54
which tool are you using? i'd start with a plot of squared residuals, and then go from there. there isn't "the right test" as you probably figured by now. it all depends on the context – Aksakal Apr 8 '14 at 18:56
Would you mind using a capital letter at the beginning of your sentences & capitalizing I (1st person singular)? We want to build a permanent record of high-quality statistical information & proper grammar / typesetting is part of it (even if secondary). – gung Apr 8 '14 at 19:44
@gung, I don't usually capitalize in writing, but will do it here – Aksakal Apr 8 '14 at 20:28