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I am trying to find the best predictor for Leaf Area Index (LAI, a plant growth indicator) among several spectral indices (these are calculated from reflectances measured in different spectral wave lengths). I am using exponential model like this: $y=a \exp(b x)$. Two of the spectral indices gave me relatively good results but I do not know which is better.

The problem is that the first spectral index has smaller residual standard error but the $a$ coefficient is insignificant; the second spectral index has significant $a$ but the residual standard error is higher. For both models $b$ is significant. Graphs of the two model fits are shown below. As I have some very low LAI measurements I think that it is not unusual to have “a” that is not significantly different of zero.

So my question is should I reject (in this situation) the model with the lower residual standard error just because the $a$ coefficient is not significant.

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So you are comparing two models of the same algebraic form, the only difference is the two different predictors. You don't say how they were estimated, I will assume nonlinear least squares.

Why do you think it is a problem that the $a$ coefficient is not statistically significant? It is of course practically significant, since $a=0$ for this model implies $y=0$ (or, with an additive error term, $y=\text{pure noise}$). So a hypothesis test of $H_0\colon a=0$ is practically meaningless!

A glance at your two plots is enough: The first plot has clearly better fit (and lower residual standard error), the second plot has a terrible fit at the lower left part, close to zero. In that region it will give systematically biased (to high) predictions, so you should prefer the first model.

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    $\begingroup$ I am using nonlinear least squares indeed (the nls function in R). Thanks for the answer. $\endgroup$ – ABC Nov 28 '18 at 11:40

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