Hi all,
I would like to estimated the elevation at which species richness is expected to be maximal.
To this, I used a quadratic relationship, y = a*x^2 + b*x + c
Where y is the species richness and x the elevation
Thus, if a < 0, the species richness is expected to be maximal at X=x[y=max(y)]=-b/2a
Now I would like to know what is the incertitude on X=x[y=max(y)]? i.e. the standard deviation of X, SD(X)
Somebody have an idea? I work with R.
Best,
Thomas
You get it through error-propagation from the standard errors of the coefficients (a, b), see https://en.wikipedia.org/wiki/Propagation_of_uncertainty
Or you use bootstrapping (more straight.foreward but reqiuires a reasonable sample size [say, n>20 or so] to be useful):
Use the residual variance from the data to generate new data. Fit the model to the new data and calculate the x-coordinate if the maximum (X). Repeat this many times and save the results. Finally, the standard deviation of the results is an estimate of the standard error of X, and the range from the 0.025-quantile to the 0.975-quantile is the 95% confidence interval for X.
Dear Jochen,
Thank for answer.
I did it using propagation of uncertainity using the "delta method".
Best
By the way, if you'd like to use a more flexible model (without restricting to a quadratic regression function), you may try convex regression estimation (assuming that you have relatively large sample size). E.g.,assume Y = m(x) + e, where m(.) is a strictly convex function and e denotes errors with zero means. You can first estimate the function m(x) and then obtain x that maximizes estimated m(x). Next, to obtain the uncertainty of argument of x that maximizes m(x), you can follow the bootstrap method (as explained above but fitting convex regression each time).
Here is a paper that provides the details of estimating a convex (and other shape restricted) regression curves. The method is relatively easy to implement in R. Contact me if you need codes.
http://www.sciencedirect.com/science/article/pii/S0167947312001004
Article Shape restricted nonparametric regression with Bernstein polynomials
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