Quantifying Uncertainty in Ecosystem Studies

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### Propagate both the PI and the CI?

Propagate both the PI and the CI
From Isabelle Auger, the statistician in Canada via Sylvie Tremblay (Oct 21, 2011)This email to Ruth Yanai includes references to the equations in the Ecosystems paper by Yanai et al. (2010) (available from the QUEST web site or from Ecosystems, open access) and to the Excel sheet in which the calculations were made, also available under “Sample Code.”

Applying the mean error of ŷ at a specified value of x (sm, eq. 5) to each tree is equivalent to estimate the error of the equation.
However, 1) it underestimates the error of the equation, and 2) it doesn't include the error of a prediction.

1) it underestimates the error of the equation
Isabelle calculated the error of an equation, developed with 853 observations, by two ways: 1) with sm, like you, and 2) with the error of the parameters equation. She concluded that using sm underestimates the error of the equation, because the error of the parameters is function of MSE/fct(X) (compare to MSE/n). The same conclusion was obtained with a smaller sample (n=30).

2) it doesn't include the error of a prediction
Once the error of the equation is estimated, it remains to estimate the prediction error for each tree. This error is small when the number of estimated trees is large, because the errors tend to cancel out, but it can also be larger when there is only few trees per plot.

Isabelle thinks that the way you calculate the error in your Excel sheet mixed up those two sources of error. In an equation, there is two sources of error: one from the parameters (constant error for all trees) et one from the residual error (different for each tree). The first one should be simulated from a normal multivariate distribution with the parameter errors and correlation between them. If this information is not available, an error from a N(0,MSE) could be use (constant for each tree), see 1). This error does not depend on the Xs. To this first error, a second one should be added, the one from the residual error. This error applies at the tree level, so depend on the Xs. This error should come from a N(0, Var), where Var = MSE*sqrt(1/n+(xi-xbar)^2/sum(xi-xbar)^2). In your Excel sheet, you used : prediction + N(0,MSE)*sqrt( 1/n+(xi-xbar)^2/sum(xi-xbar)^2). It’s strange to replace the sigma2 in a formula by a random number. The variance formula should be use to generate a random number. Isabelle think that the formula should be : prediction + N(0,MSE) + N(0,Var), where the random number from N(0,MSE) is the same for all tree and the one from N(0,Var) is different for each tree.

For these two reasons, Isabelle suggests that the error applied to each tree in a simulation should be: 1) a same equation error +  2) a different prediction error.

## 6/11/2020 10:46 am  #2

### Re: Propagate both the PI and the CI?

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