You can use the well-known Taylor series approximation to the variance of an arbitrary function:
Var( f(X) ) ~= Sum( s[i]^2*D2[i] ) + 2*Sum( Sum( s[i,j]*D[i]*D[j] ) ) where D2[i] is the second partial derivative of f(x) with respect to the ith parameter and D[j] is the first partial derivative of f(x) with respect to the jth parameter. The indices on the summations for 2*Sum( Sum( ... ) ) are i=1 to (p-1) and j>i, respectively, where p denotes the total number of parameters in the model. Also, s[i]^2 denotes the ith diagonal element of the variance-covariance matrix for the model, and s[i,j] denotes an off-diagonal element of the same matrix. You should be able to use vcov( ) to extract the variance-covariance matrix of your fitted model. This approximation will estimate the functional form of the variance of f(X). To get the approximate variance of f(X) for a specific value of X simply plug in X=x. After this, you will need to add the estimated variance of the residuals and take the square root to obtain the standard error used in calculating prediction intervals. I have used this approach for some highly nonlinear functions in the past, and the approximation is only "good" when it is reasonable to assume that E(f(x)) ~= f(E(x)). [This assumption is present in the derivation of the Taylor approximation.] When this is not a reasonable assumption, the approximation can be horrible. In other words, the more locally linear your nonlinear function is, the better the approximation will work. Hope this helps, david paul -----Original Message----- From: Enrique Portilla [mailto:[EMAIL PROTECTED] Sent: Thursday, August 14, 2003 9:28 AM To: [EMAIL PROTECTED] Subject: [R] nls confidence intervals Hi, Does anyone know how to compute the confidence prediction intervals for a nonlinear least squares models (nls)? I was trying to use the function 'predict' as I usually do for other models fitting (glm, lm, gams...), but it seems that se.fit, and interval computation is not implemented for the nls... Cheers Enrique ~~~~~~~~~~~~~~~~~~~~~~~~~~~ Fisheries Research Services, Marine Laboratory, Victoria Road, Torry, Aberdeen, UK. Tel. 44 (0) 1224 295314 ~~~~~~~~~~~~~~~~~~~~~~~~~~~ ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help ______________________________________________ [EMAIL PROTECTED] mailing list https://www.stat.math.ethz.ch/mailman/listinfo/r-help
