[R] nls confidence intervals

Spencer Graves spencer.graves at pdf.com
Thu Aug 14 18:14:01 CEST 2003

Regarding the accuracy of the Taylor series approximation, my favorite 
reference is Bates & Watts (1988) Nonlinear Regression Analysis and Its 
Applications (Wiley, esp. pp. 255-260).  Recently, Brian Ripley also 
Chambers & Hastie (1992) Statistical Models in S (Wadsworth, ch. 10) and 
Venables & Ripley, Modern Applied Statistics in S.  Unfortunately, I 
don't see predict.nls (or anything similar) in the index of any of these 

hope this helps.  spencer graves

Paul, David A wrote:
> 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:portillae at marlab.ac.uk] 
> Sent: Thursday, August 14, 2003 9:28 AM
> To: R-help at stat.math.ethz.ch
> 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
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