[R] weighted nls and confidence intervals

Joerg van den Hoff j.van_den_hoff at fzd.de
Thu Aug 23 15:41:30 CEST 2007

for unweighted fits using `nls' I compute confidence intervals for the
fitted model function by using:

se.fit <- sqrt(apply(rr$m$gradient(), 1, function(x) sum(vcov(rr)*outer(x,x))))
luconf <- yfit + outer(se.fit, qnorm(c(probex, 1 - probex)))

where `rr' contains an `nls' object, `x' is the independent variable vector,
`yfit' the corresponding model prediction (`fitted(rr)'), `se.fit' the
corresponding standard error and `luconf' the lower and upper confidence
limits at some level specified by `probex'.

when using the same approach with weighted fits (even if all weights are equal
(but not equal to one)), I noted that the confidence limits depend on the
weights in an counter-intuitive, probably erroneous way:

consider the following simple example (yes, I know that this actually is a linar
model :-)):

probex <- 0.05
x <- 1:10
y <- rnorm(x, x, .8)
w1 <- rep(1, 10)
w2 <- w1; w2[7:10] <- 0.01 * w2[7:10]

rr1 <- nls(y ~ a*x + b, data = list(x = x, y = y), start = list(a = 1, b = 0), weights = w1)
rr2 <- nls(y ~ a*x + b, data = list(x = x, y = y), start = list(a = 1, b = 0), weights = w2)
yfit1 <- fitted(rr1)
yfit2 <- fitted(rr2)
se.fit1 <- sqrt(apply(rr1$m$gradient(), 1, function(x) sum(vcov(rr1)*outer(x,x))))
luconf1 <- yfit1 + outer(se.fit1, qnorm(c(probex, 1 - probex)))

se.fit2 <- sqrt(apply(rr2$m$gradient(), 1, function(x) sum(vcov(rr2)*outer(x,x))))
luconf2 <- yfit2 + outer(se.fit2, qnorm(c(probex, 1 - probex)))

op <- par(mfrow = c(2,1))
matplot(x, cbind(y, yfit1,luconf1), type = 'blll', pch = 1, col = c(1,2,4,4), lty = 1)
matplot(x, cbind(y, yfit2,luconf2), type = 'blll', pch = 1, col = c(1,2,4,4), lty = 1)
par(op, no.readonly = T)

the second fit uses unequal weights where the last 4 points are next to
unimportant for the fit. the confidence intervals in this case are fine up to
point no. 6, but nonsense afterwards I believe.

I have tracked this down to the fact that `m$gradient' does _not_ contain the 
actual gradients w.r.t. the parameters but rather the gradients multiplied by
the sqrt of the weights. without trying to understand the inner workings of the
`nls*' functions in detail my questions are:

is the fact that `m$gradient' in an `nls' object does contain the scaled
gradients documented somewhere (I did'nt find it)? if yes, where is it, if no,
can someone please give me a hint what the rationale behind this approach is?

am I right to understand, that the above approach to compute `se.fit'
(essentially in this compact form proposed by p. daalgaard on r-help some time ago)
is erroneous (for weighted nls) in computing the confidence limits and that
I have to undo the multiplication by sqrt(weights) which is included 
in `m$gradient' (or compute the gradients of my model function myself,
e.g. with `deriv')?



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