Hi:
You're fitting y as a function of x; as in any regression model, the x's are
assumed to be conditionally fixed. If you want to model x as a function of
y, that's a calibration problem.
There are several issues at play:
1. In the model, you have assumed x is fixed, but afterward, you want to
treat it as random.
2. You're implicitly assuming that the relationship between Y and X is
invertible. If this were a math problem, you would be right: y = a exp(b *
x) => x = (1/b) log(y/a). Unfortunately, this is a statistical problem,
and several problems arise, among them:
- the X and Y scales are very different, which suggests
rather strongly that the distribution associated with
random variation in Y is not the same as the distribution
associated with random variation in X.
- the distribution of random errors in Y is quite likely not
going to have the same relationship to the distribution
of random errors in X as the functional relationship
between x and y above. More succinctly, the problem is a
lack of invariance, so you can't simply invert the problem
at the end.
3. Predictions are only as valid/reliable as the underlying model and its
attendant assumptions. Your request to predict x as a function of y is
essentially acknowledging that the prediction is useless before it is ever
made, by point (1).
Chemometrics deals with calibration problems regularly - I presume that many
calibration functions are nonlinear, so you may find something useful by
hunting in that area. In econometrics, a related (but not equivalent)
problem is the so-called 'errors in variables', in which both y and x are
assumed to be random. Of course, you could model the problem from a Bayesian
perspective and hope that MCMC comes to the rescue :)
The bottom line is that a statistical relationship between y and x does not
translate to a corresponding relationship between x and y (through
mathematical inversion). Basically, they are treated as separate problems,
which makes sense because the conditional distribution of Y|X = x is not the
same as that of X|Y = y.
HTH,
Dennis
2010/9/8 åÌÅÎÁ âÅÌÙÈ <da...@rambler.ru>
>
> Dear colleagues!
> Is it possible to make predictions in R?
> there is an exponential relationship detween y and x
> x<-c(0.001,0.003,0.01,0.16,0.3,0.7,0.9)
> y<-c(38.8,41.5,44.2,27,26.9,6.9,3)
> f<-function(x,a,b){a*exp(b*x)}
> fm<-nls(y~f(x,a,b), start=c(a=1,b=1))
>
> How one can predict x when y=10 and is it possible to calculate standard
> error of x?
> The task is equal to function ED in drc package, but it use logistic
> regression only.
>
> Best regards,
> Elena.
>
> --
> da...@rambler.ru.
>
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