Predicting x from y

Dear list members,

I have just a quick question:

I fitted a non-linear model y=a/x+b to describe my data
(x=temperature and y=damage in %) and it works really nicely
(see example below). I have 7 different species and 8 individuals
per species. I measured damage for each individual per species
at 4 different temperatures (e.g. -5, -10, -20, -40).
Using the individuals per species, I fitted one model per species.
Now I'd like to use the fitted model to go back and predict the
temperature that causes 50% damage (and it's error). Basically,
it pretty easy by just rearranging the formula to x=a/(y-b).
But that way I don't get a measure of that temperature's error,
do I? Can I use the residual standard error that R gave me for
the non-linear model fit? Or do I have to fit 8 lines (each
individual) per species, calculate x based on the 8 individuals
and then take the mean?

Unfortunately, dose.p from the MASS package doesn't work for
non-linear models. When I take the log(abs(x)) the relationship
becomes not satisfactory linear either.

Any suggestions are highly appreciated!

Thank you!
Stefan

EXAMPLE for species #1:

y.damage<-c(5.7388985,1.7813519,3.7321461,2.9671031,
0.3223196,0.3207941,-1.4197658,-5.3472160,
41.1826677,29.3115243,31.3208841,35.3934115,
58.5848778,31.1541049,42.2983479,27.0615648,
64.1037728,54.7003353,66.7317044,65.4725881,
72.5755056,67.2683495,64.8717942,65.9603073,
75.0762273,56.7041960,60.0049429,70.0286506,
73.2801947,72.7015642,75.0944694,81.0361280)

x.temp<-c(-5,-5,-5,-5,-5,-5,-5,-5,-10,-10,-10,-10,-10,-10,-10,
-10,-20,-20,-20,-20,-20,-20,-20,-20,-40,-40,-40,-40,-40,
-40,-40,-40)

nls(y.damage~a/x.temp+b,start=list(a=400,b=80))
plot(y.damage~x.temp,xlab='Temperature',ylab='Damage [%]')
curve(409.61/x+81.84,from=min(x.temp),to=max(x.temp),add=T)

On 11-Nov-11 14:51:19, Schreiber, Stefan wrote:

Dear list members,

I have just a quick question:

I fitted a non-linear model y=a/x+b to describe my data
(x=temperature and y=damage in %) and it works really nicely
(see example below). I have 7 different species and 8 individuals
per species. I measured damage for each individual per species
at 4 different temperatures (e.g. -5, -10, -20, -40).
Using the individuals per species, I fitted one model per species.
Now I'd like to use the fitted model to go back and predict the
temperature that causes 50% damage (and it's error). Basically,
it pretty easy by just rearranging the formula to x=a/(y-b).
But that way I don't get a measure of that temperature's error,
do I? Can I use the residual standard error that R gave me for
the non-linear model fit? Or do I have to fit 8 lines (each
individual) per species, calculate x based on the 8 individuals
and then take the mean?

Unfortunately, dose.p from the MASS package doesn't work for
non-linear models. When I take the log(abs(x)) the relationship
becomes not satisfactory linear either.

Any suggestions are highly appreciated!

Thank you!
Stefan

EXAMPLE for species #1:

y.damage<-c(5.7388985,1.7813519,3.7321461,2.9671031,
0.3223196,0.3207941,-1.4197658,-5.3472160,
41.1826677,29.3115243,31.3208841,35.3934115,
58.5848778,31.1541049,42.2983479,27.0615648,
64.1037728,54.7003353,66.7317044,65.4725881,
72.5755056,67.2683495,64.8717942,65.9603073,
75.0762273,56.7041960,60.0049429,70.0286506,
73.2801947,72.7015642,75.0944694,81.0361280)

x.temp<-c(-5,-5,-5,-5,-5,-5,-5,-5,-10,-10,-10,-10,-10,-10,-10,
-10,-20,-20,-20,-20,-20,-20,-20,-20,-40,-40,-40,-40,-40,
-40,-40,-40)

nls(y.damage~a/x.temp+b,start=list(a=400,b=80))
plot(y.damage~x.temp,xlab='Temperature',ylab='Damage [%]')
curve(409.61/x+81.84,from=min(x.temp),to=max(x.temp),add=T)

A couple of comments.

First, in general it is not straightforward to estimate
the value of a covariate (here temperature) by inverting
the regression of a response (here damage) on that covariate.
This the "inverse regression" or "calibration" problem,
(and it is problematic)! For instance, in linear regression
the estimate obtained by inversion has (theoretically)
no expectation, and has infinite variance. For an outline,
and a few references, see the Wikipedia article:

  http://en.wikipedia.org/wiki/Calibration_(statistics)

Second, I would be inclined to try nls() on a reformulated
version of the problem. Let T50 denote the temperature for
50% damage, and introduce this as a parameter (displacing
your parameter "a"):

  y = 50*(b + T50)/(b + x)

where T50 = a/50 - b in terms of your original parameters
"a" and "b". With this formula for the non-linear dependence
of damage on temperature, it is no longer necessAry to invert
the regression equation, since the parameter you want is
already there and will be estimated directly.

Hoping this helps,
Ted.