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(EDIT: the question has been modified just a little bit to be more specific)

I want to fit a multivariate polynomial regression that accounts for measurement errors (an Error-in-Variables model).

As an example, my input data is like: 

y      sd_y      x       sd_x       z       sd_z

9.55   0.26     6.74     0.71      0.25     0.02
8.31   0.19     5.93     0.33      -0.40    0.05
...    ...      ...      ...        ...     ...

where sd_y, sd_x, sd_z are the standard deviations of each variable, and 

 wx <- 1/(sd_x)**2 ; wy <- 1/(sd_y)**2 ; wz <- 1/(sd_z)**2

would be the weights for each variable.

If I use a standard regression model (where predictors are supposed to have been measured exactly or without error) my function or fit, in R, would be:

p <- lm(y~polym(x, z, degree = 2, raw=TRUE))

Is there a method/package in R that allows to deal with "error-in-variables models"? If so, how I would write my fit p when using the supposed package?

LRD
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  • Your question is not entirely clear. How would these weights work exactly? What is the function being optimized? – Glen_b Jun 30 '15 at 09:37
  • I want to obtain the fit I called *p*, this is my function to optimise. I've just added the kind of weights would like to take into account when looking for the best fit (wx, wy, wz). I want my polynomial to be fitted giving more or less importance (weights) to each observation depending on its weight (since the input it's observational data, I think it's logical to have these standard deviations into account). – LRD Jun 30 '15 at 09:54
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    An *observation* is a point in $\mathbb{R}^3$, $(x,z,y)$. A single point should have *one* weight. How are you combining the weights? Further, note that if $x$ and $z$ are random (the values have non-zero standard deviations) then `lm` is the wrong tool for fitting a relationship with $y$; least squares regression assumes they're fixed, not random. e.g. see [here](https://en.wikipedia.org/wiki/Errors-in-variables_models) – Glen_b Jun 30 '15 at 09:59
  • ok, I see the point about the least squares regression and the errors-in-variables models. Thanks for the link, it really helped me to learn more about that. So, is there any package or method in R which allows to deal with this "error-in-variables models" problem? I've read things about *ODRpack* or *leiv*, but I'm not sure of their usage. Are they applicable to my multivariate problem? If you have knowledge about that, maybe an example will help me a lot, thanks – LRD Jun 30 '15 at 12:44
  • Such an answer wouldn't really respond to your present question. – Glen_b Jun 30 '15 at 16:11
  • I've modified a bit the question so that now I guess that, if someone knows how to deal with this *errors-in-variables* problem in R, they can help me – LRD Jul 01 '15 at 09:18

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