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Questions:

  • Even if there is no "widely accepted" technique, is there a useful-and-above-average technique for estimating goodness of fit in orthogonal regressions?
  • What are the pros/cons of this technique?

Background and Motivation: I recently discovered the orthogonal regression (=total least square regression, i.e. Deming with ratio of variances set to 1). Basically, I have $x$ and $y$, which are disease symptoms corresponding to two steps of a disease.

$$x = x^* + \mathrm{error}$$ $$y = y^* + \mathrm{error}$$

Here $(x,y)$ are the observed variables (symptoms measured visually, including error in disease assessment: same error for both) and $(x^*,y^*)$ are the latent variables ("true" symptoms). Note that $x^*$ and $y^*$ could be measured directly, e.g. by taking pictures (HD pictures then image analysis; close to no error in disease assessment); not done here because it is very long.

See here for the graphic (and first discussion on which regression to use). I made an orthogonal regression to have the relationship between $x$ and $y$. I would like to measure the goodness of fit for my orthogonal regression:

  • explained differently, I would like to know how much $x$ could help in predicting $y$ (= how much visual disease assessment of a symptom helps in predicting the visual assessment of the other symptom).
  • if not possible, knowing how much $x^*$ could help in predicting $y^*$ (= how much one symptom measured without error helps in predicting the other symptom measured without error; also, it would help in understanding the behaviour of the disease).

I asked for hints on R functions/packages on SO, on obtained this answer from @Gaurav:

There are many proposed methods to calculate goodness of fit and tolerance intervals for Deming Regression but none of them widely accepted. The conventional methods we use for OLS regression may not make sense. This is an area of active research. I don't think there many R-packages which will help you compute that since not many mathematicians agree on any particular method.

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  • $\begingroup$ If you are predicting $Y$ from $X$, then it is not clear why you would want to use orthogonal regression. How does your situation differ from a standard regression of $Y$ against $X$? $\endgroup$
    – whuber
    Aug 19 '15 at 16:49
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    $\begingroup$ @amoeba No, it's not. When $Y$ is to be predicted from a measurement $X$, then what is of interest is precisely that: given the observed value of $X$, what can be said about $Y$? That's what regression methods do. A Deming regression fit is useless for that purpose. (It tries to uncover information about an underlying linear association between the unobserved values.) $\endgroup$
    – whuber
    Aug 19 '15 at 18:27
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    $\begingroup$ @amoeba Take a look at the model specification. The parameter estimates relate the response value to the unobserved values of the regressors. You don't have those unobserved values available when you're making predictions. The whole point of Deming regression is to uncover some theoretical relationship among the unobserved values, but it's not to make predictions in the sense of asserting what $y$ is likely to be given an observed $x$. $\endgroup$
    – whuber
    Aug 19 '15 at 21:18
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    $\begingroup$ @whuber: thanks for your answer. Actually, $x*$ could be measured, with a technique that does not make mistake (e.g., instead a visual assessment of disease symptoms that gives $x$, HD picture + image analysis: possible but very long; the error would not be null, but arguably negligible if done the right way). So yes, they are practical applications (apart from a basic understanding of what drives the disease, which helps in understanding others results properly). $\endgroup$
    – NOTM
    Aug 20 '15 at 14:31
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    $\begingroup$ That is a revealing comment; it really clarifies your situation. Please include that information in the question itself. $\endgroup$
    – whuber
    Aug 20 '15 at 14:33

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