# When should I use errors-in-variables?

I have been reading about errors-in-variables (also called regression dilution and attenuation) but I've found it hard to decide whether it is appropriate for my setting.

I want to calculate correlations, and I've read that due to measurement errors, the computed correlations will be underestimated.

On Wikipedia, it says that for two random variables $\beta, \theta$ with estimates

$\hat{\beta} = \beta + \epsilon_\beta, \;\;\; \hat{\theta} = \theta + \epsilon_\theta,$

where $\epsilon_\beta$ and $\epsilon_\theta$ are measurement errors associated with the estimates, the correlation can be expressed as

${\rm corr}(\hat{\beta},\hat{\theta}) = \rho \sqrt{R_\beta R_\theta},$

where $R_\beta$ is calculated by

$R_\beta = \displaystyle \frac{{\rm var}(\beta)}{{\rm var}(\beta) + {\rm var}(\epsilon_\beta)} =\displaystyle \frac{{\rm var}(\hat{\beta}) - {\rm var}(\epsilon_\beta)}{{\rm var}(\hat{\beta})},$

and $R_\theta$ is calculated equivalently. Thus the disattenuated $\rho$ is found by

$\rho = \displaystyle \frac{{\rm corr}(\hat{\beta},\hat{\theta})}{\sqrt{R_\beta R_\theta}}.$

My setting is that I am doing several measurements of three quantities (distance to, angle from reference line, height) at different locations.

Can I use this formula for calculating correlations, substituting the manufacturer's given uncertainty in measurements as ${\rm var}(\epsilon_\beta)$?

Edit: The measurements are done in land surveying. The measured quantities are distance, angle and height, $d, \theta, z$, measured from a measuring station, which location is considered known, to a number of reflector stations with unknown location. That is, the measuring station has a reference location, and the distance to the reflector stations, as well as height at the stations, and angle difference from some reference angle.

The three quantities are measured by laser. Several measurements are made according to given procedures.

I read in this report that measurements made with the same instrument may be correlated due to measurement uncertainty. Because of this I suspect that height, distance and angle measured from the measuring station to one given reflector station might be correlated. Also, I suspect that the measurement procedure can give rise to correlations between the angles measured at different stations (locations). If this interpretation is wrong, please let me know!

The data is to be presented and further analysed in a Cartesian coordinate system, $(x, y, z')$, (while the measurements are in a spherical coordinate system) so to find the standard deviation in, say, $x = d \sin(\theta)$ I need to calculate covariances.

• Could you explain why you want to calculate correlations? That, after all, is not usually anybody's real objective: it's just a procedure for characterizing data. What are you really trying to learn about length, angle, and height? – whuber Jun 28 '13 at 14:19
• Thank you for the question, @whuber! I've edited my question, hopefully it is more clear now. Please shout out if you think I am on the wrong track calculating correlations. – Gunnhild Jun 28 '13 at 14:35
• Presumably you meant to write $z=d\sin(\theta)$ (instead of $x$). Do you have any reason to believe the measurements of $d$ and $\theta$ may be correlated? If not, you can work out the distribution of $d\sin(\theta)$ from the distributions of $d$ and $\theta$. – whuber Jun 28 '13 at 15:10
• No, I meant $x$, I mean that the data will be transformed to a different coordinate system than the one they are measured in, so to say. The reason I believe they are correlated stem from measurement technicalities and the fact that the three quantities are measured at the same time. I read in a JCGM report that it could cause correlations – Gunnhild Jun 30 '13 at 18:10
• @Gunnhild, your measurements are unlikely to be correlated, but likely to have the same bias. The bias may be correlated with the surveyor, – Aksakal Apr 20 '14 at 15:50

## 1 Answer

Describing the nature and extent of measurement error is a very good practice for statisticians. The next important implication for practice is to interpret the results and findings in a contextually appropriate manner, which may confuse the reader somewhat, but doesn't convey false associations. For instance, suppose we observe in-patient admits to a mental hospital and measure pain sensitivity and successful rehabilitation from addictive substances. In survey items about the frequency and intensity of pain, we are careful to refer to associations as describing self reported pain. If such patients are being considered as eligible for benzos or opiods, we might expect exaggeration of the pain response among those less likely to response to rehab therapies.

In epidemiology, the above example could be referred to as differential misclassification. Even non-differential misclassification has implications on the measured associations. In particular, attentuation is a problem. However, in most circumstances, associations can be measured in the presence of attenuation. Hence we are capable of making inference on associations, but predicted values in non-linear trends are biased, and confidence intervals for effects may be too large. For inference, attenuation is less of a problem than it seems.

If we had an estimate of the measurement error, then it's possible to extend maximum likelihood methods to obtain unbiased estimates of associations. In particular, this would involve the use of the EM Algorithm. It turns out this is equivalent to a latent variable approach where a latent variable is added between your measurement-errored variable and y.