# Can the use of dummy variables reduce measurement error?

If the continuous variables are measured with error, can the use of dummy variables mitigate the problem? For instance, IQ measures intelligence with error. So will using a dummy of high, medium, low IQ mitigate the measurement error problem? Thanks!

• How do you dichotomize your data? Using the noisy measurement? If so, isn't that just hiding the noise (in your dichotomizing decision) rather than actually mitigating it? The error doesn't go away just because you rearrange things. When you feed your crisp labels (0/1) into a method that knows nothing about how you came to 0/1, these methods will naturally reflect a low error because they "believe" your crisp 0/1. – Wayne Mar 10 '14 at 13:29

Dichotomizing predictor variables actually reduces power to detect relationships between a continuous predictor and the response variable. Royston (2006) is one of many articles citing this as a reason why dichotomizing is a bad idea.

You can see @gung's answer to this question highlighting even more problems, such as hiding potential nonlinear relationships, among others.

• Thank you very much for your kind response. I really appreciate it. I can now see dichotomizing predictor variables can be problematic. However, what if the continuous variables themselves are very noisy measures? For instance, they might be the residuals of a model. These residuals might result from factors other than what they are meant to proxy. In this case, will use a dummy variable, for instance 1 for positive residuals and 0 for negative residuals be better than using the residuals themselves? Thanks! – Tammy Feb 14 '14 at 18:06
• I'm not sure what you mean by continuous variables as residuals. I have heard the belief that dichotomizing could mitigate noisiness, though I don't know good references backing this up. I'm against dichotomizing regardless. Assuming a linear relation, if you get a significant result with dichotomies, but a nonsignificant result using the continuous variable, you have almost assuredly capitalized on chance in choosing the split. Also, making a dichotomy will induce an artifact (usually spurious significance) in other variables in a model that are correlated with the dichotomized variable. – dmartin Feb 14 '14 at 18:59

What matters is the magnitude of an error multiplied by its likelihood. When a noisy continuous variable is dichotomized, the magnitude of an error is huge because the error is to put someone in the wrong category - a 100% error. http://biostat.mc.vanderbilt.edu/wiki/pub/Main/BioMod/catgNoise.r is a script that can be run in RStudio (it requires the R Hmisc package also) that provides an interactive demonstration of the fact that no amount of noise added to a predictor can make dichotomization have better power than analyzing the variable continuously - even when the relationship is nonlinear (but monotonic) and one improperly uses a linear fit.

The idea isn't even intuitively appealing (at least to me): using a discretized version of an accurate measure might be preferable to using a continuous version of an inaccurate one; but how on earth is discretizing an inaccurate measure supposed to improve it? The situations in which it might are those in which the truth is discrete, or close to it: in your example, that people really are dumb, or ordinary, or smart, & nothing in between; & that IQ is a measure from which you can predict membership of those three classes. Note that still wouldn't justify binning IQ scores into top third, middle third, & bottom third; you'd need an idea of the proportions in each class, & the whereabouts of the cut-offs. The general message from DeCoster et al.'s simulations is that discretization is only ever justified by quite specific prior knowledge about the distribution of the underlying variable being measured.

MacCallum et al. (2002), "On the Practice of Dichotomization of Quantitative Variables", Psychological Methods, 7, 1

DeCoster et al. (2009), "A Conceptual and Empirical Examination of Justifications for Dichotomization", Psychological Methods, 14, 4

It really depends what you mean by mitigate. It is not always a bad idea to bin the data. You are throwing away information when you do this but sometimes that is made up for by allowing for a more tractable model. An example would be binning data so that you can use the Chi-square test or a Chi-square minimization routine or are somehow trying to take advantage of the central limit theorem to mitigate the fact that you may not have a good handle on the distribution of the noise.

However if you do this, be aware that you are throwing away some information and that there are often other ways of using the data without binning. But if you cannot figure out how to model things without binning then go ahead and do this. If you think of a better, more sophisticated method later then revisit it.

• If you can't figure out how to model things properly, & the model's to be used for something important; consult a statistician. – Scortchi - Reinstate Monica Mar 10 '14 at 12:53
• There are plenty of examples where a Statistician is not going to be able to help and where binning really is the best solution. – Dave31415 Mar 10 '14 at 16:59
• Binning is not the best solution. Binning makes the model more complex, not less. This happens because binning creates unaccounted-for residual variation in $Y$ that must be made up by adding more variables to the model, and to accurately model continuous variables requires more bins than degrees of freedom in a smooth continuous effect. – Frank Harrell Jan 19 '15 at 13:22
• Binning can be a fine solution. Depends on the problem. Binning does not necessarily mean turning them into categorical variables. The problem as stated doesn't give enough information to decide here. Many Bayesian problems do not have a closed form solution but if you bin the data (even slightly), you can come up with a good closed form solution. – Dave31415 Jan 22 '15 at 4:01