Dose-Response Curves

One limitation of linear regression models, explains Judea Pearl in his book The Book of Why is that they are unable to accurately model interactions or relationships that don’t follow linear relationships. This lesson was hammered into my head by a statistics professor at the University of Nevada, Reno when discussing binomial variables. For variables where there are only two possible options, such as yes or no, a linear regression model doesn’t work. When the Challenger Shuttle’s O-ring failed, it was because the team had run a linear regression model to determine a binomial variable, the O-ring fails or it’s integrity holds. However, there are other situations where a linear regression becomes problematic.
 
 
In the book, Pearl writes, “linear models cannot represent dose-response curves that are not straight lines. They cannot represent threshold effects, such as a drug that has increasing effects up to a certain dosage and then no further effect.”
 
 
Linear relationship models become problematic when the effect of a variable is not constant over dosage. In the field of study that I was trained in, political science, this isn’t a big deal. In my field, simply demonstrating that there is a mostly consistent connection between ratings of trust in public institutions and receipt of GI benefits, for example, is usually sufficient. However, in fields like medicine or nuclear physics, it is important to recognize that a linear regression model might be ill suited to the actual reality of the variable.
 
 
A drug that is ineffective at small doses, becomes effective at moderate doses, but quickly becomes deadly at high doses shouldn’t be modeled with a linear regression model. This type of drug is one that the general public needs to be especially careful with, since so many individuals approach medicine with a “if some is good then more is better” mindset. Within physics, as was seen in the Challenger example, the outcomes can also be a matter of life. If a particular rubber for tires holds its strength but fails at a given threshold, if a rubber seal fails at a low temperature, or if a nuclear cooling pool will flash boil at a certain heat, then linear regression models will be inadequate for making predictions about the true nature of variables.
 
 
This is an important thing for us to think about when we consider the way that science is used in general discussion. We should recognize that people assume a linear relationship based on an experimental study, and we should look for binomial variables or potential non-linear relationships when thinking about a study and its conclusions. Improving our thinking about linear regression and dose-response curves can help us be smarter when it comes to things that matter like global pandemics and even more general discussions about what we think the government should or should not do.

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